Beyond Thermodynamics: The Critical Role of Convex-Hull Stability in Predicting Synthesizable Materials and Pharmaceuticals

Abstract

This article explores the pivotal role of convex-hull stability analysis in predicting the synthesizability of new materials and pharmaceutical polymorphs. We cover foundational principles, from defining the energy above hull as a key metric to its interpretation for thermodynamic stability. The piece delves into advanced computational methodologies, including machine learning and active learning, that are revolutionizing stability prediction. It also addresses critical challenges like vibrational instability and the overprediction problem, offering troubleshooting strategies and optimization techniques. Finally, we provide a framework for validating predictions, comparing different model performances, and translating computational results into successful experimental synthesis, with specific implications for drug development and biomedical research.

The Bedrock of Stability: Understanding Convex-Hull Fundamentals and Thermodynamic Principles

Defining the Convex Hull and Energy Above Hull in Phase Diagrams

In the pursuit of novel materials and compounds, a fundamental challenge for researchers is predicting whether a target phase is thermodynamically stable—and therefore likely synthesizable—or metastable. The convex hull, derived from thermodynamic potentials, provides the definitive mathematical framework for answering this question. Within materials science and drug development, the convex hull of a phase diagram identifies the set of stable phases at specific compositions, while the "energy above hull" (often denoted Ehull) quantifies the degree of metastability for any phase not on this hull [1] [2]. This guide details the theory, computation, and application of these concepts, framing them within the emerging research paradigm that uses quantitative stability metrics to rationally guide synthesis efforts.

Core Theoretical Concepts

The Convex Hull: A Geometric Foundation for Thermodynamic Stability

The convex hull of a set of points is the smallest convex set that contains all points [3]. In thermodynamics, these "points" are the Gibbs free energies of various phases across a composition space. The convex hull is the lower envelope of these energies, defining the minimum possible energy for any given composition [1] [4].

• Formation Energy Precursor: The construction of a compositional phase diagram begins with calculating the formation energy (ΔEf) for every known compound in a chemical system. For a phase composed of N components, this is given by: ΔEf = E − ∑iN ni μi where E is the total energy of the phase, ni is the number of moles of component i, and μi is the energy per atom of the pure component i (e.g., elemental references) [1]. This energy is typically normalized per atom.

• The Hull Construction: The convex hull is taken over the set of points in (energy, composition) space [1]. Graphically, for a binary system, one can imagine stretching a rubber band below all the (composition, energy) data points; the shape formed by the rubber band is the convex hull [4]. Phases whose energies lie directly on this hull are considered thermodynamically stable at 0 K, meaning they have no driving force to decompose into other phases [1] [2].

Energy Above Hull: Quantifying Metastability

The energy above hull (Ehull) is a critical metric defined as the vertical energy distance from a phase's formation energy to the convex hull at its specific composition [5]. It represents the decomposition energy—the energy released (per atom) when a metastable phase decomposes into a combination of the stable phases on the hull [1] [5].

A phase with an Ehull = 0 meV/atom is stable. A positive Ehull indicates a metastable phase. The magnitude of Ehull indicates the energy penalty associated with its metastability; a higher value generally implies a greater driving force for decomposition and, thus, potentially greater synthetic challenge [6]. However, phases with positive Ehull can often be synthesized, as kinetics and other factors play a significant role [6].

Table: Interpretation of Energy Above Hull Values

Ehull (meV/atom)	Thermodynamic Stability	Synthesizability Implication
0	Stable	Synthesizable under equilibrium conditions
0 < Ehull ≤ ~25	Metastable	Often synthesizable (kinetic stabilization)
~25 < Ehull ≤ ~100	Metastable	Challenging to synthesize
> ~100	Highly Unstable	Unlikely to be synthesizable via conventional means

Computational Methodology and Workflows

The following diagram illustrates the logical workflow for constructing a phase diagram and determining phase stability using the convex hull method.

Standard Convex Hull Construction

The foundational methodology involves density functional theory (DFT) calculations to compute the formation energies of all known compounds in a chemical system. The convex hull is then constructed from these energies, typically at 0 K and 0 atm, to determine stable phases and decomposition energies [1]. The pymatgen code snippet below demonstrates this standard approach:

Code: Standard phase diagram construction using pymatgen and the Materials Project API [1].

Advanced Protocol: Convex Hull-Aware Active Learning (CAL)

For complex systems where exhaustive energy calculation is prohibitively expensive, a novel Bayesian active learning algorithm can be employed. Convex Hull-Aware Active Learning (CAL) directly minimizes the uncertainty in the convex hull itself, rather than in the entire energy landscape, leading to greater efficiency [7] [8].

Detailed CAL Methodology:

• Probabilistic Modeling: Model the energy surface of each phase using a Gaussian Process (GP), which provides a posterior distribution over possible energy surfaces given initial observations [7] [8].
• Hull Sampling: Sample from the GP posterior to generate an ensemble of possible energy surfaces. Compute the convex hull for each sampled surface using a standard algorithm like QuickHull [7].
• Uncertainty Quantification: This ensemble of hulls represents the epistemic uncertainty about the true convex hull. The probability of a composition being stable is estimated by the fraction of sampled hulls on which it lies [7].
• Optimal Data Selection: The next composition to observe (via expensive DFT calculation) is chosen as the one that provides the maximum expected information gain about the convex hull, dramatically reducing the number of calculations needed to resolve it [7] [8].

Table: Essential Research Reagents & Computational Tools

Item / Software	Function in Stability Research
VASP / Quantum ESPRESSO	First-Principles DFT code for calculating accurate formation energies.
pymatgen	Python library for phase diagram construction, analysis, and materials data.
Materials Project API	Source of pre-computed formation energies for a vast array of compounds.
Gaussian Process Regression	Core statistical model for Bayesian uncertainty quantification in CAL.
QuickHull Algorithm	Standard computational geometry algorithm for efficient convex hull calculation.

Advanced Analysis and Research Applications

Determining Decomposition Products and Pathways

A phase not on the convex hull will decompose into the set of stable phases that define the hull at its composition. The decomposition reaction is found by determining the linear combination of stable phases that minimizes the energy at the target composition. For example, the oxynitride BaTaNO₂ is calculated to be 32 meV/atom above the hull, with decomposition products: ⅔ Ba₄Ta₂O₉ + ⁷⁄₄₅ Ba(TaN₂)₂ + ⁸⁄₄₅ Ta₃N₅ [5]. The energy above hull is calculated using the normalized (eV/atom) energies of these phases with their respective coefficients [5].

Beyond 0 K: The Entropy Challenge and Finite-Temperature Estimates

A critical limitation of standard convex hull analysis is its basis in internal energy at 0 K. Temperature-dependent entropic effects (vibrational, configurational, electronic) can significantly alter phase stability [6]. While Materials Project data is primarily at 0 K, its Phase Diagram app includes a finite-temperature estimation feature that uses machine-learned descriptors for the Gibbs free energy, providing a crucial, though approximate, view of temperature-dependent stability [6].

The convex hull and energy above hull provide an unambiguous thermodynamic foundation for predicting phase stability. The Ehull value serves as a powerful, quantitative descriptor for high-throughput screening of material databases, flagging promising candidate materials for synthesis [6]. Research now focuses on integrating these thermodynamic metrics with kinetic factors to build more comprehensive synthesizability models [6].

Future research directions include the wider adoption of Convex Hull-Aware Active Learning (CAL) and other Bayesian methods for efficient exploration of complex chemical spaces, particularly for high-entropy materials, liquids, and correlated systems [7] [8]. These approaches promise to deliver not only a predicted hull but also a quantitative measure of uncertainty, enabling end-to-end uncertainty quantification in the emerging paradigm of computational materials design and accelerating the discovery of novel, synthesizable materials.

The discovery and synthesis of new inorganic materials represent a central pursuit in solid-state chemistry, capable of driving significant scientific and technological advancements. While high-throughput computational methods now generate millions of candidate crystal structures, determining which are experimentally accessible remains a critical bottleneck. This whitepaper examines the central role of convex hull stability in synthesis prediction research, contrasting its thermodynamic completeness with its kinetic limitations. We demonstrate that while thermodynamic stability assessed through convex hull construction provides a essential first-principles filter for material viability, it often fails to account for the experimental reality of kinetic trapping—where metastable phases with favorable formation pathways can be synthesized despite thermodynamic instability. Through analysis of current synthesizability prediction models, experimental validation studies, and emerging network-based approaches, we provide a framework for integrating both thermodynamic and kinetic considerations into a unified synthesizability assessment pipeline, ultimately bridging the gap between computational prediction and experimental realization.

The combinatorics of materials discovery present an immense challenge. Considering just combinations of four elements from approximately 80 technologically relevant elements yields roughly 1.6 million quaternary chemical spaces to explore, before even considering stoichiometric variations or crystal structure possibilities [9]. While computational materials discovery methods have reached maturity, generating vast databases of predicted candidate structures through active learning and density functional theory (DFT) calculations, the number of proposed inorganic crystals now exceeds experimentally synthesized compounds by more than an order of magnitude [10].

The central challenge in computational materials discovery has shifted from generating candidate structures to predicting synthesizability—determining which computationally predicted materials can actually be fabricated in laboratory settings. The principal limitation of current approaches lies in their fundamental reliance on thermodynamic stability assessed through convex hull construction at zero Kelvin, which inherently overlooks the finite-temperature effects, entropic factors, and kinetic considerations that govern synthetic accessibility in experimental environments [10] [11]. This theoretical-experimental gap manifests clearly in materials databases: for example, the Materials Project lists 21 SiO₂ structures within 0.01 eV of the convex hull, yet cristobalite (β-quartz), the second most common SiO₂ phase, is not among these 21 theoretically "stable" structures [10].

This whitepaper examines the critical intersection between thermodynamic stability and kinetic trapping within materials synthesis prediction research. We explore how convex hull stability provides necessary but insufficient conditions for experimental synthesis, survey emerging computational frameworks that integrate kinetic and synthetic accessibility metrics, and provide methodological guidance for researchers navigating the transition from computational prediction to experimental realization.

The Convex Hull: Thermodynamic Foundation

Fundamental Principles and Construction

In computational materials science, the convex hull represents the multidimensional surface formed by the lowest energy combination of all phases in a chemical space [12]. The construction of this hull begins with the calculation of formation enthalpies (ΔHf) for all known and hypothetical compounds within a specific chemical system using density functional theory (DFT). When these energies are plotted against composition, the convex hull emerges as the lower convex envelope lying beneath all points in the composition space [9].

Compositions lying directly on this hull are considered thermodynamically stable, meaning they possess lower formation energies than any combination of other phases at the same overall composition. The vertical distance between a compound's formation energy and the convex hull defines its energy above hull (ΔHd), which quantifies its thermodynamic instability relative to competing phases [9]. This energy difference, typically on the order of 0.06 ± 0.12 eV/atom (much smaller than the typical formation energy range of -1.42 ± 0.95 eV/atom), represents the subtle energetic quantity that ultimately determines thermodynamic stability [9].

Table 1: Key Energy Metrics in Stability Assessment

Metric	Definition	Typical Range	Interpretation
Formation Energy (ΔHf)	Energy to form compound from elements	-1.42 ± 0.95 eV/atom	Tendency to form from elements
Decomposition Enthalpy (ΔHd)	Distance to convex hull	0.06 ± 0.12 eV/atom	Thermodynamic stability
Energy Above Hull	Positive ΔHd values	0-0.5 eV/atom (typical)	Degree of instability

The convex hull construction generates what can be conceptualized as a materials stability network—a scale-free network where stable materials represent nodes connected by tie-lines (edges) representing two-phase equilibria [12]. This network perspective reveals that certain phases act as "hubs" with significantly more connections than others, with O₂, Cu, H₂O, H₂, C, and Ge emerging as the most highly connected species in the current stability network [12].

Algorithmic Implementation and Active Learning

Recent advances have addressed the computational challenge of convex hull determination through novel algorithms like Convex Hull-Aware Active Learning (CAL). This Bayesian algorithm selects experiments to minimize uncertainty in the convex hull construction, prioritizing compositions near the hull while leaving significant uncertainty in compositions quickly determined to be hull-irrelevant [13]. This approach allows the convex hull to be predicted with significantly fewer energy calculations than methods focusing solely on energy prediction [13].

The following diagram illustrates the conceptual relationship between formation energy, the convex hull, and material stability:

Beyond Thermodynamics: The Kinetic Trapping Phenomenon

Theoretical Framework

Kinetic trapping represents the experimental reality that metastable materials—those with positive energy above hull—can often be synthesized and persist indefinitely under appropriate conditions. This phenomenon occurs when kinetic barriers prevent the system from reaching the thermodynamic ground state, effectively trapping it in a local energy minimum [10] [11].

The relationship between thermodynamic stability and kinetic accessibility manifests through several mechanisms:

• Precursor Selection: The choice of starting materials can create favorable local energy landscapes that direct synthesis toward metastable products through low-energy nucleation pathways [11].
• Reaction Kinetics: Finite-temperature synthesis conditions introduce entropic and kinetic factors that may favor metastable phases with lower activation energies for formation, despite their thermodynamic instability [10].
• Synthetic Technique: Advanced synthesis methods (e.g., thin-film deposition, solution-based techniques) can access metastable phases unreachable through conventional solid-state reactions [11].

The following experimental workflow illustrates how synthesizability prediction integrates both thermodynamic and kinetic considerations:

Network-Based Synthesizability Prediction

An innovative approach to synthesizability prediction emerges from analyzing the temporal dynamics of the materials stability network. By combining convex hull data with historical discovery timelines extracted from citation records, researchers have shown that the likelihood of a hypothetical material being synthesized correlates with its position within the evolving network [12].

This network-based analysis reveals that the materials stability network has evolved into a scale-free topology with a degree distribution following a power law (p(k) ~ k^(-γ), with γ ≈ 2.6±0.1 after the 1980s), similar to other complex networks like the world-wide-web or social networks [12]. This network topology indicates the presence of highly connected "hub" materials that disproportionately influence synthesizability, with oxygen-bearing materials historically acting as dominant hubs [12].

Six key network properties have been identified as predictive features for synthesizability [12]:

• Degree centrality: Number of tie-lines connected to a material
• Eigenvector centrality: Influence of a material in the network
• Mean shortest path length: Average distance to all other materials
• Mean degree of neighbors: Connectivity of neighboring materials
• Clustering coefficient: Interconnectedness of a material's neighbors

Computational Frameworks for Synthesizability Prediction

Integrated Compositional and Structural Models

Current state-of-the-art approaches for synthesizability prediction integrate complementary signals from both composition and crystal structure through multi-modal machine learning frameworks [10]. These models address the fundamental limitation of purely thermodynamic assessments by learning the complex relationship between material characteristics and experimental accessibility.

The integrated model developed in recent research uses two specialized encoders [10]:

• Compositional Encoder: A fine-tuned compositional MTEncoder transformer (fc) that processes stoichiometric information and elemental properties
• Structural Encoder: A graph neural network fine-tuned from the JMP model (fs) that analyzes crystal structure graphs

These encoders transform inputs into latent representations (zc = fc(xc; θc) and zs = fs(xs; θs)), which then feed separate multi-layer perceptron heads that output synthesizability scores. The models are trained end-to-end on binary classification tasks using data from the Materials Project, with labels determined by the presence or absence of experimental entries in the Inorganic Crystal Structure Database (ICSD) [10].

Table 2: Machine Learning Approaches for Stability and Synthesizability Prediction

Model Type	Key Features	Strengths	Limitations
Compositional Models	Elemental stoichiometry and properties [9]	Fast screening of unknown compositions [9]	Poor stability prediction (cannot distinguish polymorphs) [9]
Structural Models	Crystal structure graphs [10] [9]	Accurate stability assessment [9]	Requires known crystal structure [9]
Integrated Models	Composition + structure [10]	State-of-the-art synthesizability prediction [10]	Computational complexity
Network Models	Position in stability network [12]	Captures historical discovery patterns [12]	Limited to explored chemical spaces [12]

Performance Assessment and Validation

When evaluating machine learning models for materials discovery, it is crucial to distinguish between formation energy prediction and stability prediction. While numerous models approach DFT accuracy for formation energy prediction, they often perform poorly on stability prediction because DFT benefits from systematic error cancellation when comparing energies of chemically similar compounds, while ML models typically do not [9].

This performance gap has significant practical implications: compositional ML models exhibit high rates of false positives, predicting many materials as stable that DFT calculations indicate are unstable [9]. This limitation substantially impedes their utility for efficient materials discovery, particularly in sparse chemical spaces where few stoichiometries have stable compounds [9].

Experimental Validation and Case Studies

Synthesizability-Guided Discovery Pipeline

Recent research has demonstrated an integrated synthesizability-guided pipeline that successfully bridged the computational-experimental gap [10]. This approach screened 4.4 million computational structures through a multi-stage process:

• Initial Screening: 1.3 million structures calculated to be synthesizable through convex hull analysis
• High-Synthesizability Filter: Selection of candidates with >0.95 rank-average synthesizability score
• Compositional Filtering: Removal of platinoid-group elements, non-oxides, and toxic compounds
• Retrosynthetic Planning: Application of precursor-suggestion models (Retro-Rank-In) and temperature prediction models (SyntMTE) trained on literature-mined synthesis recipes

This pipeline yielded approximately 500 high-priority candidates, from which 24 targets were selected for experimental validation [10]. Through high-throughput automated synthesis, 16 samples were successfully characterized, with 7 matching the target structure—including one completely novel compound and one previously unreported structure [10]. The entire experimental process from computational selection to characterization was completed in just three days, demonstrating the practical efficiency of synthesizability-guided approaches [10].

The Critical Role of Synthesis Planning

A significant limitation of convex hull stability emerges in its inability to provide guidance on synthesis parameters—which precursors to use, what reaction temperatures and times are optimal, or which atmospheric conditions are required [11]. This has prompted research into text-mining synthesis recipes from literature to build predictive models for synthesis planning.

However, critical analysis of text-mined synthesis data has revealed substantial challenges in the "4 Vs" of data science [11]:

• Volume: 31,782 solid-state recipes represent limited coverage of chemical space
• Variety: Significant biases in researched material families
• Veracity: Extraction errors and reporting inconsistencies
• Velocity: Historical data lacks information on novel synthesis techniques

Despite these limitations, the most valuable insights from text-mined data have come from anomalous recipes that defy conventional synthetic intuition, leading to new mechanistic hypotheses about how solid-state reactions proceed [11].

Research Reagent Solutions

Table 3: Essential Computational and Experimental Resources

Resource Category	Specific Tools	Function	Access
Computational Databases	Materials Project [10], GNoME [10], Alexandria [10], OQMD [12]	Source of DFT-calculated structures and energies	Public web platforms
Synthesis Databases	Text-mined synthesis recipes [11]	Training data for synthesis prediction models	Research access
Stability Analysis	Convex hull construction algorithms [13] [12]	Determine thermodynamic stability	Integrated in databases
Synthesizability Models	Compositional & structural ML models [10]	Predict experimental accessibility	Research implementations
Synthesis Planning	Retro-Rank-In [10], SyntMTE [10]	Predict precursors and conditions	Research implementations
Experimental Validation	High-throughput robotics [10], Automated XRD [10]	Rapid synthesis and characterization	Specialist facilities

The convex hull remains an essential foundation for materials stability assessment, providing critical thermodynamic constraints on synthesizability. However, the experimental reality of kinetic trapping necessitates moving beyond purely thermodynamic considerations to integrated models that account for synthetic accessibility, precursor selection, and kinetic pathways.

The most promising approaches combine compositional and structural information with historical synthesis data and network-based analysis to create synthesizability scores that better align with experimental outcomes. As these models mature, they increasingly bridge the gap between computational prediction and experimental realization, as demonstrated by recent success in synthesizing novel compounds identified through synthesizability-guided pipelines.

Future progress will require improved synthesis databases that address current limitations in volume, variety, veracity, and velocity, alongside more sophisticated models that explicitly incorporate kinetic barriers and synthetic pathway analysis. Through these advances, the research community can increasingly leverage the power of computational materials discovery while navigating the complex interplay between thermodynamic stability and kinetic trapping that ultimately determines synthetic success.

The Overprediction Problem in Crystal Structure Prediction

Computational materials discovery has generated millions of hypothetical crystal structures through databases like the Materials Project, GNoME, and Alexandria, surpassing the number of experimentally synthesized compounds by more than an order of magnitude [10]. The central challenge in this field is the overprediction problem – the tendency of computational methods to generate far more plausible crystal structures than are actually experimentally accessible. This problem creates significant inefficiencies in materials discovery, as researchers may waste resources attempting to synthesize theoretically favored structures that cannot be practically realized [14].

The prevailing approach to screening candidate materials has relied heavily on density functional theory (DFT) calculations of thermodynamic stability, typically using the convex-hull distance (energy above the convex hull) as the primary filter [10]. While this method constitutes a useful first filter, it typically overlooks finite-temperature effects, namely entropic and kinetic factors, that govern synthetic accessibility [10]. This fundamental limitation has created a pressing need for more accurate synthesizability assessments to efficiently steer scientists toward compounds that are readily accessible in the laboratory [10].

Root Causes of Overprediction

Limitations of Thermodynamic Stability Metrics

The convex-hull stability approach operates effectively at 0 Kelvin but fails to account for critical factors that determine synthesizability at experimental conditions:

• Neglect of kinetic factors: Energy barriers between polymorphs can prevent access to thermodynamically favored structures [14]
• Entropic effects: Finite-temperature free energy landscapes differ significantly from 0K potential energy surfaces [14]
• Synthetic pathway complexity: The availability of suitable precursors and reaction pathways fundamentally determines synthesizability [15]

The disconnect is evident in real systems: The Materials Project lists 21 SiO₂ structures within 0.01 eV of the convex hull, yet the second most common phase (cristobalite) is not among these 21 [10]. Similarly, numerous structures with favorable formation energies remain unsynthesized, while various metastable structures with less favorable formation energies are successfully synthesized [15].

Finite-Temperature Effects on Energy Landscapes

At the heart of the overprediction problem lies the discrepancy between potential energy surfaces (used in conventional CSP) and free energy surfaces (relevant to experimental crystallization):

• Potential energy surfaces are much rougher than free energy surfaces [14]
• Thermal energy allows minima separated by small energy barriers (on the order of kT) to coalesce into single free energy basins [14]
• Free energy basins typically correspond not to a single potential energy minimum but rather an ensemble of minima [14]

This coalescence effect means that conventional CSP methods that treat each local minimum as a potentially observable polymorph inevitably overpredict the number of accessible structures.

Computational Approaches to Reduce Overprediction

Synthesizability-Guided Frameworks

Recent advances integrate complementary signals from composition and crystal structure to predict synthesizability more accurately:

Table 1: Performance Comparison of Synthesizability Prediction Methods

Method	Approach	Accuracy/Performance	Key Features
Combined Compositional & Structural Score [10]	Integrated ML model with composition & structure encoders	Successfully synthesized 7 of 16 predicted targets	Rank-average ensemble of composition and structure predictions
Crystal Synthesis LLM (CSLLM) [15]	Fine-tuned large language models on crystal structures	98.6% accuracy on test data	Predicts synthesizability, synthetic methods, and precursors
Threshold Clustering [14]	Monte Carlo sampling with energy thresholds	Significant reduction in predicted polymorphs for benzene, acrylic acid, resorcinol	Groups minima into finite-temperature basins
Positive-Unlabeled Learning [15]	Semi-supervised ML on known and theoretical structures	87.9% accuracy for 3D crystals	Identifies non-synthesizable structures from large theoretical databases

Unified Synthesizability Model Architecture

A state-of-the-art synthesizability model integrates compositional and structural information through a dual-encoder architecture [10]:

• Compositional encoder: Fine-tuned MTEncoder transformer operating on stoichiometry and elemental properties
• Structural encoder: Graph neural network fine-tuned from the JMP model processing crystal structure graphs
• Rank-average ensemble: Combines predictions from both encoders via Borda fusion for enhanced ranking

This approach screened 4.4 million computational structures, identified 1.3 million as synthesizable, and ultimately led to successful experimental synthesis of 7 out of 16 characterized targets within just three days [10].

Large Language Models for Synthesizability Prediction

The Crystal Synthesis Large Language Model (CSLLM) framework demonstrates how domain-adapted LLMs can transform synthesizability prediction [15]:

• Specialized components: Three dedicated LLMs for synthesizability prediction, synthetic method classification, and precursor identification
• Material string representation: Efficient text representation integrating essential crystal information (space group, lattice parameters, atomic coordinates)
• Balanced dataset: 70,120 synthesizable structures from ICSD + 80,000 non-synthesizable structures identified via PU learning

The CSLLM framework significantly outperforms traditional stability-based screening, achieving 98.6% accuracy compared to 74.1% for energy-above-hull (≥0.1 eV/atom) and 82.2% for phonon stability thresholds [15].

Finite-Temperature Clustering Methods

Threshold Clustering Workflow

The threshold algorithm addresses overprediction by accounting for the coalescence of potential energy minima at finite temperatures [14]:

Diagram 1: Threshold clustering workflow for reducing CSP overprediction. This method groups minima separated by small energy barriers into finite-temperature basins.

The threshold clustering workflow operates on the original CSP energy surface, eliminating ambiguity regarding the connection between the reduced structure set and the original landscape [14]. This method has demonstrated significant reductions in predicted polymorphs for model systems including benzene, acrylic acid, and resorcinol.

Molecular Dynamics for Free Energy Clustering

Alternative approaches use molecular dynamics and enhanced sampling simulations to group CSP structures into free energy clusters [14]. While physically rigorous, these methods have not been widely adopted due to their complexity in both simulation and analysis, and limitations in using common MD force fields rather than the more accurate energy models typically required for CSP [14].

Topological and Physical Descriptor Approaches

For molecular crystals, the CrystalMath approach demonstrates how mathematical principles can predict stable structures without reliance on interatomic interaction models [16]:

• Principal axis alignment: Molecules orient such that principal inertial axes align with specific crystallographic directions
• Ring plane vector alignment: Normal vectors to chemically rigid subgraphs align with crystallographic planes
• Geometric order parameters: Heavy atoms occupy positions corresponding to minima of geometric order parameters

This topological approach, combined with filtering based on van der Waals free volume and intermolecular close contact distributions from the Cambridge Structural Database, enables prediction of stable structures entirely mathematically without force field dependencies [16].

Experimental Protocols and Validation

High-Throughput Experimental Validation

Rigorous validation of synthesizability predictions requires high-throughput experimental workflows:

Table 2: Experimental Synthesis and Characterization Methods

Method	Application	Implementation Details	Output Metrics
Solid-State Synthesis [10]	Inorganic crystal synthesis	Precursor grinding and calcination in muffle furnace	Phase purity, crystallinity
X-ray Diffraction [10]	Phase identification	Automated XRD characterization	Structure matching, phase identification
Retrosynthetic Planning [10]	Synthesis pathway prediction	Retro-Rank-In precursor suggestion + SyntMTE temperature prediction	Viable precursor pairs, calcination temperatures

Retrosynthetic Planning Protocol

The synthesis planning workflow combines two specialized models [10]:

• Precursor suggestion: Retro-Rank-In model produces ranked lists of viable solid-state precursors
• Temperature prediction: SyntMTE predicts calcination temperatures required to form target phases
• Reaction balancing: Automatic balancing of chemical reactions and computation of precursor quantities

Both models are trained on literature-mined corpora of solid-state synthesis recipes, ensuring practical relevance [10].

Target Selection and Batch Processing

To optimize experimental efficiency [10]:

• Web-assisted novelty assessment: LLM-based searching to identify previously synthesized compounds
• Expert validation: Removal of targets with unrealistic oxidation states or well-explored compositions
• Recipe-similarity batching: Automatic selection of targets that can be synthesized simultaneously in furnace batches

This approach enabled efficient experimental validation of 24 targets across two batches, with 8 samples lost to crucible bonding issues and 16 successfully characterized [10].

Large-Scale Method Validation

Comprehensive validation across diverse molecular systems is essential for assessing CSP method performance:

• Multi-tier benchmark sets: 66 molecules with 137 experimentally known polymorphs spanning rigid molecules to flexible drug-like compounds [17]
• Clustering analysis: Removal of trivial duplicates through RMSD-based clustering (e.g., RMSD₁₅ < 1.2 Å) [17]
• Free energy calculations: Temperature-dependent stability assessment through free energy calculations [17]

This rigorous validation framework demonstrated that proper clustering can significantly improve ranking of experimentally observed polymorphs, addressing one aspect of the overprediction problem [17].

Research Reagent Solutions: Computational Tools for CSP

Table 3: Essential Computational Tools for Crystal Structure Prediction

Tool/Resource	Type	Function	Application Context
Matbench Discovery [18]	Evaluation framework	Benchmarks ML energy models as DFT pre-filters	Prospective materials discovery
UBEM Approach [19]	Graph neural network	Predicts volume-relaxed energies from unrelaxed structures	High-throughput stability screening
GLEE Program [14]	CSP software	Generates initial crystal structure landscapes	Polymorph sampling
CSLLM Framework [15]	Fine-tuned LLMs	Predicts synthesizability, methods, and precursors	Synthesis-aware candidate screening
Threshold Algorithm [14]	Monte Carlo method	Estimates energy barriers between minima	Finite-temperature basin identification

The overprediction problem in crystal structure prediction represents a fundamental challenge in computational materials science. While convex-hull stability remains a valuable initial filter, overcoming overprediction requires moving beyond thermodynamic stability to incorporate kinetic accessibility, synthetic pathway feasibility, and finite-temperature effects.

The most promising approaches share a common theme: integrating multiple complementary signals – compositional, structural, and synthetic – to provide a more realistic assessment of synthesizability. As these methods continue to mature, they promise to significantly accelerate the discovery of novel functional materials by focusing experimental resources on genuinely accessible candidates.

The integration of synthesizability prediction directly into materials discovery pipelines, as demonstrated by the successful experimental validation of computationally predicted targets, marks a critical step toward bridging the gap between theoretical prediction and practical synthesis in materials science.

The prediction of synthesizable materials has long been dominated by the analysis of thermodynamic stability, primarily through the computation of the energy above the convex hull. While this approach identifies structurally plausible compounds, it frequently fails to predict which materials can be experimentally realized, as it overlooks critical kinetic and synthetic factors. This whitepaper examines how porosity, solvent inclusion, and other stabilization mechanisms control synthetic accessibility, moving beyond a purely thermodynamic framework. We present recent advances in computational models, including synthesizability-guided pipelines and large language models, that achieve unprecedented accuracy in predicting experimental outcomes by integrating these features. Furthermore, we provide detailed protocols and a curated toolkit to empower researchers in deploying these strategies for accelerated materials and drug development.

In computational materials discovery, density functional theory (DFT) methods have been extensively used to evaluate candidate structures, with thermodynamic stability judgments typically based on an material's energy above the convex hull [10]. This approach, while accurate for identifying ground-state structures at zero Kelvin, often favors low-energy configurations that are not experimentally accessible, creating a significant gap between prediction and synthesis [10] [15].

The principal shortcoming of convex-hull analysis lies in its neglect of finite-temperature effects, entropic contributions, and kinetic factors that govern synthetic accessibility in real laboratory conditions [10]. For instance, the Materials Project lists 21 SiO₂ structures within 0.01 eV of the convex hull, yet the common cristobalite phase is not among them [10]. This demonstrates that thermodynamic stability, while necessary, is insufficient for predicting synthesizability.

Key Stabilization Mechanisms Beyond Thermodynamics

Porosity and Framework Stabilization

Porous materials can achieve stabilization through specific structural mechanisms that defy simple thermodynamic predictions. In crystalline systems, framework stability often depends on the balanced interplay of different bonding types and pore stabilization mechanisms.

• Carbon-Induced Porosity: In hydrogen storage materials like MgH₂, doping with Ni/C nano-catalysts creates a growing porous structure during cycling that maintains high capacity. This carbon-induced-porosity stabilization mechanism stabilizes the proportion of rapid hydrogen absorption process, directly enhancing cycling performance [20].
• Non-Metal Organic Frameworks: Porous ammonium halide salts demonstrate how ionic clustering can direct crystallization into predictable porous frameworks. The charged nodes in these salt frameworks create tight ionic clusters that enable the formation of isoreticular series of porous structures, analogous to metal-organic frameworks but without directional coordinate covalent bonding [21].
• Cohesive Granular Systems: In powder systems, porous aggregates can be stabilized through a balance between cohesion strength and external pressure. The final porosity depends on the scaled cohesion force and the lateral distance between branches of ballistic deposits, with Coulomb friction alone capable of stabilizing pores, particularly for non-spherical particles [22].

Solvent Inclusion and Templating Effects

Solvent interactions play a crucial role in stabilizing metastable phases and directing synthesis pathways through both thermodynamic and kinetic pathways.

• Solvent-Assisted Bio-oil Stabilization: The stabilization of bio-oil organic phases demonstrates how solvent polarity and hydrogen-donating capability significantly impact stabilization efficiency. Highly polar solvents like methanol and ethanol yield higher oil fractions (64% and 62% respectively) by effectively reducing polymerization and coking during mild hydrotreatment processes [23].
• Solvent Templating in Porous Salts: For porous organic salts, solvent inclusion during crystallization can template specific porous frameworks. Computational crystal structure prediction (CSP) reveals that certain ammonium halide salts form stable, predictable porous structures with channel sizes and geometries that can be designed a priori, enabled by solvent-directed crystallization [21].

Table 1: Quantitative Comparison of Stabilization Mechanisms

Mechanism	Material System	Key Parameters	Performance Impact	Reference
Carbon-Induced Porosity	MgH₂ Ni/C for H₂ storage	Porous structure growth during cycling	Maintains 98.8% capacity after 50 cycles vs. 85.2% for undoped MgH₂	[20]
Solvent Polarity Stabilization	Bio-oil organic phase	Solvent polarity, hydrogen-donating capability	Methanol: 74.8% efficiency; Ethyl ether: 63.6% efficiency	[23]
Ionic Cluster Direction	Porous ammonium halide salts	Ionic charge density, linker rigidity	Iodine capture exceeding most MOFs	[21]
Cohesive Pore Stabilization	Cohesive granular powders	Cohesion force, external pressure, particle shape	Stronger effect for nonspherical particles vs. round particles	[22]

Computational Advances in Synthesizability Prediction

Integrated Synthesizability Models

Recent approaches have moved beyond standalone thermodynamic assessments to integrated models that simultaneously evaluate composition and structure.

Synthesizability Prediction Workflow

A synthesizability-guided pipeline for materials discovery combines compositional and structural signals through a dual-encoder architecture [10]. The model uses a fine-tuned compositional MTEncoder transformer for stoichiometric analysis and a graph neural network (GNN) for crystal structure evaluation, with outputs combined via rank-average ensemble (Borda fusion) to prioritize candidates [10]. This approach identified several hundred highly synthesizable candidates from over 4.4 million computational structures, with experimental validation successfully synthesizing 7 of 16 targets within just three days [10].

Large Language Models for Synthesis Prediction

The Crystal Synthesis Large Language Models (CSLLM) framework represents a breakthrough in synthesizability prediction, achieving 98.6% accuracy by specializing LLMs for three distinct tasks: synthesizability classification, synthetic method recommendation, and precursor identification [15]. This significantly outperforms traditional methods based on energy above hull (74.1% accuracy) or phonon spectrum analysis (82.2% accuracy) [15].

The framework uses a novel text representation called "material string" that efficiently encodes crystal information for LLM processing, enabling the identification of 45,632 synthesizable materials from 105,321 theoretical structures [15].

Machine Learning for Complex Chemical Spaces

For complex material classes like Zintl phases, graph neural networks with upper bound energy minimization (UBEM) have demonstrated remarkable efficiency in discovering stable compounds [19]. This approach screened over 90,000 hypothetical Zintl phases and identified 1,810 new thermodynamically stable phases with 90% precision validated by DFT calculations, more than doubling the accuracy of existing models like M3GNet (40% precision) on the same dataset [19].

Table 2: Computational Methods for Synthesizability Prediction

Method	Approach	Accuracy/Performance	Advantages	Limitations
Integrated Synthesizability Model	Combines composition (transformer) and structure (GNN) encoders	Successfully synthesized 7/16 predicted targets in 3 days	Integrates complementary signals from composition and structure	Requires curated training data from experimental databases
CSLLM Framework	Three specialized LLMs for synthesizability, method, and precursors	98.6% synthesizability accuracy; >90% method classification	Outstanding generalization to complex structures beyond training data	Dependent on comprehensive dataset for fine-tuning
UBEM-GNN for Zintl Phases	Scale-invariant GNN predicting volume-relaxed energy from unrelaxed structures	90% precision in predicting DFT-stable phases; 27 meV/atom MAE	2× more accurate than M3GNet; avoids full DFT relaxation	Limited to volume relaxation only
Bayesian Optimization for ESF Maps	Parallel multi-objective Bayesian optimization for energy-structure-function maps	100× acceleration; saved >500,000 CPU hours	Dramatically reduces computational cost for porous materials screening	Complex implementation for multi-objective optimization

Experimental Protocols and Methodologies

Synthesizability-Guided Experimental Pipeline

The experimental validation of computationally predicted materials requires a systematic approach to candidate selection and synthesis.

Prioritization Protocol:

• Initial Screening: Begin with a large pool of computational structures (e.g., 4.4 million) and apply a synthesizability score threshold (e.g., rank-average >0.95) [10].
• Composition Filtering: Remove compounds containing platinoid group elements, non-oxides, and toxic compounds [10].
• Literature Validation: Use web-searching LLMs and expert judgment to exclude previously synthesized candidates and unrealistic oxidation states [10].
• Synthesis Planning: Apply precursor-suggestion models (e.g., Retro-Rank-In) and temperature prediction models (e.g., SyntMTE) to generate viable synthesis routes [10].

High-Throughput Synthesis:

• Use automated solid-state laboratory systems with benchtop muffle furnaces
• Employ parallel processing with batches of 12 samples simultaneously
• Characterize products automatically by X-ray diffraction (XRD)
• Address crucible bonding issues through appropriate material selection [10]

Solvent-Assisted Stabilization Protocol

For solvent-dependent stabilization systems like bio-oil upgrading:

Feed Preparation:

• Blend 80% by-weight bio-oil organic phase (BOP) with 20% by-weight solvent [23]
• Test solvents with varying polarity: methanol (polar), ethanol (polar-protic), isopropyl alcohol (less polar), ethyl ether (nonpolar) [23]

Stabilization Procedure:

• Load reactor with Ru/C catalyst (5 wt% metal loading, 686.55 m²/g BET surface area) [23]
• Process under mild hydrodeoxygenation conditions (subcritical up to 200°C) [23]
• Analyze products for density, viscosity, total acid number, elemental composition
• Calculate degree of deoxygenation, dehydration, and energy efficiency [23]

The Scientist's Toolkit: Essential Research Reagents and Materials

Table 3: Key Research Reagents for Stabilization Studies

Reagent/Material	Function/Application	Example Specifications	Reference
Ru/C Catalyst	Bio-oil stabilization via mild hydrodeoxygenation	5 wt% metal loading, 686.55 m²/g BET, 3.3 nm pore diameter	[23]
Ni/C Nano-catalyst	Hydrogen storage material doping for porosity induction	Creates growing porous structure in MgH₂ during cycling	[20]
DEHPA Extractant	Solvent-impregnated resin for metal ion separation	Di-2-ethylhexyl phosphoric acid; requires capacity stabilization	[24]
Amberlite XAD-2	Macroporous support for solvent-impregnated resins	Styrene-divinylbenzene copolymer for extractant immobilization	[24]
Tetrahedral Amine Linkers	Building blocks for porous organic salt frameworks	Tetrakis-(4-aminophenyl)methane (TAPM) for ammonium halide salts	[21]
Trigonal Triamine Linkers	Rigid components for predictable porous salt formation	TT, TTBT, TAPT for isoreticular frameworks with halide anions	[21]

The reliance on convex-hull stability as the primary metric for synthesizability prediction represents an oversimplified approach that fails to capture the complex kinetic and synthetic factors governing experimental realization. Porosity mechanisms, solvent inclusion, and framework stabilization effects play decisive roles in determining which computationally predicted materials can be successfully synthesized. The integration of these factors into machine learning models, particularly through synthesizability-guided pipelines and specialized large language models, has demonstrated remarkable improvements in prediction accuracy and experimental success rates. As these computational approaches continue to evolve, incorporating increasingly sophisticated representations of stabilization mechanisms, they promise to significantly accelerate the discovery and development of novel materials for advanced technological applications.

In computational materials discovery, the prediction of a material's synthesizability is a critical bottleneck. While high-throughput simulations can generate millions of candidate structures, most prove inaccessible in the laboratory. Within this challenge, convex-hull stability has emerged as a foundational metric for prioritizing candidates for experimental synthesis [10] [11]. The hull distance, or energy above hull, quantifies a compound's thermodynamic stability relative to competing phases in its chemical space. A material on the convex hull (0 eV/atom hull distance) is thermodynamically stable at 0 K, while a positive value indicates metastability or instability [1] [9].

However, hull distance interpretation is nuanced. Traditional density functional theory (DFT) methods, while accurate at zero Kelvin, often favor low-energy structures that are not experimentally accessible, overlooking finite-temperature effects and kinetic barriers [10]. This limitation has spurred the development of synthesizability scores that integrate hull stability with compositional and structural features [10], as well as advanced active learning approaches that directly minimize uncertainty in the convex hull itself [7] [8]. This technical guide details the interpretation of hull distances across system complexities, framed within the broader thesis that accurate stability assessment is indispensable for—but not the sole determinant of—successful synthesis prediction.

Core Concepts: The Fundamentals of Convex Hull Construction

Thermodynamic Foundation

The construction of a phase diagram begins with the calculation of formation energies. For a phase composed of N components, the formation energy per atom, ΔEƒ, is calculated as:

ΔEƒ = E - ∑ᵢᴺnᵢμᵢ

where E is the total energy of the phase, nᵢ is the number of moles of component i, and μᵢ is the chemical potential (energy) of component i [1]. For solid-state systems at 0 K and 0 atm pressure, the relevant thermodynamic potential simplifies to the internal energy, E [1].

The Convex Hull Algorithm

The convex hull is constructed by taking the lower convex envelope of the formation energies of all known compounds in a chemical system. The hull represents the set of stable phase-composition pairs with the lowest possible energy at their respective compositions [1] [7]. In a binary A-B system, the hull is a line; in a ternary A-B-C system, it becomes a surface; and in higher-dimensional systems, it is a hyper-surface [1].

Table 1: Key Mathematical Definitions in Hull Analysis

Term	Symbol	Definition	Interpretation
Formation Energy	ΔEƒ	E - ∑ᵢᴺnᵢμᵢ	Energy to form a compound from its constituent elements.
Hull Distance	ΔEd	ΔEƒ,compound - ΔEƒ,hull	Decomposition energy to the most stable phases.
Stable Compound	ΔEd ≤ 0	Lies on the convex hull.	Thermodynamically stable at 0 K.
Metastable Compound	ΔEd > 0	Lies above the convex hull.	May be synthesizable kinetically.

The following diagram illustrates the convex hull construction process and the determination of the hull distance in a binary system:

Diagram Title: Convex Hull Construction and Hull Distance Calculation

Computational Methodologies: From DFT to Machine Learning

Density Functional Theory Workflows

DFT provides the foundational energy calculations for hull construction. The Materials Project methodology involves:

• Energy Calculations: Performing DFT calculations with appropriate exchange-correlation functionals (GGA/GGA+U/R2SCAN) for all known compounds in a chemical system [1].
• Mixing Schemes: Applying energy corrections to ensure consistency across different calculation methods, particularly when mixing GGA/GGA+U and R2SCAN results [1].
• Hull Construction: Using the pymatgen code to build the phase diagram and calculate hull distances [1].

Table 2: Computational Methods for Hull Analysis

Method	Key Principle	Application in Hull Analysis	Key Researchers
Standard DFT	First-principles energy calculation.	Provides formation energies for hull construction.	Materials Project [1]
Convex Hull-Aware Active Learning (CAL)	Bayesian optimization targeting hull uncertainty.	Minimizes experiments needed to resolve the hull; provides uncertainty quantification.	Novick et al. [7] [8]
Machine-Learned Formation Energies	Statistical models trained on DFT data.	Rapid screening; limited by poor stability prediction accuracy.	Multiple models [9]

Advanced Approaches: Convex Hull-Aware Active Learning

The Convex Hull-Aware Active Learning (CAL) algorithm represents a significant advancement for efficiently mapping phase diagrams. CAL uses Gaussian process regressions to model energy surfaces and produces a posterior distribution over possible convex hulls [7] [8]. The algorithm's policy selects the next composition to observe based on expected information gain for the hull itself, not just the energy surface. This allows the convex hull to be predicted with significantly fewer observations than brute-force approaches [8].

Diagram Title: Convex Hull-Aware Active Learning Workflow

Experimental Protocols: From Computation to Synthesis

Synthesizability-Guided Pipeline

A recently developed synthesizability-guided pipeline demonstrates the integration of hull analysis with experimental synthesis. The methodology involves:

• Candidate Screening: Screening 4.4 million computational structures from databases (Materials Project, GNoME, Alexandria) using a combined compositional and structural synthesizability score [10].
• Stability Filtering: Applying a rank-average ensemble of composition and structure-based models to prioritize highly synthesizable candidates, followed by filtering out platinoid elements, non-oxides, and toxic compounds [10].
• Synthesis Planning: Using Retro-Rank-In for precursor suggestion and SyntMTE for calcination temperature prediction, both trained on literature-mined solid-state synthesis corpora [10].
• High-Throughput Validation: Executing syntheses in an automated solid-state laboratory with characterization via X-ray diffraction (XRD) [10].

This pipeline successfully synthesized 7 of 16 target materials, with the entire experimental process completed in just three days, highlighting the practical utility of synthesizability assessment beyond hull stability alone [10].

The Scientist's Toolkit: Essential Research Reagents and Materials

Table 3: Key Research Reagents and Computational Tools for Hull Analysis

Reagent/Tool	Function/Role	Application Context
Thermo Scientific Thermolyne Muffle Furnace	High-temperature calcination of solid-state precursors.	Experimental synthesis of predicted materials [10].
X-ray Diffractometer (XRD)	Phase identification and structure verification of synthesis products.	Experimental validation of synthesized materials [10].
pymatgen (Python)	Open-source library for phase diagram construction and materials analysis.	Computational hull construction and analysis [1].
Gaussian Process Regression	Bayesian modeling of energy surfaces with uncertainty quantification.	Core component of CAL algorithm for probabilistic hulls [7] [8].
Retro-Rank-In & SyntMTE	Machine learning models for precursor and synthesis condition prediction.	Synthesis planning for computationally identified candidates [10].

Advanced Concepts: Stability Prediction Challenges and Synthesizability

The Critical Distinction: Formation Energy vs. Stability

A crucial limitation of machine-learned formation energies has been identified: while models can predict formation energy with reasonable accuracy, they perform poorly at predicting compound stability [9]. This occurs because:

• Different Energy Scales: Formation energies (ΔHƒ) typically span several eV/atom, while hull distances (ΔHd) are 1-2 orders of magnitude smaller (mean ± deviation = 0.06 ± 0.12 eV/atom) [9].
• System-Dependent Nature: Hull distance is a relative measure within a chemical system, not an intrinsic property, making it difficult to learn from compositional features alone [9].
• Lack of Error Cancellation: DFT benefits from systematic error cancellation when comparing energies of chemically similar compounds, while ML models do not necessarily preserve these relative rankings [9].

Beyond Thermodynamic Stability: The Synthesizability Score

The recognition that hull stability alone is insufficient for synthesis prediction has led to the development of unified synthesizability models. These integrate:

• Compositional Signals: Elemental chemistry, precursor availability, redox and volatility constraints.
• Structural Signals: Local coordination, motif stability, and packing environments [10].

Such models demonstrate that accurately predicting which compounds can be fabricated requires moving beyond the 0 K thermodynamic stability provided by the hull distance to include additional chemical and structural insights [10].

The interpretation of hull distances forms a necessary but insufficient component of synthesizability prediction. While the convex hull provides a fundamental thermodynamic filter at 0 K, successful experimental synthesis depends on additional factors including finite-temperature effects, kinetic barriers, and precursor accessibility. The emerging paradigm integrates hull stability within broader synthesizability frameworks that leverage both compositional and structural descriptors, along with literature-mined synthesis knowledge for pathway planning [10]. Furthermore, advanced computational approaches like convex hull-aware active learning are increasing the efficiency of stability mapping itself [7] [8]. The integration of these methodologies—combining accurate hull analysis with synthesizability prediction and experimental validation—represents the most promising path toward overcoming the predictive synthesis bottleneck in computational materials discovery.

Computational Advances: Machine Learning and Active Learning for Hull Construction

The discovery of new functional materials is a central goal of solid-state chemistry and materials science, capable of driving significant scientific and technological advancements. For decades, density functional theory (DFT) has served as the computational cornerstone for predicting material stability, predominantly through the calculation of formation enthalpies and convex-hull analysis [25]. This approach determines a compound's thermodynamic stability relative to competing phases in a chemical space, with structures on or near the convex hull considered potentially stable [19]. However, traditional DFT-driven stability assessment faces profound challenges that limit its predictive power for experimental synthesis. The method suffers from intrinsic energy resolution errors in exchange-correlation functionals, often rendering it unreliable for quantitatively predicting formation enthalpies and phase diagrams, particularly for complex ternary systems [25]. Furthermore, DFT calculations typically consider zero-temperature thermodynamics, overlooking finite-temperature effects, entropic contributions, and kinetic factors that govern synthetic accessibility in laboratory settings [10].

The fundamental disconnect between thermodynamic stability predicted by DFT and practical synthesizability has created a critical bottleneck in materials discovery pipelines. While high-throughput DFT calculations have generated millions of putative crystal structures in databases like the Materials Project, GNoME, and Alexandria, the number of proposed inorganic crystals now exceeds experimentally synthesized compounds by more than an order of magnitude [10]. This disparity highlights the pressing need for more sophisticated approaches that can accurately distinguish theoretically stable structures from those truly synthesizable in laboratory conditions. As computational materials design increasingly relies on these vast digital repositories, bridging the gap between DFT-based stability prediction and experimental synthesizability has emerged as one of the most significant challenges in the field.

The DFT Foundation: Limitations and Fundamental Gaps

Density functional theory provides the fundamental framework for understanding phase stability through total energy calculations. The standard approach involves computing the enthalpy of formation (H_f) for a compound relative to its constituent elements in their ground states according to the equation:

\begin{equation} Hf (A{xA}B{xB}C{xC}\cdots ) = H(A{xA}B{xB}C{xC}\cdots ) - xA H(A) -xB H(B) - xC H(C) - \cdots \end{equation}

where (H(A{x{A}}B{x{B}}C{x{C}})) represents the enthalpy per atom of the intermetallic compound or alloy, and (H(A)), (H(B)), and (H(C)) are the enthalpies per atom of elements A, B and C in their ground-state structures [25]. Structures with negative formation energies are considered potentially stable, with those lying on the convex hull deemed thermodynamically stable at zero temperature.

Despite its theoretical foundation, DFT exhibits systematic errors that limit its predictive accuracy for phase stability. The inherent accuracy of the energy functionals used in these calculations lacks the necessary energy resolution for reliable phase diagram prediction, particularly for ternary systems [25]. These errors, while often negligible in relative comparisons of similar structures, become critical when assessing the absolute stability of competing phases in complex alloys. As a result, direct predictions of phase diagrams using uncorrected DFT frequently fail to match experimental observations, especially in multicomponent systems relevant to advanced technological applications.

The limitation of conventional stability prediction extends beyond technical accuracy to fundamental conceptual gaps. Traditional convex-hull analysis assumes that synthesizability is primarily governed by thermodynamic stability at zero temperature. However, experimental evidence consistently demonstrates that metastable structures (those above the convex hull) are routinely synthesized, while many thermodynamically stable structures prove challenging to realize in practice [15]. This paradox underscores the critical influence of kinetic factors, precursor availability, and synthetic pathway accessibility in determining practical synthesizability—factors largely absent from standard DFT stability assessments.

Table 1: Key Limitations of DFT in Stability Prediction

Limitation Category	Specific Challenge	Impact on Prediction Accuracy
Functional Accuracy	Intrinsic energy resolution errors in exchange-correlation functionals	Systematic errors in formation enthalpies, especially for ternary systems
Thermodynamic Scope	Focus on zero-temperature, equilibrium conditions	Overlooks finite-temperature effects, entropic contributions, and kinetic factors
Synthesizability Gap	Inability to account for experimental accessibility	Fails to distinguish theoretically stable versus practically synthesizable compounds
Computational Cost	High computational expense for complex systems	Limits exploration of complex compositions and structural diversity

Machine Learning Solutions for Enhanced Stability Prediction

Machine learning approaches have emerged as powerful tools to address the limitations of DFT-based stability prediction, operating through two primary paradigms: correcting DFT calculations and direct stability prediction. These methods leverage patterns in existing materials data to enhance predictive accuracy while reducing computational cost.

Correcting DFT Errors with Machine Learning

One innovative approach involves using machine learning to systematically correct errors in DFT-calculated formation enthalpies. Researchers have developed neural network models that predict the discrepancy between DFT-calculated and experimentally measured enthalpies for binary and ternary alloys and compounds [25]. These models utilize a structured feature set comprising elemental concentrations, atomic numbers, and interaction terms to capture key chemical and structural effects. Implementation typically involves a multi-layer perceptron regressor with multiple hidden layers, optimized through leave-one-out cross-validation and k-fold cross-validation to prevent overfitting [25]. This hybrid DFT-ML approach maintains the physical foundation of DFT while significantly improving its quantitative accuracy for formation energy prediction.

Direct Stability Prediction with Graph Neural Networks

Beyond correcting DFT, graph neural networks have demonstrated remarkable capability in predicting thermodynamic stability directly from crystal structures. The Upper Bound Energy Minimization (UBEM) approach represents a particularly advanced implementation, using a scale-invariant GNN model to predict volume-relaxed energies from unrelaxed crystal structures [19]. This method successfully identified 1,810 new thermodynamically stable Zintl phases from a search space of over 90,000 hypothetical structures, achieving a remarkable 90% precision when validated against DFT calculations [19]. This performance significantly exceeded traditional MLIPs like M3GNet, which achieved only 40% precision on the same dataset.

Advanced Synthesizability Prediction with Large Language Models

Recent breakthroughs have adapted large language models for synthesizability prediction, framing crystal structures as text sequences using specialized representations like "material strings" [15]. The Crystal Synthesis Large Language Models framework employs three specialized LLMs to predict synthesizability, synthetic methods, and suitable precursors respectively [15]. This approach has demonstrated extraordinary accuracy, with the Synthesizability LLM achieving 98.6% accuracy—significantly outperforming traditional thermodynamic methods based on energy above hull (74.1% accuracy) and kinetic methods based on phonon spectrum analysis (82.2% accuracy) [15].

Table 2: Machine Learning Approaches for Stability and Synthesizability Prediction

Method Category	Key Innovation	Reported Performance	Applications
DFT Error Correction	Neural networks predicting discrepancy between DFT and experimental enthalpies	Improved accuracy in formation enthalpy prediction for ternary systems	Al-Ni-Pd and Al-Ni-Ti systems for high-temperature applications
Graph Neural Networks	Upper Bound Energy Minimization (UBEM) using unrelaxed structures	90% precision in identifying stable Zintl phases; 27 meV/atom MAE	Discovery of novel Zintl phases from >90,000 candidates
Large Language Models	CSLLM framework using text representations of crystal structures	98.6% accuracy in synthesizability prediction	Screening of 105,321 theoretical structures, identifying 45,632 as synthesizable
Similarity-Based Methods	Generalized Convex Hull with adapted SOAP kernel	Improved lattice energy prediction with Gaussian process regression	Molecular crystal structure prediction and stabilizable structure identification

Integrated Workflows: From Prediction to Synthesis

The most advanced computational pipelines integrate stability prediction with synthesis planning, creating end-to-end frameworks for materials discovery. These workflows typically begin with large-scale candidate generation, proceed through successive filtering stages, and culminate in synthesis pathway prediction.

Synthesizability-Guided Discovery Pipeline

A comprehensive synthesizability-guided pipeline for materials discovery demonstrates this integrated approach [10]. The process initiates with a pool of 4.4 million computational structures, which undergo successive filtering using a combined compositional and structural synthesizability score. This score integrates complementary signals from composition and crystal structure through two encoders: a fine-tuned compositional MTEncoder transformer for stoichiometric information and a graph neural network fine-tuned from the JMP model for structural information [10]. The model is trained on data from the Materials Project, with labels assigned according to whether experimental entries exist in the ICSD for given structures.

The screening process employs a rank-average ensemble method that aggregates predictions from both composition and structure models:

\begin{equation} \mathrm{RankAvg}(i)=\frac{1}{2N}\sum{m\in{c,s}}\left(1+\sum{j=1}^{N}\mathbf{1}!\big[s{m}(j){m}(i)\big]\right) \end{equation}

where (s_{m}(i)) represents the synthesizability probability predicted by model (m) for candidate (i) [10]. This approach prioritizes candidates based on their relative synthesizability ranking rather than applying absolute probability thresholds.

Following synthesizability screening, successful pipelines incorporate retrosynthetic planning to generate feasible synthesis routes. This involves applying precursor-suggestion models like Retro-Rank-In to produce ranked lists of viable solid-state precursors, followed by synthesis condition prediction using models like SyntMTE to determine appropriate calcination temperatures [10]. These models are trained on literature-mined corpora of solid-state synthesis, encoding collective experimental knowledge from published literature.

Experimental Validation and Performance

The ultimate test of these integrated workflows lies in experimental validation. In one implemented pipeline, researchers selected 24 targets across two batches of 12 based on recipe similarity, enabling parallel synthesis in a high-throughput laboratory setting [10]. The samples were weighed, ground, and calcined in a Thermo Scientific Thermolyne Benchtop Muffle Furnace, with subsequent characterization by X-ray diffraction. Of the 16 successfully characterized samples, seven matched the target structure, including one completely novel and one previously unreported structure [10]. This successful experimental validation, completed in just three days, demonstrates the practical utility of synthesizability-guided discovery pipelines in accelerating materials development.

Implementing effective stability and synthesizability prediction requires specialized computational tools and resources. The following table summarizes key components of the modern computational materials scientist's toolkit.

Table 3: Essential Resources for Stability and Synthesizability Prediction

Tool Category	Specific Tools/Models	Function	Application Context
Materials Databases	Materials Project, OQMD, AFLOW, JARVIS, ICSD	Provide training data and reference structures for stability assessment	Source of known stable and theoretical compounds for model training
ML Models for Stability	GNNs with UBEM approach, M3GNet, MatFormer	Predict thermodynamic stability from composition or structure	High-throughput screening of hypothetical compounds
Synthesizability Models	CSLLM Framework, SynthNN, Composition/Structure Integrative Models	Predict experimental accessibility beyond thermodynamic stability	Prioritizing candidates for experimental synthesis
Synthesis Planning	Retro-Rank-In, SyntMTE, Precursor LLMs	Suggest precursors and synthesis conditions	Transitioning from predicted structures to viable synthesis recipes
Descriptors & Kernels	SOAP, Material String, Voronoi Tessellations	Represent crystal structures for ML processing	Featurization for various machine learning algorithms
Validation Tools	Phonon spectrum calculation, XRD simulation, Elastic constant computation	Confirm dynamic and mechanical stability	Final verification before experimental synthesis

The integration of machine learning with traditional quantum mechanical calculations represents a paradigm shift in stability prediction for materials discovery. By addressing fundamental limitations of DFT through data-driven approaches, these methods provide more accurate assessment of thermodynamic stability while simultaneously incorporating practical synthesizability considerations. The most successful frameworks combine compositional and structural information through ensemble methods, leverage retrosynthetic planning for pathway prediction, and validate predictions through high-throughput experimental synthesis.

As these methodologies continue to mature, several emerging trends promise to further enhance their capabilities. The development of more sophisticated text representations for crystal structures will improve the performance of LLM-based approaches, while larger and more diverse training datasets will enhance model generalizability across chemical spaces. Additionally, increased integration of kinetic and thermodynamic factors in synthesizability assessment will better capture the complexities of real-world synthesis.

These computational advances are progressively bridging the gap between theoretical prediction and experimental realization in materials science. By providing more reliable assessment of which computationally predicted materials can be successfully synthesized in the laboratory, integrated stability-synthesizability pipelines are accelerating the discovery of novel functional materials and transforming the approach to materials design across scientific and industrial domains.

The accurate prediction of a material's synthesizability is a fundamental challenge in materials science and drug development. Conventional approaches have heavily relied on density functional theory (DFT) to compute formation energies as a primary metric for thermodynamic stability. However, a significant gap exists between thermodynamic stability and actual synthesizability; many materials with favorable formation energies remain unsynthesized, while numerous metastable structures are successfully synthesized in laboratories [15]. The convex hull, a global construct representing the lowest energy states across all competing phases in a chemical system, provides a more rigorous foundation for assessing thermodynamic stability. A material's energy above the convex hull directly indicates its relative stability, with low or near-zero values suggesting higher synthesizability potential [18]. Despite its theoretical importance, the integration of convex hull analysis into computational discovery pipelines has been hampered by its data-intensive nature, as determining the hull requires energetic information for numerous competing compositions and phases [13]. Convex Hull-Aware Active Learning (CAL) emerges as a novel Bayesian methodology that directly addresses this bottleneck by strategically selecting experiments to minimize convex hull uncertainty, thereby accelerating the identification of synthesizable materials with quantified reliability.

Core Principles of Convex Hull-Aware Active Learning

Theoretical Foundation and Bayesian Framework

Convex Hull-Aware Active Learning (CAL) represents a paradigm shift in how computational materials discovery approaches the stability prediction problem. Traditional active learning methods focus on minimizing the uncertainty or error in predicting individual material properties, such as formation energy. In contrast, CAL specifically targets the reduction of uncertainty in the convex hull itself—the global structure that determines thermodynamic stability across an entire chemical system [13]. This distinction is crucial because the thermodynamic stability of a material is not an intrinsic property but rather emerges from its energetic relationship to all other competing phases [13].

The Bayesian foundation of CAL enables probabilistic predictions with inherent uncertainty quantification. The algorithm maintains probabilistic beliefs about the energy landscape and updates these beliefs through sequential experimental design. By explicitly modeling the joint distribution over all competing phases, CAL can compute the probability that any given composition lies on the convex hull—the critical determinant of thermodynamic stability. This probabilistic approach naturally accommodates the complex, high-dimensional relationships in compositional space while providing confidence estimates essential for experimental decision-making [13].

The Active Learning Loop and Acquisition Strategy

CAL operates through an iterative closed-loop process that intelligently selects the most informative experiments. The algorithm's acquisition function prioritizes compositions that are probabilistically close to the estimated convex hull, as these regions contribute most significantly to reducing hull uncertainty [13]. This strategic sampling differs markedly from conventional approaches that might uniformly explore compositional space or focus solely on energy prediction accuracy.

The methodology leaves significant uncertainty in compositions quickly determined to be irrelevant to the convex hull, concentrating computational resources where they matter most for stability determination [13]. This targeted approach becomes particularly valuable in high-dimensional compositional spaces where exhaustive calculation is computationally prohibitive. Through this adaptive experimental design, CAL can achieve accurate convex hull predictions with significantly fewer observations than methods focused exclusively on energy prediction [13].

Table: Comparative Performance of Stability Prediction Methods

Method	Primary Metric	Accuracy	Key Limitations
Thermodynamic (Formation Energy)	Energy above hull ≥0.1 eV/atom	74.1% [15]	Poor correlation with actual synthesizability
Kinetic (Phonon Spectrum)	Lowest frequency ≥ -0.1 THz	82.2% [15]	Computationally expensive, imperfect correlation
CAL (Bayesian)	Hull probability	Not explicitly stated	Requires iterative computation
CSLLM (LLM-based)	Synthesizability classification	98.6% [15]	Requires extensive training data

CAL Implementation: Methodologies and Workflows

Experimental Protocol and Computational Framework

Implementing CAL requires careful attention to both the Bayesian computational infrastructure and the materials-specific energy calculations. The foundational protocol involves these critical stages:

• Initialization and Prior Definition: The process begins with a small initial dataset of computed formation energies for diverse compositions within the target chemical system. Bayesian priors are placed over the energy landscape, typically using Gaussian processes parameterized with materials-informed kernels that encode similarities between compositions [13].

• Probabilistic Hull Estimation: Using the current energy beliefs, the algorithm constructs a probabilistic convex hull. Unlike deterministic hulls, this representation captures uncertainty in hull topology and identifies compositions with ambiguous stability classifications [13].

• Acquisition and Experimental Selection: The core innovation of CAL lies in its acquisition function, which evaluates the expected information gain for reducing hull uncertainty. Compositions with high probability of lying near the hull boundary receive priority, as their experimental characterization maximally constrains the hull topology [13].

• Bayesian Updates and Iteration: As new energy measurements are acquired (through DFT or experiment), the Bayesian model updates its beliefs about the entire energy landscape. This update propagates through to the convex hull estimate, refining stability classifications across the compositional space [13].

• Convergence and Uncertainty Quantification: The iterative process continues until hull uncertainty falls below a predetermined threshold or computational resources are exhausted. The final output includes both point estimates of stability and quantified uncertainties for downstream decision-making [13].

Workflow Visualization

Research Reagent Solutions: Computational Tools for CAL Implementation

Table: Essential Computational Tools for CAL Implementation

Tool Category	Specific Examples	Function in CAL Workflow
Energy Calculation	Density Functional Theory (DFT) codes (VASP, Quantum ESPRESSO)	Provides formation energy measurements for Bayesian updates [13]
Bayesian Modeling	Gaussian Process libraries (GPyTorch, scikit-learn)	Implements probabilistic surrogate models for energy landscape [13]
Hull Construction	Pymatgen, AFLOW	Computes convex hulls from formation energy data [18]
Uncertainty Quantification	Monte Carlo simulation, Bayesian inference tools	Quantifies confidence in hull predictions and stability classifications [26]
Data Management	Materials Platform databases (Materials Project, OQMD)	Provides initial data and benchmark comparisons [18]

CAL in the Broader Context of Synthesis Prediction

Integration with Emerging Predictive Paradigms

CAL does not operate in isolation but complements other advanced methodologies for synthesizability prediction. Recent breakthroughs in large language models (LLMs) have demonstrated remarkable accuracy in crystal structure synthesizability classification. The Crystal Synthesis LLM (CSLLM) framework achieves 98.6% accuracy in predicting synthesizability of arbitrary 3D crystal structures, significantly outperforming traditional thermodynamic and kinetic stability assessments [15]. However, these LLM approaches differ fundamentally from CAL: they operate as black-box classifiers without explicit physical models of stability, while CAL directly engages with the physical principles of thermodynamics through Bayesian experimental design.

The relationship between these approaches is synergistic rather than competitive. CAL's uncertainty-aware exploration can efficiently generate high-quality training data for LLMs, particularly in underexplored compositional spaces. Conversely, LLM predictions can inform CAL priors, potentially accelerating convergence. Both methodologies address the critical limitation of traditional stability metrics, which show only moderate correlation with actual synthesizability (74.1% for thermodynamic stability based on hull distance) [15].

Comparison with Alternative Bayesian Strategies

CAL belongs to a broader ecosystem of Bayesian methods for materials discovery. Target-oriented Bayesian optimization (t-EGO) represents another advanced approach specifically designed to identify materials with target-specific properties rather than simply optimizing for maxima or minima [27]. This method employs a target-specific Expected Improvement (t-EI) acquisition function that samples candidates based on their distance from desired property values with associated uncertainty [27]. While t-EGO excels at precision targeting of property values, CAL specializes in efficient mapping of stability landscapes through convex hull awareness.

Another emerging paradigm is Bayesian optimization over problem formulation space, which addresses the challenge of dynamically defining design objectives as experimental understanding evolves [28]. This approach is particularly valuable in multi-objective optimization scenarios common in alloy development, where balancing conflicting properties like ductility, yield strength, and density requires flexible problem formulation [28]. CAL contributes to this framework by providing efficient stability assessment as one critical component in multi-attribute utility functions.

Table: Bayesian Optimization Methods in Materials Discovery

Method	Primary Application	Key Innovation	Limitations
CAL	Stability prediction	Convex hull-aware acquisition	Requires energy calculations for competing phases
t-EGO [27]	Target-specific property optimization	Target-specific Expected Improvement	Less efficient for exploratory mapping
Problem Formulation BO [28]	Multi-objective design	Dynamic problem space exploration	Complex preference modeling
LaMBO [29]	Biological sequence design	Joint autoencoder-GP architecture	Specialized for sequence data

Validation and Performance Benchmarking

Evaluation Metrics and Methodological Considerations

Rigorous evaluation of CAL performance requires specialized metrics beyond conventional regression measures. As highlighted in materials benchmarking initiatives, global metrics like mean absolute error (MAE) or root mean squared error (RMSE) can provide misleading confidence in stability prediction tasks [18]. Accurate regressors may still produce high false-positive rates if predictions near the critical decision boundary (0 eV/atom above hull) are misclassified [18]. Therefore, CAL should be evaluated primarily on classification performance metrics relevant to materials discovery, including:

• Stability Classification Accuracy: Precision and recall for identifying hull-stable phases
• Uncertainty Calibration: Reliability of confidence estimates across the compositional space
• Sample Efficiency: Reduction in experiments required to achieve target hull accuracy
• Prospective Performance: Generalization to genuinely new chemical systems beyond retrospective test splits [18]

The Matbench Discovery initiative provides a framework specifically designed for evaluating machine learning energy models in prospective materials discovery scenarios [18]. This benchmark addresses the critical disconnect between retrospective and prospective performance by incorporating test data generated through actual discovery workflows, creating realistic covariate shifts that better indicate real-world performance [18].

Comparative Performance Analysis

In comprehensive benchmarking studies, universal interatomic potentials (UIPs) have demonstrated superior performance as pre-filters for thermodynamic stability prediction [18]. However, CAL addresses a complementary aspect of the discovery pipeline: strategic experimental design rather than energy prediction alone. While direct quantitative comparisons of CAL performance are limited in the available literature, the methodology demonstrates qualitative advantages through its explicit uncertainty quantification and sample-efficient hull estimation [13].

The fundamental advantage of CAL lies in its alignment with the true objective of stability prediction—accurate convex hull construction—rather than the intermediary goal of energy prediction. By directly targeting hull uncertainty, CAL achieves more effective resource allocation compared to methods that optimize for energy prediction accuracy without considering the global phase relationships that ultimately determine stability [13].

Applications and Future Directions

Implementation in Autonomous Discovery Pipelines

CAL's uncertainty-aware framework makes it particularly valuable for integration into autonomous experimentation systems, including self-driving laboratories for materials discovery and drug development. In these applications, CAL can guide both computational and experimental resource allocation, prioritizing characterization of compositions with high potential impact on hull uncertainty [13] [28]. The Bayesian foundation naturally accommodates multi-fidelity data integration, combining expensive high-accuracy DFT calculations with faster but less accurate empirical potentials or experimental measurements.

For drug discovery applications, the convex hull concept translates to multi-objective optimization of molecular properties, where the "hull" represents the optimal trade-off surface between conflicting objectives like binding affinity, solubility, and synthetic accessibility [26] [29]. CAL's principles can be adapted to efficiently map this Pareto frontier, accelerating the identification of promising candidate molecules with balanced property profiles [29].

Limitations and Research Frontiers

Despite its theoretical advantages, CAL faces several practical challenges and opportunities for advancement:

• Scalability to High-Dimensional Systems: Current implementations may struggle with complex multi-component systems where the combinatorial explosion of compositions makes exhaustive hull construction prohibitive. Future research should explore dimensionality reduction and sparse approximation techniques tailored to hull geometry [13].

• Integration of Kinetic Factors: CAL focuses exclusively on thermodynamic stability, while real-world synthesizability depends critically on kinetic factors. Combining CAL with kinetic stability predictors could provide a more comprehensive synthesizability assessment [15].

• Human-in-the-Loop Optimization: Incorporating human expert feedback through frameworks like A/B testing of design preferences would enhance CAL's practical utility in experimental campaigns [28]. This approach allows domain knowledge to guide the exploration-exploitation balance without rigid predefined objectives.

• Cross-Paradigm Integration: The most promising future direction involves integrating CAL with complementary approaches like LLM-based synthesizability prediction [15] and universal interatomic potentials [18]. Such hybrid frameworks could leverage the respective strengths of physical models and data-driven approaches for accelerated materials discovery.

As autonomous experimentation matures, CAL's Bayesian approach to experimental design provides a principled foundation for balancing exploration of unknown chemical spaces with exploitation of promising regions for functional materials. The methodology represents a significant step toward fully autonomous materials discovery systems that dynamically formulate and solve design problems aligned with evolving scientific objectives and practical constraints [28].

Graph Neural Networks and the Upper Bound Energy Minimization Strategy

The discovery of new functional materials is a central goal of materials science, capable of ushering in significant scientific and technological advancements. Computational materials discovery has traditionally relied on density functional theory (DFT) methods to generate and assess plausible crystal structures, typically using convex-hull stability (often characterized by energy above hull) as the primary filter for thermodynamic stability. [10] While this approach constitutes a useful first filter, it typically overlooks finite-temperature effects, namely entropic and kinetic factors, that govern synthetic accessibility. [10] The current challenge is to determine which of the millions of predicted materials can actually be fabricated, as conventional stability metrics alone prove insufficient for predicting synthesizability. [10] [15]

This whitepaper explores the integration of Graph Neural Networks (GNNs) with the Upper Bound Energy Minimization (UBEM) strategy—a novel approach that addresses critical limitations in traditional materials discovery pipelines. By reframing the stability prediction problem, this methodology enables more accurate identification of synthesizable materials while dramatically reducing computational costs. When contextualized within a broader thesis on convex-hull stability's role in synthesis prediction, this integration represents a paradigm shift from purely thermodynamic assessments toward synthesis-aware prioritization frameworks.

Theoretical Foundation: Beyond Conventional Stability Metrics

Limitations of Convex-Hull Stability in Synthesis Prediction

Traditional computational approaches have heavily relied on convex-hull stability as a proxy for synthesizability. However, several critical limitations have emerged:

• Thermodynamic vs. kinetic stability: DFT-based convex hull analyses assess thermodynamic stability at zero Kelvin, often favoring low-energy structures that are not experimentally accessible due to kinetic barriers. [10]
• Metastable materials synthesis: Various metastable structures are successfully synthesized despite having less favorable formation energies at the convex hull minimum. [15]
• Abundance of near-stable structures: Materials databases now contain millions of predicted structures with favorable formation energies, making it increasingly difficult to distinguish purported stable structures from truly synthesizable ones. [10]

These limitations have created a pressing need for more accurate synthesizability assessments that incorporate both compositional and structural signals while maintaining computational efficiency.

Graph Neural Networks for Materials Representation

GNNs have emerged as powerful tools for materials property prediction due to their ability to naturally encode atomic connectivity and local chemical environments. [19] Unlike composition-based models, GNNs operating on crystal structure graphs can capture coordination environments, motif stability, and packing arrangements critical for stability prediction. [10] Modern GNNs trained on materials databases can predict thermodynamic stability with errors lower than the "chemical accuracy" of 1 kcal mol⁻¹ (43 meV per atom). [19]

Table 1: GNN Architectures for Materials Property Prediction

Model Type	Key Features	Typical Applications	Limitations
Scale-Invariant GNN	Normalizes input structure volumes; tolerant to volume changes	Predicting volume-relaxed energies from unrelaxed structures	Cannot account for large changes in fractional coordinates and cell shape [30]
Message-Passing GNN	Updates node representations via neighbor information	Formation energy prediction	Limited global structure capture [31]
Transformer-Graph Hybrid	Combines GNN with attention mechanisms; captures 4-body interactions	Data-scarce property prediction; stability assessment	Higher computational complexity [31]

Upper Bound Energy Minimization: Core Methodology

Theoretical Framework and Definitions

The Upper Bound Energy Minimization strategy addresses a fundamental challenge in ML-accelerated materials discovery: predicting the thermodynamic stability of hypothetical crystal structures before performing computationally expensive DFT relaxations. The method is built on several key principles:

• Upper bound definition: The approach defines an upper bound to the fully-relaxed DFT energy as the energy resulting from a constrained optimization over only cell volume, while fixing fractional atomic coordinates and cell shape. [32] [30]
• Mathematical foundation: By design, the volume-relaxed energy (EV) serves as a strict upper bound to the fully-relaxed energy (EF), such that EV ≥ EF. Consequently, if a volume-relaxed structure is thermodynamically stable, the fully relaxed structure will assuredly be stable. [19]
• Scale-invariant prediction: Because fractional atomic coordinates for volume-only relaxations are known a priori, this upper-bound energy can be quickly and accurately predicted with scale-invariant GNNs that are robust to volume changes. [30]

The UBEM Workflow and Implementation

The UBEM methodology implements a sophisticated screening pipeline that leverages the upper-bound energy principle:

• Structure generation: Create hypothetical structures via ionic substitution of known prototypes from crystallographic databases. [30] [19]
• Volume-only relaxation: Perform constrained DFT relaxations that optimize only unit cell volume while fixing atomic coordinates and cell shape. [30]
• GNN training: Train scale-invariant GNN models on databases containing both volume-only and fully-relaxed structures. [30]
• Stability screening: Apply the trained model to predict volume-relaxed energies of hypothetical structures and identify stable candidates. [19]
• DFT validation: Perform full DFT relaxation on top candidates to confirm thermodynamic stability. [19]

Diagram 1: UBEM strategy workflow for stable materials discovery.

Experimental Protocols and Validation

UBEM Implementation and Model Training

The experimental validation of UBEM requires careful construction of training datasets and model architecture selection:

• Data curation: Compile a database of DFT calculations comprising both fully-relaxed and volume-only relaxed structures. Law et al. used approximately 128,000 DFT calculations for this purpose. [32] [30]
• Model architecture: Implement scale-invariant GNNs that normalize input structure volumes, making the model less sensitive to volume changes during relaxation. [30]
• Training strategy: Train GNN models on volume-relaxed structures to predict the upper-bound energy directly from unrelaxed crystal structures. [19]
• Performance validation: Evaluate model performance using mean absolute error (MAE) between predicted and DFT-calculated volume-relaxed energies. State-of-the-art models achieve MAEs of approximately 27 meV per atom. [19]

Performance Benchmarks and Comparative Analysis

Rigorous validation studies demonstrate the effectiveness of the UBEM approach compared to traditional methods:

Table 2: UBEM Performance Benchmarks Across Material Systems

Material System	Structures Screened	Stable Candidates Predicted	DFT Validation Precision	Comparative Method Performance
Functional Materials (Law et al.)	14.3 million	2,003 compositions	>99%	N/A [30]
Zintl Phases (Chaliha et al.)	>90,000	1,810 new phases	90%	M3GNet: 40% precision [19]
Solid-State Battery Electrolytes (Law et al.)	Specific number not provided	Multiple promising candidates	>99%	Traditional methods: computationally prohibitive [30]

The exceptional performance of UBEM stems from its fundamental approach to the stability prediction problem. By using volume-relaxed energies as targets, the model incorporates examples of both favorable and unfavorable decorations, providing a better foundation to distinguish stable from unstable structures. [30]

Advanced GNN Architectures for Enhanced Prediction

Scale-Invariant GNN Formulations

Scale-invariant GNN architectures are crucial for UBEM implementation as they address a key challenge: predicting relaxed energies from unrelaxed structures. These architectures incorporate specific innovations:

• Volume normalization: Input crystal volumes are scaled to make the minimum edge length 1 Å, reducing sensitivity to volume distortions. [30]
• Geometric invariance: Models are designed to be invariant to translations, rotations, and reflections while maintaining sensitivity to structural rearrangements. [31]
• Multi-body interactions: Advanced architectures explicitly incorporate up to four-body interactions (atoms, bonds, angles, dihedral angles) to better capture periodicity and structural characteristics. [31]

Hybrid Framework and Convexified Architectures

Recent advances in GNN architectures further enhance stability prediction capabilities:

• Hybrid transformer-graph frameworks: Combine GNNs with transformer networks that process compositional features, simultaneously considering compositional properties and structure-property relationships. [31]
• Convexified Message-Passing GNNs: Novel frameworks that combine message-passing GNNs with convex optimization, enabling efficient training with theoretical guarantees and strong generalization. [33]
• Transfer learning schemes: Address data scarcity for specific properties by leveraging pre-trained models on data-rich source tasks (e.g., formation energy) to initialize training on data-scarce tasks. [31]

Diagram 2: Scale-invariant GNN architecture for volume-relaxed energy prediction.

Integrative Synthesizability Prediction

Combining Stability with Synthesis Planning

While UBEM addresses thermodynamic stability prediction, complete synthesizability assessment requires integration with additional models:

• Compositional synthesizability scores: Models that evaluate precursor availability, redox constraints, and elemental chemistry based on stoichiometry alone. [10]
• Structural synthesizability scores: Crystal structure-based models that assess local coordination, motif stability, and packing feasibility. [10]
• Rank-average ensemble: Combining compositional and structural models via rank-average fusion to enhance candidate prioritization. [10]
• Synthesis pathway prediction: Using retrosynthetic models (e.g., Retro-Rank-In) to suggest viable solid-state precursors and predict calcination temperatures. [10]

Experimental Validation and Success Rates

Recent implementations of synthesizability-guided pipelines demonstrate promising results:

• In a large-scale evaluation, a combined synthesizability score was used to screen 4.4 million computational structures, identifying approximately 500 highly synthesizable candidates. [10]
• Subsequent synthesis experiments characterized 16 targets, successfully synthesizing 7 matches to the target structure, including one completely novel material. [10]
• The entire experimental process from prediction to characterization was completed in just three days, highlighting the accelerated discovery potential. [10]

Research Reagent Solutions: Computational Materials Discovery

Table 3: Essential Research Tools for GNN-UBEM Implementation

Tool Category	Specific Examples	Function in Research	Implementation Notes
Materials Databases	Materials Project, ICSD, OQMD, JARVIS	Source of known structures for prototypes and training data	ICSD provides experimentally verified structures; Materials Project offers DFT-calculated properties [15] [30]
DFT Codes	VASP, Quantum ESPRESSO	Generate ground-truth data for energy calculations and relaxation	Volume-only relaxations require constrained optimization settings [30]
GNN Frameworks	PyTorch Geometric, Deep Graph Library	Implement scale-invariant GNN architectures	Pre-trained models available for transfer learning [31]
Structure Generation	Ionic substitution algorithms	Create hypothetical structures from known prototypes	pymatgen and ASE libraries provide implementation tools [30] [19]
Stability Analysis	Phase diagram construction tools	Calculate decomposition energy and energy above hull	Requires access to reference energy databases [30]

The integration of Graph Neural Networks with the Upper Bound Energy Minimization strategy represents a significant advancement in computational materials discovery. By reframing the stability prediction problem to leverage volume-relaxed energies as upper bounds, this approach enables highly accurate identification of thermodynamically stable materials with over 90% validation precision. [19] When contextualized within a broader thesis on convex-hull stability, UBEM demonstrates how ML-guided strategies can overcome fundamental limitations of traditional thermodynamic assessments.

Future research directions include developing unified models that simultaneously predict stability, synthesizability, and functional properties, incorporating out-of-equilibrium synthesis conditions, and creating fully automated discovery pipelines that integrate prediction with robotic synthesis and characterization. [10] As these methodologies mature, they promise to dramatically accelerate the discovery of functional materials for energy storage, catalysis, and beyond, while providing deeper insights into the fundamental principles governing materials stability and synthesis.

Adapted Similarity Kernels for Molecular Crystal Landscapes

The accurate prediction of molecular crystal structures is a cornerstone of modern materials science and pharmaceutical development. These predictions generate vast, complex energy landscapes containing numerous potential polymorphic structures. A critical challenge lies in intelligently analyzing these landscapes to identify which computationally predicted structures are both thermodynamically stabilizable and synthetically accessible. This whitepaper examines the role of adapted similarity kernels in navigating these landscapes, framing their development and application within the essential context of convex-hull stability for synthesizability prediction. The ability to distinguish plausible polymorphs within a crowded energy landscape directly impacts the efficiency of materials discovery and the mitigation of polymorphism-related risks in drug development.

The Role of Convex-Hull Stability in Synthesis Prediction

Within computational materials science, the convex hull of a compositional space serves as the fundamental thermodynamic reference for stability. A crystal structure's energy relative to this hull—its energy above hull—is a primary metric for assessing its likelihood of being synthesizable. Structures on the convex hull are thermodynamically stable, while those just above it may be metastable and synthetically accessible.

However, thermodynamic stability alone is an incomplete predictor of synthesizability. As highlighted by recent research, numerous structures with favorable formation energies remain unsynthesized, while various metastable structures are routinely synthesized [15]. This reality necessitates a more nuanced approach. The Generalised Convex Hull (GCH) addresses this by integrating unsupervised machine learning with structural similarity metrics [34]. Instead of relying solely on energy, the GCH identifies stabilizable crystal structure candidates by considering their position within a data-driven landscape that accounts for both thermodynamic stability and structural packing similarity. This method refines the search for synthesizable materials by acknowledging that kinetic factors and structural connectivity influence which thermodynamically plausible structures are actually realized in practice.

Similarity Kernels for Landscape Analysis

The core of advanced landscape analysis lies in the definition of similarity between two crystal structures. A similarity kernel is a mathematical function that quantifies this resemblance, and its choice profoundly influences the interpretation of the crystal energy landscape.

The Smooth Overlap of Atomic Positions (SOAP) Kernel

The Smooth Overlap of Atomic Positions (SOAP) kernel is a powerful and widely used method for comparing atomic environments and, by extension, periodic crystal structures [34]. It provides a robust, multi-body descriptor of local atomic neighborhoods, offering a rigorous foundation for assessing structural similarity.

Kernel Adaptation for Molecular Crystals

The development of a global SOAP kernel for molecular crystals involves integrating the SOAP descriptors from all atomic environments into a single, holistic measure of similarity between two full crystal structures. Research has demonstrated that the specific construction of this global kernel—how the local descriptors are aggregated—is not a trivial decision [34]. Different aggregation methods can lead to varying interpretations of the same crystal energy landscape.

Comparative studies have shown that the choice of kernel construction impacts several key performance metrics essential for materials discovery [34]:

• Effectiveness in Identifying Stabilizable Candidates: The kernel's ability to group structures in a chemically meaningful way directly affects the GCH's performance.
• Interpretability of Machine-Learned Descriptors: Some kernel constructions lead to features that are more easily understood by human experts, fostering trust and providing chemical insight.
• Utility in Machine Learning of Energies: The accuracy of predictive models for crystal energetics can depend on the underlying similarity metric used.

The sensitivity of landscape analysis to kernel construction underscores that there is no universally superior kernel. The "best" kernel often depends on the specific molecule and the goals of the CSP study [34].

Quantitative Analysis of Kernel Performance

Evaluating the performance of different similarity kernels requires a structured assessment across multiple criteria. The table below summarizes key quantitative and qualitative metrics for comparison, as derived from research in the field.

Table 1: Performance Metrics for Similarity Kernels in Crystal Landscape Analysis

Metric	Description	Impact on Materials Discovery
Identification of Stabilizable Candidates	Effectiveness of the GCH or similar methods in correctly classifying structures as stabilizable using the kernel.	Directly influences the success rate of predicting synthesizable polymorphs [34].
Energy Prediction Accuracy	The utility of the kernel in machine learning models for predicting crystal lattice energies.	Affects the accuracy of the final ranked list of predicted polymorphs [34].
Descriptor Interpretability	The degree to which the ML descriptors derived from the kernel can be linked to chemical or structural features.	Provides crucial chemical insight and validates the machine learning model's reasoning [34].
Landscape Connectivity	The kernel's sensitivity to small structural changes, affecting the perceived connectivity of energy minima.	Impacts the analysis of polymorphism and the depth of energy minima, relevant for kinetic stability [35].

Experimental Protocols for Kernel Validation

Validating an adapted similarity kernel is a multi-stage process that ties its performance directly to the goal of stability and synthesizability prediction. The following protocol outlines the key experimental steps.

Protocol: Validation of an Adapted Similarity Kernel

Objective: To develop and validate a new global SOAP kernel for the analysis of molecular crystal energy landscapes, assessing its performance against established metrics.

Input Requirements:

• A set of candidate crystal structures for a target molecule, generated through a CSP procedure.
• High-fidelity lattice energies for all candidate structures (e.g., calculated using DFT-D or NNPs).

Methodology:

• Kernel Construction: Define the new method for aggregating local SOAP descriptors into a global kernel similarity score between two crystal structures.
• Landscape Dimensionality Reduction: Use the kernel to compute a pairwise similarity matrix for all candidate structures in the CSP set. Apply dimensionality reduction techniques (e.g., t-SNE, PCA) to project the high-dimensional landscape into a 2D or 3D visualization.
• Generalised Convex Hull Analysis: Apply the GCH algorithm using the new kernel to identify a subset of stabilizable crystal structure candidates from the full CSP set.
• Performance Benchmarking: Compare the results obtained with the new kernel against those from a baseline (e.g., a simple average SOAP kernel) using the metrics in Table 1.
  • Stabilizable Candidate Identification: Compare the GCH-selected sets to known experimental polymorphs or a expert-curated ground truth.
  • Energy Model Utility: Train a kernel-based machine learning model (e.g., Gaussian Process Regression) to predict CSP energies and evaluate its accuracy.
  • Descriptor Interpretability: Use techniques like SHAP analysis to probe the trained energy model and determine which structural features most influence its predictions [34].

Output: A validated similarity kernel and an analyzed crystal energy landscape, with a curated list of stabilizable candidate structures ranked by their likelihood of being synthesizable.

Visualization of Workflows and Relationships

The following diagrams illustrate the logical workflow for kernel validation and its place within the broader context of crystal structure prediction.

Diagram 1: Kernel Validation Workflow. This diagram outlines the protocol for developing and testing an adapted similarity kernel, showing how convex-hull stability data guides the identification of synthesizable candidates.

Diagram 2: CSP Landscape Analysis Logic. This diagram shows the role of the similarity kernel in connecting raw CSP data to synthesizability predictions via the Generalised Convex Hull, which is informed by thermodynamic stability.

The Scientist's Toolkit: Essential Research Reagents and Materials

The computational research described relies on a suite of software tools, datasets, and algorithms. The following table details these essential "research reagents."

Table 2: Key Research Reagents and Computational Tools

Tool / Reagent	Type	Function in Research
CSP Dataset	Data	A curated set of predicted crystal structures for a target molecule; the foundational input for all landscape analysis [34].
SOAP Descriptor	Software Algorithm	Generates a mathematical representation of local atomic environments, serving as the building block for structural similarity comparisons [34].
Generalised Convex Hull (GCH)	Software Algorithm	An unsupervised machine learning method that uses a similarity kernel to identify stabilizable crystal structures from a CSP dataset [34].
Disconnectivity Graph	Analysis/Visualization	A tool for representing the global energy landscape, showing energy minima and the barriers between them, providing insight into kinetic stability [35].
Neural Network Potentials (NNPs)	Software Algorithm	Machine-learned force fields that enable accurate energy evaluations at a fraction of the cost of DFT, facilitating larger-scale CSP studies [36].
t-SNE / PCA	Software Algorithm	Dimensionality reduction techniques used to visualize high-dimensional CSP landscapes in 2D or 3D based on kernel-derived similarities [37].

The predictability and success of synthesizing new solid-state materials, whether for life-saving drugs or advanced technological applications, fundamentally rely on understanding thermodynamic stability. Convex hull stability analysis serves as the cornerstone of this understanding, providing a rigorous mathematical framework to determine which solid form is the most stable under given conditions or if a new material is likely to form at all. This whitepaper explores the pivotal role of convex hulls in synthesis prediction through two distinct but parallel case studies: the control of active pharmaceutical ingredient (API) polymorphs in drug development and the discovery of new Zintl-phase materials for optoelectronics. By examining practical applications and detailing experimental protocols, we provide researchers and scientists with a comprehensive guide to integrating stability prediction into modern research and development workflows.

The Critical Role of Convex Hulls in Predicting Solid Form Stability

A convex hull, in the context of materials science, is a global construct that defines the set of thermodynamically stable phases at zero temperature by connecting the points representing the lowest formation energies across a composition space. A compound or polymorph is considered thermodynamically stable if its formation energy lies on this hull, and metastable if it lies above it. The vertical distance from a phase's energy to the hull indicates its driving force for decomposition into more stable phases.

Traditional convex hull construction requires exhaustive calculation of formation energies for all competing phases—a computationally expensive and often impractical endeavor for complex systems. Convex Hull-Aware Active Learning (CAL), a novel Bayesian algorithm, addresses this by strategically selecting which experiments or calculations to perform to minimize uncertainty in the hull itself. CAL uses Gaussian process regressions to model energy surfaces and produces a posterior belief over possible convex hulls, prioritizing measurements for compositions close to the hull boundary and dramatically reducing the number of observations needed to predict stability [7] [8].

This framework is universally applicable, underpinning stability predictions in diverse areas from drug solubility and polymer blends to metallic alloys and battery materials [7] [8]. The following case studies demonstrate its practical implementation across domains.

Case Study 1: Pharmaceutical Polymorph Screening and Control

Problem: Inconsistent API Lot Quality

A major pharmaceutical company encountered inconsistent flow properties in different production lots of an API for an antibiotic drug that had been on the market for over twenty years. The issue originated at a third-party contract manufacturer. While crystal morphologies appeared similar and X-ray powder diffraction (XRPD) analysis suggested the same form was present in all lots, problem batches contained a large number of fine particles, impacting manufacturing consistency [38].

Investigation and Polymorph Screening Methodology

A comprehensive polymorph screen was initiated to resolve the inconsistency. The core methodology is designed to probe the solid-form landscape extensively.

Table 1: Key Analytical Techniques in Polymorph Screening

Technique	Acronym	Primary Function in Screening
X-Ray Powder Diffraction	XRPD	Confirms novelty of a crystal form; provides a fingerprint for identification [39] [40].
Differential Scanning Calorimetry	DSC	Determines thermal profile, including melting point and transition energies [39].
Thermogravimetric Analysis	TGA	Assesss hydration or solvation levels [39].
Dynamic Vapour Sorption	DVS	Measures hygroscopicity and potential for hydrate formation/dehydration [39].
Nuclear Magnetic Resonance	NMR	Confirms chemical integrity and can quantify solvent content [39].

Experimental Workflow for Polymorph Screening:

• Sample Preparation: Generation of a suitable base material, often amorphous, using methods like lyophilization, spray drying, or melt quench cooling [39].
• Crystallization Experiments: The base material is subjected to a wide array of solvents and experimental conditions (e.g., varying temperature, evaporation rates) to maximize the discovery of crystalline forms [39]. This includes conditions the API might encounter during scale-up and manufacture.
• Solid Form Analysis: Solids generated from the screen are primarily analyzed using XRPD. Any novel forms are characterized further using the techniques listed in Table 1 [39].
• Form Selection: The most suitable form is nominated for development based on criteria including stability, solubility, dissolution profile, and manufacturability [39].

Figure 1: Workflow for experimental polymorph screening.

Solution and Outcome: A Manufacturing Flaw Revealed

The polymorph screening conducted by Aptuit revealed that the client's API existed as multiple solid forms, not a single form as previously believed. The crystallization process used by the supplier was inefficient and unable to consistently produce only the desired polymorph. The variation in fine particles between lots was a direct result of this uncontrolled process. By implementing a revised manufacturing solution that controlled for the specific crystallization parameters of the desired form, Aptuit achieved clear efficiencies, resulting in less waste, reduced cost, and improved production time [38]. This case underscores that routine XRPD analysis can sometimes miss subtle polymorphic impurities and that a thorough, stability-based screen is essential for robust process control.

Case Study 2: Discovery of Novel Zintl Phases

Objective: Systematic Exploration of a Vast Chemical Space

Zintl phases are intermetallic compounds with a combination of ionic, covalent, and metallic bonding, leading to a wide array of functional properties for optoelectronics and thermoelectrics. Traditional discovery has relied on empirical knowledge and serendipity, leaving a vast chemical space largely unexplored. A research team set out to systematically discover new, thermodynamically stable Zintl phases from a space of over 90,000 hypothetical compounds [19].

Methodology: Machine Learning and the Upper Bound Energy Minimization (UBEM) Approach

The research team employed a computationally efficient strategy combining graph neural networks (GNNs) with the Upper Bound Energy Minimization (UBEM) approach to navigate the immense compositional space.

Key Computational and Experimental Protocols:

• Dataset Curation: A dataset of 824 pnictide-based Zintl prototypes was extracted from the Inorganic Crystal Structure Database (ICSD) [19].
• Chemical Substitution (Decoration): These parent structures were systematically decorated with elements from Groups 1, 2, 12, 13, 14, and 15, generating >90,000 candidate structures [19].
• GNN Model Training: A scale-invariant GNN model was trained to predict the DFT volume-relaxed energy from an unrelaxed crystal structure input. This model achieved a low test mean absolute error (MAE) of 27 meV per atom [19].
• UBEM Stability Prediction: The trained GNN predicted the volume-relaxed energy for all candidates. The UBEM approach relies on the principle that the volume-relaxed energy is an upper bound to the fully-relaxed DFT energy. Therefore, if a volume-relaxed structure is predicted to be stable (on the convex hull), its fully relaxed counterpart is guaranteed to be stable [19].
• DFT Validation: Predicted stable phases were validated using high-fidelity density functional theory (DFT) calculations to confirm their stability on the convex hull [19].

Table 2: Performance Metrics for Zintl Phase Discovery via GNN/UBEM

Metric	Value	Comparison to M3GNet Model
Stable Phases Discovered	1810 new phases	N/A
Validation Precision	90%	More than 2x more accurate (M3GNet precision: 40%)
Model Mean Absolute Error (MAE)	27 meV/atom	Below "chemical accuracy" (43 meV/atom)

Figure 2: Workflow for computational discovery of Zintl phases using GNN and UBEM.

Outcome and Practical Synthesis: BaCd₂P₂ Quantum Dots

The ML-driven discovery framework identified 1810 new thermodynamically stable Zintl phases with 90% precision. In a parallel experimental study, researchers synthesized and characterized a specific Zintl phase, BaCd₂P₂, as colloidal quantum dots.

Synthesis Protocol for BaCd₂P₂ Quantum Dots:

• Method: A hot injection method was used, where a phosphorus precursor was rapidly injected into a heated ligand-solubilized mixture containing barium and cadmium [41].
• Size Control: The size of the quantum dots, which determines their optoelectronic properties through quantum confinement, was controlled by adjusting the temperature during growth [41].
• Characterization: The team used selected area electron diffraction, X-ray diffraction, Raman spectroscopy, and X-ray fluorescence to confirm the crystal structure and composition matched the bulk material [41].

The resulting quantum dots exhibited a brilliant photoluminescence with a quantum yield of 21% without complex surface treatments, highlighting their defect tolerance and potential for use in LEDs, displays, and solar panels [41].

The Scientist's Toolkit: Essential Research Reagents and Materials

Successful solid-form research relies on a suite of analytical techniques and computational tools. The following table details key resources and their functions.

Table 3: Essential Research Tools for Solid-State Science

Tool / Resource	Category	Primary Function
X-Ray Powder Diffraction (XRPD)	Analytical	Definitive identification of crystalline phases and determination of unit cell parameters [38] [39].
Differential Scanning Calorimetry (DSC)	Analytical	Measurement of thermal events (melting, crystallization, solid-solid transitions) and their enthalpies [39].
Graph Neural Network (GNN) Models	Computational	Predicts material properties (e.g., formation energy) directly from crystal structure, enabling high-throughput screening [19].
Density Functional Theory (DFT)	Computational	High-accuracy quantum-mechanical calculation of electronic structure and total energy for stability assessment [19].
Supercritical Fluid Technology (e.g., mSAS)	Experimental	An enhanced polymorph screening technique effective at isolating stable, metastable, and novel polymorphs under high pressure [40].
Selected Area Electron Diffraction (SAED)	Analytical	Provides structural and phase information from nanoscale regions of a sample, crucial for characterizing quantum dots [41].

The case studies presented herein demonstrate that a deep understanding of convex hull stability is not a mere academic exercise but a critical, practical tool for predicting and controlling synthesis across disparate fields. In pharmaceuticals, it enables the robust manufacturing of APIs by ensuring consistent polymorphic form, thereby safeguarding product performance and patient safety. In materials science, it empowers the accelerated discovery of new functional materials, such as Zintl-phase quantum dots, by efficiently guiding researchers toward stable compositions in a vast chemical space. The integration of advanced computational methods like convex hull-aware active learning and graph neural networks with traditional experimental techniques represents the forefront of solid-state research. This synergy creates a powerful paradigm for future discovery, reducing development time and cost while increasing the reliability and success of synthesizing new solid forms.

Navigating Pitfalls: Addressing Vibrational Instability and Prediction Challenges

The energy above the convex hull (E𝐻) has long served as the primary metric for assessing thermodynamic stability and predicting the synthesizability of new materials. A low E𝐻 (typically < 100 meV/atom) indicates that a material is stable against decomposition into other phases in its chemical space. However, a growing body of evidence reveals that this metric alone is insufficient. A significant number of materials with low E𝐻 are vibrationally unstable, meaning they possess imaginary phonon modes that prevent them from existing on a minimum of the potential energy surface. This whitepaper details the critical necessity of supplementing convex-hull analysis with a vibrational stability filter, a practice rapidly becoming essential for reliable computational predictions in synthesis and materials design.

In computational materials science, the convex hull of a chemical space represents the set of the most thermodynamically stable phases. The energy above the convex hull (E𝐻) for a compound measures its energy relative to this stable set; it is the energy penalty for decomposing into the most stable combination of other phases on the hull. For years, an E𝐻 below a threshold—often 100 meV/atom—has been considered a strong indicator that a material could be synthesized [42].

However, thermodynamic stability is only one facet of synthesizability. A material must also be dynamically stable. This means that when atoms are displaced slightly from their equilibrium positions, the restoring forces should bring them back, not push them further away. In quantum mechanical terms, the collective atomic vibrations (phonons) must have real, positive frequencies. The presence of imaginary frequencies (negative values on a phonon dispersion plot) indicates vibrational instability, signifying that the structure is not at a true energy minimum but rather at a saddle point, and will distort to find a lower-energy configuration [42] [43].

Numerous examples from online material databases underscore this disconnect. Compounds like LiZnPS₄ (mp-11175), SiC (mp-11713), and Ca₃PN (mp-11824) all possess an E𝐻 of 0 meV or nearly 0 meV, yet each is vibrationally unstable [42]. This demonstrates that convex hull information cannot be taken at face value; a secondary filter for vibrational stability is required to enhance the predictive accuracy of synthesizability assessments.

The Computational Workflow for Stability Assessment

A robust computational assessment of a material's stability requires a two-step verification process, combining thermodynamic and dynamic stability checks. The following workflow visualizes this integrated protocol:

Figure 1: A two-step computational workflow for assessing material stability, combining thermodynamic (convex hull) and dynamic (phonon) analysis.

Core Theoretical Concepts

The Convex Hull and E_H

The convex hull in materials stability is constructed by plotting the formation energies of all known compounds in a given chemical space. The phases with the lowest formation energies form the vertices of the hull. The energy above the hull (E𝐻) for any compound not on the hull is its energy distance above the tie-line connecting the most stable decomposition phases [42].

Phonons and Vibrational Stability

Within the harmonic approximation, the vibrational dynamics of a crystal are described by the dynamical matrix. Solving its eigenvalue problem yields the phonon frequencies (ω) and their corresponding polarization vectors. A dynamically stable structure will have only real, positive phonon frequencies for all wavevectors in the Brillouin zone. The presence of any imaginary frequency indicates vibrational instability [43]. The vibrational free energy, a key component of a material's stability at finite temperatures, is given by: [ F{vib}(T) = \int0^\infty \left[ \frac{\hbar\omega}{2} + kB T \ln\left(1 - e^{-\frac{\hbar\omega}{kB T}}\right) \right] g(\omega) d\omega ] where (g(\omega)) is the phonon density of states. This contribution can be significant, on the order of 1 eV per atom in complex disordered solids like the garnet electrolyte Li₇La₃Zr₂O₁₂ (LLZO), and can critically influence phase stability [44].

Machine Learning as a Vibrational Stability Filter

The primary obstacle to routine vibrational stability checking is computational cost. Density Functional Perturbation Theory (DFPT) or finite-difference supercell approaches for phonon calculations are prohibitively expensive for high-throughput screening. This is where machine learning (ML) offers a transformative solution.

ML Classifier for Vibrational Stability

A study trained a Random Forest (RF) classifier on a dataset of vibrational stability for approximately 3,100 materials from the Materials Project [42]. The goal was to distinguish between vibrationally stable and unstable materials based on structural and compositional features.

Key Methodology [42]:

• Dataset: ~3100 materials with known vibrational stability labels (stable/unstable).
• Features: 1145 initial features were generated, including:
  • BACD (Bond Angle Concentration Descriptors): Describe the local chemical environment.
  • ROSA (Radial Site Statistics): Capture radial distribution statistics.
  • SG (Space Group) features: Symmetry-related information.
  • Specific elemental properties like std_average_anionic_radius and metals_fraction were consistently identified as highly important.
• Model Training: A Random Forest model was trained on the dataset. To address class imbalance (unstable materials were the minority), synthetic data was introduced into the training folds using SMOTE and mixup augmentation techniques. The model was evaluated via cross-validation.

Performance: The model achieved an average f1-score of 63% for the unstable class (minority class), with recall increasing to 68% from 42% without augmentation. When the model's operation was restricted to predictions with a confidence level of 0.65 or higher, its performance for the unstable class improved to an average f1-score of 70%, while still covering about 65% of the data points [42]. This demonstrates its potential as an effective pre-screening filter.

Table 1: Performance metrics of the Random Forest vibrational stability classifier across different confidence thresholds [42].

Confidence Threshold	Avg. Recall (Unstable)	Avg. Precision (Unstable)	Avg. F1-Score (Unstable)	Data Coverage
0.50	0.68	0.59	0.63	100%
0.65	0.71	0.70	0.70	~65%

Feature Importance in the ML Model

Analysis revealed that the top 30 features carried almost all the predictive information. A model trained on only these 30 features performed similarly to the model using all 1145 features. The most significant feature categories were [42]:

• BACD (Bond Angle Concentration Descriptors)
• ROSA (Radial Site Statistics)
• SG (Space Group) features

This suggests that the local chemical environment and symmetry play a more critical role in determining vibrational stability than the specific chemical identity of the elements alone.

Advanced AI-Driven Methods for Vibrational Analysis

Beyond classification, AI is revolutionizing the entire computational pipeline for vibrational properties. Machine learning interatomic potentials (MLIPs) are a key innovation, enabling accurate and rapid molecular dynamics simulations that capture anharmonic effects.

Case Study: Disordered Solid LLZO with MLIPs

The garnet electrolyte cubic Li₇La₃Zr₂O₁₂ (c-LLZO) presents a monumental challenge for traditional DFT. Its Li-sublattice is disordered, with an estimated 7×10³⁴ possible configurations in a single unit cell [44]. Sampling the configurational and vibrational entropy is computationally intractable with DFT alone.

Research Protocol [44]:

• Machine-Learned Forcefield (MLFF): Researchers developed an accurate forcefield for LLZO using an equivariant message-passing neural network (SO3KRATES). This model was trained on a diverse set of DFT calculations to learn the potential energy surface.
• High-Throughput Sampling: The trained MLFF was used to perform structural optimizations and molecular dynamics (MD) simulations at 300 K and 1500 K for a subset of 70,120 unique configurations of c-LLZO.
• Free Energy Calculation: For each configuration, the MLFF-enabled MD simulations provided the data needed to compute the vibrational contribution to the Helmholtz free energy, (F_{vib}(T)), with high efficiency.

Finding: The study deterministically showed that the vibrational contributions to the total configurational free energy at 1500 K are significant (on the order of 1 eV per atom) and are essential for correctly ordering the stability of cubic LLZO over its tetragonal counterpart [44]. This underscores that neglecting vibrational energy can lead to incorrect predictions of phase stability, even after accounting for configurational entropy.

Table 2: A comparison of computational methods for assessing vibrational stability and their respective trade-offs.

Method	Accuracy	Computational Cost	Key Application
DFT Phonons (DFPT)	High	Prohibitively High	Small systems, final validation
Classical Forcefields	Low to Medium	Low	Large systems, limited transferability
Machine Learning Interatomic Potentials (MLIPs)	High (if well-trained)	Medium (High after training)	Complex, disordered solids (e.g., LLZO)
ML Stability Classifier	Medium	Very Low	High-throughput pre-screening

For researchers embarking on stability analysis, the following tools and databases are indispensable.

Table 3: Key resources and computational "reagents" for stability assessment research.

Resource / Tool	Type	Primary Function	Relevance to Stability
Materials Project [42]	Database	Provides computed E𝐻 and structures for >140,000 materials.	Source for initial candidate materials and training data.
JARVIS-DFT [42]	Database	Includes DFT-computed properties, including phonons for some materials.	Source for validation data and benchmark calculations.
Random Forest Classifier	ML Model	Classifies materials as vibrationally stable/unstable.	Fast, pre-screening filter before expensive phonon calculations [42].
SO3KRATES / M3GNet	MLIP	Generates machine-learned forcefields from DFT data.	Enables molecular dynamics and free energy calculations for complex materials [44].
VASP, Quantum ESPRESSO	DFT Code	Performs first-principles electronic structure calculations.	The "gold standard" for computing E𝐻 and generating training data for MLIPs.

The energy above the convex hull is a necessary but insufficient metric for predicting viable, synthesizable materials. The integration of a vibrational stability filter is no longer a niche consideration but a critical component of a robust computational materials discovery workflow. As demonstrated, machine learning offers powerful tools to implement this filter, both through classifiers for high-throughput screening and through advanced forcefields that enable the precise calculation of vibrational free energies in fantastically complex materials. Ignoring dynamic stability risks the continued prediction of "theoretically stable" materials that cannot exist in practice. The future of accurate synthesis prediction lies in a holistic approach that rigorously accounts for both thermodynamic and vibrational stability.

Machine Learning Model Calibration for Imbalanced Stability Datasets

The prediction of material stability via convex hull construction represents a critical challenge in synthesis prediction research. This in-depth technical guide examines the convergence of class imbalance and probability calibration within this domain. We demonstrate that accurate stability prediction requires specialized machine learning approaches that address both the inherent data skew in stable compounds and the need for well-calibrated probabilistic outputs. By integrating convex hull-aware active learning with advanced calibration techniques, researchers can achieve more reliable stability assessments while significantly reducing computational costs. This whitepaper provides experimental protocols, metrics, and practical frameworks to advance the field of computational materials discovery and drug development.

In materials science and drug development, thermodynamic stability is determined through convex hull construction in formation energy-composition space [9]. A material's stability is not an intrinsic property but emerges from global competition with all other competing phases and compositions. The convex hull defines the set of stable phase-composition pairs, with stable compounds lying on the hull and unstable compounds lying above it [7] [8]. This global nature creates fundamental challenges for machine learning:

• Needle-in-a-haystack Problem: Stable materials are exceptionally rare within vast compositional spaces [9]
• Non-linear Sensitivity: Formation energy predictions require extraordinary precision (often <0.1 eV/atom) to accurately determine stability via convex hull position [9]
• Compositional vs. Structural Models: Composition-only models struggle with stability prediction despite reasonable formation energy accuracy, while structural models show superior performance but require prior structural knowledge [9]

The intersection of convex hull analysis with imbalanced learning creates a unique research challenge where standard machine learning approaches frequently fail, necessitating specialized methodologies for model calibration and evaluation.

The Imbalance Challenge in Stability Datasets

Fundamental Tensions in Stability Prediction

Machine learning for material stability operates under dual constraints: extreme class imbalance and subtle energy differentials. The combinatorial complexity of materials discovery means that for each stable composition, numerous unstable possibilities exist [9]. This imbalance is not merely statistical but structural—the convex hull construction ensures that only compositions forming the lower envelope contribute positively to stability classification.

Quantifying the Challenge: Experimental analyses reveal that while formation energies (ΔHf) span a wide range (-1.42 ± 0.95 eV/atom), the decisive decomposition enthalpies (ΔHd) operate at much finer scales (0.06 ± 0.12 eV/atom) [9]. This energy sensitivity, combined with sparse stability distribution, creates a uniquely challenging machine learning environment where traditional accuracy metrics become virtually meaningless.

Limitations of Standard Approaches

Common compositional machine learning models exhibit critical limitations in stability prediction:

• Error Propagation: Small errors in formation energy prediction amplify significantly in hull position determination [9]
• False Stability Predictions: Models frequently misclassify unstable compounds as stable, impeding materials discovery efficiency [9]
• Data Scarcity: For rare stable compounds, insufficient examples exist for robust pattern recognition using conventional methods

Table 1: Performance Comparison of Compositional ML Models on Stability Prediction

Model Type	Formation Energy MAE (eV/atom)	Stability Prediction Accuracy	Critical Limitations
ElFrac (Baseline)	0.43	Poor	Limited feature representation
Magpie	0.24	Moderate	Improved but insufficient for discovery
ElemNet	0.11	Moderate	Good ΔHf prediction, poor ΔHd accuracy
Structural Models	0.09-0.15	High	Require known crystal structure

Calibration Methodologies for Imbalanced Stability Data

Data-Level Strategies

Data-based methods modify dataset distribution before model training, directly addressing class representation:

Advanced Oversampling Techniques:

• SMOTE Variants: Generate synthetic minority samples through interpolation [45]
  • K-Means SMOTE: Applies clustering before oversampling to maintain natural data structure [45]
  • SVM-SMOTE: Focuses synthetic sample generation near decision boundaries [45]
• GAN-Based Oversampling: Uses conditional Generative Adversarial Networks to create realistic synthetic samples for complex data spaces [45]

Undersampling Techniques:

• Edited Nearest Neighbors (ENN): Removes majority class samples misclassified by nearest neighbors [45]
• Tomek Links: Eliminates borderline majority samples to improve class separation [45]

Hybrid Approaches:

• SMOTE + ENN: Combines synthetic minority generation with majority class cleaning [45]
• Stratified Splitting: Maintains original class distribution across data splits to prevent information leakage [45]

Algorithmic-Level Approaches

These methods adapt machine learning algorithms to emphasize minority classes through modified objective functions and specialized architectures:

Convex Hull-Aware Active Learning (CAL): A novel Bayesian approach that directly addresses the global nature of convex hull stability [7] [8]. CAL employs Gaussian processes to model energy surfaces and selects experiments to minimize convex hull uncertainty rather than energy prediction uncertainty.

CAL Experimental Protocol:

• Initialization: Begin with limited observations of formation energies across composition space
• Gaussian Process Modeling: Create separate GP regressions for each phase's energy surface
• Hull Sampling: Generate posterior samples of possible convex hulls from GP predictions
• Information Gain Calculation: Compute expected reduction in hull uncertainty for candidate compositions
• Iterative Selection: Choose next composition for evaluation based on maximum expected information gain [7] [8]

Focal Loss Adaptation: Reshapes standard cross-entropy to focus learning on hard-to-classify examples, particularly relevant for stable compounds near the convex hull boundary [45]: L = -α(1-pₜ)ᵞlog(pₜ) where α balances class contributions and γ focuses training on challenging samples.

Ensemble Methods:

• Boosting Variants: SMOTEBoost and RUSBoost integrate sampling with sequential model training [45]
• Class Weighting: Modern frameworks (XGBoost, LightGBM) support explicit class weights in loss functions [45]

Probability Calibration Techniques

Model calibration ensures predicted probabilities reflect true likelihoods of stability—critical for reliable materials discovery decisions [46].

Calibration Methods:

• Platt Scaling: Applies logistic regression to classifier scores, effective for sigmoid-shaped distortions [46]
• Isotonic Regression: Non-parametric approach that corrects monotonic distortions, suitable for larger datasets [46]

Imbalance-Specific Considerations:

• Resampling techniques can distort probability estimates, requiring post-processing calibration [46]
• Bayesian methods naturally incorporate uncertainty quantification beneficial for low-data scenarios [7]

Table 2: Calibration Methods Comparison for Stability Prediction

Method	Best For	Data Requirements	Considerations for Stability Data
Platt Scaling	SVM, Neural Networks	Smaller datasets	May underperform with complex distortions
Isotonic Regression	Any classifier	Larger datasets	Can overfit with limited stable examples
Bayesian Calibration	Probabilistic models	Variable	Natural uncertainty quantification
Ensemble Calibration	Multiple classifier types	Moderate to large	Combines strengths of individual methods

Evaluation Metrics for Imbalanced Stability Prediction

Standard accuracy metrics fail completely with imbalanced stability datasets. Specialized evaluation frameworks are essential:

Stability-Specific Metrics:

• Precision-Recall Curves (AUC-PR): More informative than ROC curves for imbalanced data [45]
• Matthews Correlation Coefficient (MCC): Comprehensive metric considering all confusion matrix categories [45]
• Brier Score: Measures calibration quality of probabilistic predictions [46]
• Energy Above Hull Error: Quantitative measure of stability prediction accuracy [9]

Critical Diagnostic Tools:

• Calibration Curves: Visualize alignment between predicted probabilities and actual outcomes [46]
• Confusion Matrix Analysis: Identify specific failure modes in stability classification [45]
• Convex Hull Entropy: Measure uncertainty in hull predictions within CAL framework [7]

Experimental Protocols and Workflows

Convex Hull-Aware Active Learning Protocol

Objective: Minimize computational resources required to determine convex hull while maximizing stability prediction accuracy.

Materials and Software Requirements:

• Gaussian Process regression framework
• Convex hull computation algorithm (e.g., QuickHull)
• Formation energy dataset with compositional diversity

Methodology:

• Initial Phase:
  • Select diverse initial compositions covering chemical space
  • Compute formation energies via DFT or experimental measurement
  • Establish baseline convex hull

• Iterative Active Learning Phase:

  • Train separate Gaussian Processes for each phase's energy surface
  • Sample possible energy surfaces from posterior distributions
  • Compute convex hull for each sample
  • Calculate entropy of hull distribution as uncertainty measure
  • Select next composition evaluation to maximize expected information gain about hull
  • Iterate until hull uncertainty falls below threshold

• Validation Phase:

  • Compare predicted stabilities with ground truth
  • Assess calibration using reliability diagrams
  • Compute specialized metrics for imbalanced data

Model Calibration Experimental Protocol

Objective: Achieve well-calibrated probability estimates for stability predictions despite class imbalance.

Methodology:

• Base Model Training:
  • Train stability classifier using imbalance-aware techniques
  • Generate initial probability estimates

• Calibration Set Application:

  • Reserve representative validation set maintaining imbalance
  • Apply Platt Scaling or Isotonic Regression
  • Learn mapping from raw scores to calibrated probabilities

• Evaluation:

  • Assess calibration using reliability diagrams
  • Compute Brier score and log loss
  • Verify maintained discrimination performance

Table 3: Research Reagent Solutions for Stability Prediction

Tool/Category	Specific Examples	Function in Stability Research
Data Sources	Materials Project, OQMD	Provide formation energy and stability data for model training
ML Frameworks	Scikit-learn, XGBoost, PyTorch	Implement classification, calibration, and active learning
Sampling Algorithms	SMOTE variants, GAN-based oversampling	Address class imbalance in training data
Calibration Methods	Platt Scaling, Isotonic Regression	Improve reliability of probabilistic predictions
Hull Computation	QuickHull, PHull	Determine stability from formation energies
Evaluation Metrics	AUC-PR, MCC, Brier Score	Assess model performance on imbalanced data
Active Learning	CAL implementation	Efficiently explore composition space
Uncertainty Quantification	Gaussian Processes, Bayesian Neural Networks	Propagate and quantify prediction uncertainty

Machine learning calibration for imbalanced stability datasets represents a critical frontier in computational materials discovery and drug development. The integration of convex hull-aware active learning with advanced calibration techniques enables researchers to navigate the challenges of extreme class imbalance while maintaining probabilistic reliability. Future advancements will likely focus on:

• Integration with Multi-fidelity Data: Combining high-cost computational data with abundant low-fidelity sources
• Transfer Learning Approaches: Leveraging knowledge from data-rich chemical spaces to inform sparse regions
• Automated Workflows: End-to-end pipelines integrating active learning, model training, and calibration
• Regulatory Acceptance: Establishing standards for model validation in pharmaceutical applications [47]

As these methodologies mature, they promise to accelerate the discovery of novel materials and therapeutic compounds while reducing computational costs—ultimately bridging the gap between computational prediction and experimental synthesis.

Optimizing Global Kernels for Molecular Crystal Structure Comparison

Within the field of computational materials science, accurately comparing predicted crystal structures is a cornerstone of reliable Crystal Structure Prediction (CSP). The ability to quantify similarity between two different molecular packing arrangements directly enables the construction of the crystal energy landscape—a map of all plausible polymorphs for a given compound. This landscape is the foundation for determining the convex-hull stability of crystal structures, a critical metric for predicting which polymorphs are synthesizable under specific thermodynamic conditions. A structure is considered thermodynamically stable and potentially synthesizable if it lies on the convex hull, meaning no linear combination of other structures has a lower free energy at a given composition.

Recent advances have demonstrated that the sensitivity of similarity kernel-based landscape analysis methods is highly dependent on kernel construction [48]. An ill-defined kernel can lead to misclassification of structures, erroneous deduplication of candidate crystals, and ultimately, an incorrect convex hull. This technical guide details the methodology for optimizing a global kernel for molecular crystal structure comparison, framing it as an essential prerequisite for accurate synthesis prediction research.

Theoretical Foundations: Kernels, Convex Hulls, and Crystal Stability

The Convex Hull in Synthesis Prediction

The generalized convex hull (GCH) is a mathematical construct used to identify stabilizable crystal structures from large prediction sets [48]. In the context of CSP, the vertical axis of the hull represents the lattice energy of a crystal structure. A structure lies on the convex hull if its energy per molecule is lower than that of any physical mixture of other predicted structures. Structures on the convex hull are thermodynamically stable at zero Kelvin and are primary candidates for experimental synthesis, while those above it are metastable. The accuracy of this hull is therefore paramount, as it guides experimental efforts in polymorph screening and drug development.

Kernel Methods for Structural Comparison

A kernel function acts as a similarity measure between two data points in a high-dimensional space. For crystal structures, a well-designed kernel quantifies the similarity between two periodic atomic arrangements. The core challenge lies in creating a kernel that is sensitive to subtle atomic displacements and molecular orientations that differentiate polymorphs, yet robust enough to identify identical structures despite different unit cell choices.

The Smooth Overlap of Atomic Positions (SOAP) kernel is a leading approach for this task. It provides a rigorous, rotationally invariant similarity measure between local atomic environments [48]. However, its standard formulation may not fully capture the unique packing motifs and intermolecular interactions in molecular crystals, necessitating adaptation for optimal performance.

Optimizing the SOAP Kernel for Molecular Crystals

Limitations of the Standard SOAP Kernel

The standard SOAP kernel treats all atoms equivalently, which can be suboptimal for molecular crystals where specific functional groups drive packing through directed interactions like hydrogen bonds or π-π stacking. Furthermore, it may not adequately prioritize the long-range order that characterizes crystalline materials over the short-range order found in liquids or amorphous solids.

The Adapted Similarity Kernel Approach

Recent research has adapted the SOAP kernel to define the similarity of molecular crystal structures in a more physically motivated way [48]. The key adaptations include:

• Molecular Awareness: Modifying the kernel to respect molecular boundaries, ensuring that intramolecular geometry (which is largely fixed) does not disproportionately influence the similarity score compared to critical intermolecular packing parameters.
• Interaction-Specific Weighting: Incorporating chemical intelligence to assign greater weight to specific atomic pairs involved in key intermolecular interactions (e.g., O-H...N for hydrogen bonding).
• Global Descriptor Integration: Combining the local environment descriptors of SOAP with global crystal descriptors, such as unit cell parameters and space group symmetry, to better capture long-range periodicity.

This adapted kernel has demonstrated improved interpretability of the resulting machine-learned descriptors and yields better performance in predicting lattice energies using Gaussian process regression [48]. The enhanced physical motivation directly translates to a more reliable construction of the crystal energy landscape and its associated convex hull.

Experimental Protocols for Kernel Validation

Validating an optimized kernel requires a rigorous experimental pipeline to assess its performance against both computational and empirical benchmarks.

Protocol 1: Benchmarking against Known Crystal Landscapes

• Objective: To quantify the kernel's ability to correctly map the energy landscape of a molecule with well-characterized polymorphs.
• Methodology:
  • Select a target molecule with multiple experimentally known and computationally predicted polymorphs (e.g., from the Cambridge Structural Database, CSD).
  • Generate a set of putative crystal structures using a random structure generator like Genarris 3.0, which employs a "Rigid Press" algorithm to produce physically realistic, close-packed trial structures [49].
  • Relax all generated structures using a high-accuracy method, such as a universal Machine Learning Interatomic Potential (MLIP) like the Universal Model for Atoms (UMA) [50] or dispersion-inclusive Density Functional Theory (DFT).
  • Compute the pairwise similarity matrix for all low-energy, non-duplicate structures using both the standard and adapted SOAP kernels.
  • Use dimensionality reduction (e.g., Principal Component Analysis or t-SNE) to visualize the landscape. A superior kernel will show clear clustering of structurally similar polymorphs and adequate separation between distinct polymorphs.
• Metrics for Success: The adapted kernel should lead to a convex hull where all known experimental polymorphs are correctly identified as lying on or near the hull, with a clear separation from higher-energy, less plausible structures.

Protocol 2: Success Rate in Crystal Structure Prediction Workflows

• Objective: To test the kernel's performance in a practical, end-to-end CSP workflow.
• Methodology:
  • Integrate the adapted kernel into a CSP workflow, such as the SPaDe-CSP or FastCSP framework [51] [50]. In these workflows, the kernel is used for deduplication after structure generation and relaxation.
  • Run the CSP workflow for a diverse set of organic molecules, typically rigid molecules with known experimental structures for validation.
  • The workflow's success is measured by its ability to generate and correctly rank the experimental structure as the global minimum or within a few kJ/mol of it.
• Key Results: A study on 20 organic molecules showed that an ML-guided workflow (SPaDe-CSP) achieved an 80% success rate in predicting experimental structures, double that of a random CSP approach [51] [52]. While this workflow used ML for space group and density prediction, the deduplication and final landscape analysis rely heavily on an accurate similarity kernel.

Table 1: Quantitative Performance of a Modern ML-Driven CSP Workflow [51]

Metric	Random CSP	ML-Guided CSP (SPaDe-CSP)
Overall Success Rate	~40%	80%
Key Limitation	Generates many low-density, unstable structures	Effectively narrows search space
Critical Dependency	--	Accurate similarity kernel for deduplication & clustering

Integration with Broader CSP Workflows

An optimized global kernel does not operate in isolation but is a critical component in a larger computational pipeline. The diagram below illustrates how kernel-based comparison is integrated into a state-of-the-art CSP workflow.

Workflow Logic and Kernel Role

The workflow begins with an input molecule, from which one or more low-energy conformers are generated. These conformers are then packed into random crystal structures using algorithms like Genarris 3.0 [49]. The resulting thousands to millions of candidate structures are relaxed to their local energy minimum using a fast and accurate method, typically a universal MLIP like UMA [50] or DFT. The Kernel-Based Comparison & Deduplication step is crucial here; it uses the optimized global kernel to identify and remove duplicate structures, ensuring the diversity of the candidate pool. The unique, low-energy structures are then ranked by their lattice energy, allowing for the construction of the convex hull. The final output is a set of potentially synthesizable crystal structures on the convex hull, which directs experimental synthesis efforts.

The Scientist's Toolkit: Essential Research Reagents

The following table details key computational tools and materials essential for implementing the described kernel optimization and CSP protocols.

Table 2: Key Research Reagents and Computational Tools for Kernel-Optimized CSP

Item Name	Function/Brief Explanation	Example/Reference
SOAP Descriptor	Generates a rotationally invariant mathematical representation of local atomic environments, serving as the foundation for the similarity kernel.	Central to the adapted kernel in [48].
Generalized Convex Hull (GCH)	Identifies thermodynamically stable crystal structures from a large set of predictions; the target of the analysis.	Defined in [48].
Genarris 3.0	An open-source Python package for generating random, physically plausible molecular crystal structures for initial sampling.	Used for structure generation in [50] [49].
Universal Model for Atoms (UMA)	A machine learning interatomic potential for highly accelerated and accurate geometry relaxation of crystal candidates.	MLIP used in the FastCSP workflow [50].
Cambridge Structural Database (CSD)	A repository of experimentally determined organic and metal-organic crystal structures used for training and validation.	Source of data for training ML models in [51].
LightGBM	A gradient boosting framework used for building machine learning models, such as space group and density predictors.	Used in the SPaDe-CSP workflow [51] [52].

The optimization of global kernels for molecular crystal structure comparison is not merely a technical exercise in machine learning. It is a fundamental step that directly impacts the reliability of the crystal energy landscape and the subsequent identification of synthesizable materials via convex-hull analysis. By adapting kernels like SOAP to be more chemically aware and physically motivated, researchers can achieve a more accurate and interpretable mapping of polymorphic space. This progress, when integrated into robust CSP workflows powered by universal MLIPs and efficient structure generators, significantly accelerates the design and discovery of new functional materials and pharmaceutical compounds, making the goal of predictive materials synthesis increasingly attainable.

In the pursuit of new functional materials, computational materials discovery has generated millions of candidate crystal structures. The prevailing strategy for prioritizing these candidates has long relied on a simple thermodynamic rule: a material is considered promising if its calculated energy above the convex hull (Eₕₕ) is within a narrow window, often set at 0 eV/atom. This metric, while useful for an initial filter, has proven to be a significant source of false positives, leading to wasted computational and experimental resources on compounds that are not synthetically accessible. This whitepaper examines the critical shortcomings of using Eₕₕ as a sole synthesizability metric and details the advanced, multi-faceted computational frameworks that are emerging to address this challenge, thereby refining the materials discovery pipeline.

The Problem: Why the Convex Hull is an Inadequate Filter

The energy above the convex hull represents the thermodynamic stability of a compound relative to other phases in its chemical system. An Eₕₕ of 0 eV/atom indicates that a material is thermodynamically stable at 0 Kelvin, a condition that rarely, if ever, exists in a real laboratory. This fundamental disconnect is the primary cause of high false-positive rates in discovery campaigns [10] [18].

• Omission of Kinetic and Entropic Factors: Thermodynamic stability does not account for the kinetic barriers or finite-temperature effects that govern real synthesis. A compound might be thermodynamically stable yet have an insurmountable kinetic barrier to its formation, or conversely, a metastable compound (Eₕₕ > 0) might be readily synthesized due to favorable kinetics or entropic stabilization [10] [18].
• Susceptibility Near the Decision Boundary: Machine learning (ML) models that achieve accurate regression for formation energy can still produce unexpectedly high false-positive rates. Even small prediction errors near the critical 0 eV/atom boundary can cause a model to incorrectly classify a metastable structure as stable [18]. This highlights a misalignment between traditional regression metrics (e.g., MAE) and the more relevant classification metrics for discovery.
• Empirical Evidence of Failure: Conventional stability-based screening methods show limited accuracy in predicting actual synthesizability. For instance, one study demonstrated that using Eₕₕ ≥ 0.1 eV/atom as a filter achieved only 74.1% accuracy, while a kinetic method based on phonon spectra (lowest frequency ≥ -0.1 THz) achieved 82.2% accuracy. Both are substantially outperformed by modern ML models [15].

Beyond Thermodynamics: Modern Frameworks for Synthesizability Prediction

To overcome the limitations of the convex hull, the field is shifting towards integrated models that directly predict synthesizability by learning from experimental data. These approaches consider a richer set of features, including composition, crystal structure, and historical synthesis data. The table below summarizes the performance of various modern approaches compared to traditional methods.

Table 1: Comparison of Synthesizability and Stability Prediction Methods

Method / Model	Core Approach	Reported Accuracy/Performance	Key Advantage
Convex Hull (Eₕₕ) [18]	Thermodynamic stability via DFT	74.1% accuracy in synthesizability prediction	Well-established, physically intuitive
Phonon Stability [15]	Kinetic stability via phonon spectrum analysis	82.2% accuracy in synthesizability prediction	Accounts for dynamical stability
CSLLM (Synthesizability LLM) [15]	Fine-tuned Large Language Model on material strings	98.6% accuracy	High accuracy and generalizability; also predicts methods and precursors
UBEM-GNN [19]	Graph Neural Network with Upper Bound Energy Minimization	90% precision in discovering stable Zintl phases	High precision validated by DFT; uses unrelaxed structures
Universal MLIPs (e.g., eSEN, ORB-v2) [53]	Machine Learning Interatomic Potentials for energy/force prediction	Errors in energy < 10 meV/atom across all dimensionalities	Near-DFT accuracy at a fraction of the cost; good for pre-screening

Integrated Compositional and Structural Models

One advanced pipeline employs a dual-encoder model that integrates both compositional and structural signals to predict synthesizability [10].

• Model Architecture: A compositional transformer (MTEncoder) is combined with a structural graph neural network (fine-tuned from JMP). These feed separate multi-layer perceptron (MLP) heads to output a synthesizability probability [10].
• Training Data: Models are trained on data from the Materials Project, where a composition is labeled as synthesizable (y=1) if any of its polymorphs has a counterpart in the experimental ICSD database, and unsynthesizable (y=0) otherwise [10].
• Rank-Average Ensemble: Instead of using raw probability thresholds, the final ranking of candidates uses a Borda fusion method that aggregates the ranks from both the composition and structure models, providing a more robust prioritization scheme [10].

Large Language Models for Synthesis Planning

The Crystal Synthesis Large Language Models (CSLLM) framework represents a paradigm shift, treating synthesizability and synthesis planning as a text-based reasoning task [15].

• Input Representation: Crystal structures are converted into a concise "material string" that includes space group, lattice parameters, and a minimal set of atomic sites with their Wyckoff positions, providing a comprehensive yet efficient text representation [15].
• Multi-Task Framework: CSLLM uses three specialized LLMs: a Synthesizability LLM to judge if a structure can be made, a Method LLM to classify the synthetic pathway (e.g., solid-state or solution), and a Precursor LLM to identify suitable precursor materials [15].

The following diagram illustrates a synthesizability-guided pipeline that integrates these modern approaches, from initial screening to experimental synthesis.

Figure 1: A synthesizability-guided discovery pipeline. This workflow successfully synthesized 7 out of 16 target materials in just three days by moving beyond simple convex hull stability [10].

Experimental Protocols for Model Validation

The ultimate test for any synthesizability model is its performance in guiding real experimental synthesis. The following protocol, derived from a successful validation study, provides a template for such validation [10].

High-Throughput Synthesis and Characterization

• Candidate Selection: From a pool of ~500 highly synthesizable candidates, final targets are selected using an LLM to check for prior synthesis and expert judgment to remove candidates with unrealistic oxidation states or overly common formulas [10].
• Synthesis Planning: For each target, a precursor-suggestion model (e.g., Retro-Rank-In) generates a ranked list of viable solid-state precursors. A separate model (e.g., SyntMTE) then predicts the required calcination temperature. The reaction is balanced, and precursor quantities are computed [10].
• Automated Synthesis: Precursors are weighed, ground, and calcined in a benchtop muffle furnace. To maximize efficiency, targets are batched by recipe similarity so that 12 experiments can be run simultaneously in the same furnace [10].
• Product Verification: The synthesis products are automatically characterized using X-ray diffraction (XRD). The experimental diffraction pattern is compared to the pattern of the target structure to confirm successful synthesis [10].

The Scientist's Toolkit: Key Research Reagents & Solutions

In both computational and experimental synthesizability research, a common set of tools and data sources forms the foundation of the workflow.

Table 2: Essential Research Reagents and Resources for Synthesizability Research

Resource / Tool	Type	Primary Function in Research
Materials Project (MP) [10] [18]	Computational Database	Source of DFT-calculated crystal structures and formation energies for training and benchmarking.
Inorganic Crystal Structure Database (ICSD) [10] [15]	Experimental Database	Curated source of experimentally synthesized crystal structures, used as positive examples for model training.
Graph Neural Networks (GNNs) [10] [19]	Computational Model	Architecture for learning from crystal structure graphs to predict stability and properties.
Universal ML Interatomic Potentials (MLIPs) [18] [53]	Computational Model	Provides near-DFT accuracy for energy and forces at low cost, enabling large-scale stability pre-screening.
CIF / POSCAR Format [15]	Data Standard	Standard file formats for representing crystal structure information.
Retro-Rank-In / SyntMTE [10]	Computational Model	Predicts viable solid-state precursors and synthesis conditions (e.g., temperature) for a target material.

The 0 eV/atom boundary on the convex hull is a useful but fundamentally limited concept for predicting synthesizability. Its high false-positive rate stems from an oversimplified view of material stability that ignores the kinetic and entropic realities of synthesis. The future of efficient materials discovery lies in sophisticated, data-driven models that directly learn the complex relationship between composition, structure, and experimental synthesizability. Frameworks that integrate compositional and structural insights, such as hybrid ML models and specialized LLMs, along with powerful universal MLIPs for pre-screening, are proving to be vastly more effective. By adopting these tools, researchers can finally move beyond the brittle thermodynamic boundary and significantly accelerate the journey from in-silico prediction to realized material.

Data Augmentation and Feature Selection for Improved Classification Performance

In computational materials science and drug discovery, accurately predicting the stability and synthesizability of new compounds is a fundamental challenge. The core of this challenge lies in the convex-hull stability of a material's phase diagram, a global determinant of thermodynamic stability that indicates whether a compound can exist in equilibrium with its elemental components [7] [18]. The integration of data augmentation and feature selection into machine learning (ML) workflows directly enhances our ability to classify whether a candidate material will be stable, thereby accelerating the discovery process.

Data augmentation techniques address the critical issue of data scarcity and imbalance often encountered in experimental and computational materials datasets. By artificially expanding training data, these methods improve model robustness and generalization [54] [55]. Concurrently, feature selection optimizes model performance by identifying the most predictive descriptors of stability, reducing computational cost, and mitigating the risk of overfitting [56]. When strategically combined, these methodologies significantly boost the performance of classifiers tasked with distinguishing stable, synthesizable materials within the vast compositional and structural space, creating a more efficient and reliable pipeline for materials discovery [18] [10].

Background and Theoretical Framework

The Critical Role of Convex-Hull Stability

In materials science, thermodynamic stability is not an intrinsic property of a single compound but is determined through competition with all other possible phases in a chemical system. This competition is mathematically represented by the convex hull of formation energies. A material is deemed thermodynamically stable at 0 K if its formation energy lies on this convex hull, a state defined as being "at-hull" with an energy difference (Ehull) of 0 meV/atom [7] [18]. Conversely, a positive Ehull signifies a metastable or unstable compound.

The convex hull formalism provides a powerful tool for predicting stability not only under standard conditions but also in response to external perturbations such as temperature, pressure, and applied voltage [7] [8]. Consequently, the primary classification task in computational discovery is to predict whether a candidate material's Ehull is zero, a binary decision that guides high-throughput screening. However, a significant disconnect exists between a model's regression accuracy on formation energy and its performance on this critical classification task. Models with low mean absolute error can still produce high false-positive rates if their predictions for unstable materials lie close to the Ehull = 0 decision boundary [18].

Data Augmentation and Feature Selection as Enablers

Machine learning models for stability prediction face two major data-related challenges, which are also prevalent in related fields like drug discovery [55] [56]:

• Limited and Costly Data: Experimental and first-principles computational data (e.g., from Density Functional Theory) are resource-intensive to acquire, leading to small datasets that are insufficient for training complex models like deep neural networks.
• High-Dimensional Feature Space: Materials can be described by a vast array of compositional, structural, and electronic features. Including irrelevant or redundant features can degrade model performance and interpretability.

Data augmentation and feature selection directly address these limitations. Augmentation expands the effective training set, while feature selection prunes the feature space, together enabling the development of more accurate, robust, and efficient classifiers for stability prediction [54] [56].

Data Augmentation Techniques and Protocols

Data augmentation encompasses a suite of techniques designed to increase the size and diversity of a dataset by creating modified versions of existing data instances. In the context of materials and molecular science, these techniques can be applied across different data representations.

Table 1: Common Data Augmentation Techniques for Scientific Data

Category	Technique	Description	Impact on Model Performance
Geometric Transformations	Random Rotation, Flipping, Scaling	Alters spatial orientation and perspective of image-based data (e.g., microscopy, spectroscopy) [54].	Improves invariance to object orientation; crucial for image-based classification [54].
Color & Value Adjustments	Color Jitter, Brightness/Contrast Adjustment, Gaussian Noise	Modifies pixel intensities or numerical values to simulate variations in lighting and sensor noise [54].	Enhances model generalization and robustness to noise in real-world data acquisition [54].
SMILES-Based Augmentation	Multiple SMILES String Generation	Leverages the inherent permutation rules of Simplified Molecular Input Line Entry System (SMILES) notation to create different string-based representations of the same molecule [55].	Enriches molecular datasets for NLP-based models; improves learning of structural relationships and model robustness [55].
Synthetic Data Generation	Generative Adversarial Networks (GANs)	Generates entirely new, synthetic data samples that mimic the statistical distribution of the original training data [56].	Effectively addresses class imbalance; can enhance dataset size and variability, improving accuracy on minority classes [56].

Experimental Protocol for SMILES and GAN Augmentation

SMILES Augmentation for Molecular Property Prediction: This protocol is used to build models for predicting molecular properties, such as inhibitor activity [55].

• Data Preparation: Compile a dataset of molecular structures represented as canonical SMILES strings alongside their target properties (e.g., IC₅₀ values for alpha-glucosidase inhibition).
• Augmentation Execution: For each SMILES string in the dataset, generate multiple equivalent representations by:
  • Enumerating different starting atoms in the molecular graph.
  • Writing branches in different orders.
  • This can be achieved using chemoinformatics libraries like RDKit.
• Model Training: Use the augmented set of SMILES strings to train a deep learning model, such as a Bidirectional Encoder Representations from Transformers (BERT) model adapted for molecular data. The model learns that different SMILES sequences correspond to the same underlying structure and property [55].

GAN Augmentation for Imbalanced Clinical/Materials Data: This protocol is designed for tabular data where the class of interest (e.g., stable materials, asthmatic patients) is underrepresented [56].

• Data Preprocessing: Normalize or standardize all numerical features in the original imbalanced dataset.
• GAN Training: Train a Generative Adversarial Network on the samples from the minority class. The generator learns to produce synthetic feature vectors, while the discriminator learns to distinguish between real and synthetic samples.
• Synthetic Data Generation: After training, use the generator to create a sufficient number of synthetic samples for the minority class to balance the dataset.
• Classifier Training: Train the final classifier (e.g., XGBoost) on the combined dataset of original majority class samples and synthetic minority class samples. This prevents the model from being biased toward the majority class [56].

Feature Selection Methodologies

Feature selection enhances model performance by identifying and retaining the most relevant input variables, thereby reducing overfitting, improving computational efficiency, and increasing model interpretability.

Table 2: Categories and Applications of Feature Selection Methods

Category	Method Examples	Mechanism	Advantages
Filter Methods	Correlation-based (e.g., with Ehull), Mutual Information	Selects features based on statistical measures of their relationship with the target variable, independent of the classifier.	Fast and computationally efficient; scalable to very high-dimensional spaces.
Wrapper Methods	Recursive Feature Elimination (RFE)	Uses the performance of a specific classifier to evaluate and select feature subsets.	Considers feature interactions; typically yields high-performing feature sets.
Embedded Methods	Tree-based (e.g., XGBoost), L1 Regularization (Lasso)	Performs feature selection as an integral part of the model training process.	Balances efficiency and performance; no separate training step required.

Experimental Protocol for Embedded Feature Selection with XGBoost

The Extreme Gradient Boosting (XGBoost) algorithm is a powerful embedded method for feature selection due to its built-in mechanism for calculating feature importance [56].

• Model Training: Train an XGBoost classifier on the (optionally augmented) training dataset. It is crucial to use a separate validation set or cross-validation to tune hyperparameters and avoid overfitting during this step.
• Importance Calculation: After training, extract the "feature importance" scores from the model. XGBoost typically provides importance based on the average gain of splits that use the feature.
• Feature Ranking and Selection: Rank all features by their importance scores in descending order. The number of features to retain can be determined by:
  • Setting a threshold on the importance score.
  • Selecting the top-k features.
  • Using a scree plot of cumulative importance to identify the point of diminishing returns.
• Final Model Validation: Retrain the final classification model (which could be XGBoost again or a different classifier) using only the selected subset of features. Evaluate its performance on a held-out test set to confirm the effectiveness of the feature selection.

Integrated Framework for Stability Classification

The true power of data augmentation and feature selection is realized when they are strategically integrated into a cohesive workflow for material stability classification. This framework explicitly connects these techniques to the overarching goal of accurate convex-hull prediction.

A Unified Workflow for Discovery

The diagram below illustrates a proposed pipeline that integrates data augmentation and feature selection to improve the classification of convex-hull stability.

Case Study: Connecting Augmentation to Hull-Aware Active Learning

The Convex hull-Aware Active Learning (CAL) algorithm provides a compelling use case for the role of uncertainty in guiding discovery [7] [8]. CAL uses Gaussian Processes (GPs) to model energy surfaces, producing a probabilistic convex hull where every composition has a probability of being stable.

• Role of Feature Selection: In a high-dimensional feature space (e.g., including elemental properties, structural descriptors), feature selection can be used to identify the most salient descriptors for the GP model's covariance function, leading to a more accurate and tractable energy surface model.
• Role of a Probabilistic Hull: The output of CAL is not a single hull, but a distribution of possible hulls. This provides a natural, uncertainty-informed ranking for experimental validation, prioritizing compositions with high probability of being stable (e.g., P(stable) > 0.95) [7].
• Synergy with Augmentation: The probabilistic framework of CAL can be extended. For systems where DFT data is sparse, data augmentation techniques could be explored to enrich the training set for the GP, potentially leading to a better-informed posterior distribution of the hull and faster convergence in the active learning loop.

This approach directly minimizes the number of expensive energy calculations required to resolve the convex hull, as it focuses computational effort on compositions whose stability is most uncertain and most relevant to the hull's structure [7] [8].

The Scientist's Toolkit: Essential Research Reagents & Solutions

This section details key computational tools and data resources that form the foundation of modern data-driven discovery pipelines.

Table 3: Key Research Reagents and Computational Tools

Tool/Resource Name	Type	Function in the Pipeline
PyTorch / TensorFlow [54] [55]	Deep Learning Library	Provides the framework for building and training models for data augmentation (GANs), NLP (BERT on SMILES), and other complex tasks.
RDKit	Chemoinformatics Library	Facilitates the manipulation and augmentation of molecular structures, including SMILES enumeration and molecular descriptor calculation.
XGBoost [56]	Machine Learning Algorithm	Serves as a powerful classifier and an embedded method for feature selection via its built-in feature importance metrics.
Hugging Face Transformers [55]	NLP Library	Provides access to pre-trained BERT and other transformer models that can be fine-tuned on molecular SMILES data for property prediction.
Materials Project (MP) [18] [10]	Materials Database	A primary source of DFT-calculated data, including formation energies and convex hull distances, used for training and benchmarking stability prediction models.
Gaussian Process (GP) Regression [7] [8]	Statistical Model	The core engine of CAL for modeling energy surfaces and quantifying uncertainty, which is propagated to the convex hull.

The integration of data augmentation and feature selection presents a powerful paradigm for enhancing classification performance in the critical task of material stability prediction. By systematically addressing the challenges of data scarcity and high-dimensional feature spaces, these methodologies enable the construction of more robust, accurate, and efficient machine learning models. When framed within the context of convex-hull stability, this integrated approach directly contributes to a more rational and accelerated materials and drug discovery pipeline. The resulting workflows, such as those incorporating hull-aware active learning, provide not only predictions but also crucial uncertainty quantification, allowing researchers to prioritize experimental efforts intelligently and navigate the vast chemical space with greater confidence and success.

Benchmarks and Reality Checks: Validating Predictions and Comparing Methodologies

Prospective vs. Retrospective Benchmarking in Materials Discovery

The acceleration of materials discovery through machine learning (ML) has created a critical need for robust model evaluation frameworks. Benchmarking practices, which assess the performance and predictive power of these models, are broadly categorized into two distinct paradigms: retrospective and prospective benchmarking. The choice between these paradigms has profound implications for how we gauge a model's utility in a real-world discovery campaign, especially when the research goal involves predicting the thermodynamic stability of new materials via metrics like the convex-hull stability.

The convex hull of a chemical system represents the lowest-energy mixture of phases at given compositions. The energy above the convex hull (Eₕᵤₗₗ) is a key metric of thermodynamic stability, indicating the energy penalty for a compound not being on this hull [5]. A compound with Eₕᵤₗₗ = 0 eV/atom is considered thermodynamically stable, while one with Eₕᵤₗₗ > 0 is metastable or unstable. Despite its importance, a low Eₕᵤₗₗ is not a definitive predictor of synthesizability, as kinetic and entropic barriers also play a crucial role [10]. This complex relationship between computational stability and experimental realizability sits at the heart of modern materials discovery, framing the critical need for meaningful benchmarking.

Defining the Benchmarking Paradigms

Retrospective Benchmarking

Retrospective benchmarking evaluates ML models using existing, historically acquired datasets. The model is trained on a subset of known data and its predictions are validated against a held-out test set from the same data distribution.

• Core Principle: Measures a model's ability to interpolate or generalize within a known chemical space.
• Common Practice: Uses standardized dataset collections, such as Matbench, which aggregate data from sources like the Materials Project (MP) for tasks like property prediction [18].
• Key Limitation: It operates under the assumption that the training and test data are independently and identically distributed, a condition often violated in real discovery campaigns aimed at finding truly novel materials outside known databases [18].

Prospective Benchmarking

Prospective benchmarking tests a model's performance in a simulated or live discovery workflow where the test data is generated after the model is trained, often as a direct result of the model's own predictions.

• Core Principle: Measures a model's ability to guide an exploration process and make successful predictions on genuinely new data, often facing a realistic covariate shift [18] [57].
• Core Principle: Requires the model to have "skin in the game," meaning its predictions directly influence the selection of candidates for subsequent (and costly) validation via computation or experiment [57].
• Key Strength: Provides a more realistic estimate of how much a model can accelerate discovery by reducing the number of failed trials, as it tests the complete closed-loop discovery pipeline [58].

Table 1: Comparative Overview of Benchmarking Paradigms

Feature	Retrospective Benchmarking	Prospective Benchmarking
Data Split	Random or clustered split of existing data [18]	Temporal split or data generated by the discovery workflow [18]
Performance Estimate	Optimistic, measures interpolation	Realistic, measures extrapolation and guidance capability
Cost	Lower, uses existing data	Higher, requires new calculations or experiments
Primary Goal	Compare model architectures on standardized tasks [57]	Evaluate a model's efficacy in a real discovery campaign [18]
Context in Materials Discovery	Useful for initial model screening and development	Essential for justifying experimental validation efforts [18]

The Critical Need for Prospective Frameworks

The disconnect between retrospective model performance and real-world utility has driven the development of prospective evaluation frameworks like Matbench Discovery [18]. This initiative addresses four key challenges in benchmarking ML for materials discovery:

• Prospective vs. Retrospective Performance: Idealized retrospective splits can create an "illusion of utility" [18]. Models exhibiting low error on a held-out test set may still produce high false-positive rates when their (nominally accurate) predictions lie close to the critical decision boundary—for example, an Eₕᵤₗₗ of 0 eV/atom [18]. This can lead to significant opportunity costs from pursuing unstable candidates.

• Relevant Prediction Targets: While formation energy is a common regression target, the energy above the convex hull (Eₕᵤₗₗ) is a more direct indicator of thermodynamic stability [18] [5]. Framing the discovery task as a classification problem (e.g., stable vs. unstable based on an Eₕᵤₗₗ threshold) is often more aligned with the end goal than a pure regression task.

• Informative Metrics: Global regression metrics like Mean Absolute Error (MAE) can be misleading. A more task-relevant evaluation assesses a model's performance as a classifier, focusing on metrics like precision and recall in identifying stable materials [18].

• Scalability and Chemical Diversity: Effective benchmarks must operate at a scale where the test set is larger than the training set to mimic true deployment. They must also encompass broad chemical diversity to test a model's ability to generalize across compositional space [18].

The following diagram illustrates the conceptual and procedural differences between these two benchmarking approaches within a materials discovery workflow.

Quantitative Benchmarks and Experimental Protocols

Prospective benchmarking quantitatively measures how much a model can accelerate discovery compared to a baseline method, such as random search.

Benchmarking Sequential Learning for Catalysts

A landmark study benchmarked Sequential Learning (SL) for discovering oxygen evolution reaction (OER) catalysts [58] [59]. The experimental protocol was as follows:

• Dataset: Four distinct chemical spaces, each containing 2121 unique metal oxide compositions derived from six-element sets. The performance metric was the electrocatalytic overpotential [58].
• SL Workflow: An ML model was iteratively retrained. In each cycle, it prioritized the most promising candidates from the entire search space for subsequent experimental synthesis and testing, based on its predictions.
• Research Goals: The study evaluated SL performance against three different objectives:
  • Discovery of any "good" material (top-percentile performance).
  • Discovery of all "good" materials.
  • Development of a globally accurate predictive model.
• Results: The acceleration factor provided by SL was highly dependent on the chosen goal and model. While SL accelerated the discovery of any good material by up to a factor of 20, it could sometimes decelerate progress for other goals, highlighting the need for goal-aware benchmarking [58].

Table 2: Benchmarking Results for Sequential Learning in Catalyst Discovery

Research Goal	Best-Performing Model	Acceleration Factor vs. Random	Key Finding
Find any 'good' material	Random Forest (RF)	Up to 20x	Effective for rapid initial discovery [58]
Find all 'good' materials	Gaussian Process (GP) / Linear Ensemble (LE)	Variable, can decelerate	Requires exploration; model choice is critical [58]
Build a globally accurate model	Gaussian Process (GP)	Low acceleration	Conflicts with pure optimization objective [58]

Prospective Discovery of Stable Crystals

Large-scale projects like GNoME (Graph Networks for Materials Exploration) exemplify prospective discovery of crystals with low convex-hull energies [60].

• Protocol:
  • Generation: Diverse candidate structures are generated through symmetry-aware substitutions and random searches.
  • Filtration: A deep learning model (GNN) predicts the stability (decomposition energy) of millions of candidates.
  • Validation: The most promising candidates are validated using Density Functional Theory (DFT) calculations.
  • Active Learning: The DFT-validated data is fed back into the training set, improving the model in subsequent rounds [60].
• Results: This prospective, active-learning-driven approach led to the discovery of over 2.2 million new crystal structures stable with respect to the prior convex hull, expanding the number of known stable materials by nearly an order of magnitude [60]. The "hit rate" (precision of stable predictions) improved from 6% to over 80% through this iterative process.

A Synthesizability-Guided Pipeline

Recognizing that convex-hull stability alone is an insufficient predictor of experimental success, recent research has integrated synthesizability prediction directly into the discovery pipeline [10].

• Protocol:
  • Synthesizability Score: A model integrates both compositional and structural features to predict the likelihood that a computationally stable compound can be synthesized in the laboratory. This moves beyond the Eₕᵤₗₗ by considering finite-temperature effects and kinetic barriers [10].
  • Candidate Prioritization: From a pool of 4.4 million computed structures, candidates are ranked by their synthesizability score and Eₕᵤḷₗ.
  • Synthesis Planning: A second model predicts viable solid-state precursors and calcination temperatures by learning from literature-mined synthesis recipes.
  • Experimental Validation: The prioritized candidates and their predicted synthesis routes are executed in a high-throughput laboratory [10].
• Results: In a prospective test, this pipeline identified 80 target materials. Of 16 characterized, 7 were successfully synthesized, matching the target structure, including one novel and one previously unreported compound [10]. This demonstrates the power of coupling stability prediction with synthesizability assessment.

The workflow below outlines the key stages of this integrated, synthesizability-guided approach to materials discovery.

Table 3: Essential Computational and Experimental Resources for Stability and Synthesizability Research

Tool / Resource	Type	Primary Function
Density Functional Theory (DFT)	Computational Method	Provides a quantum-mechanical estimate of a crystal's energy, used to calculate formation energy and Eₕᵤₗₗ [18] [60].
Universal Interatomic Potentials (UIPs)	Machine Learning Model	Acts as a fast, approximate force field to screen thousands of candidate structures before running more expensive DFT [18].
High-Throughput Experimentation (HTE)	Experimental Platform	Enables the parallel synthesis and electrochemical testing of thousands of material compositions (e.g., 2121 catalysts) to generate benchmarking datasets [58].
Synthesizability Model	Machine Learning Model	Predicts the likelihood of successful laboratory synthesis based on composition and crystal structure, going beyond Eₕᵤₗₗ [10].
Retrosynthetic Planning Model	Machine Learning Model	Suggests viable solid-state precursors and reaction conditions (e.g., temperature) for a target material [10].

The transition from retrospective to prospective benchmarking marks a maturation of machine learning's role in materials discovery. Retrospective benchmarks remain valuable for initial model development and architectural comparisons. However, only prospective benchmarking, which tests the full, closed-loop discovery pipeline, can truly measure a model's capacity to accelerate the finding of new, stable, and synthesizable materials. The integration of convex-hull stability calculations with data-driven synthesizability predictors represents the forefront of this effort, creating a more holistic and effective framework for guiding experimental synthesis. For researchers, the imperative is clear: to validate a model's real-world impact, it must be tested prospectively with "skin in the game," ultimately leading to the successful synthesis of novel compounds.

The accelerated discovery of new inorganic crystals is a critical driver of technological progress, paving the way for more efficient solar cells, lighter batteries, and smaller transistors [18]. The combinatorial space of possible materials is vast, with an estimated ~10^10 possible quaternary compounds allowed by simple chemical rules, yet only a minuscule fraction have been synthesized or simulated [18] [61]. Traditional computational discovery, primarily using Kohn-Sham density functional theory (DFT), provides a favorable compromise between accuracy and computational cost but remains resource-intensive, consuming up to 45% of core hours on some national supercomputers [18] [61].

Machine learning (ML) promises to accelerate this process by acting as a rapid pre-filter for more expensive, higher-fidelity DFT calculations [18]. However, the rapid proliferation of ML models has created a critical need for standardized evaluation frameworks to assess their real-world utility in discovery campaigns. The Matbench Discovery framework, introduced in 2025, addresses this need by providing a robust, task-oriented benchmark for evaluating ML models on their ability to predict thermodynamic stability from unrelaxed crystal structures, thereby simulating a realistic high-throughput discovery pipeline [18] [62].

This framework is situated within a broader research context where accurately predicting a material's convex-hull stability—a more reliable indicator of synthesizability than formation energy alone—is paramount for successful computational materials discovery [18] [61]. By focusing on this key determinant, Matbench Discovery provides a more meaningful assessment of a model's potential to accelerate real materials innovation.

Core Challenges in ML-Guided Materials Discovery

Matbench Discovery was designed to overcome four fundamental limitations that have historically plagued the evaluation of ML models for materials science [18] [61].

The Need for Prospective Benchmarking

Idealized benchmarks often fail to reflect real-world challenges. Matbench Discovery adopts a prospective benchmarking approach where the test data is generated by the intended discovery workflow itself. This creates a substantial but realistic covariate shift between training and test distributions, providing a more reliable indicator of model performance in actual deployment compared to retrospective data splits that may test artificial use cases [18].

Defining Relevant Prediction Targets

A significant disconnect exists between commonly used regression targets and actual materials stability. While DFT formation energies are widely used as ML targets, they do not directly indicate thermodynamic stability [18]. The true stability of a material depends on its energetic competition with other phases in the same chemical system, quantified by its distance to the convex hull in the phase diagram [18] [61]. This energy above the convex hull represents the leading-order term predictive of (meta-)stability at standard conditions, making it a more suitable target property despite other factors like kinetic and entropic effects that influence real-world stability [61].

Selecting Informative Performance Metrics

Global regression metrics like Mean Absolute Error (MAE) and Root Mean Squared Error (RMSE) can provide misleading confidence in model reliability [18] [61]. Accurate regressors can still produce high false-positive rates if their nominally accurate predictions lie close to the decision boundary at 0 eV/atom above hull, where many materials reside [61]. These failed predictions incur high opportunity costs through wasted laboratory resources. Matbench Discovery therefore emphasizes classification performance based on a model's ability to facilitate correct decision-making in a discovery pipeline, not just regression accuracy [18].

Ensuring Scalability to Real Discovery Campaigns

Future materials discovery requires models that perform well in large data regimes with broad chemical diversity. Small benchmarks can obscure poor scaling relations or weak out-of-distribution performance [18]. Matbench Discovery constructs tasks where the test set is larger than the training set to mimic true deployment at scale, differentiating models capable of leveraging representation learning in large-data environments from those that cannot [18].

Framework Methodology and Experimental Design

Task Definition and Workflow

The core task in Matbench Discovery requires ML models to predict the energy above the convex hull of the relaxed structure using only the unrelaxed (initial) crystal structure as input [61] [63]. This setup avoids circular dependencies, as obtaining relaxed structures traditionally requires expensive DFT calculations—the very process the ML models are meant to accelerate [63]. In the framework, the convex hull used for final evaluation is constructed from DFT reference energies, not model predictions [62].

Benchmarking Protocol

The framework employs a rigorous benchmarking protocol that simulates a high-throughput screening campaign [18] [62]. Key aspects include:

• Input Constraints: Models can train on any available data, but must make predictions using only unrelaxed structures at test time [63].
• Stability Classification: The continuous energy above hull is converted into a binary stability classification using the threshold of 0 eV/atom [61].
• Model Agnosticism: The benchmark accommodates diverse ML methodologies including random forests, graph neural networks, one-shot predictors, iterative Bayesian optimizers, and universal interatomic potentials [18].
• Comprehensive Evaluation: Performance is assessed using both regression metrics (MAE, RMSE) and task-relevant classification metrics (F1 score, precision, recall) [18] [61].

Quantitative Results and Model Performance

Performance Metrics and Leaderboard

Matbench Discovery maintains an online leaderboard that ranks models by various metrics, allowing researchers to prioritize those most relevant to their needs [18] [62]. The F1 score for stability classification emerges as a particularly informative metric, balancing precision and recall in the identification of stable crystals [61].

Table 1: Model Performance Rankings by Methodology Type (adapted from [61])

Rank	Model	Methodology	F1 Score	Discovery Acceleration Factor
1	EquiformerV2 + DeNS	Universal Interatomic Potential	0.82	~6x
2	Orb	Universal Interatomic Potential	High	N/A
3	SevenNet	Universal Interatomic Potential	High	N/A
4	MACE	Universal Interatomic Potential	0.6-0.82	Up to 5x
5	CHGNet	Universal Interatomic Potential	0.6-0.82	Up to 5x
6	M3GNet	Universal Interatomic Potential	0.6-0.82	Up to 5x
7	ALIGNN	Graph Neural Network	Moderate	N/A
8	MEGNet	Graph Neural Network	Moderate	N/A
9	CGCNN	Graph Neural Network	Moderate	N/A
10	Wrenformer	One-shot Predictor	Moderate	N/A
11	BOWSR	Iterative Bayesian Optimizer	Low	N/A
12	Voronoi Random Forest	Random Forest	Low	N/A

Key Findings: Universal Interatomic Potentials Dominate

The most significant finding from Matbench Discovery is the clear superiority of Universal Interatomic Potentials (UIPs), which occupy all top positions in the rankings [61] [63]. These models achieve F1 scores of 0.57–0.82 for crystal stability classification and discovery acceleration factors (DAF) of up to 6× compared to random selection in the first 10k most stable predictions [61].

UIPs represent a significant advancement over earlier approaches because they learn the underlying density functional theory potential energy surface and can perform rapid crystal structure relaxations during inference, providing more accurate energy estimates [18]. This capability explains their superior performance compared to methods that use fixed input features or do not explicitly model atomic interactions.

Table 2: Comparative Analysis of ML Methodologies for Materials Discovery

Methodology	Representative Models	Strengths	Limitations
Universal Interatomic Potentials	MACE, CHGNet, M3GNet	Highest accuracy; Explicit physics modeling; Best F1 scores	Computationally intensive training; Complex architecture
Graph Neural Networks	ALIGNN, MEGNet, CGCNN	Strong representation learning; Good scalability	Lower accuracy than UIPs; No explicit relaxation capability
One-shot Predictors	Wrenformer	Fast inference; Simple architecture	Limited accuracy; No atomic position optimization
Iterative Bayesian Optimizers	BOWSR	Uncertainty quantification; Sequential decision-making	Computationally expensive; Poor performance in benchmark
Random Forests	Voronoi Fingerprint RF	Interpretable; Fast training	Poor performance on large datasets; Limited representation power

Experimental Protocols and Implementation

Model Training and Evaluation Protocol

For researchers implementing Matbench Discovery benchmarks, the following protocol ensures consistent evaluation [18] [61]:

• Data Acquisition: Access the benchmark datasets through the official Matbench Discovery Python package or GitHub repository [62].
• Training Data: Models may be trained on any publicly available data, with the Materials Project, AFLOW, and Open Quantum Materials Database being common sources [61].
• Input Featurization: Each methodology employs its own featurization scheme:
  • UIPs: Use atomic numbers, positions, and lattice vectors directly
  • GNNs: Construct crystal graphs from atomic connections
  • Random Forests: Use engineered features like Voronoi fingerprints [61]
• Prediction: Models predict the energy above hull for unrelaxed structures from the test set.
• Evaluation: Compare predictions against DFT-calculated convex hull energies using both regression and classification metrics.

Integration with High-Throughput Workflows

The framework is specifically designed to simulate realistic discovery workflows [18]:

Table 3: Essential Computational Tools for ML-Guided Materials Discovery

Tool/Resource	Type	Function	Access
Matbench Discovery	Benchmark Framework	Standardized evaluation of ML models	Python package [62]
Automatminer	AutoML Pipeline	Automated feature generation and model selection	Python package [64] [65]
Matminer	Featurization Library	Materials-specific feature generation	Python package [65]
Materials Project	Database	DFT-calculated materials properties	Public API [65]
AFLOW	Database	High-throughput computational data	Public access [61]
Open Quantum Materials Database	Database	DFT-calculated formation energies	Public access [61]

The Matbench Discovery framework represents a significant advancement in standardizing the evaluation of machine learning models for materials discovery. By addressing key challenges in prospective benchmarking, relevant target selection, informative metrics, and scalability, it provides a realistic assessment of model utility in real-world discovery campaigns [18].

The framework's findings have substantial implications for synthesis prediction research, particularly in establishing the convex-hull stability as the critical prediction target for thermodynamic stability assessment [18] [61]. The clear emergence of Universal Interatomic Potentials as the dominant methodology indicates that explicit modeling of atomic interactions and relaxation processes is essential for accurate stability prediction [61] [63].

For researchers engaged in computational materials discovery, Matbench Discovery offers a standardized platform to validate new models and methodologies. The publicly available leaderboard, Python package, and growing repository of model implementations create a foundation for continued innovation in ML-guided materials discovery [62]. As the field progresses, this framework will enable more efficient allocation of computational resources toward promising candidate materials, ultimately accelerating the discovery of new functional materials for energy, electronics, and sustainability applications.

The accurate prediction of a material's synthesizability, guided by its thermodynamic stability on the convex hull, is a critical bottleneck in the computational materials discovery pipeline. While density functional theory (DFT) provides a reliable foundation for calculating formation energies and constructing these hulls, its computational expense renders the exhaustive screening of vast chemical spaces prohibitive. Machine learning (ML) models have emerged as powerful surrogates to accelerate this process. Two dominant paradigms in this field are universal interatomic potentials (uMLIPs), which use atomic structure as input, and compositional models, which rely solely on chemical formula. This whitepaper provides a technical comparison of these approaches, evaluating their performance in predicting stability and other properties within the context of synthesizability prediction. Recent benchmarks indicate that uMLIPs, by explicitly modeling atomic interactions, have advanced sufficiently to effectively and cheaply pre-screen for thermodynamically stable materials, thereby addressing a key challenge in the discovery pipeline [18].

Universal Machine Learning Interatomic Potentials (uMLIPs)

uMLIPs are foundational models trained to approximate the potential energy surface (PES) of a wide range of materials with near-DFT accuracy but at a fraction of the computational cost [66]. They take the full crystallographic structure—atomic positions, lattice vectors, and species—as input and produce the total energy, forces, and stresses as outputs. The energy can then be used to compute the formation energy and, consequently, the distance to the convex hull.

A key architectural advancement in modern uMLIPs is the incorporation of geometric equivariance. This design ensures that the model's internal feature representations transform correctly under rotational and translational symmetries (the E(3) Euclidean group), guaranteeing that scalar outputs like energy are invariant, while vector outputs like forces are equivariant [66] [67]. This principle is crucial for accurately capturing the physics of atomic systems.

Several state-of-the-art architectures exemplify this progress:

• MACE (Multiscale Atomic Cluster Expansion): Utilizes explicit body-ordered messages, embedding a hierarchy of local many-body interactions directly into each message-passing step. This allows it to achieve high accuracy with fewer layers [68].
• CHGNet (Crystal Hamiltonian Graph Neural Network): Incorporates electronic structure information into the latent space via magnetic moment constraints, allowing it to distinguish between different ionic states and electronic configurations [68].
• E2GNN (Efficient Equivariant Graph Neural Network): Employs a scalar-vector dual representation to maintain equivariance without relying on computationally expensive higher-order tensor operations, achieving a strong balance of accuracy and efficiency [67].

The general workflow of a uMLIP involves representing the crystal structure as a graph, passing messages between atoms to build a representation of the local atomic environment, and finally aggregating these representations to predict the total energy.

Compositional Models

In contrast to uMLIPs, compositional models rely solely on the chemical stoichiometry of a compound as input. They bypass the need for explicit atomic coordinates, making them computationally very lightweight. These models operate on the premise that composition alone can be a strong predictor of average material properties.

These models typically use:

• Fixed Descriptors: Pre-computed features based on stoichiometry and elemental properties (e.g., atomic radius, electronegativity, valence electron counts).
• Learned Representations: Embeddings of the chemical formula, often using transformer architectures, that are learned from large datasets [10].

While exceptionally fast, these models lack any explicit knowledge of atomic arrangement. They cannot distinguish between polymorphs (different crystal structures with the same composition) and are inherently limited in predicting properties that are highly sensitive to structure, such as elastic constants and phonon spectra.

Quantitative Performance Benchmarking

Performance on Stability and Energy Prediction

The core task in synthesizability prediction is the accurate identification of thermodynamically stable materials, typically defined as those lying on or very near the convex hull of formation energies. Systematic benchmarking on this task reveals a significant performance gap between uMLIPs and compositional models.

The Matbench Discovery benchmark, an evaluation framework designed to simulate a real-world materials discovery campaign, has shown that uMLIPs consistently outperform other methodologies. A key finding is that accurate regression of formation energy alone is insufficient; models must be evaluated on their ability to correctly classify materials as stable or unstable, as even models with low mean absolute error (MAE) can produce high false-positive rates near the stability decision boundary [18]. In this rigorous, prospective testing environment, universal interatomic potentials have been found to surpass all other methodologies, including compositional models, in both accuracy and robustness [18].

Table 1: Performance Comparison on Material Property Prediction

Model Type	Example Models	Key Strengths	Key Limitations	Stability Prediction Performance
Universal Interatomic Potentials (uMLIPs)	MACE, CHGNet, MatterSim, SevenNet [68] [69]	High accuracy for energies, forces, and stresses; Can predict a wide range of mechanical and dynamical properties; Polymorph-aware [18] [69]	High computational cost; Requires atomic coordinates; Performance depends on training data fidelity [66]	Superior; Effectively pre-screens stable hypothetical materials [18]
Compositional Models	MTEncoder, random forests, graph networks on composition [10]	Extremely fast inference; Simple input (formula only); Useful for high-level compositional screening [10]	Cannot distinguish polymorphs; Limited accuracy for properties beyond simple heuristics; No force/stress output [10]	Limited; Useful for initial screening but insufficient for reliable discovery [18]

Performance on Mechanical and Dynamical Properties

The advantage of uMLIPs becomes even more pronounced for properties that depend on the curvature of the potential energy surface, which are entirely inaccessible to compositional models.

• Elastic Properties: A systematic benchmark of four uMLIPs (MatterSim, MACE, SevenNet, CHGNet) on nearly 11,000 elastically stable materials from the Materials Project found that SevenNet achieved the highest accuracy, while MACE and MatterSim offered a good balance of accuracy and efficiency [68]. Accurately predicting elastic constants is a strict test of a model's ability to capture second derivatives of the PES.
• Phonon Properties: The prediction of harmonic phonon spectra is another critical test, as phonons determine thermodynamic stability and thermal properties. A benchmark of seven uMLIPs revealed that while some models like MACE-MP-0 and SevenNet-0 achieve high accuracy, others show substantial inaccuracies despite performing well on energy and force predictions for equilibrium structures [69]. This highlights that excellent performance on primary targets (energy/forces) does not automatically guarantee accuracy for second-order properties.

Table 2: Specialized Property Prediction Accuracy of uMLIPs

Property Type	Top-Performing uMLIPs	Reported Performance / Notes	Importance for Synthesis
Elastic Constants	SevenNet, MACE, MatterSim [68]	SevenNet achieves highest accuracy; MACE & MatterSim balance accuracy with efficiency.	Indicates mechanical stability and hardness; crucial for structural materials.
Phonon Spectra	MACE-MP-0, SevenNet-0 [69]	High accuracy for harmonic properties; some models (e.g., CHGNet) show larger errors.	Determines dynamical stability and finite-temperature thermodynamic stability.
Synthesizability Score	Integrated models (composition + structure) [10]	Combined models successfully identified and guided the synthesis of 7 new materials from a pool of candidates [10].	Directly predicts experimental feasibility, going beyond convex-hull stability.

Experimental Protocols and Workflows

uMLIP Workflow for Stability Assessment

The standard protocol for using a uMLIP to assess thermodynamic stability involves a structure relaxation step, which compositional models omit.

Diagram 1: uMLIP stability assessment workflow.

• Input: The process begins with an initial, often unrelaxed, crystal structure (atomic coordinates and lattice vectors).
• Energy/Force Evaluation: The uMLIP is used to compute the total energy and atomic forces for the structure.
• Geometry Optimization: An optimization algorithm (e.g., L-BFGS) uses the predicted forces to iteratively adjust atomic positions and cell parameters until the forces are minimized, finding the local energy minimum. This step is critical as the stability is assessed for the relaxed ground-state structure.
• Formation Energy Calculation: The total energy of the relaxed structure is used to calculate its formation energy from the constituent elements.
• Convex-Hull Construction & Analysis: The formation energy of the candidate material is compared against the formation energies of all other competing phases in its chemical space to compute its energy above the convex hull (Ehull). A material is typically considered potentially synthesizable if its Ehull is very small or zero [18] [10].

Workflow for Integrated Synthesizability Prediction

Recent research demonstrates that convex-hull stability alone is an incomplete metric for synthesizability. A more advanced workflow integrates stability with other data-driven synthesizability scores.

Diagram 2: Integrated synthesizability prediction pipeline.

This pipeline, as implemented in a synthesizability-guided discovery effort [10], involves:

• Initial Pool Screening: A large pool of computational structures (e.g., from the Materials Project or GNoME) is first filtered using convex-hull stability calculated by DFT or uMLIPs.
• Multi-Faceted Synthesizability Scoring: The stable candidates are evaluated by a model that integrates two streams of information:
  • Compositional Signals: A transformer model (MTEncoder) processes the chemical formula to account for precursor chemistry and elemental constraints [10].
  • Structural Signals: A graph neural network processes the relaxed crystal structure to capture local coordination and motif stability [10].
• Ranking and Synthesis Planning: The scores from both encoders are aggregated (e.g., via rank-average ensemble) to produce a final prioritized list. For the top candidates, precursor suggestion and reaction condition models (trained on text-mined literature data) are used to plan viable synthesis routes [10].
• Experimental Validation: The planned syntheses are executed in a high-throughput laboratory. This pipeline successfully synthesized 7 out of 16 target compounds, including novel materials, demonstrating the practical utility of moving beyond hull stability alone [10].

The Scientist's Toolkit

Table 3: Essential Resources for Computational Stability and Synthesizability Prediction

Resource / Tool	Type	Function in Research	Example Use Case
Materials Project Database [68]	Computational Database	Source of DFT-calculated crystal structures, formation energies, and elastic properties for training and benchmarking.	Providing a benchmark dataset of 10,994 structures with elastic properties [68].
Matbench Discovery [18]	Benchmarking Framework	An evaluation framework to rank ML models on their ability to identify stable inorganic crystals prospectively.	Identifying that uMLIPs are the top-performing methodology for materials discovery [18].
VASP (Vienna Ab initio Simulation Package) [70]	Quantum Mechanics Code	Generates high-fidelity training data (energies, forces, stresses) for uMLIPs and provides ground-truth validation.	Performing DFT calculations to relax structures and compute reference formation energies [70].
MACE [68] [69]	Universal MLIP	A state-of-the-art equivariant model for accurate energy, force, and stress prediction, suitable for property prediction.	Benchmarking phonon and elastic properties [68] [69].
CHGNet [68] [69]	Universal MLIP	A pretrained graph neural network potential that incorporates charge information for crystal structures.	Used in comparative benchmarks for elastic constants and phonons [68] [69].
DeePMD-kit [66]	MLIP Training/Inference	A popular open-source platform for training and running deep potential molecular dynamics simulations.	Enabling large-scale molecular dynamics simulations with near-DFT accuracy [66].
Text-Mined Synthesis Datasets [11] [10]	Training Data Corpus	Databases of solid-state and solution-based synthesis recipes extracted from scientific literature for training synthesis models.	Training precursor-suggestion models (Retro-Rank-In) and temperature prediction models (SyntMTE) [10].

The comparative analysis between universal interatomic potentials and compositional models clearly demonstrates that uMLIPs offer superior performance for predicting thermodynamic stability and a wide array of mechanical and dynamical properties essential for assessing synthesizability. Their ability to explicitly model atomic interactions and relax crystal structures provides a fundamental advantage over the coarse, composition-only approach. However, the frontier of predictive synthesis is advancing beyond the sole reliance on convex-hull stability from uMLIPs. The most promising pipelines now integrate uMLIP-derived stability data with complementary, data-driven synthesizability scores that fuse compositional and structural insights, ultimately guiding retrosynthetic planning. This hybrid methodology, which successfully transitions from in-silico prediction to experimental synthesis, represents the current state-of-the-art in accelerating the discovery of novel, manufacturable materials.

The discovery of new functional materials often relies on computational screening of candidate structures, where density functional theory (DFT) provides formation energies used to assess thermodynamic stability via convex hull construction. While machine learning (ML) models have demonstrated remarkable accuracy in predicting formation energies, this capability does not translate reliably to correct stability classification. This whitepaper examines the fundamental disconnect between formation energy regression and stability classification, analyzes quantitative performance gaps across ML approaches, and presents methodological frameworks that directly address stability prediction for synthesizability assessment in materials discovery research.

The Fundamental Disconnect: Formation Energy vs. Stability

Thermodynamic Stability Through Convex Hull Construction

The thermodynamic stability of a material is not determined by its formation energy alone, but rather by its energy relative to all other competing phases in the same chemical space. This relative stability is quantified through decomposition enthalpy (ΔHd), obtained via convex hull construction in formation enthalpy-composition space [9].

As illustrated in Figure 1, the convex hull represents the lower convex enthalpy envelope beneath all points in the composition space. Stable compositions lie directly on this hull, while unstable compositions lie above it. The value |ΔHd| represents the energy penalty for a compound's instability—the minimum amount its formation energy must decrease to become stable, or the maximum amount it can increase while remaining stable [9].

The Statistical Challenge

This convex hull relationship creates fundamental challenges for ML stability prediction:

• Different energy scales: Formation energies (ΔHf) typically span several eV/atom (mean ± AAD = -1.42 ± 0.95 eV/atom), while decomposition energies (ΔHd) operate on a much finer scale (0.06 ± 0.12 eV/atom) [9]
• Non-linear threshold behavior: Stability exhibits a sharp, non-linear transition around ΔHd = 0, where small energy errors can flip classification outcomes
• Global dependence: A material's stability depends not only on its own energy but on all other compounds in its chemical space, making it inherently contextual

Table 1: Energy Range Comparison Between Formation and Decomposition Enthalpies [9]

Energy Metric	Typical Range (eV/atom)	Prediction Challenge	Stability Impact
Formation Enthalpy (ΔHf)	-1.42 ± 0.95	Broad energy scale	Indirect
Decomposition Enthalpy (ΔHd)	0.06 ± 0.12	Fine energy scale	Direct threshold

Quantitative Evidence: The Regression-Classification Gap

Performance Discrepancies in Compositional Models

Composition-based ML models demonstrate competent formation energy prediction but perform poorly on stability classification. Tests on 85,014 inorganic crystals from the Materials Project reveal that while formation energy MAE values appear reasonable, the models generate unacceptable false-positive rates when classifying stability [9].

The core issue is error distribution relative to the decision boundary. Even models with excellent overall MAE can have poorly distributed errors near ΔHd = 0, causing correct formation energy predictions to yield incorrect stability classifications due to the absence of systematic error cancellation that benefits DFT [9].

Table 2: Performance Comparison of ML Approaches for Stability Prediction [18] [19]

Method Type	Representative Models	Formation Energy MAE (eV/atom)	Stability Prediction Precision	Key Limitations
Composition-only	Magpie, ElemNet, Roost	0.08-0.15	Low (High FPR)	Same prediction for all polymorphs
Structure-aware (GNN)	M3GNet, JMP	0.03-0.08	Moderate	Requires relaxed structures
Universal Interatomic Potentials	CHGNet, MACE	0.02-0.05	High	Training data limitations
UBEM-GNN (Volume-relaxed)	Custom Zintl GNN	0.027	90% (validated)	Upper-bound approach

Case Study: Zintl Phase Discovery with UBEM-GNN

The Upper Bound Energy Minimization (UBEM) approach exemplifies specialized methodology for accurate stability prediction. Applied to Zintl phases, this method uses a scale-invariant GNN to predict volume-relaxed energies from unrelaxed structures, providing an energy upper bound [19].

Experimental Protocol:

• Dataset Curation: 824 pnictide-based Zintl phases from ICSD as structural prototypes
• Chemical Decoration: Systematic element substitution across Groups 1, 2, 12, 13, 14, and 15, generating >90,000 hypothetical structures
• GNN Training: Model trained on 6,571 DFT volume-relaxed structures to predict volume-relaxed energies from unrelaxed inputs
• Stability Assessment: Minimum energy structures identified for each composition, with decomposition energies computed relative to competing phases
• DFT Validation: Predicted stable phases validated with full DFT calculations

Results: The UBEM-GNN approach identified 1,810 new thermodynamically stable Zintl phases with 90% precision, significantly outperforming M3GNet (40% precision) on the same dataset [19].

Methodological Solutions for Stability Prediction

Integrated Synthesizability Assessment

Beyond thermodynamic stability, practical synthesizability requires additional assessment. A synthesizability-guided pipeline combines compositional and structural models to prioritize experimentally accessible materials [10].

Figure 1: Integrated pipeline combining computational screening with experimental validation. The rank-average ensemble synthesizability score filters candidates before retrosynthetic planning and high-throughput experimental validation [10].

The Matbench Discovery Framework

The Matbench Discovery framework addresses key evaluation challenges in ML-for-materials-discovery through four principles [18]:

• Prospective Benchmarking: Using test data generated from the intended discovery workflow rather than artificial splits
• Relevant Targets: Focusing on distance to convex hull rather than formation energy
• Informative Metrics: Emphasizing classification performance over regression accuracy
• Scalability: Testing on problems where the test set exceeds the training set

This framework reveals that universal interatomic potentials currently provide the most effective pre-screening for thermodynamic stability, outperforming composition-only and structure-aware models that don't require full relaxation [18].

Convex Hull-Aware Active Learning

The global nature of convex hull construction presents challenges for traditional active learning. Convex Hull-Aware Active Learning (CAL) addresses this by using Bayesian methods to prioritize experiments that minimize hull uncertainty [13].

Experimental Protocol:

• Probabilistic Modeling: Represent uncertainty in predicted energies for all compounds
• Hull Uncertainty Quantification: Compute how each energy uncertainty propagates to hull uncertainty
• Experimental Selection: Choose compositions whose precise energy determination would most reduce hull uncertainty
• Iterative Refinement: Update probabilities and uncertainties after each new measurement

This approach minimizes the number of experiments needed to determine phase diagrams by focusing resources on compositions near the hull boundary [13].

Research Reagent Solutions

Table 3: Essential Computational and Experimental Resources for Stability Prediction Research

Category	Tool/Resource	Function	Application Context
Data Sources	Materials Project	DFT-calculated formation energies & structures	Training data for ML models
	ICSD	Experimentally confirmed structures	Positive samples for synthesizability
	OQMD	High-throughput DFT data	Expanded chemical space coverage
ML Models	GNN Surrogates (e.g., JMP)	Structure-property prediction	Stability prediction from crystal structures
	UBEM-GNN	Volume-relaxed energy prediction	Efficient screening without full relaxation
	CSLLM Framework	Synthesizability & precursor prediction	LLM-based synthesis assessment
Experimental	High-Throughput Furnace	Parallel synthesis	Experimental validation of candidates
	Automated XRD	Structure characterization	Phase identification of synthesis products
Software	Matbench Discovery	Model evaluation & benchmarking	Standardized performance assessment
	pymatgen	Materials analysis	Structure manipulation & analysis

The disconnect between accurate formation energy regression and reliable stability classification stems from fundamental differences between these tasks. Formation energy prediction operates on an eV scale with direct learning targets, while stability classification depends on fine-grained energy differences relative to a sharp decision threshold in a multi-compound context. Successful synthesizability prediction requires specialized approaches that directly address the convex hull construction problem, integrate complementary signals from composition and structure, and focus evaluation on classification metrics relevant to real discovery campaigns. Methodologies like UBEM-GNN, convex hull-aware active learning, and integrated synthesizability pipelines represent promising directions that explicitly model the relationship between computational prediction and experimental realization.

In computational materials and drug discovery, predicting a candidate's stability is a critical first step, but it is the subsequent experimental validation that ultimately bridges the gap between digital predictions and tangible outcomes. Thermodynamic stability, commonly quantified by the energy above the convex hull (E({}{\text{hull}})), serves as a primary initial filter for synthesizability [18]. A material on the convex hull (E({}{\text{hull}}) = 0) is thermodynamically stable against decomposition into other phases, while those with a small positive E({}_{\text{hull}}) may be metastable and synthesizable under kinetic control [11]. However, this stability metric, calculated from first-principles density functional theory (DFT) at 0 Kelvin, often overlooks finite-temperature effects, entropic factors, and kinetic barriers that govern real-world synthetic accessibility [10]. Consequently, a growing paradigm emphasizes that computational predictions, including convex-hull stability, must be rigorously tested through experimental validation to demonstrate practical utility and verify reported results [71]. This guide details the methodologies and frameworks for effectively uniting computational stability predictions with experimental synthesis, focusing on the critical role of validation within a discovery pipeline.

Computational Frameworks for Stability Prediction

Advanced Methods for Convex Hull Construction

Accurately determining a material's stability requires constructing a convex hull phase diagram, which defines the set of stable phase-composition pairs. Traditional methods can be computationally expensive, as they require exhaustive energy evaluations for all competing compositions and phases. Novel algorithms are addressing this challenge.

Convex Hull-Aware Active Learning (CAL) is a Bayesian approach that significantly accelerates stability predictions. Unlike conventional methods that focus solely on minimizing energy uncertainty, CAL uses Gaussian processes (GPs) to model energy surfaces and directly reasons about the global convex hull. It iteratively selects the next composition to evaluate by choosing the experiment expected to maximize the information gain about the hull itself, prioritizing compositions on or near the hull and quickly eliminating irrelevant phases [7] [72]. This method can predict the convex hull with significantly fewer energy observations than brute-force or energy-focused approaches [7].

The probabilistic hull generated by CAL provides uncertainty quantification for both the hull and derived properties like stability and chemical potential. This explicit uncertainty is vital for prioritizing experimental candidates, as it allows researchers to assess the confidence of stability predictions [7].

Machine Learning and Synthesizability Prediction

While convex-hull stability is a crucial first-pass filter, its limitations have spurred the development of machine learning (ML) models that predict synthesizability more directly. These models learn from existing experimental data to estimate the probability that a proposed material can be realized in a laboratory.

One advanced framework integrates complementary signals from both composition (x_c) and crystal structure (x_s) [10]:

• Compositional Encoder (f_c): Often a fine-tuned transformer model that processes stoichiometry and elemental chemistry.
• Structural Encoder (f_s): Typically a graph neural network that analyzes the crystal structure graph [10].

These encoders are trained end-to-end on a binary classification task, using data from resources like the Materials Project, where materials are labeled as synthesizable if they have experimental counterparts in databases like the Inorganic Crystal Structure Database (ICSD) [10]. During screening, predictions from both models are aggregated via a rank-average ensemble (Borda fusion) to produce a robust synthesizability score for candidate ranking [10].

Table 1: Key Metrics for Evaluating Computational Prediction Models

Model Type	Primary Metric	Typical Performance (State-of-the-Art)	Key Utility in Discovery
Universal Interatomic Potentials (UIPs) [18]	Precision/Recall for Stable Classification	Surpass other ML methodologies in accuracy & robustness [18]	Effective pre-screening of thermodynamically stable hypothetical materials
CAL (Convex Hull-Aware Learner) [7]	Reduction in Observations to Define Hull	Significantly fewer energy evaluations than baseline methods [7]	Efficiently resolves convex hull with minimal thermodynamic calculations
Composition/Structure Synthesizability Model [10]	Area Under Precision-Recall Curve (AUPRC)	State-of-the-art performance on curated test sets [10]	Ranks candidate structures by likelihood of successful laboratory synthesis

From Prediction to Synthesis: Experimental Validation Protocols

A Synthesizability-Guided Discovery Pipeline

A robust pipeline for transitioning from computational prediction to synthesized material involves multiple stages of filtering and planning, as illustrated in the workflow below.

Workflow for Material Discovery: This diagram outlines the key stages in a synthesizability-guided materials discovery pipeline, from initial computational screening to final experimental validation.

The process begins with a vast pool of computationally generated candidates (e.g., 4.4 million structures). This pool is first filtered using the synthesizability model to retain only the top-ranked candidates. Subsequent filtering applies practical constraints, such as excluding elements from the platinoid group or known toxic compounds [10]. The resulting shortlist (e.g., ~500 candidates) undergoes synthesis planning, which involves:

• Precursor Suggestion: Using models like Retro-Rank-In to generate a ranked list of viable solid-state precursors.
• Condition Prediction: Employing models like SyntMTE, trained on literature-mined data, to predict calcination temperatures and times [10].

Finally, selected targets proceed to high-throughput experimental synthesis and characterization (e.g., via X-ray diffraction) to validate the formation of the target phase [10].

Detailed Experimental Methodology for Solid-State Synthesis

The following protocol is adapted from high-throughput validation campaigns for computationally predicted inorganic crystals [10].

Objective: To synthesize and characterize a target compound from a list of computationally prioritized candidates. Key Reagent Solutions and Materials: Table 2: Essential Materials for Solid-State Synthesis

Reagent/Material	Function/Description	Example/Note
High-Purity Oxide/Carbonate Precursors	Source of cationic elements in the target material	e.g., TiO₂, Li₂CO₃, Co₃O₄; purity ≥ 99%
Mortar and Pestle or Ball Mill	Homogenization and mechanochemical mixing of precursor powders	Ensures intimate contact for solid-state reaction
Thermo Scientific Thermolyne Benchtop Muffle Furnace	High-temperature calcination to drive the solid-state reaction	Allows precise control of temperature and time
Alumina Crucibles	Containers for powder reactions during calcination	Withstand high temperatures, inert to most reactants
X-Ray Diffractometer (XRD)	Primary characterization tool for phase identification and validation	Compares diffraction pattern to computational target

Step-by-Step Procedure:

• Precursor Weighing: Accurantly weigh out solid-state precursor powders according to the stoichiometry of the balanced chemical reaction for the target compound.
• Mixing and Grinding: Transfer the powder mixture to a mortar and pestle (or ball mill) and grind for at least 20-30 minutes to achieve a homogeneous mixture.
• Calcination: Transfer the mixed powder to an alumina crucible and place it in a muffle furnace. Heat the sample to the predicted calcination temperature (e.g., as suggested by SyntMTE models, often between 700-1100°C) for a specified duration (e.g., 12-24 hours) in air.
• Characterization: After the sample cools to room temperature, gently remove it from the crucible. Grind a small portion into a fine powder for XRD analysis. Acquire the XRD pattern and compare it to the simulated pattern of the target crystal structure to confirm successful synthesis.

Validation Metrics: Successful synthesis is primarily confirmed by a strong match between the experimental XRD pattern and the computationally predicted pattern for the target phase [10].

Case Studies in Integrated Prediction and Validation

Successful Discovery of Novel Inorganic Crystals

A landmark study demonstrating this integrated pipeline applied a unified synthesizability model to over 4.4 million simulated crystal structures. The model identified 24 highly synthesizable candidates, which were then targeted for synthesis based on predicted recipes. The entire experimental process, from synthesis to characterization, was completed in just three days using a high-throughput laboratory. The outcome was the successful synthesis and characterization of 16 targets, of which 7 matched the predicted crystal structure, including one completely novel compound and one previously unreported phase [10]. This success highlights the power of combining computational screening with rapid experimental validation to accelerate discovery.

Experimental Validation in Antibody Affinity Maturation

The principle of uniting computation with experiment also proves critical in drug development, such as for antibody affinity maturation. A study aimed at enhancing the affinity of a human antibody against avian influenza virus used a multi-pronged computational approach. This involved constructing a complementarity-determining region (CDR) library, acquiring evolutionary information from sequence alignment, and developing a statistical potential methodology to calculate the binding free energy of antibody-antigen interfaces [73]. The top 10 designed antibody mutants were then subjected to experimental validation. The results confirmed that one point mutation successfully enhanced affinity by 2.5-fold, achieving a final antibody affinity of 2 nM [73]. This case underscores that even outside materials science, computational predictions require experimental confirmation to identify truly effective candidates.

Critical Limitations and Best Practices

Despite promising successes, significant challenges remain in bridging computation and experiment.

• Data Quality and Bias: Large datasets text-mined from scientific literature can suffer from limitations in volume, variety, veracity, and velocity. These arise from technical text-mining challenges and, more profoundly, from the "social, cultural, and anthropogenic biases" in how scientists have historically chosen to explore chemical spaces [11]. Models trained on such data may capture past human preferences rather than fundamental physical laws governing synthesis.
• Misalignment of Metrics: A critical disconnect exists between common regression metrics (e.g., Mean Absolute Error for formation energy) and task-relevant classification metrics for discovery. A model with excellent energy prediction accuracy can still have a high false-positive rate if its predictions cluster near the stability boundary (0 eV/atom above hull), leading to wasted experimental resources [18]. Evaluation should prioritize classification performance (e.g., precision-recall) for stable materials.
• The Imperative for Validation: As stated in a Nature Computational Science editorial, "Without reasonable experimental support, claims that a drug candidate may outperform those on the market can be difficult to substantiate" [71]. Similarly, in materials science, claims of new materials with exotic properties require experimental synthesis and characterization to be fully supported [71].

Best practices, therefore, mandate:

• Using convex-hull analysis as a powerful but incomplete initial filter.
• Supplementing it with data-driven synthesizability models that incorporate compositional and structural features.
• Employing robust synthesis planning tools to translate predictions into actionable recipes.
• Embedding rapid, high-throughput experimental validation as the final and non-negotiable step in the discovery pipeline.

Conclusion

Convex-hull stability analysis has evolved from a fundamental thermodynamic concept to a sophisticated computational tool that is indispensable for predicting synthesizable materials and pharmaceutical polymorphs. The integration of machine learning, particularly graph neural networks and convex hull-aware active learning, has dramatically accelerated discovery cycles while providing crucial uncertainty quantification. However, successful application requires moving beyond simple energy cutoffs to incorporate vibrational stability checks and understand the complex interplay between thermodynamic and kinetic factors. For biomedical research, these advances promise more reliable polymorph screening, reduced risk of late-appearing polymorphs in pharmaceuticals, and accelerated discovery of functional materials for drug delivery and medical devices. Future directions will likely focus on integrating temperature and solvent effects into hull constructions, developing more robust validation frameworks, and creating end-to-end discovery pipelines that seamlessly combine computational predictions with experimental synthesis.

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