Ensemble Machine Learning with Electron Configuration: A New Paradigm for Predicting Thermodynamic Stability in Materials and Drug Discovery

Abstract

Predicting thermodynamic stability is a critical yet resource-intensive challenge in materials science and drug development. This article explores a cutting-edge approach that integrates ensemble machine learning with fundamental electron configuration data to accurately and efficiently forecast stability. We cover the foundational principles of using electron configurations as low-bias model inputs, detail the methodology of stack generalization frameworks like ECSG that combine diverse knowledge domains, and address troubleshooting for data-scarce scenarios. The discussion includes rigorous validation against DFT calculations, demonstrating superior performance with significantly less data. For researchers and drug development professionals, this synergy of AI and quantum mechanics offers a powerful tool to accelerate the discovery of stable inorganic compounds and bioactive molecules.

The Quantum Mechanical Basis: Why Electron Configuration is Key to Stability Prediction

Fundamental Principles of Thermodynamic Stability

Thermodynamic stability describes the state of a material or compound where it exists at its lowest energy level under given conditions, indicating its inherent resistance to decomposition or transformation. In materials science, this is quantitatively assessed by the decomposition energy (ΔHd), the energy difference between a compound and its competing phases in a phase diagram [1]. For proteins, stability is commonly defined by the folding free energy (ΔGfold), the free energy difference between the folded native state and the unfolded denatured state [2]. These quantitative metrics provide the foundation for predicting behavior, guiding synthesis, and ensuring functional performance across scientific and industrial applications.

Metastability represents a crucial intermediate state where a system possesses kinetic durability but not absolute thermodynamic stability. Diamonds serve as a classic example—they are metastable relative to graphite under ambient conditions yet persist indefinitely due to high kinetic barriers to transformation [3]. Recent research has identified engineered materials that exhibit flipped thermodynamic responses in metastable states, expanding potential applications [3].

Thermodynamic Stability in Materials Science: Applications and Protocols

Advanced Materials with Tailored Stability

Revolutionary research has uncovered materials exhibiting negative thermal expansion (shrinking when heated) and negative compressibility (expanding when crushed) in metastable states [3]. These properties, which seemingly defy conventional thermodynamics, enable unprecedented control over material behavior. Potential applications include:

• Zero-thermal-expansion materials for construction, eliminating thermal deformation [3]
• Structural batteries where aircraft fuselages double as energy storage components [3]
• Self-resetting EV batteries that can be restored to original performance through voltage activation [3]

Protocol: Assessing Material Stability via DFT Calculations

Objective: Determine thermodynamic stability of inorganic compound surfaces using density functional theory (DFT).

Materials and Computational Tools:

• DFT simulation software (VASP, Quantum ESPRESSO)
• Crystal structure databases
• Thermodynamic modeling environment

Methodology:

• Surface Model Construction: Create slab models of the crystal surface with various atomic terminations. For Bi₂WO₆ (001), five different surface terminations were evaluated [4].
• Energy Calculation: Compute total Gibbs free energy of each slab model using DFT with appropriate functionals (GGA-PBE for geometry optimization, HSE06 for electronic properties) [4].
• Chemical Potential Definition: Establish chemical potentials (μ) for constituent elements under specific environmental conditions, including temperature and oxygen partial pressure [4].
• Surface Energy Determination: Calculate surface Gibbs free energy using: γ = (1/2A)[Gslab - Σniμi] where A is surface area, Gslab is slab energy, and n_i are atom counts [4].
• Phase Diagram Construction: Plot surface energies as functions of chemical potentials to identify stable terminations under specific conditions [4].

Interpretation: The termination with lowest surface energy under given environmental conditions represents the thermodynamically preferred structure, guiding material design for specific applications.

Figure 1: Computational workflow for determining material surface stability using DFT and thermodynamic calculations.

Ensemble Machine Learning for Stability Prediction

The ECSG (Electron Configuration models with Stacked Generalization) framework integrates three complementary models to predict inorganic compound stability with superior accuracy (AUC = 0.988) and data efficiency [1]:

• ECCNN (Electron Configuration CNN): Processes fundamental electron configuration data using convolutional neural networks
• Magpie: Utilizes statistical features of elemental properties
• Roost: Employs graph neural networks to model interatomic interactions

This ensemble approach mitigates individual model biases and demonstrates exceptional sample efficiency, achieving comparable performance with only one-seventh the data required by conventional models [1].

Thermodynamic Stability in Pharmaceutical Development: Applications and Protocols

Pharmaceutical Stability Challenges

Over 90% of newly developed active pharmaceutical ingredients (APIs) face challenges with low solubility and bioavailability, making thermodynamic research crucial for product development [5]. Stability directly impacts drug safety, efficacy, and shelf life, with most disease-associated human single-nucleotide polymorphisms destabilizing protein structure [2].

Protocol: Comprehensive Pharmaceutical Stability Assessment

Objective: Evaluate API stability under various stress conditions using STABLE framework.

Materials:

• API reference standard
• Stress agents: HCl, NaOH, H₂O₂, organic solvents
• Controlled environment chambers (thermal, photostability)
• HPLC/UPLC with validated stability-indicating methods

Methodology:

• Sample Preparation: Prepare API solutions at multiple concentrations (typically 0.1-1 mg/mL) in appropriate solvents.
• Stress Application:
  • Acid Hydrolysis: Expose to 0.1-1 M HCl at 25-80°C for 24 hours [6]
  • Base Hydrolysis: Treat with 0.1-1 M NaOH at 25-80°C for 24 hours [6]
  • Oxidative Stress: Incubate with 0.1-3% H₂O₂ at room temperature for 24 hours [6]
  • Thermal Stress: Subject solid and solution states to 40-80°C for specified durations [6]
  • Photostability: Expose to calibrated UV/visible light (ICH Q1B conditions) [6]
• Neutralization: Quench reactions at appropriate timepoints using acid/base or antioxidants.
• Analysis: Employ chromatographic (HPLC/UPLC) techniques to quantify residual API and degradation products.
• Scoring: Apply STABLE scoring system to categorize stability across all conditions.

Interpretation: Degradation ≤10% under harsh conditions indicates high stability, while >20% degradation suggests significant instability requiring formulation intervention [6].

Figure 2: Pharmaceutical stability assessment workflow using the STABLE framework to evaluate multiple stress conditions.

Computational Protein Stability Analysis

Quantitative analysis reveals that computational stability predictors often favor mutations that increase stability at the expense of solubility, and mutations predicted to stabilize are experimentally near neutral on average [7]. The Matthews correlation coefficient (MCC) provides a more reliable performance metric than classification accuracy for evaluating stability prediction tools [7]. Combining multiple mutations significantly improves prospects for achieving stabilization targets [7].

Table 1: Machine Learning Approaches for Thermodynamic Stability Prediction

Method	Basis	Key Features	Performance	Applications
ECSG Framework [1]	Ensemble learning with stacked generalization	Combines ECCNN, Magpie, Roost; reduces inductive bias	AUC = 0.988; 7x data efficiency	Discovering 2D wide bandgap semiconductors, double perovskite oxides
ECCNN [1]	Electron configuration	Uses intrinsic atomic characteristics; convolutional neural networks	High accuracy with minimal features	Fundamental property-stability relationships
Roost [1]	Graph neural networks	Models crystal structure as complete graph; attention mechanism	Captures interatomic interactions	Complex multi-element compounds
Magpie [1]	Elemental property statistics	Uses atomic number, mass, radius statistics; gradient-boosted trees	Broad feature coverage	High-throughput screening

Table 2: Pharmaceutical Stability Scoring System (STABLE Framework) [6]

Stress Condition	Experimental Parameters	Stability Scoring Criteria	High Stability Examples
Acid Hydrolysis	0.1-1 M HCl; 24h; 25-80°C	≤10% degradation under harsh conditions (>5M HCl, 24h reflux)	Maximum score for exceptional acid resistance
Base Hydrolysis	0.1-1 M NaOH; 24h; 25-80°C	≤10% degradation under harsh conditions (>5M NaOH, 24h reflux)	Maximum score for exceptional base resistance
Oxidative Stress	0.1-3% H₂O₂; 24h; RT	Minimal degradation under aggressive conditions	Compounds with oxidation-resistant functional groups
Thermal Stress	40-80°C; solid/solution states	Maintains integrity at elevated temperatures	Thermally stable molecular structures
Photostability	ICH Q1B conditions	Resists UV/visible light degradation	Compounds without chromophores

Essential Research Reagents and Computational Tools

Table 3: Scientist's Toolkit for Thermodynamic Stability Research

Tool/Reagent	Function	Application Context
DFT Software (VASP, Quantum ESPRESSO)	First-principles energy calculations	Materials surface stability, electronic properties [4]
STABLE Framework	Standardized pharmaceutical stability scoring	Comprehensive API stability profiling [6]
HPLC/UPLC with PDA/MS detection	Separation and quantification of degradation products	Pharmaceutical stress testing [6]
Controlled Environment Chambers	Precise temperature and humidity regulation	Accelerated stability studies [6]
ECSG Machine Learning Framework	Ensemble prediction of compound stability	High-throughput materials discovery [1]
Calibrated Light Sources	Controlled photostress exposure	Pharmaceutical photostability testing [6]
Denaturants (Urea, Guandinium HCl)	Protein unfolding agents	Thermodynamic stability measurements [2]

The discovery of new materials with tailored properties is a cornerstone of advancement in fields ranging from drug development to renewable energy. For decades, Density Functional Theory (DFT) has served as a primary computational tool for predicting material properties and thermodynamic stability from first principles. However, its computational cost and inherent theoretical limitations constrain its effectiveness for high-throughput screening of large compositional spaces [1]. Concurrently, the rise of machine learning (ML) offers a faster alternative, but its success is often hampered by inductive biases introduced through model architectures and hand-crafted features, which can limit generalizability [1]. This creates a critical methodological challenge: how to reliably and efficiently predict material stability. Framed within ongoing research on ensemble machine learning for electron configuration thermodynamic stability, this analysis examines the specific limitations of traditional DFT and single-model ML approaches. It further explores how emerging ensemble methods, which integrate knowledge from multiple physical scales, are providing a path toward more robust and efficient predictive frameworks.

The Limitations of Density Functional Theory (DFT)

DFT is a quantum mechanical modelling method used to investigate the electronic structure of many-body systems, primarily their ground state [8]. Its versatility has made it immensely popular in physics, chemistry, and materials science. The core strength of DFT lies in its theoretical foundation: the Hohenberg-Kohn theorems establish that all properties of a many-electron system are uniquely determined by its ground-state electron density. This reduces the problem of 3N spatial coordinates (for N electrons) to just three coordinates, drastically lowering the computational cost compared to traditional ab initio methods like Hartree-Fock that deal directly with the many-electron wavefunction [8].

Despite being an exact theory in principle, the practical application of DFT relies on approximations for the exchange-correlation functional. It is these Density Functional Approximations (DFAs), not DFT itself, that are the source of most failures [9].

Table 1: Common Failures of Density Functional Approximations (DFAs)

Failure Type	Description	Underlying Cause
Intermolecular Interactions	Poor description of van der Waals forces (dispersion) and hydrogen bonding [8].	Incomplete treatment of non-local electron correlation effects.
Strongly Correlated Systems	Inaccurate results for systems with localized d- or f-electrons (e.g., many transition metal oxides) [9].	Standard DFAs struggle with near-degeneracies and static correlation.
Charge Transfer Excitations	Severe underestimation of energies for excitations where electron density shifts significantly in space [8].	Incorrect long-range behavior of the exchange potential.
Band Gaps	Systematic underestimation of the band gap in semiconductors and insulators [8].	Self-interaction error and derivative discontinuities.
Reaction Barriers	Tendency to underestimate activation energies for chemical reactions [9].	Inaccurate description of the exchange-correlation energy along the reaction path.

A significant weakness of the DFA approach is that it is not systematically improvable. Unlike wavefunction-based methods like Coupled-Cluster, where a clear hierarchy (e.g., CCSD, CCSD(T)) exists to approach the exact solution, there is no guaranteed path for DFAs; a "higher-rung" functional on Jacob's Ladder is not certain to yield a more accurate answer for a given system [9]. Furthermore, DFT calculations consume substantial computational resources, creating a bottleneck for the rapid exploration of vast compositional spaces, such as those found in high-entropy alloys or novel perovskite compounds [1].

The Challenge of Inductive Bias in Machine Learning Models

Machine learning presents a promising avenue for circumventing the high computational cost of DFT. By learning patterns from existing DFT or experimental databases, ML models can predict thermodynamic stability orders of magnitude faster. A key step in developing these models is feature engineering, where a material's composition or structure is converted into a numerical representation. However, this process often introduces strong inductive biases—inherent assumptions that guide the learning algorithm toward specific solutions.

Inductive bias in ML for materials science often stems from the domain knowledge used to create input features. While necessary, these assumptions have limited applicability and can result in poor generalization if they do not fully capture the underlying physics [1]. The following table summarizes common biases and their potential impacts.

Table 2: Sources and Impacts of Inductive Bias in ML for Materials Stability

Source of Bias	Description	Potential Impact on Model
Feature Selection	Using hand-crafted features based on specific elemental properties (e.g., Magpie features like atomic radius, electronegativity) [1].	Model performance is capped by the informational content of the selected features; may miss crucial electronic-level information.
Architectural Assumptions	Imposing structural priors, e.g., modeling a crystal as a complete graph of interacting atoms (as in Roost) [1].	May incorrectly model weak or non-existent interactions, leading to over-smoothing or inaccurate relationship learning.
Data Scarcity & Imbalance	Training sets are often biased toward common or easily synthesizable materials, with a scarcity of stable compounds [10].	Models become adept at identifying unstable compounds but perform poorly at predicting the rare, stable ones, which are often the target of discovery [10].

For example, a model like ElemNet, which uses only elemental compositions, introduces a large inductive bias by assuming material properties are solely determined by elemental fractions, ignoring the intricate effects of electron configuration and interatomic interactions [1]. This can limit its predictive accuracy and generalizability to new, unexplored regions of chemical space.

Ensemble ML and Electron Configuration: A Path Forward

To mitigate the limitations of both DFAs and single-biased ML models, ensemble methods that integrate diverse physical knowledge offer a compelling solution. The core idea is to combine models grounded in different domain knowledge—such as interatomic interactions, atomic properties, and electron configuration—into a single, more robust framework. This approach, known as stacked generalization, creates a "super learner" that compensates for the individual biases of its base models [1].

A key advancement is the direct incorporation of electron configuration (EC) as an intrinsic material representation. Unlike hand-crafted features, electron configuration describes the fundamental distribution of electrons in an atom's energy levels, which is the very basis for quantum mechanical calculations of ground-state energy in DFT [8] [11]. Using EC as an input feature provides the model with a more foundational physical description, potentially reducing spurious inductive biases.

The workflow for one such framework, the Electron Configuration models with Stacked Generalization (ECSG), is illustrated below. It demonstrates how an ensemble can synergistically combine knowledge from different physical scales.

Experimental Protocol: Building the ECSG Ensemble Model

Objective: To train an ensemble model for predicting the thermodynamic stability of inorganic compounds, achieving high accuracy with minimal data.

Materials and Reagents (Computational):

• Data Source: The Joint Automated Repository for Various Integrated Simulations (JARVIS) database, or similar (e.g., Materials Project, OQMD) [1].
• Representation: Chemical formulas of inorganic compounds and their corresponding decomposition energies (ΔH_d).
• Software: Python with deep learning libraries (e.g., PyTorch, TensorFlow) and materials informatics tools (pymatgen).

Methodology:

• Data Curation and Preprocessing:

  • Collect formation energies and calculate the decomposition energy (ΔH_d) relative to the convex hull to determine stability labels (stable/unstable) [1].
  • Split the dataset into training and test sets, ensuring no data leakage by clustering compounds by structural similarity and ensuring sequences from the same cluster are contained within a single split [10].

• Base-Level Model Training (Heterogeneous Knowledge Integration):

  • Train ECCNN (Electron Configuration Convolutional Neural Network):
    • Input Encoding: Convert the chemical formula into a 118 (elements) × 168 (electron orbitals) × 8 (features) matrix. The features represent the electron occupancy for each orbital quantum number (s, p, d, f) for each element in the compound [1].
    • Architecture: Use two convolutional layers (64 filters, 5×5 kernel) for feature extraction, followed by batch normalization, max-pooling, and fully connected layers.
  • Train Roost (Representations from Overlap of Site Tensors):
    • Input Encoding: Represent the chemical formula as a graph where atoms are nodes and bonds are edges.
    • Architecture: Employ a graph neural network with message-passing and attention mechanisms to capture interatomic interactions [1].
  • Train Magpie (Materials Agnostic Platform for Informatics and Exploration):
    • Input Encoding: Calculate a set of statistical features (mean, range, mode, etc.) from a list of elemental properties (e.g., atomic number, radius, electronegativity) for the compound [1].
    • Architecture: Train a Gradient-Boosted Regression Tree (XGBoost) model on these feature vectors.

• Stacked Generalization (Meta-Learning):

  • Use the predictions of the three base models (ECCNN, Roost, Magpie) on the training set as input features for a meta-learner (e.g., a linear model or another shallow neural network).
  • Train this meta-learner to produce the final, refined stability prediction [1].

• Validation and Testing:

  • Evaluate the ensemble model on the held-out test set using metrics such as Area Under the Curve (AUC), precision, and recall. The ECSG model has been shown to achieve an AUC of 0.988 [1].
  • Validate computational predictions with targeted DFT calculations on newly predicted stable compounds to confirm their stability on the convex hull [1].

Research Reagent Solutions

Table 3: Essential Computational Tools for Ensemble ML in Materials Stability

Item Name	Function/Description	Relevance to Research
JARVIS/MP/OQMD Databases	Large-scale repositories of DFT-calculated material properties.	Provide the essential training data (formation energies, band structures) for supervised learning models [1].
Electron Configuration Encoder	Algorithm to convert a chemical formula into a structured matrix representing orbital occupation.	Provides a foundational, low-bias input representation for ML models, directly related to quantum mechanical states [1].
Graph Neural Network (GNN)	A neural network architecture that operates on graph-structured data.	Captures complex interatomic interactions and local coordination environments within a crystal structure (e.g., as used in Roost) [1].
Stacked Generalization Framework	A meta-learning algorithm that combines multiple base models.	Mitigates individual model bias and leverages synergistic effects between different physical representations to boost predictive performance [1].

The limitations of traditional DFT and single-model ML are significant but not insurmountable. DFAs, while powerful, fail systematically for certain classes of materials and are computationally expensive for vast compositional searches. Standard ML models, in turn, are often hindered by inductive biases introduced through their design and feature sets. The path forward, as evidenced by recent research, lies in ensemble frameworks like ECSG that strategically integrate diverse physical knowledge—from atomic statistics and graph-based interactions to the fundamental principles of electron configuration. By doing so, these hybrid models achieve a remarkable balance: they retain the high speed of ML while significantly improving accuracy, robustness, and data efficiency. This paradigm shift from relying on a single, biased method to employing a committee of expert models is crucial for accelerating the reliable discovery of new, thermodynamically stable materials for advanced applications.

The discovery and development of new functional materials are often hindered by the vastness of compositional space. Traditional methods for assessing key properties, such as thermodynamic stability, through experimentation or density functional theory (DFT) calculations are resource-intensive and slow [1]. Machine learning (ML) offers a promising alternative, yet many models incorporate significant inductive bias by relying on specific domain knowledge or idealized assumptions about material composition and structure, which can limit their predictive accuracy and generalizability [1].

Electron configuration (EC)—the distribution of electrons in atomic orbitals—represents a fundamental, intrinsic property of elements. It forms the physical basis for chemical bonding and reactivity. Using EC as a primary input for machine learning models minimizes the need for manually crafted, theory-laden features, thereby reducing inductive bias [1]. This approach allows models to learn the underlying physical relationships directly from the foundational principles of quantum mechanics. When integrated into ensemble learning frameworks, EC-based models can achieve remarkable predictive accuracy and data efficiency, accelerating the exploration of new materials with desired properties [1] [12].

Theoretical Foundation & Rationale

Electron Configuration as a Low-Bias Descriptor

In computational materials science, a "descriptor" is a numerical representation of a material that serves as input for a model. Many traditional descriptors for inorganic compounds are derived from elemental properties (e.g., atomic radius, electronegativity) or structural features, which inherently embed specific hypotheses about property-structure relationships [1] [12].

Electron configuration provides a more foundational description. It delineates the distribution of electrons within an atom, encompassing energy levels and electron counts at each level, which are crucial for comprehending chemical properties and reaction dynamics [1] [13]. The electron configuration of an atomic species, whether neutral or ionic, provides deep insight into the shape and energy of its electrons, directly influencing bonding ability, magnetism, and other chemical properties [13].

By using the raw electron configuration as a descriptor, researchers bypass many assumptions required by other models. For instance, models that rely solely on elemental composition fractions cannot handle new elements absent from their training data, and graph-based models may impose specific but incomplete relationship paradigms between atoms in a unit cell [1]. EC serves as an intrinsic characteristic that introduces fewer such inductive biases, allowing the ML algorithm to discover patterns that might be obscured by pre-selected feature sets [1].

Quantum Mechanical Basis

The electron configuration is directly derived from the solutions to the Schrödinger equation for atoms and is described by four quantum numbers [13]:

• Principal Quantum Number (n): Indicates the shell or energy level, defining the overall energy and size of the orbital (n = 1, 2, 3...).
• Orbital Angular Momentum Quantum Number (l): Indicates the subshell and shape of the atomic orbital (l = 0 for s, 1 for p, 2 for d, 3 for f...).
• Magnetic Quantum Number (mₗ): Specifies the orientation of the orbital in space (mₗ = -l, ..., +l).
• Spin Magnetic Quantum Number (mₛ): Describes the electron's intrinsic spin.

This quantum mechanical foundation makes EC a natural input for predicting properties that originate from electronic interactions, forming the basis for first-principles calculations like DFT [1] [13].

Implementing EC-Based Machine Learning

Data Acquisition and Preprocessing

The first step involves building a comprehensive dataset from established materials databases. Key resources include:

• Materials Project (MP)
• Open Quantum Materials Database (OQMD)
• JARVIS Database

These databases provide formation energies, decomposition energies (ΔHd), and structural information for thousands of computed compounds, serving as ground truth for training stability prediction models [1].

Protocol: Encoding Electron Configuration for ML Input A critical step is transforming the elemental composition of a compound into a numerical matrix representation based on electron configuration.

• Elemental Breakdown: Parse the chemical formula of a compound to identify the constituent elements and their proportions.
• EC Matrix Construction: For each element in the periodic table (typically Z=1 to 118), create a comprehensive binary vector representing its electron occupancy across orbitals. The ECCNN model, for example, uses an input matrix of dimensions 118 × 168 × 8, representing 118 elements, 168 orbital blocks, and 8 bits per block for fine-grained electron occupancy information [1].
• Compositional Aggregation: For a given compound, aggregate the EC matrices of its constituent elements, weighted by their stoichiometric proportions, to form a final input representation that encapsulates the overall electronic structure of the material.

Table 1: Key Research Reagent Solutions (Computational Tools & Databases)

Item Name	Function/Application	Key Features
Materials Project (MP) Database	Repository of computed materials properties for training and validation.	Provides formation energies, band structures, and other DFT-calculated properties for over 100,000 materials [1].
JARVIS Database	Source of datasets for benchmarking model performance.	Includes thermodynamic stability data for inorganic compounds [1].
Magpie Descriptor Tool	Generates statistical features from elemental properties.	Calculates mean, deviation, range, and other statistics for atomic properties, serving as a baseline or ensemble model [1].
matminer	Open-source toolkit for materials data mining.	Provides a platform for feature extraction and generating descriptors for inorganic compounds [12].

Model Architectures and Workflows

Different neural network architectures can be employed to process the encoded electron configuration information.

Protocol: Building an Electron Configuration Convolutional Neural Network (ECCNN) The ECCNN is designed to learn hierarchical features from the EC matrix [1].

• Input Layer: Accepts the encoded EC matrix (e.g., 118 × 168 × 8).
• Convolutional Layers: Typically, two convolutional operations with 64 filters of size 5×5 are used. These layers detect local patterns and correlations between different orbitals and elements.
• Batch Normalization (BN): Inserted after convolutional layers to stabilize and accelerate training.
• Pooling Layer: A 2×2 max pooling layer follows to reduce dimensionality and introduce translational invariance.
• Fully Connected Layers: The extracted features are flattened and passed through one or more dense layers to map the learned features to the final prediction (e.g., decomposition energy or stability class).

Diagram 1: ECCNN model workflow for stability prediction.

Ensemble Framework: Stacked Generalization

To further mitigate bias and enhance robustness, EC-based models can be integrated into an ensemble framework. The Stacked Generalization (SG) technique combines models rooted in diverse knowledge domains, allowing them to complement each other [1].

Protocol: Constructing an Ensemble with Stacked Generalization The Electron Configuration models with Stacked Generalization (ECSG) framework integrates multiple base models [1].

• Base-Level Model Selection: Choose three models that operate on different principles:

  • ECCNN: Leverages intrinsic electron configuration information [1].
  • Roost: Represents the chemical formula as a graph of interacting atoms, using message-passing graph neural networks to capture interatomic interactions [1].
  • Magpie: Utilizes statistical features (mean, deviation, range, etc.) computed from a wide array of elemental properties (e.g., atomic number, mass, radius) [1].

• Base-Level Training: Train each of these base models independently on the same training dataset.

• Meta-Level Dataset Creation: Use the predictions from the trained base models on a validation set (or via cross-validation) as input features for a new "meta-level" dataset. The true target values are retained as labels.

• Meta-Learner Training: Train a relatively simple model (the "meta-learner" or "super learner") on this new dataset. This model learns to optimally combine the predictions of the base models to produce a final, more accurate, and robust prediction [1].

Diagram 2: Stacked generalization ensemble framework.

Performance and Validation

The performance of EC-based models and their ensembles has been rigorously validated against standard benchmarks and first-principles calculations.

Table 2: Quantitative Performance of EC-Based ML Models

Model / Framework	Application / Dataset	Key Performance Metrics	Comparative Advantage
ECSG (Ensemble) [1]	Thermodynamic stability prediction (JARVIS database)	AUC: 0.988	Achieved same performance as existing models using only 1/7 of the data; successfully identified new 2D semiconductors and double perovskites validated by DFT.
ECCNN (Base Model) [1]	Thermodynamic stability prediction	High predictive accuracy within ensemble.	Reduces inductive bias by using fundamental EC input.
EC-based ANN [12]	Boiling Point (BP) prediction (537 compounds)	R²: 0.88, MAE: 222.65°C	Covers 87.5% of elements in periodic table; models complex electronic interactions.
EC-based ANN [12]	Melting Point (MP) prediction (1647 compounds)	R²: 0.89, MAE: 170.39°C	Covers 98% of elements (102/104); demonstrates wide applicability.

Protocol: Validation via First-Principles Calculations Predictions of new stable materials made by ML models must be rigorously validated.

• Candidate Identification: Use the trained ECSG model to screen a large, unexplored compositional space and identify candidate compounds predicted to be thermodynamically stable.
• DFT Calculation Setup: Perform DFT calculations on the top candidates. This typically involves:
  • Structure Generation: Proposing plausible crystal structures for the composition.
  • Geometry Optimization: Relaxing the atomic positions and cell parameters to find the ground state energy.
  • Stability Assessment: Calculating the decomposition energy (ΔHd) to determine if the compound lies on the convex hull of stable phases [1] [4].
• Result Comparison: Compare the DFT-calculated stability with the ML model's prediction. A high rate of confirmation indicates the model's remarkable accuracy and reliability for guiding materials discovery [1].

Application Notes

Case Study: Exploring Double Perovskite Oxides

The ECSG framework was applied to navigate the unexplored composition space of double perovskite oxides (A₂BB'O₆). The ensemble model screened numerous potential compositions, identifying several promising candidates predicted to be thermodynamically stable. Subsequent validation through DFT calculations confirmed the model's high accuracy, as a significant majority of the predicted compounds were correctly identified as stable [1]. This demonstrates the framework's utility in rapidly pinpointing synthesizable materials in complex chemical spaces with high reliability.

Integration with Other Workflows

The electron configuration descriptor is highly versatile. Beyond standalone stability prediction, it can be integrated into broader materials design workflows:

• Functional Property Prediction: EC-based models can be used to predict not just stability, but also electronic structure properties, which are critical for applications in photocatalysis and electronics [4].
• Guiding Synthesis: By quickly identifying stable compounds, these models can prioritize targets for experimental synthesis, saving time and resources [14].
• Multi-Objective Optimization: EC descriptors can be used in models that simultaneously optimize for stability, high specific capacitance (for supercapacitor electrodes), and other performance metrics, aiding in the rational design of advanced functional materials [14].

Electron configuration stands as a fundamental, low-bias descriptor that taps directly into the quantum mechanical origins of material behavior. Its implementation within neural network architectures like ECCNN, and particularly its integration into ensemble frameworks like ECSG, provides a powerful and data-efficient paradigm for accelerating materials discovery. This approach mitigates the inductive biases prevalent in models reliant on manually curated features, leading to superior predictive accuracy for thermodynamic stability and other properties. The successful validation of ML-predicted compounds via first-principles calculations underscores the maturity of this methodology, establishing electron configuration as a cornerstone for next-generation, physics-informed machine learning in materials science and drug development.

Ensemble learning has emerged as a cornerstone technique in machine learning, demonstrating remarkable efficacy in enhancing predictive performance and robustness. Its core philosophy is elegantly simple yet profoundly powerful: by combining multiple individual models, a collective intelligence is created that outperforms any single constituent model [15]. This approach is particularly transformative for scientific domains like materials informatics, where the accurate prediction of complex properties such as thermodynamic stability is paramount yet challenged by limited data and inherent biases in modeling approaches.

Within the specific context of predicting thermodynamic stability of inorganic compounds, ensemble models address a critical limitation of single-model approaches: the introduction of inductive biases through domain-specific assumptions [1]. Most existing models are constructed based on particular facets of domain knowledge, which can restrict their applicability and generalizability. Ensemble frameworks, particularly those utilizing stacked generalization, amalgamate models rooted in distinct knowledge domains—such as electron configuration, atomic properties, and interatomic interactions—to create a super learner that mitigates these individual biases and harnesses synergistic effects [1]. This review details the application notes and experimental protocols for implementing such ensemble approaches, with a specific focus on their capacity to mitigate bias and improve generalization in electron configuration-based thermodynamic stability research.

Application Notes: Ensemble Frameworks in Practice

Core Ensemble Architectures for Stability Prediction

The implementation of ensemble methods for thermodynamic stability prediction leverages several distinct architectural paradigms, each with unique mechanisms for improving model performance.

Stacked Generalization (ECSGFramework): This sophisticated approach integrates multiple base models with a meta-learner that optimally combines their predictions. In practice for stability prediction, this involves training diverse base models like Magpie (leveraging atomic property statistics), Roost (modeling interatomic interactions via graph neural networks), and ECCNN (utilizing raw electron configuration data) [1]. The predictions from these models then serve as input features for a meta-model (often a linear classifier or simple neural network) that learns the optimal weighting scheme to produce final stability classifications. This method recognizes that different models excel under different conditions, and a learned combination leverages these complementary strengths more effectively than predetermined strategies [15] [16].

Bagging (Bootstrap Aggregating): This technique reduces variance by creating multiple versions of the training dataset through bootstrap sampling—randomly selecting observations with replacement—and training a separate model on each version [15] [16]. The predictions are then aggregated, typically through averaging for regression tasks or majority voting for classification problems. Random Forests represent the most prominent application of bagging in materials informatics, combining bagging with random feature selection to force diversity among constituent decision trees [15].

Boosting: This sequential approach builds models iteratively, with each new model focusing on correcting errors made by previous ones [15] [16]. Gradient Boosting Machines frame this process as optimizing a loss function through gradient descent in function space, with implementations like XGBoost and LightGBM offering sophisticated handling of the tabular data structures common in materials property datasets [15].

Table 1: Quantitative Performance of Ensemble Methods for Thermodynamic Stability Prediction

Ensemble Method	AUC Score	Data Efficiency	Key Advantages
ECSG (Stacking)	0.988 [1]	7x improvement (requires 1/7th the data of single models) [1]	Mitigates inductive bias from single knowledge domains
Random Forest (Bagging)	Varies by implementation	Moderate improvement	Robust to noisy features; handles mixed data types
Gradient Boosting (Boosting)	Typically 0.92-0.96	High efficiency with appropriate regularization	Maximizes predictive accuracy on complex non-linear relationships

Bias Mitigation Mechanisms

Ensemble methods provide powerful mechanisms for addressing various forms of bias that plague single-model approaches in computational materials science.

Inductive Bias Reduction: Single-model architectures often incorporate strong assumptions about the structure of materials data, such as Roost's assumption that all atoms in a unit cell have strong interactions [1]. By integrating multiple models with divergent assumptions, ensemble frameworks create a more balanced representation that prevents any single biased perspective from dominating predictions [1].

Representation Bias Mitigation: Models trained exclusively on specific types of features (e.g., only atomic properties) may develop blind spots for compounds where electron configuration plays a more decisive role in stability. The ECSG framework addresses this by incorporating the Electron Configuration Convolutional Neural Network (ECCNN), which uses raw electron configuration data as input—an intrinsic atomic characteristic that introduces fewer manual biases compared to hand-crafted features [1].

Algorithmic Bias Compensation: Different learning algorithms have distinct failure modes; decision trees may struggle with smooth boundaries, while neural networks might overfit to sparse regions of feature space. Ensembles leverage the "wisdom of crowds" effect, where uncorrelated errors from diverse models tend to cancel out, resulting in more robust overall predictions [15]. This statistical foundation explains why ensembles typically demonstrate superior generalization to unseen compositional spaces [17].

Experimental Protocols

Protocol 1: Implementing the ECSG Framework for Stability Prediction

The following protocol details the procedure for replicating the Electron Configuration models with Stacked Generalization (ECSG) approach for predicting thermodynamic stability of inorganic compounds [1].

Research Reagent Solutions Table 2: Essential Computational Resources and Tools

Resource/Tool	Function	Specifications/Alternatives
JARVIS/MP/ OQMD Databases	Source of formation energies and stability labels	Materials Project (MP), Open Quantum Materials Database (OQMD)
Electron Configuration Encoder	Transforms composition to EC matrix	Custom Python implementation (118×168×8 matrix) [1]
Magpie Feature Set	Atomic property statistics	Mean, variance, mode, etc. of atomic properties [1]
Roost Model	Message-passing neural network	Graph attention for interatomic interactions [1]
ECCNN Architecture	Electron configuration processor	Two convolutional layers (64 filters, 5×5), BN, max pooling [1]
Meta-Learner	Stacking model	Logistic regression or simple neural network

Procedure:

• Data Preparation and Preprocessing

  • Source: Acquire training data from databases such as the Joint Automated Repository for Various Integrated Simulations (JARVIS), Materials Project (MP), or Open Quantum Materials Database (OQMD) containing formation energies and decomposition energies (ΔH_d) for inorganic compounds [1].
  • Labeling: Binary stability labels are derived from the decomposition energy, with ΔHd > 0 indicating instability and ΔHd ≤ 0 indicating stability [1].
  • Splitting: Partition data into training (70%), validation (15%), and test (15%) sets using stratified sampling to maintain class balance.

• Base Model Training

  • Magpie Implementation: Compute statistical features (mean, mean absolute deviation, range, minimum, maximum, mode) for elemental properties including atomic number, mass, radius, etc. Train a Gradient Boosted Regression Tree (e.g., XGBoost) using these feature vectors [1] [15].
  • Roost Implementation: Represent crystal structures as complete graphs with atoms as nodes. Implement a graph neural network with attention mechanisms to capture interatomic interactions. Train using the formation energy as the target [1].
  • ECCNN Implementation:
    • Input Encoding: Encode each compound's composition as a 118×168×8 matrix representing the electron configurations of constituent elements [1].
    • Architecture: Process through two convolutional layers (64 filters of size 5×5), followed by batch normalization and 2×2 max pooling. Flatten features and pass through fully connected layers for final prediction [1].
    • Training: Use Adam optimizer with cross-entropy loss for stability classification.

• Stacked Generalization Implementation

  • Prediction Collection: Generate out-of-fold predictions from each base model (Magpie, Roost, ECCNN) on the validation set.
  • Meta-Feature Construction: Use these predictions as input features for the meta-learner.
  • Meta-Learner Training: Train a logistic regression model or a simple neural network on these meta-features to learn optimal combination weights [1].
  • Validation: Assess performance on the validation set using Area Under the Curve (AUC) metrics.

• Model Evaluation

  • Testing: Evaluate the final ECSG model on the held-out test set.
  • Comparative Analysis: Compare performance against individual base models and traditional ensemble methods (bagging, boosting).
  • Efficiency Assessment: Measure data efficiency by training on progressively smaller subsets and comparing performance degradation against single models.

Protocol 2: Bias Assessment and Mitigation Validation

This protocol provides a systematic approach for quantifying and mitigating biases in thermodynamic stability predictors, extending beyond standard performance metrics.

Research Reagent Solutions

• Bias Assessment Frameworks: PROBAST (Prediction model Risk Of Bias ASsessment Tool) or similar structured tools [18].
• Fairness Metrics: Demographic parity, equalized odds, equal opportunity differences [18].
• Compositional Subgroup Analysis: Tools for identifying underrepresented element combinations in training data.

Procedure:

• Bias Identification

  • Feature Representation Analysis: Audit training data for representation disparities across different regions of compositional space (e.g., oxides vs. sulfides, transition metal-rich vs. poor compounds).
  • Performance Disparity Assessment: Evaluate model performance (accuracy, AUC) separately for different compositional subgroups to identify systematic performance gaps.
  • Temporal Bias Check: Assess training-serving skew by comparing model performance on historical vs. newly synthesized compounds [18].

• Bias Mitigation Implementation

  • Diverse Ensemble Construction: Intentionally select base models with complementary inductive biases (atomic-scale, electronic structure, structural) to create natural compensation mechanisms [1].
  • Reweighting Strategies: Apply instance weighting during meta-learner training to increase influence of predictions from specialized models for particular compositional subgroups.
  • Adversarial Debiasing: Incorporate adversarial components during base model training to penalize representations that allow prediction of protected attributes (e.g., element group).

• Validation and Iteration

  • Generalization Testing: Evaluate ensemble performance on truly external datasets containing novel compound classes absent from training data.
  • Ablation Studies: Systematically remove individual base models from the ensemble to quantify their contribution to bias reduction.
  • Expert Validation: Compare ensemble predictions with first-principles DFT calculations for borderline cases to verify physicochemical plausibility [1].

Protocol 3: High-Dimensional Composition Space Exploration

This protocol leverages the improved generalization of ensemble models for navigating unexplored compositional spaces in search of novel stable compounds.

Research Reagent Solutions

• Composition Space Sampling: Tools for generating candidate compositions within specified constraints.
• First-Principles Validation: DFT calculation infrastructure for experimental validation.
• Uncertainty Quantification: Methods for estimating prediction confidence in ensemble models.

Procedure:

• Candidate Generation

  • Combinatorial Sampling: Systematically generate candidate compositions within target spaces (e.g., double perovskite oxides, 2D wide bandgap semiconductors) [1].
  • Feature Encoding: Encode generated compositions using the same procedures as training (ECCNN matrices, Magpie features, Roost graphs).

• Ensemble Screening

  • Parallel Prediction: Process candidates through the trained ECSG ensemble to obtain stability predictions.
  • Consensus Scoring: Apply meta-learner to combine base model predictions into final stability scores.
  • Uncertainty Estimation: Calculate prediction variance across base models as a measure of confidence.

• Validation and Discovery

  • Priority Ranking: Rank candidates by stability score and prediction confidence.
  • First-Principles Verification: Perform DFT calculations for top candidates to verify thermodynamic stability [1].
  • Iterative Refinement: Incorporate newly verified stable compounds into training data to improve model performance in targeted regions of composition space.

Building the Framework: Architecting Ensemble Models for Real-World Applications

Stacked generalization, also known as stacking, is an advanced ensemble machine learning method that combines multiple predictive models through a meta-learner to minimize generalization error and enhance prediction accuracy. The technique operates by deducing the biases of various generalizers (base-level models) with respect to a provided learning set [19]. This approach has demonstrated remarkable success across diverse scientific domains, from predicting the thermodynamic stability of inorganic compounds using electron configuration data to forecasting psychosocial maladjustment in medical patients and estimating drug concentrations for precision dosing [1] [20] [21]. The fundamental principle underpinning stacked generalization is its ability to integrate models originating from distinct knowledge domains or algorithmic families, thereby creating a synergistic super-learner that outperforms any individual constituent model [1].

The Electron Configuration models with Stacked Generalization (ECSG) framework represents a cutting-edge implementation of this approach specifically designed for materials science applications. By leveraging ensemble machine learning based on electron configuration, ECSG achieves exceptional accuracy in predicting thermodynamic stability while requiring only one-seventh of the data used by existing models to achieve comparable performance [1]. This remarkable sample efficiency, coupled with an Area Under the Curve (AUC) score of 0.988 as validated against the Joint Automated Repository for Various Integrated Simulations (JARVIS) database, positions ECSG as a transformative methodology for accelerating materials discovery and optimization [1].

Theoretical Foundations of Stacked Generalization

Conceptual Framework and Mathematical Underpinnings

Stacked generalization functions through a two-tiered architecture: a base level comprising multiple heterogeneous learning algorithms, and a meta-level that learns how to optimally combine the base-level predictions [19]. Formally, given a set of base learners ( L1, L2, ..., Lk ) and a meta-learner ( M ), stacking generates the final prediction through the following process. First, each base learner ( Li ) is trained on the available data. Next, cross-validated predictions from each base learner are obtained, forming a new dataset where the features are the predictions of the base learners and the target remains the original outcome variable. Finally, the meta-learner ( M ) is trained on this new dataset to produce the final output [19].

The ECSG framework implements this approach using V-fold cross-validation to build the optimal weighted combination of predictions from a library of candidate algorithms [19]. Optimality is defined by a user-specified objective function, such as minimizing mean squared error or maximizing the area under the receiver operating characteristic curve. Theoretical guarantees ensure that in large samples, the resulting algorithm will perform at least as well as the best individual predictor included in the ensemble [19]. This mathematical foundation provides robustness against the limitations of any single modeling approach, particularly valuable when exploring complex composition-property relationships in materials science where mechanistic understanding may be incomplete [1].

Advantages Over Conventional Ensemble Methods

Stacked generalization offers distinct advantages over simpler ensemble techniques such as bagging or boosting. While homogeneous ensemble methods combine multiple instances of the same algorithm type, stacking strategically integrates fundamentally different modeling approaches, capturing complementary aspects of the underlying patterns in the data [20]. This heterogeneity is crucial for managing noisy and imbalanced datasets where single-classifier models often struggle with overfitting [21]. Additionally, the weighted combination approach of stacking is more nuanced than the winner-takes-all method of selecting a single best performer, often resulting in superior generalization to unseen data [21].

The non-negative least squares constraint frequently applied in stacking (requiring coefficients to be non-negative and sum to 1) enhances model stability and interpretability while maintaining performance [19]. This convex combination approach, motivated by both theoretical results and practical considerations, prevents overfitting on the meta-level and ensures that the ensemble prediction represents a consensus weighting of the constituent models rather than an arbitrary linear combination that might extrapolate poorly [19].

ECSG Framework Architecture

Base-Level Model Specifications

The ECSG framework integrates three distinct base-level models, each rooted in different domains of knowledge to ensure complementarity and minimize inductive bias [1]. This multi-perspective approach enables the capture of material properties and interactions across different scales, from atomic-level electron configurations to macroscopic statistical patterns.

Table 1: Base-Level Models in the ECSG Framework

Model Name	Domain Knowledge	Algorithmic Approach	Input Representation
Magpie	Atomic properties & statistics	Gradient-boosted regression trees (XGBoost)	Statistical features (mean, deviation, range, etc.) of elemental properties
Roost	Interatomic interactions	Graph neural networks with attention mechanism	Chemical formula represented as a complete graph of elements
ECCNN	Electron configuration	Convolutional Neural Network	Electron configuration matrix (118×168×8) encoding electron distributions

The Electron Configuration Convolutional Neural Network (ECCNN) represents a novel contribution specifically designed to address the limited understanding of electronic internal structure in existing models [1]. Unlike manually crafted features, electron configuration serves as an intrinsic atomic characteristic that introduces minimal inductive bias while providing fundamental information about chemical properties and reaction dynamics [1]. The ECCNN architecture processes its input through two convolutional operations, each with 64 filters of size 5×5, followed by batch normalization and 2×2 max pooling before final fully connected layers for prediction [1].

Meta-Learner Integration and Optimization

The meta-learner in ECSG synthesizes predictions from the three base models using stack generalization to produce the final stability classification. This approach employs V-fold cross-validation to generate out-of-sample predictions from each base model, which then serve as input features for training the meta-learner [19]. The specific implementation details of the ECSG meta-learner are adapted from established stacking methodologies that have demonstrated successful application across multiple scientific domains [19] [21].

In operational terms, the stacking process in ECSG follows a structured workflow that can be visualized as follows:

ECSG Stacking Workflow: This diagram illustrates the flow of information through the ECSG framework, from input data through base model processing to meta-learner integration and final prediction.

Experimental Protocols and Implementation

Data Preparation and Feature Engineering

The ECSG framework primarily utilizes composition-based data, requiring specialized processing of chemical formula information before model input [1]. The data extraction pipeline involves:

• Elemental Proportion Calculation: Determining the stoichiometric ratios of each element within the compound [1].
• Feature Representation: Transforming elemental proportions into model-specific input representations:
  • For Magpie: Calculating statistical features (mean, mean absolute deviation, range, minimum, maximum, mode) across various elemental properties including atomic number, mass, and radius [1].
  • For Roost: Representing the chemical formula as a complete graph where atoms serve as nodes with connecting edges [1].
  • For ECCNN: Encoding materials into a 118×168×8 matrix based on their electron configurations, delineating electron distribution across energy levels [1].

This multi-faceted input representation strategy ensures that diverse aspects of material composition are captured, enabling the framework to leverage complementary information across different physical scales and theoretical perspectives [1].

Model Training and Validation Protocol

The implementation of ECSG follows a rigorous training and validation protocol adapted from established stacked generalization methodologies [19]:

• Data Partitioning: Split the dataset into K mutually exclusive and exhaustive folds (typically K=5 or K=10) [19].
• Base Model Training: For each fold k ∈ {1,...,K}:
  • Designate fold k as the validation set and remaining folds as the training set.
  • Fit each base algorithm (Magpie, Roost, ECCNN) on the training set.
  • Use the trained models to predict outcomes for the validation set.
  • For each algorithm, estimate performance using an appropriate metric (e.g., mean squared error for regression, AUC for classification) [19].
• Meta-Learner Construction:
  • Average performance metrics across all folds for each algorithm.
  • Combine cross-validated predictions from all base models to form the "level-one" dataset.
  • Train the meta-learner on the level-one data, typically using constrained regression (non-negative coefficients summing to 1) to determine optimal combination weights [19].
• Final Model Generation:
  • Re-fit all base models on the complete dataset.
  • Generate predictions from these fully-trained base models.
  • Combine these predictions using the weights learned by the meta-learner to produce the final ECSG model [19].

This protocol ensures robust performance estimation while minimizing overfitting, as each base model's predictions used for meta-learner training are based on data not used for model fitting [19].

Performance Evaluation Metrics

The ECSG framework employs comprehensive evaluation metrics to assess predictive performance across multiple dimensions:

Table 2: Performance Metrics for Stability Prediction

Metric	ECSG Performance	Comparative Advantage	Application Context
Area Under Curve (AUC)	0.988 [1]	Superior discriminative ability	Binary classification (stable/unstable)
Sample Efficiency	1/7 data requirement [1]	Reduced computational resource needs	Data-scarce environments
First-Principles Validation	Remarkable accuracy [1]	Experimental verification	Real-world materials discovery

These metrics demonstrate the compelling advantages of the ECSG approach, particularly its exceptional data efficiency which enables accurate predictions with substantially smaller training datasets compared to conventional methods [1].

Research Reagent Solutions

Implementing the ECSG framework requires specific computational tools and data resources that constitute the essential "research reagents" for reproducibility and extension of the methodology.

Table 3: Essential Research Reagents for ECSG Implementation

Reagent Category	Specific Instances	Function/Purpose	Access Method
Computational Libraries	XGBoost, PyTorch/TensorFlow, Graph Neural Network libraries	Implement base models and meta-learning	Open-source platforms
Materials Databases	Materials Project (MP), Open Quantum Materials Database (OQMD), JARVIS	Provide training data and benchmark comparisons	Publicly accessible databases
Validation Tools	Density Functional Theory (DFT) codes (VASP, Quantum ESPRESSO)	First-principles validation of predictions [1]	Academic licenses
Electron Configuration Encoder	Custom matrix transformation (118×168×8)	Convert composition to ECCNN input format [1]	Implementation from source

These computational reagents represent the essential infrastructure for implementing the ECSG framework, with particular importance placed on the materials databases for training and the DFT validation tools for confirming predictions [1].

Application Notes for Materials Research

Case Study: Two-Dimensional Wide Bandgap Semiconductors

The ECSG framework has been successfully applied to navigate unexplored composition spaces, including the discovery of new two-dimensional wide bandgap semiconductors [1]. The implementation protocol for this application involves:

• Problem Formulation: Define the target material class (2D wide bandgap semiconductors) and desired electronic properties.
• Composition Space Sampling: Systematically generate candidate compositions within the defined chemical space.
• High-Throughput Screening: Apply the pre-trained ECSG model to rapidly evaluate thermodynamic stability across the composition space.
• Candidate Selection: Identify promising compounds with predicted high stability and appropriate electronic properties.
• Experimental Validation: Confirm stability and properties through first-principles calculations, with reported results indicating "remarkable accuracy" in correctly identifying stable compounds [1].

This workflow demonstrates the practical utility of ECSG in accelerating materials discovery by prioritizing synthesis efforts toward the most promising candidates, substantially reducing the experimental resources required for exploration of new material systems [1].

Implementation Considerations for Different Material Classes

The ECSG framework exhibits versatility across diverse material systems, with implementation nuances for different classes:

Materials Discovery Pipeline: This workflow illustrates the generalized materials discovery process enhanced by the ECSG framework's stability predictions.

For perovskite materials, particularly lead-free variants being explored for next-generation applications, ECSG provides critical stability assessment capabilities [22]. The framework's ability to predict thermodynamic stability from composition alone is especially valuable for these systems, where stability represents a major bottleneck for practical implementation [22]. Similar advantages extend to other material classes including double perovskite oxides, which have been successfully investigated using the ECSG methodology [1].

The ECSG framework represents a significant advancement in computational materials science, integrating ensemble machine learning with electron configuration theory to achieve unprecedented accuracy and efficiency in predicting thermodynamic stability. By leveraging stacked generalization across complementary modeling approaches, ECSG effectively mitigates the inductive biases inherent in single-perspective models while providing robust predictions validated against first-principles calculations [1]. The framework's demonstrated success in identifying novel two-dimensional wide bandgap semiconductors and double perovskite oxides underscores its practical utility in accelerating materials discovery [1].

Future developments will likely focus on expanding the framework's applicability to dynamic stability assessment under non-equilibrium conditions, integrating kinetic factors alongside thermodynamic stability, and extending the approach to predict functional properties beyond stability. As materials databases continue to grow and computational methods evolve, the ECSG methodology provides a adaptable foundation for next-generation materials informatics, positioning stacked generalization as a cornerstone technique in the digital transformation of materials research and development.

The discovery and development of new materials with specific properties represent a significant challenge in materials science, primarily due to the vast, unexplored compositional space of potential compounds [1]. Conventional methods for determining key properties, such as thermodynamic stability, rely on inefficient experimental investigation or computationally intensive density functional theory (DFT) calculations [1]. Machine learning (ML) offers a promising avenue for expediting this discovery process, providing significant advantages in time and resource efficiency [1].

However, many existing ML models are constructed based on specific, idealized domain knowledge, which can introduce large inductive biases and limit their predictive performance and generalizability [1]. For instance, models that assume material performance is determined solely by elemental composition may overlook critical structural or electronic factors [1]. This application note details a robust ensemble framework that integrates diverse knowledge sources—from classical Magpie features to the relational graph-based approach of Roost—to mitigate these limitations. This methodology is contextualized within a broader research thesis on using ensemble machine learning, anchored by electron configuration data, for predicting thermodynamic stability.

Background and Core Concepts

The Challenge of Isolated Models

Training a machine learning model can be likened to a search for ground truth within the model's parameter space [1]. When a model is built on a single hypothesis or a narrow set of features, the ground truth may lie outside its searchable parameter space, leading to suboptimal accuracy [1]. This is particularly true in materials science, where the relationship between composition, structure, and properties is complex and not fully understood [1].

Our ensemble framework, termed Electron Configuration Stacked Generalization (ECSG), synergistically combines three distinct knowledge paradigms to create a more comprehensive and accurate super learner [1] [23]. The three base models are:

• Magpie: This model emphasizes statistical features derived from diverse elemental properties (e.g., atomic number, mass, radius, electronegativity). It uses statistical summaries (mean, deviation, range, etc.) across the elements in a compound and is typically implemented with gradient-boosted regression trees (XGBoost) [1]. It operates on a coarse, elemental-property level.
• Roost (Representation Learning from Stoichiometry): This model conceptualizes a chemical formula as a graph, where atoms are nodes and their interactions are edges. It employs graph neural networks (GNNs) with an attention mechanism to capture complex interatomic interactions and message-passing processes that are critical for thermodynamic stability [1].
• ECCNN (Electron Configuration Convolutional Neural Network): Developed to address the limited understanding of electronic internal structure in existing models, this model uses electron configuration as its foundational input [1]. Electron configuration is an intrinsic atomic property that delineates the distribution of electrons within an atom, providing crucial information for understanding chemical properties and reaction dynamics with minimal inductive bias [1].

Methodology and Experimental Protocols

The ECSG Ensemble Framework

The core of our approach is the stacked generalization technique [1]. The framework does not simply average the predictions of the base models but uses them as inputs to a meta-learner. The experimental protocol is as follows:

• Base Model Training: The three foundational models (Magpie, Roost, ECCNN) are trained independently on the same dataset. Each model learns to predict thermodynamic stability from its unique perspective.
• Prediction Generation: The trained base models are used to generate predictions on a hold-out validation set or via cross-validation.
• Meta-Model Training: The predictions from the base models are used as input features for a meta-level model (the super learner), which is trained to produce the final, refined prediction [1].
• Validation: The integrated ECSG model is validated against benchmark datasets, such as those from the Joint Automated Repository for Various Integrated Simulations (JARVIS) or Materials Project (MP), to evaluate its performance metrics, including Area Under the Curve (AUC) [1] [23].

This framework effectively mitigates the limitations of individual models by harnessing a synergy that diminishes inductive biases [1].

Protocol for Composition-Based Stability Prediction

This protocol is used when the crystal structure of a compound is unknown, and prediction must be based solely on its chemical formula.

• Input Data Preparation: Provide a CSV file containing at least two columns: material-id (a unique identifier) and composition (the chemical formula, e.g., Fe2O3) [23].
• Feature Processing:
  • Option A (Runtime Processing): The ECSG code processes the CSV file and generates features for each model at runtime [23].
  • Option B (Preprocessed Features): For large datasets, features can be precomputed and saved using the provided feature.py script to reduce computation time during cross-validation [23].
• Model Execution:
  • Software Environment: Install the ECSG package in a Python 3.8.0 environment with PyTorch 1.13.0 and required dependencies (e.g., pymatgen, matminer). Specific wheels are provided for torch-scatter [23].
  • Prediction Command: Use the command python predict.py --path your_data.csv to generate stability predictions. Results are saved in results/meta/[name]_predict_results.csv, with the stability outcome in the target column [23].

Protocol for Structure-Enhanced Stability Prediction

For higher accuracy when crystal structure information is available, the ECSG framework can be extended to incorporate structural data.

• Input Data Preparation:
  • Prepare a folder containing CIF (Crystallographic Information File) files for the materials to be predicted.
  • Within this folder, include an id_prop.csv file listing the IDs of the CIF files.
  • Ensure an atom_init.json file is present for atom embedding [23].
• Model Execution:
  • Download and place pre-trained structure-based models (e.g., CGCNN models) in the models folder [23].
  • Run the prediction script: python predict_with_cifs.py --cif_path /path/to/your/cif/folder [23].

Performance and Validation

Experimental results have validated the efficacy of the ECSG framework. The model achieved an Area Under the Curve (AUC) score of 0.988 in predicting compound stability within the JARVIS database, demonstrating high predictive accuracy [1] [23].

A key advantage of ECSG is its remarkable sample efficiency. The model attained performance equivalent to existing models using only one-seventh of the training data, which dramatically reduces the computational resources required for training [1].

The model's versatility was further demonstrated through practical case studies, where it facilitated the exploration of new two-dimensional wide bandgap semiconductors and double perovskite oxides. Subsequent validation using first-principles calculations confirmed the model's high reliability in correctly identifying stable compounds [1].

Table 1: Quantitative Performance Summary of the ECSG Framework

Metric	Performance	Context & Comparison
Predictive Accuracy (AUC)	0.988	Achieved on the JARVIS database for thermodynamic stability prediction [1].
Data Efficiency	Uses ~1/7 of the data	Requires only one-seventh of the data used by existing models to achieve the same performance [1].
Validation Method	First-Principles Calculations	Applied to discovered compounds (e.g., 2D semiconductors, perovskites), confirming high reliability [1].

The ECSG Workflow Visualization

The following diagram illustrates the complete workflow of the ECSG framework, from data input to the final ensemble prediction.

To implement the ECSG framework and reproduce the experiments, the following software and data resources are essential.

Table 2: Key Research Reagent Solutions for ECSG Implementation

Item Name	Type	Function & Application	Source / Example
Magpie Feature Set	Software Library / Feature Generator	Generates a vector of statistical summaries (mean, deviation, range, etc.) from a list of elemental properties for a given chemical composition [1].	Matminer library [1].
Roost Model	Graph Neural Network Model	Represents a chemical formula as a graph and uses message-passing with attention to model interatomic interactions for property prediction [1].	Original Roost implementation [1].
ECCNN Model	Convolutional Neural Network Model	Uses electron configuration matrices as input to capture intrinsic electronic structure information with minimal manual feature engineering [1].	ECSG GitHub repository [23].
JARVIS/MP Databases	Data Repository	Provides large-scale, curated datasets of inorganic materials with computed properties (e.g., formation energy, stability) for training and benchmarking ML models [1].	JARVIS Database; Materials Project (MP) [1].
Pymatgen	Software Library	A robust, open-source Python library for materials analysis, used for parsing CIF files, handling compositions, and core materials algorithms [23].	Pymatgen Python package [23].

The integration of diverse knowledge sources—from the classical elemental statistics of Magpie to the advanced relational learning of Roost GNNs—within the ECSG ensemble framework represents a significant advancement in the machine learning-based prediction of materials properties. By effectively mitigating the inductive biases inherent in single-hypothesis models, this approach achieves superior predictive accuracy and exceptional data efficiency. The provided application notes and detailed protocols equip researchers and scientists with the tools to apply this powerful framework to their own investigations, accelerating the discovery of novel, thermodynamically stable materials for applications ranging from drug development to advanced semiconductors.

The accurate prediction of thermodynamic stability is a cornerstone of materials discovery and drug development. Traditional methods, whether experimental or computational, are often resource-intensive, creating a bottleneck in the exploration of novel chemical spaces. Machine learning (ML) offers a promising alternative; however, many models introduce significant inductive bias by relying on idealized assumptions or limited domain knowledge. The Electron Configuration Convolutional Neural Network (ECCNN) addresses this gap by using the fundamental electron configuration (EC) of atoms as a primary input feature. This approach is designed to minimize manual feature engineering, thereby reducing bias and capturing intrinsic atomic properties that are critically important for stability. Integrated within a stacked generalization framework, ECCNN contributes to a robust super learner for predicting compound decomposition energy ((\Delta H_{d})), achieving state-of-the-art performance with remarkable sample efficiency [1].

This application note details the architecture of ECCNN and provides a comprehensive protocol for encoding chemical compositions into its required input format, serving as an essential guide for researchers and scientists.

ECCNN Architecture and Workflow

The ECCNN is a composition-based model, meaning it requires only the chemical formula of a compound as its input. Its architecture is specifically designed to process the spatially structured data of encoded electron configurations.

Model Architecture Specifications

The ECCNN processes the input through a series of feature extraction and transformation layers, summarized in Table 1.

Table 1: ECCNN Architecture Specifications

Layer Type	Specifications	Output Shape (Conceptual)	Key Parameters & Notes
Input Layer	Accepts encoded electron configuration matrix	118 (elements) × 168 (energy levels/orbitals) × 8 (features)	Fixed input size; see Section 3 for encoding details.
Convolutional Block 1	2D Convolution + Activation	118 × 168 × 64	64 filters, kernel size 5×5, ReLU activation.
Convolutional Block 2	2D Convolution + Batch Normalization + Max Pooling + Activation	~59 × 84 × 64	64 filters, kernel size 5×5; Batch Normalization; 2×2 Max Pooling; ReLU activation.
Feature Flattening	Flatten Layer	1D Vector (e.g., ~59 * 84 * 64 features)	Converts 2D feature maps into a 1D vector for dense layers.
Prediction Head	Fully Connected (Dense) Layers	Scalar (ΔH_d prediction)	Maps flattened features to the final stability prediction.

End-to-End Workflow

The following diagram illustrates the complete workflow from chemical composition to thermodynamic stability prediction, highlighting the role of ECCNN within the larger ensemble framework.

Figure 1: Workflow of the ECSG framework. The chemical formula is encoded into an electron configuration matrix, which is processed by the ECCNN. Its prediction, along with those from other base models, is used by a meta-learner to produce the final, robust stability estimate.

Input Encoding: From Chemical Formula to ECCNN Input

The unique aspect of ECCNN is its input representation, which is based on the fundamental electron structure of the constituent elements.

Input Matrix Structure

The input to ECCNN is a 3D tensor with dimensions 118 × 168 × 8. The encoding process, detailed in the protocol below, transforms a chemical formula into this structured format.

Table 2: ECCNN Input Matrix Dimensions

Dimension	Size	Description
Elements	118	Corresponds to the 118 elements in the periodic table. Each row represents a different chemical element.
Energy Levels/Orbitals	168	Represents a comprehensive set of possible atomic orbitals and energy levels for electron occupation.
Feature Channels	8	Encodes different properties of the electron configuration at each orbital, such as occupation number, spin, etc.

Step-by-Step Encoding Protocol

Protocol 1: Encoding a Chemical Formula for ECCNN

Objective: To convert a chemical formula (e.g., TiO₂) into the standardized 118×168×8 input matrix for the ECCNN model. Reagents & Materials: A periodic table database with full electron configuration data for all 118 elements.

Step	Procedure	Critical Notes
1. Formula Parsing	Parse the input chemical formula to identify the constituent elements and their stoichiometric ratios.	For TiO₂, the elements are Ti (1 atom) and O (2 atoms).
2. Element Mapping	For each of the 118 elements on the periodic table, retrieve its complete electron configuration.	The electron configuration defines the distribution of electrons in atomic orbitals.
3. Orbital Mapping	Map the electron configuration of each element onto a standardized set of 168 energy levels/orbitals.	This creates a consistent feature vector length for all elements, regardless of their atomic number.
4. Feature Population	For each element and its corresponding orbitals, populate the 8 feature channels with relevant electronic structure information.	Features may include electron occupation number, energy level, orbital angular momentum, spin, etc.
5. Stoichiometric Integration	Incorporate the stoichiometric information from Step 1 into the feature representation for the compound.	This step is crucial to differentiate, for example, CO from CO₂. The specific mathematical operation may vary.
6. Matrix Assembly	Assemble the feature vectors for all 118 elements into the final 118 (elements) × 168 (orbitals) × 8 (features) tensor.	For elements not present in the compound, their rows are typically populated with zeros.

The Scientist's Toolkit

Table 3: Essential Research Reagents and Computational Tools

Item / Resource	Function / Description	Relevance to ECCNN Protocol
Periodic Table Database	A computational database containing atomic properties, including the full electron configuration for all 118 elements.	Essential for performing the input encoding (Steps 2-4 in Protocol 1).
JARVIS Database	A comprehensive materials database (Joint Automated Repository for Various Integrated Simulations).	Serves as a key source of training data (stable and unstable compounds) and for benchmark testing.
Stacked Generalization Framework	An ensemble machine learning technique that combines multiple models.	ECCNN functions as a base model within this framework, which integrates its predictions with those of Magpie and Roost to form the final ECSG super learner.
Convolutional Block Attention Module	A neural network component that helps the model focus on informative parts of the input.	While not explicitly in the base ECCNN, such attention mechanisms are a common extension to improve featurization of complex inputs like electronic structure data [24].

Integration in Ensemble Stability Prediction

The ECCNN is not used in isolation. Its strength is leveraged in an ensemble model called ECSG (Electron Configuration models with Stacked Generalization). In this framework, ECCNN serves as one of three base models, each grounded in different domain knowledge:

• Magpie: Relies on statistical features of elemental properties.
• Roost: Models the chemical formula as a graph of interacting atoms.
• ECCNN: Provides a first-principles perspective via electron configurations.

The predictions from these base models are then used as input features for a meta-learner, which is trained to produce the final, highly accurate, and robust prediction of thermodynamic stability. This approach mitigates the inductive bias inherent in any single model [1].

The discovery of novel functional materials is pivotal for advancing technologies in photovoltaics, optoelectronics, and energy storage. Traditional methods relying on trial-and-error or computationally intensive first-principles calculations struggle to efficiently navigate vast chemical spaces. Ensemble machine learning (ML) frameworks, particularly those integrating electron configuration data, have emerged as powerful tools for predicting thermodynamic stability and functional properties, dramatically accelerating the screening and discovery of materials such as two-dimensional (2D) semiconductors and double perovskite oxides [25]. These approaches demonstrate remarkable efficiency, achieving high predictive accuracy with significantly smaller datasets compared to conventional models [25] [26].

This application note details specific case studies and protocols for applying ensemble ML to screen 2D semiconductors and double perovskite oxides, providing researchers with practical methodologies for materials discovery.

Ensemble ML Framework for Stability Prediction

Core Architecture and Workflow

The ECSG (Electron Configuration models with Stacked Generalization) framework mitigates the inductive bias inherent in single-hypothesis models by amalgamating knowledge from different physical scales [25]. Its super-learner architecture integrates three base models—Magpie, Roost, and ECCNN—followed by a meta-learner that synthesizes their predictions.

• Magpie: Utilizes statistical features (mean, deviation, range, etc.) from elemental properties like atomic number, mass, and radius, trained with gradient-boosted regression trees (XGBoost) [25].
• Roost: Conceptualizes a chemical formula as a complete graph of elements, employing graph neural networks with an attention mechanism to capture interatomic interactions [25].
• ECCNN (Electron Configuration Convolutional Neural Network): Processes raw electron configuration data, an intrinsic atomic property, using convolutional neural networks to learn representations directly from electron orbital distributions [25].

The workflow involves training these base models on existing stability data (e.g., decomposition energy, ΔHd), using their outputs as features for a final meta-learner to produce supervening predictions [25].

Fig. 1 Ensemble ML workflow for stability prediction integrating multiple knowledge domains

Performance Metrics

The ECSG framework achieves state-of-the-art performance in thermodynamic stability prediction, as summarized in Table 1.

Table 1: Performance metrics of the ensemble ML framework for stability prediction

Metric	Performance	Context
Area Under the Curve (AUC)	0.988 [25]	Predicts compound stability within the JARVIS database
Data Efficiency	Uses ~1/7 of data [25]	Achieves performance parity with existing models using significantly less data
Accuracy in Case Studies	Remarkable accuracy [25]	Validated via first-principles calculations for 2D wide-bandgap semiconductors and double perovskite oxides

Case Study 1: Screening 2D Semiconductors

Application Protocol

Objective: Identify novel, thermodynamically stable 2D semiconductors with wide bandgaps suitable for UV photodetection and advanced optoelectronics [27] [25].

Challenges: Traditional synthesis of 2D perovskite oxides like Ca₂Nb₃O₁₀ (CNO) faces harsh conditions and defect chemistry, limiting large-scale production and integration [27].

ML-Guided Screening Workflow:

• Define Target Space: Focus on compositional space of 2D materials, particularly perovskite oxides and related wide-bandgap semiconductors [25].
• Generate Stability Predictions: Apply the trained ECSG model to predict the thermodynamic stability (decomposition energy, ΔHd) of candidate compositions [25].
• Down-Select Stable Candidates: Filter the list to retain only compounds predicted to be stable with high confidence.
• Experimental Validation: Synthesize predicted stable materials using advanced fabrication techniques, such as the charge-assisted oriented assembly (COAF) process for wafer-scale integration [27].

Experimental Synthesis & Validation

For experimentally validating ML-predicted 2D semiconductors, the COAF process enables wafer-scale production of high-quality films [27].

Protocol: Charge-Assisted Oriented Assembly Film-Formation (COAF) [27]

• Material Exfoliation: Obtain 2D perovskite oxide nanosheets (e.g., Ca₂Nb₃O₁₀) via a top-down process involving high-temperature solid-phase calcination, ion exchange, and liquid-phase exfoliation.
• Solution Preparation: Disperse exfoliated nanosheets in a green water-ethanol cosolvent system. This reduces surface energy, enhances wettability on substrates, and controls solvent volatilization rate.
• Spray Deposition: Uniformly spray micro-droplets of the nanosheet dispersion onto a substrate (e.g., a 6-inch n-type Si wafer). Intrinsic negative charges on nanosheets create repulsive forces, preventing aggregation and promoting dispersion.
• Oriented Assembly: Upon contact with the substrate, non-uniform repulsive forces between nanosheets in the droplet and those on the substrate induce rotation to an approximately parallel orientation, leading to highly ordered stacking as the solvent evaporates.
• Film Formation: Micro-droplets coalesce, and nanosheets connect across the substrate, forming a continuous, dense, and highly oriented film. Smaller nanosheets fill voids between larger ones.

Characterization and Performance:

• Structural: XRD analysis confirms highly oriented film structure, showing only equivalent (00l) crystal planes [27].
• Optoelectronic: Photoluminescence (PL) characterization shows quenched PL intensity and longer lifetime in assembled films, indicating reduced defect density and suppressed carrier recombination [27].
• Device Integration: Enables fabrication of ultra-flexible 256-pixel imaging arrays and motion recognition systems with over 99.8% accuracy [27].

Case Study 2: Screening Double Perovskite Oxides

Hierarchical Screening for Band Gap and Stability

Objective: Discover novel double perovskite oxides (A₂BB′O₆) that are thermodynamically stable and possess targeted electronic properties, particularly wide and/or direct band gaps, for applications in photovoltaics, electrocatalysis, and supercapacitors [28] [29] [30].

ML-Guided Screening Workflow: This hierarchical approach sequentially applies specialized ML models to efficiently down-select candidates, as visualized in Fig. 2.

• Stability and Formability Screening: Use ensemble ML models (e.g., ECSG) to screen candidate compositions from databases, predicting thermodynamic stability and formability to generate an initial candidate list [25] [28].
• Band Gap Classification: Apply a trained classification model (e.g., Cost-Sensitive Extreme Gradient Boosting) to separate candidates with wide band gaps (E₉ ≥ 0.5 eV) from those with narrow or zero band gaps [28] [30].
• Band Gap Regression: Apply a regression model to quantitatively predict the band gap values of the wide-band-gap candidates identified in the previous step [28].
• First-Principles Validation: Perform DFT calculations on the final shortlisted compounds to confirm their stability and electronic properties before experimental synthesis [25] [28].

Fig. 2 Hierarchical ML screening workflow for double perovskite oxides

Screening Outcomes and Validation

This workflow has successfully identified numerous promising double perovskite compositions.

Table 2: Outcomes of ML-guided screening for double perovskite oxides

Screening Focus	ML Model Used	Key Outcomes
General Stable DPs	ECSG (Stability) [25]	Identification of numerous novel, stable double perovskite oxide structures, validated by DFT.
Wide-Bandgap DPs	Hierarchical RF [28]	Down-selection of 13,589 cubic compositions predicted as stable and wide-bandgap (E₉ ≥ 0.5 eV); 310 identified as high-confidence candidates.
Direct-Bandgap DPs	Cost-Sensitive XGBoost [30]	Identification of 2,027 direct-bandgap perovskites from 21,021 formable candidates, optimal for photovoltaics.

Example: Prediction of Ba₂ScXO₆ (X = As, Sb) First-principles calculations validated the ML predictions for these scandium-based double perovskites [31]:

• Stability: Confirmed via negative formation energy, satisfied Born-Huang stability criterion, and absence of imaginary phonon frequencies.
• Electronic Properties: Identified as indirect band gap insulators with band gaps of 3.829 eV (Ba₂ScAsO₆) and 4.796 eV (Ba₂ScSbO₆).
• Optical Properties: High light absorption intensity in the 100–200 nm wavelength range, suggesting potential for deep-UV optoelectronic devices [31].

The Scientist's Toolkit

Table 3: Essential research reagents and materials for synthesis and characterization

Reagent/Material	Function/Application	Key Details
Ca₂Nb₃O₁₀ (CNO) Precursors	Base material for 2D perovskite oxide nanosheets	Synthesized via high-temperature solid-phase calcination [27].
Water-Ethanol Cosolvent	Dispersion medium for nanosheet exfoliation and assembly	Reduces surface energy, enhances wettability, controls evaporation [27].
A₂BB′O₆ Precursors	Starting materials for double perovskite oxide synthesis	A-site: Alkaline/rare-earth metals. B/B'-site: Transition metals [29].
Organic Spacers (Amines)	Templates for 2D HOIP structure formation	Linear/cyclic monovalent or divalent cations; steric/topological properties critical [32].

Ensemble machine learning frameworks rooted in electron configuration provide a robust and efficient pathway for discovering and designing advanced functional materials. The case studies outlined demonstrate their successful application in screening 2D semiconductors and double perovskite oxides for thermodynamic stability and targeted optoelectronic properties. The provided experimental protocols offer researchers detailed methodologies for synthesizing and characterizing ML-predicted materials, bridging the gap between computational prediction and experimental realization. This integrated approach significantly accelerates the development of next-generation materials for energy, electronics, and catalysis.

The successful application of ensemble machine learning (ML) to predict thermodynamic stability from electron configuration in materials science presents a compelling paradigm for the field of molecular property prediction in drug discovery. The core principle—using stacked generalization to harmonize models based on diverse physical knowledge—demonstrates significant potential for overcoming analogous challenges in predicting molecular behavior. This document details the translation of this ensemble framework into practical protocols for predicting critical molecular properties, leveraging and adapting the "Electron Configuration models with Stacked Generalization" (ECSG) approach [1]. The procedures herein are designed for researchers and development professionals aiming to enhance the accuracy and efficiency of in-silico drug design.

Ensemble Framework and Comparative Performance

The ECSG framework mitigates inductive bias by integrating multiple base models, each rooted in distinct domains of knowledge, with their predictions serving as input for a final meta-learner [1]. This approach is directly applicable to molecular property prediction. Table 1 summarizes the performance of various modeling approaches on benchmark tasks, demonstrating the superior accuracy of the ensemble method.

Table 1: Comparative Performance of Machine Learning Models on Property Prediction Tasks

Model / Framework	Prediction Task	Performance Metric	Score	Key Advantage
ECSG (Ensemble) [1]	Thermodynamic Stability	AUC (Area Under Curve)	0.988	Mitigates inductive bias
Gradient Boosting [33]	Aqueous Solubility (logS)	R² (Test Set)	0.87	Manages complex descriptor interactions
CatBoost [34]	Reactive Species Generation	Accuracy	0.936	Effective for dual-task learning
PET-MAD-DOS [35]	Electronic Density of States	MAE (Mean Absolute Error)	< 0.2 (on most structures)	Universal model across chemical space
ECCNN (Base Model) [1]	Thermodynamic Stability	Sample Efficiency	7x more efficient	Learns directly from electron configuration

Detailed Experimental Protocols

Protocol 1: Implementing an Ensemble Framework for Molecular Property Prediction

This protocol adapts the ECSG framework for general molecular property prediction, such as solubility or toxicity.

I. Data Preparation and Feature Encoding

• Data Curation: Compile a dataset of molecular structures (e.g., SMILES strings) and their corresponding experimental property values. Employ stringent data curation to remove errors and duplicates [33].
• Feature Generation (Base Models): Encode the molecular data into three distinct feature sets to train the base models:
  • Domain Knowledge 1 (Electron Configuration): For the ECCNN model, represent a molecule as a matrix that encodes the electron configuration of its constituent atoms. This treats the molecule as a "bag of atoms" and captures intrinsic electronic structure [1].
  • Domain Knowledge 2 (Atomic Properties): For a model like Magpie, calculate a set of statistical features (mean, variance, etc.) from a list of elemental properties (e.g., atomic radius, electronegativity) for all atoms in the molecule [1].
  • Domain Knowledge 3 (Interatomic Interactions): For a graph-based model like Roost, represent the molecule as a graph where atoms are nodes and bonds are edges. This allows a Graph Neural Network to learn from the topology and interactions within the molecule [1].

II. Base-Level Model Training

• Independently train the three base models (ECCNN, Magpie, and Roost) using their respective feature sets and the target molecular property.
• Perform hyperparameter tuning for each model using a validation set.

III. Stacked Generalization (Meta-Learning)

• Meta-Feature Generation: Use the trained base models to generate predictions on the training data via cross-validation (to avoid data leakage) and on a hold-out test set.
• Meta-Dataset Construction: Create a new dataset where these predictions are the input features (meta-features) and the original experimental property values are the targets.
• Meta-Learner Training: Train a relatively simple model (e.g., linear model, XGBoost) on this meta-dataset to learn how to best combine the base models' predictions [1].

IV. Validation and Application

• Validate the final ECSG model on the held-out test set, comparing its performance against the individual base models using metrics like R² or AUC.
• Use the trained ECSG framework to predict properties for new, uncharacterized molecules.

The following workflow diagram illustrates the complete ensemble framework protocol.

Protocol 2: Predicting Aqueous Solubility Using MD-Derived Properties

This protocol provides a specific application for predicting aqueous solubility (logS), a critical property in drug development, using properties derived from Molecular Dynamics (MD) simulations.

I. Dataset Construction

• Source experimental aqueous solubility data (logS) for a diverse set of drug-like molecules, such as the Huuskonen dataset (211 drugs) [33].
• Collect or calculate the octanol-water partition coefficient (logP) for each molecule from literature.

II. Molecular Dynamics Simulations

• Setup: Use software like GROMACS 5.1.1 with the GROMOS 54a7 force field. Place a single solute molecule in a cubic box filled with water molecules (e.g., SPC water model). Run energy minimization before the main simulation [33].
• Production Run: Perform simulations in the NPT ensemble (constant Number of particles, Pressure, and Temperature) for a sufficient duration (e.g., 10-50 ns) to ensure equilibrium [33].
• Property Extraction: From the simulation trajectories, calculate the following key properties for each molecule:
  • SASA (Solvent Accessible Surface Area): Quantifies the molecule's surface exposed to the solvent.
  • Coulombic and LJ (Lennard-Jones) Interaction Energies: Measure the electrostatic and van der Waals interactions between the solute and water.
  • DGSolv (Estimated Solvation Free Energy): The free energy change associated with solvation.
  • RMSD (Root Mean Square Deviation): Measures the conformational stability of the solute in solution.
  • AvgShell (Average number of solvents in Solvation Shell): Describes the local solvent environment around the solute [33].

III. Model Training and Validation

• Feature Selection: Combine the extracted MD properties with logP to form the feature set.
• Train Ensemble Models: Apply ensemble algorithms like Gradient Boosting, Random Forest, or XGBoost to predict logS from the MD features.
• Validation: Validate the model on a held-out test set. The Gradient Boosting algorithm, for instance, has achieved a test R² of 0.87 using this approach [33].

Applications in Drug Discovery

The outlined frameworks enable several advanced applications in drug discovery and materials science.

• Accelerated High-Throughput Screening: Models like the universal PET-MAD-DOS, which predicts the electronic density of states for molecules and materials, can screen thousands of compounds for target electronic properties at near-quantum accuracy but drastically reduced computational cost [35]. This is invaluable for identifying novel semiconductors or photovoltaic materials.

• Interpretable Design of Functional Molecules: ML models can move beyond prediction to provide interpretable design rules. For example, in designing photocatalysts for antibiotic degradation, SHAP analysis revealed that the d-electron count of metal elements is a critical threshold governing the generation of specific reactive species [34]. This allows for the targeted design of molecules with desired functions.

• De-Risking Synthesis with Stability Prediction: The ECSG framework's high accuracy (AUC = 0.988) in predicting thermodynamic stability can be applied to molecular systems to prioritize synthetically accessible and stable compounds, thereby reducing late-stage attrition in drug development [1].

The Scientist's Toolkit: Essential Research Reagents

Table 2 catalogs key computational tools and their functions, forming the essential toolkit for implementing the protocols described in this document.

Table 2: Key Research Reagent Solutions for Computational Experiments

Tool / Resource Name	Type	Primary Function in Research	Application Context
Mordred [36]	Molecular Descriptor Calculator	Generates 1,800+ 2D and 3D molecular descriptors from structure.	QSPR modeling for properties like boiling point or solubility.
GROMACS [33]	Molecular Dynamics Simulator	Simulates physical movements of atoms and molecules over time.	Deriving properties like SASA and solvation free energy for solubility prediction.
ChemXploreML [37]	Desktop Application	User-friendly, offline tool for predicting molecular properties without coding.	Rapid prototyping and prediction for chemists lacking deep programming expertise.
AlvaDesc / Dragon [36]	Molecular Descriptor Calculator	Alternative software for generating large sets of molecular descriptors (5,000+).	Creating comprehensive feature sets for machine learning models.
ECCNN [1]	Machine Learning Model	Neural network that uses electron configuration as direct input for prediction.	Serving as a base model in an ensemble framework to capture electronic properties.
VICGAE [37]	Molecular Embedder	Creates compact numerical representations (embeddings) of molecular structures.	Fast featurization of molecules for machine learning pipelines.
SHAP (SHapley Additive exPlanations) [34]	Model Interpretation Tool	Explains the output of any ML model by quantifying feature importance.	Interpreting model predictions to gain physicochemical insights (e.g., identifying d-electron count as critical).

Overcoming Practical Hurdles: Data Efficiency, Feature Selection, and Model Interpretation

In scientific fields such as materials science and drug development, the high cost and difficulty of acquiring labeled data often severely constrain research progress. Experimental synthesis and characterization typically demand expert knowledge, expensive equipment, and time-consuming procedures, making large datasets a luxury. This application note details practical strategies and protocols for maximizing model performance under such stringent data budgets, with a specific focus on applications in thermodynamic stability prediction of inorganic compounds using ensemble machine learning based on electron configuration. The methodologies outlined herein are designed to help researchers and scientists achieve high accuracy with minimal data investment.

Core Strategies for Enhanced Sample Efficiency

Ensemble Learning with Electron Configuration

Integrating multiple models through ensemble methods, particularly stacked generalization, effectively mitigates the inductive bias inherent in single models and significantly enhances sample efficiency [1].

• Key Implementation: The Electron Configuration models with Stacked Generalization (ECSG) framework synergistically combines three base models: an Electron Configuration Convolutional Neural Network (ECCNN), a graph-based Roost model, and a feature-based Magpie model [1].
• Documented Efficiency: This ensemble approach has been experimentally validated to achieve state-of-the-art performance in predicting thermodynamic stability using only one-seventh of the data required by existing models to achieve the same performance level [1].
• Protocol: The input for the ECCNN component is a matrix (118×168×8) encoded from the electron configurations of the constituent elements, which serves as an intrinsic, less-biased material descriptor. Subsequent convolutional layers extract hierarchical features for the final ensemble prediction [1].

Active Learning for Strategic Data Acquisition

Active Learning (AL) is a supervised machine learning approach that strategically selects the most informative data points for labeling, thereby minimizing labeling costs while maximizing model performance [38].

• Operational Workflow: AL operates through an iterative loop: (1) training an initial model on a small labeled set, (2) using a query strategy to select the most informative unlabeled samples, (3) having a human expert label these samples, and (4) updating the model with the newly labeled data [38].
• Empirical Performance: Benchmark studies in materials science regression tasks show that uncertainty-driven and diversity-hybrid AL strategies (e.g., LCMD, Tree-based-R, RD-GS) significantly outperform random sampling and geometry-only heuristics, especially in the early, data-scarce phases of a project [39].
• Query Strategies:
  • Uncertainty Sampling: Selects data points where the model's prediction confidence is lowest [38].
  • Diversity Sampling: Selects data points that are most dissimilar to those already in the labeled set, ensuring broad coverage of the feature space [38].
  • Query-by-Committee: Selects data points that cause the most disagreement among an ensemble of models [38].

Table 1: Comparison of Active Learning Query Strategies

Strategy	Core Principle	Best-Suited Scenario	Key Advantage
Uncertainty Sampling	Selects instances with lowest prediction confidence	Single-model settings, clear probabilistic outputs	Directly targets model's points of confusion
Diversity Sampling	Maximizes coverage of the input feature space	Initial phases, ensuring dataset representativeness	Mitigates risk of sampling bias
Query-by-Committee	Selects points with highest disagreement among model ensemble	When a model ensemble is available	Leverages multiple hypotheses
Stream-Based Selective	Evaluates and queries data points one-by-one from a continuous stream	Real-time data generation or online learning	Computationally efficient, adaptable

Integration with Automated Machine Learning (AutoML)

Combining Active Learning with Automated Machine Learning creates a powerful, adaptive pipeline for small-data scenarios. AutoML automates the selection and hyperparameter tuning of machine learning models, which is crucial when the optimal model family is unknown a priori [39].

• Benchmark Finding: In an AutoML context, where the underlying surrogate model can change across iterations, uncertainty-driven and diversity-hybrid AL strategies have demonstrated robustness and superior performance, making them ideal for such dynamic environments [39].
• Protocol: The recommended workflow involves using an AutoML framework (e.g., AutoSklearn, TPOT) as the core learner within an AL loop. The AutoML system is retrained in each AL cycle after the query strategy selects and acquires labels for the most informative new samples [39].

Detailed Experimental Protocols

Protocol 1: Building an Ensemble Model for Stability Prediction

This protocol outlines the steps to replicate the ECSG framework for predicting thermodynamic stability of inorganic compounds [1].

• Data Encoding and Input Preparation:

  • Feature Set 1 (ECCNN): Encode the chemical composition into an electron configuration matrix. For each element in the compound, generate a representation of its electron distribution across energy levels. The final input is a 3D tensor (118 elements × 168 features × 8 channels) for the entire compound [1].
  • Feature Set 2 (Roost): Represent the chemical formula as a graph, where atoms are nodes and bonds are edges, to model interatomic interactions [1].
  • Feature Set 3 (Magpie): Calculate a set of statistical features (mean, standard deviation, minimum, maximum, etc.) from a wide range of elemental properties (e.g., atomic number, mass, radius) for the composition [1].

• Base Model Training:

  • Train the three base models (ECCNN, Roost, Magpie) independently on the same small, labeled training dataset using their respective feature sets.

• Stacked Generalization:

  • Use the predictions of the three base models on a hold-out validation set as input features for a meta-learner (e.g., a linear model or another simple classifier).
  • Train this meta-learner to produce the final, combined prediction.

• Validation:

  • Validate the super learner's performance on a separate test set using metrics such as Area Under the Curve (AUC) and accuracy. The ECSG framework achieved an AUC of 0.988 in predicting compound stability [1].

The workflow for this ensemble framework is depicted in the diagram below.

Protocol 2: Implementing an Active Learning Cycle with AutoML

This protocol describes how to set up an iterative AL cycle to build a high-accuracy model with minimal labeled data [39] [38].

• Initialization:

  • Start with a very small, randomly sampled set of labeled data ( L = {(xi, yi)}{i=1}^l ) and a large pool of unlabeled data ( U = {xi}_{i=l+1}^n ) [39].

• Model Training & Query:

  • Train: Fit an AutoML regression/classification model on the current labeled set ( L ). The AutoML system will automatically search and optimize the best model and hyperparameters [39].
  • Query: Use the trained AutoML model to evaluate all instances in the unlabeled pool ( U ). Apply an AL query strategy (e.g., uncertainty sampling based on prediction variance) to select the most informative sample ( x^* ) [39] [38].

• Expert Annotation & Loop:

  • Annotate: The selected sample ( x^* ) is presented to a human expert (or an experimental protocol) for labeling, obtaining its true label ( y^* ) [38].
  • Update: Augment the labeled set ( L = L \cup {(x^, y^)} ) and remove ( x^* ) from ( U ) [39].
  • Iterate: Repeat steps 2 and 3 until a predefined stopping criterion is met (e.g., a performance plateau or exhaustion of the labeling budget) [39].

The following diagram illustrates this iterative cycle.

The Scientist's Toolkit: Research Reagent Solutions

Table 2: Essential Computational Tools and Frameworks

Tool / Solution	Category	Function in Research
ECCNN Model	Custom Ensemble Component	Provides electron configuration-based features, reducing inductive bias in material stability prediction [1].
AutoML Frameworks (e.g., AutoSklearn, TPOT)	Model Automation	Automates the selection and hyperparameter tuning of machine learning models, optimizing performance with limited manual intervention [39].
Active Learning Libraries (e.g., modAL, ALiPy)	Strategic Sampling	Provides implemented query strategies (uncertainty, diversity) to efficiently select data for labeling [38].
JARVIS, Materials Project	Materials Database	Provides foundational data on material compositions and properties for initial model training and feature engineering [1].
Density Functional Theory (DFT)	Validation Tool	Used as a computational method to validate the predictions of machine learning models (e.g., thermodynamic stability) [1].

Achieving high accuracy with limited data is a critical challenge in scientific research and development. The synergistic application of ensemble learning, active learning, and AutoML provides a robust and sample-efficient methodology. By leveraging intrinsic data representations like electron configuration, strategically querying for informative samples, and automating the model optimization process, researchers can dramatically reduce the experimental and computational costs associated with materials discovery and drug development, thereby accelerating the pace of innovation.

The discovery and development of new functional materials are crucial for advancing technologies in energy storage, electronics, and healthcare. A critical step in this process is the accurate prediction of material properties, particularly thermodynamic stability, which determines whether a proposed material can be synthesized and remain stable under operational conditions. Traditionally, this has been accomplished through two complementary computational approaches: composition-based and structure-based models.

Composition-based models predict material properties solely from their chemical formula, enabling rapid screening of vast chemical spaces where structural data is unavailable. In contrast, structure-based models incorporate detailed crystallographic information, typically achieving higher accuracy but requiring complete atomic coordinates that are often unknown for novel materials. Within this context, ensemble machine learning frameworks have emerged as powerful tools that integrate both approaches, leveraging their complementary strengths to mitigate individual limitations and enhance predictive performance.

Recent research has demonstrated that ensemble methods combining models from different knowledge domains can achieve remarkable accuracy in stability prediction. The ECSG framework, for instance, has achieved an Area Under the Curve (AUC) score of 0.988 on benchmark datasets while requiring only one-seventh of the training data compared to existing models to achieve equivalent performance [1]. Such advances highlight the potential of sophisticated ML approaches to accelerate materials discovery.

Comparative Analysis: Composition-Based vs. Structure-Based Modeling

The choice between composition-based and structure-based modeling approaches involves fundamental trade-offs between computational efficiency, data requirements, and predictive accuracy. Understanding these trade-offs is essential for selecting the appropriate methodology for a given research objective.

Table 1: Fundamental Trade-offs Between Modeling Approaches

Characteristic	Composition-Based Models	Structure-Based Models
Primary Input Data	Chemical formula (elemental proportions)	Crystallographic structure (atomic coordinates)
Data Availability	Widely available for known and hypothetical compounds	Often limited for novel, unsynthesized materials
Computational Cost	Low (rapid prediction enabling high-throughput screening)	High (requires complex calculations or data acquisition)
Exploration Capability	Excellent for exploring uncharted chemical spaces	Limited to materials with known or predicted structures
Predictive Accuracy	Generally lower, but improving with advanced ML	Typically higher due to richer input information
Key Limitations	Limited by lack of structural information; potential inductive bias	Challenging application to novel compounds without known structures

Composition-Based Models: Expanding Exploratory Horizons

Composition-based models operate on chemical formulas alone, making them uniquely suited for initial screening of hypothetical materials where structural data is nonexistent. These models transform elemental compositions into machine-readable inputs using various strategies, from simple elemental fractions to sophisticated representations derived from domain knowledge [1].

Advanced composition-based models have moved beyond manual feature engineering. Deep learning approaches like ElemNet utilize deep neural networks on elemental fractions, while later successors incorporated pretrained element embeddings and attention mechanisms [40]. More recently, Chemical Language Models (CLMs) have reframed composition-based prediction as a sequence modeling task, demonstrating significant performance improvements [40].

The principal advantage of composition-based approaches is their ability to rapidly evaluate enormous swathes of chemical space. However, their performance is inherently constrained by the lack of structural information, which can be a critical determinant of material properties and stability.

Structure-Based Models: Enhancing Predictive Fidelity

Structure-based models leverage the complete crystallographic information of materials, typically representing crystal structures as graphs where atoms serve as nodes and bonds as edges. These models, particularly Graph Neural Networks (GNNs), have consistently demonstrated superior performance for property prediction when structural data is available [40].

Recent advancements in structure-based modeling have incorporated increasingly sophisticated architectural improvements. The Crystal Graph Convolutional Neural Network (CGCNN) introduced convolution operations on crystal graphs, while subsequent innovations included learnable bond embeddings, many-body interactions, and neighbor equalization techniques [40]. The emerging frontier involves multimodal architectures that incorporate data beyond spatial atom arrangements, such as electronic structure information [40].

The primary limitation of structure-based models remains their dependency on known crystal structures, which presents a fundamental barrier for exploring truly novel compounds without prior structural knowledge.

Ensemble Approaches: Integrating Multiple Knowledge Domains

Ensemble machine learning methods have emerged as a powerful strategy to mitigate the limitations of individual modeling approaches. By combining models grounded in distinct domains of knowledge, ensemble frameworks can reduce inductive biases and enhance overall predictive performance.

The ECSG Framework: A Case Study in Ensemble Integration

The Electron Configuration models with Stacked Generalization (ECSG) framework exemplifies the ensemble approach to thermodynamic stability prediction. This framework integrates three distinct models based on complementary knowledge domains [1]:

• Magpie: Utilizes statistical features derived from various elemental properties (atomic number, mass, radius, etc.) and employs gradient-boosted regression trees for prediction.
• Roost: Conceptualizes chemical formulas as graphs of elements, using graph neural networks with attention mechanisms to capture interatomic interactions.
• ECCNN (Electron Configuration Convolutional Neural Network): A novel model that uses electron configuration matrices as input, processed through convolutional layers to capture electronic structure information crucial for stability determination.

This ensemble approach leverages stacked generalization, where the outputs of these base-level models serve as inputs to a meta-learner that produces the final prediction. This methodology demonstrates how integrating diverse representations can create a more robust and accurate predictive system [1].

Cross-Modal Knowledge Transfer: Bridging Composition and Structure

Another innovative ensemble strategy involves cross-modal knowledge transfer, which enhances composition-based predictions by incorporating structural intelligence through indirect means. Two primary formulations have been proposed [40]:

• Implicit Transfer (imKT): Involves pretraining chemical language models on multimodal embeddings that align compositional information with structural and electronic representations.
• Explicit Transfer (exKT): Generates hypothetical crystal structures from composition using predictive models like CrystaLLM, then applies structure-aware predictors to the generated crystals.

These approaches have demonstrated substantial performance improvements, achieving state-of-the-art results in 25 out of 32 benchmark tasks and reducing mean absolute error by up to 39.6% for certain properties like total energy prediction [40].

Diagram 1: ECSG ensemble framework integrating three model types.

Experimental Protocols for Model Development and Validation

Protocol: Developing an Ensemble Model for Thermodynamic Stability Prediction

Objective: To develop and validate an ensemble machine learning model for predicting thermodynamic stability of inorganic compounds using composition and electron configuration features.

Materials and Computational Resources:

• Dataset Source: Materials Project (MP) or Joint Automated Repository for Various Integrated Simulations (JARVIS) databases [1]
• Software Environment: Python with specialized libraries (scikit-learn, PyTorch/TensorFlow, MatDeepLearn)
• Computational Hardware: GPU-accelerated computing resources for deep learning model training

Procedure:

• Data Acquisition and Curation:

  • Download formation energies and decomposition energies (ΔHd) for inorganic compounds from materials databases.
  • Define stability classification labels (stable/unstable) based on energy above the convex hull.
  • Perform train-test-validation split (typical ratio: 70-15-15) with stratified sampling to maintain class distribution.

• Feature Engineering and Input Representation:

  • For Magpie model: Calculate statistical features (mean, variance, mode, range) for elemental properties across composition [1].
  • For Roost model: Represent composition as stoichiometrically weighted element sets for graph construction [1].
  • For ECCNN model: Encode electron configurations as 118×168×8 matrices representing electron distributions across energy levels [1].

• Base Model Training:

  • Train Magpie model using gradient boosting trees (XGBoost) on statistical features.
  • Train Roost model using graph neural networks with message passing and attention mechanisms.
  • Train ECCNN model using convolutional neural networks with two convolutional layers (64 filters, 5×5 kernel), batch normalization, and max pooling, followed by fully connected layers [1].

• Ensemble Integration via Stacked Generalization:

  • Generate predictions from all three base models on validation set.
  • Use these predictions as meta-features to train a meta-learner (logistic regression or neural network).
  • Validate ensemble performance using nested cross-validation to prevent data leakage.

• Model Evaluation:

  • Evaluate final model using area under the receiver operating characteristic curve (AUC-ROC), precision-recall curves, and accuracy metrics.
  • Compare ensemble performance against individual base models and existing benchmarks.

Protocol: Experimental Validation of Computational Predictions

Objective: To experimentally validate computationally predicted materials using synthesis and electrochemical characterization.

Materials:

• Precursors: Transition metal salts (e.g., NiCl₂·6H₂O, CoCl₂·6H₂O) [14]
• Reactants: Sodium hydroxide (NaOH), sodium hypophosphite (NaH₂PO₂) [14]
• Solvents: Ethanol-water mixture [14]
• Electrode Components: Conductive additives (carbon black), binders (PVDF), current collectors (nickel foam) [14]

Procedure:

• Material Synthesis:

  • Synthesize predicted stable compositions using modified coprecipitation in ethanol-water medium [14].
  • Maintain precise stoichiometric ratios (e.g., varying Ni:Co ratios) as guided by ML predictions.
  • Perform thorough washing and drying of synthesized materials.

• Structural and Compositional Characterization:

  • Conduct X-ray diffraction (XRD) to confirm phase formation and crystal structure.
  • Perform X-ray photoelectron spectroscopy (XPS) to determine elemental oxidation states (e.g., Ni²⁺/Ni³⁺, Co²⁺/Co³⁺) [14].
  • Analyze morphology using transmission electron microscopy (TEM) to determine particle size and distribution.

• Electrochemical Performance Validation:

  • Prepare electrode inks by mixing active material, conductive carbon, and binder in appropriate ratios.
  • Fabricate working electrodes using nickel foam or similar current collectors.
  • Perform cyclic voltammetry and galvanostatic charge-discharge measurements in three-electrode configuration.
  • Calculate specific capacitance from discharge curves and compare with ML predictions.
  • Evaluate rate capability and cycling stability over thousands of cycles.

• Performance Agreement Analysis:

  • Calculate percentage error between predicted and experimental specific capacitance, rate retention, and cyclic stability.
  • Validate ML predictions with experimental results, with typical errors ranging from 2.5-8.5% for capacitance and 0.03-19.6% for stability metrics [14].

Diagram 2: Experimental validation workflow for ML predictions.

Table 2: Research Reagent Solutions for Experimental Validation

Reagent/Material	Function/Application	Specifications
Transition Metal Salts (NiCl₂·6H₂O, CoCl₂·6H₂O)	Precursors for active electrode material synthesis	High purity (≥99%), analytical grade [14]
Sodium Hydroxide (NaOH)	Precipitation agent for hydroxide formation	Pellet form, ≥98% purity [14]
Sodium Hypophosphite (NaH₂PO₂)	Phosphorus source for phosphate incorporation	≥99% purity [14]
Ethanol-Water Mixture	Reaction medium for coprecipitation synthesis	Ethanol purity 99.9% [14]
Conductive Carbon Additives	Enhancing electrical conductivity of electrodes	Carbon black, graphene, or carbon nanotubes
Polyvinylidene Fluoride (PVDF)	Binder for electrode fabrication	Dissolved in N-methyl-2-pyrrolidone (NMP)
Nickel Foam	Current collector for supercapacitor electrodes	High porosity (>95%) for electrolyte access

Performance Metrics and Validation

Rigorous validation is essential for assessing model performance and practical utility. Ensemble models for thermodynamic stability prediction should be evaluated using multiple complementary metrics and experimental corroboration.

Table 3: Quantitative Performance Comparison of Modeling Approaches

Model/Approach	Dataset	Key Performance Metrics	Experimental Agreement
ECSG (Ensemble)	JARVIS [1]	AUC: 0.988, High sample efficiency (1/7 data required)	N/A
Cross-Modal Knowledge Transfer	LLM4Mat-Bench [40]	MAE reduction: 15.7% avg. (up to 39.6% for total energy)	N/A
Ensemble ML for Electrodes	Experimental NCP compositions [14]	Prediction errors: 2.48-8.46% (capacitance), 0.03-19.3% (rate capability)	Specific capacitance: 2247.6 F g⁻¹ at 3 A g⁻¹
ElemNet	Materials Project [1]	Baseline for formation energy prediction	N/A

For electrochemical applications, experimental validation has demonstrated remarkable agreement with ML predictions. Recent studies on transition metal-based electrodes have shown percentage errors as low as 2.48-8.46% for specific capacitance, 0.03-19.30% for rate capability, and 3.95-19.64% for cyclic stability between predicted and experimentally measured values [14]. This close agreement underscores the growing reliability of ML approaches in guiding materials development.

The integration of composition-based and structure-based modeling approaches through ensemble methods represents a paradigm shift in computational materials discovery. By leveraging the complementary strengths of both approaches—the exploratory power of composition-based models and the predictive accuracy of structure-based models—researchers can more effectively navigate the complex trade-offs inherent in materials design.

The ECSG framework demonstrates how integrating models based on electron configuration, elemental properties, and interatomic interactions can achieve exceptional predictive accuracy for thermodynamic stability while dramatically improving sample efficiency. Similarly, cross-modal knowledge transfer approaches show how implicit and explicit integration of structural information can enhance composition-based predictions. These ensemble strategies are particularly valuable in early-stage discovery when structural data is limited.

As machine learning methodologies continue to evolve, the distinction between composition-based and structure-based approaches is likely to blur further through advanced transfer learning and multimodal integration. These developments will accelerate the discovery of novel materials with tailored properties for applications ranging from energy storage to drug development, ultimately reducing the time and cost associated with traditional experimental approaches.

The discovery and development of advanced energetic materials (EMs) has historically been constrained by a fundamental challenge: the inherently small data regimes characteristic of this field. Unlike domains with abundant data, EM research faces practical limitations in data collection due to the high costs, safety concerns, and substantial time investments required for experimental synthesis and testing [41]. This data scarcity creates a significant bottleneck for traditional machine learning approaches, which typically require large datasets to develop accurate predictive models. Consequently, researchers have developed sophisticated strategies to maximize information extraction from limited data points, transforming how we approach materials discovery in data-sparse environments [42].

Within this context, this application note explores cutting-edge methodologies for addressing small data challenges, with particular emphasis on their integration with ensemble machine learning frameworks built upon electron configuration features. These approaches enable researchers to extract meaningful patterns and relationships from limited datasets, accelerating the discovery of novel energetic compounds with targeted properties.

The table below summarizes the primary strategies employed to overcome data limitations in energetic materials research, along with their key implementations and performance metrics.

Table 1: Methodologies for Addressing Small Data Challenges in Energetic Materials Research

Methodology	Key Implementation	Reported Performance/Advantage	Reference
Ensemble Learning with Stacked Generalization	ECSG framework integrating Magpie, Roost, and ECCNN models	AUC of 0.988 for stability prediction; 7x improvement in data efficiency	[1] [23]
Data Augmentation	SMILES enumeration for molecular representations	Enables effective model training with limited molecular datasets	[41]
Active Learning	Gaussian Process Regression with custom acquisition function	Identifies tens of optimal nanothermites within 200 samplings vs. <10 with Latin Hypercube	[43]
Multi-Fidelity Information Fusion	Combining high-cost experimental data with lower-fidelity computational data	More optimal predictive models when high-quality data is scarce	[41]
Transfer Learning	Leveraging knowledge from related materials domains	Improves model performance on small target datasets	[42]

Experimental Protocols and Application Notes

Protocol: Implementing the ECSG Ensemble Framework

The Electron Configuration Stacked Generalization (ECSG) framework represents a significant advancement for predictive modeling in small-data regimes, achieving high accuracy in thermodynamic stability prediction with dramatically reduced data requirements [1].

Table 2: Research Reagent Solutions for Ensemble Implementation

Resource Category	Specific Tool/Platform	Function/Purpose
Computational Environment	Python 3.8.0, PyTorch 1.13.0	Core machine learning framework and computational backbone
Specialized Libraries	torch-scatter, pymatgen, matminer	Handling graph-based data structures and materials informatics
Data Sources	Materials Project (MP), JARVIS, EM Database	Providing foundational training data and benchmark compounds
Pretrained Models	ECCNN, Magpie, Roost	Base learners capturing complementary materials representations

Step-by-Step Implementation Procedure:

• Data Preparation: Compile a CSV file containing material-id and composition columns. For enhanced accuracy with known structures, include a folder of CIF files with corresponding id_prop.csv and atom_init.json files [23].

• Feature Generation:

  • Process composition data to generate electron configuration matrix inputs (118×168×8) for the ECCNN model [1].
  • Simultaneously, compute Magpie features (elemental property statistics) and Roost graph representations to capture interatomic interactions [1].

• Model Training:

  • Train the three base models (ECCNN, Magpie, Roost) using k-fold cross-validation (typically k=5) to prevent overfitting [23].
  • Generate predictions from each base model to serve as input features for the meta-learner.

• Stacked Generalization:

  • Implement a super learner that combines the predictions of all three base models, typically using linear models or gradient boosting [1].
  • Train this meta-model on hold-out validation sets to learn optimal weighting of the base model predictions.

• Validation and Prediction:

  • Evaluate model performance using AUC-ROC curves and precision-recall metrics appropriate for imbalanced datasets.
  • Deploy the trained ECSG framework to predict thermodynamic stability of novel compositions.

Protocol: Active Learning for Targeted Exploration

Active learning provides a strategic framework for navigating vast design spaces with limited experimental resources, particularly valuable for optimizing nanothermite formulations [43].

Implementation Workflow:

• Initial Design Space Definition: Characterize the multi-dimensional parameter space encompassing material composition, particle size, morphology, and synthesis conditions.

• Acquisition Function Design: Develop a customized acquisition function that combines:

  • Standard deviation from Gaussian Process Regression (uncertainty sampling)
  • Directional guidance toward user-defined regions of interest
  • Incentive function for exploring under-sampled regions [43]

• Iterative Experimentation Loop:

  • Select the most informative candidate materials based on the acquisition function.
  • Synthesize and characterize the selected candidates (highest expected information gain).
  • Update the surrogate model with new experimental data.
  • Refine predictions and repeat the selection process.

• Termination Criteria: Continue iterations until either (a) target performance thresholds are achieved, or (b) diminishing returns are observed in model improvement.

Protocol: Data Augmentation for Molecular Representations

In the small-data environment typical of energetic materials research, data augmentation techniques effectively expand limited datasets to improve model generalization [41].

SMILES Enumeration Protocol:

• Molecular Representation: Convert all molecular structures in the dataset to Simplified Molecular Input Line Entry Specification (SMILES) strings.

• Enumeration Implementation: Apply SMILES enumeration to generate multiple valid string representations for each molecule in the dataset, creating augmented training examples [41].

• Model Training: Utilize augmented datasets to train recurrent neural networks (RNNs) or other deep learning architectures that benefit from larger training sets.

• Validation: Employ rigorous cross-validation to ensure that augmented data improves generalization without introducing artifacts or biases.

Integration with Ensemble Machine Learning for Thermodynamic Stability

The small data methodologies discussed herein find natural synergy with ensemble machine learning approaches grounded in electron configuration features for predicting thermodynamic stability. The ECSG framework exemplifies this integration, demonstrating that models incorporating electron configuration information achieve superior data efficiency, requiring only one-seventh of the data to match the performance of existing models [1].

This enhanced efficiency stems from the fundamental physical relationship between electron configuration and material stability. By using electron configuration as a foundational input feature, the model incorporates intrinsic atomic-level information that directly influences bonding behavior and compound formation, thereby reducing the need for extensive training data to learn these relationships empirically [1].

Furthermore, the ensemble approach mitigates the inductive biases that plague single-model methodologies. By integrating diverse knowledge sources—from atomic properties (Magpie) to interatomic interactions (Roost) and electronic structure (ECCNN)—the stacked generalization framework creates a more robust predictive model that generalizes effectively even from limited data [1].

The methodologies outlined in this application note provide practical solutions to the pervasive challenge of small datasets in energetic materials research. The integration of ensemble methods with electron configuration features, augmented by strategic sampling and data enhancement techniques, represents a paradigm shift in how researchers can extract meaningful insights from limited experimental data.

As the field advances, promising research directions include developing more sophisticated multi-fidelity information fusion approaches, meta-learning strategies that transfer knowledge across related material classes, and semi-supervised learning techniques that leverage both labeled and unlabeled data [44]. These innovations will further empower researchers to navigate the complex design space of energetic materials with unprecedented efficiency, accelerating the discovery of next-generation compounds with tailored properties and performance characteristics.

As machine learning (ML), particularly ensemble learning, becomes integral to complex scientific domains like thermodynamic stability research, the need for model interpretability is paramount. SHapley Additive exPlanations (SHAP) is a game-theoretic approach that provides a unified measure of feature importance for any machine learning model, bridging the gap between model complexity and human understanding [45]. For researchers and drug development professionals, this translates to the ability not just to predict, for instance, the stability of a material or the efficacy of a compound, but to understand the specific atomic interactions or molecular descriptors driving that prediction. SHAP moves beyond a "black box" by quantifying the contribution of each input feature to an individual prediction, ensuring that model-driven discoveries are both actionable and trustworthy [46].

The core principle of SHAP is rooted in distributing the "payout" (a model's prediction for a specific instance) fairly among its "players" (the input features) [47]. It does this by computing the average marginal contribution of a feature value across all possible coalitions of features, providing a robust, theoretically sound foundation for explainability that satisfies properties of local accuracy, missingness, and consistency [45].

Theoretical Foundations of SHAP

The Shapley Value Framework

SHAP is built upon Shapley values, a concept from cooperative game theory. The Shapley value is the average marginal contribution of a feature value across all possible coalitions of features [47]. For a prediction model, this translates to fairly distributing the difference between the actual prediction and the average prediction among the input features.

The mathematical definition of the Shapley value for feature j is given by [48]:

$$\phij(val)=\sum{S\subseteq{1,\ldots,p} \setminus {j}}\frac{|S|!\left(p-|S|-1\right)!}{p!}\left(val\left(S\cup{j}\right)-val(S)\right)$$

where:

• S is a subset of the features used in the model
• p is the total number of features
• val(S) is the value function (prediction) for the feature subset S

This formula ensures a mathematically fair distribution of the prediction among features, satisfying key properties including efficiency (the sum of all Shapley values equals the model's output), symmetry (features contributing equally receive equal values), dummy (features with no contribution receive zero value), and additivity [48].

SHAP as an Additive Feature Attribution Method

SHAP explains a model prediction as a linear model of binary variables where each variable indicates whether a corresponding feature is included in the explanation [45]. The explanation model is defined as:

[g(\mathbf{z}')=\phi0+\sum{j=1}^M\phij zj']

where:

• (g) is the explanation model
• (\mathbf{z}' \in {0,1}^M) is the coalition vector
• (M) is the maximum coalition size
• (\phi_j \in \mathbb{R}) is the Shapley value for feature (j)

This additive formulation connects SHAP to other explanation methods while maintaining its game-theoretic foundations.

SHAP Estimation Methodologies

Algorithmic Approaches

Different SHAP estimation methods have been developed to balance computational efficiency with accuracy across various model types:

KernelSHAP is a model-agnostic method that uses specially weighted linear regression to estimate Shapley values. It involves sampling coalitions, getting predictions for these coalitions, and fitting a linear model [45]. While flexible, it can be computationally intensive for high-dimensional data.

TreeSHAP is a high-speed method specifically for tree-based models and ensemble methods (e.g., Random Forest, XGBoost, LightGBM) that computes Shapley values in polynomial time by leveraging the tree structure [45]. This makes it particularly suitable for ensemble learning applications in scientific research.

Permutation-based methods offer another model-agnostic approach, though they may struggle with correlated features [49]. Recent advances in libraries like ACV (Active Coalition of Variables) address these limitations by providing more robust Shapley value computations when features are correlated [49].

Computational Protocol

Table 1: SHAP Computation Methods and Their Characteristics

Method	Model Compatibility	Computational Complexity	Handling of Correlated Features
KernelSHAP	Model-agnostic	High (exponential in features)	Standard
TreeSHAP	Tree-based models	Low (polynomial time)	Improved
Permutation-based	Model-agnostic	Medium	Standard
ACV Tree	Tree-based models	Medium	Advanced

Application to Ensemble Learning in Thermodynamic Stability Research

Ensemble Learning Framework

Ensemble methods combine multiple machine learning models to achieve superior predictive performance. In thermodynamic stability research, a stacking ensemble approach has demonstrated significant advantages, integrating heterogeneous base learners (e.g., Random Forest, Gradient Boosting, SVM) with a meta-learner (e.g., Logistic Regression) to generate final predictions [50]. This framework effectively captures complex, nonlinear relationships in material properties while mitigating biases inherent in individual models.

The EASE-Predict framework (Ensemble-SHAP Explainable Student Prediction), while from an educational domain, exemplifies this approach with achieved 77.4% accuracy, representing a 4.3 percentage point improvement over the best individual model (Random Forest at 73.1%) [51]. Such ensemble frameworks show exceptional discriminative performance with AUC scores up to 0.930 for target class prediction [51].

SHAP Integration Protocol

Table 2: SHAP Analysis Protocol for Ensemble Models

Step	Procedure	Tools/Parameters	Output
1. Model Training	Train ensemble model using stacking with heterogeneous base learners	Scikit-learn, XGBoost; 5-10 base models	Trained ensemble model
2. SHAP Value Computation	Calculate Shapley values for each prediction	SHAP Python library; TreeExplainer for tree-based ensembles	SHAP value matrix (samples × features)
3. Global Interpretation	Analyze feature importance across dataset	shap.summary_plot(), shap.bar_plot()	Global feature rankings
4. Local Interpretation	Explain individual predictions	shap.force_plot(), shap.waterfall_plot()	Instance-level explanations
5. Dependence Analysis	Examine feature interactions	shap.dependence_plot()	Feature relationship patterns

Experimental Protocols

SHAP Analysis for Model Interpretation

Protocol 1: Global Feature Importance Analysis

• Compute SHAP Values: After training the ensemble model, use the appropriate SHAP explainer (e.g., TreeExplainer for tree-based ensembles) to compute SHAP values for a representative sample of the test dataset [51] [50].
• Aggregate Results: Calculate mean absolute SHAP values for each feature across the dataset to determine global importance.
• Visualize: Generate a bar plot of mean absolute SHAP values and a summary plot showing feature importance and impact direction [48].
• Interpret: Identify features with the largest mean absolute SHAP values as the most impactful drivers of model predictions.

Protocol 2: Local Prediction Explanation

• Select Instance: Choose a specific prediction to explain (e.g., a compound predicted to have high thermodynamic stability).
• Generate Explanation: Use shap.force_plot() or shap.waterfall_plot() to visualize how each feature contributed to pushing the model output from the base value (average prediction) to the final prediction [47] [48].
• Contextualize: Relate the top contributing features to domain knowledge to validate the explanation's scientific plausibility.

Protocol 3: Feature Dependency Analysis

• Select Target Feature: Choose a feature of scientific interest identified as important in global analysis.
• Create Dependence Plot: Use shap.dependence_plot() to visualize how the feature's value relates to its SHAP value, colored by a potentially interacting feature [48].
• Identify Interactions: Look for nonlinear relationships and interactions that provide insight into complex feature dynamics in thermodynamic stability.

Validation and Best Practices

Validation Steps:

• Compare SHAP explanations with domain knowledge and existing scientific literature [46].
• Conduct sensitivity analysis by comparing explanations across different subsets of data.
• When possible, validate identified feature relationships through controlled experiments or simulation.

Common Pitfalls and Mitigations:

• Correlated Features: SHAP can produce misleading explanations with highly correlated features. Use ACV library for improved handling or group correlated features [49].
• Categorical Features: Avoid one-hot encoding issues by using entity embeddings or targeted encoding schemes [49].
• Reference Dataset: Select a meaningful background dataset for SHAP computation that represents the population of interest [47].

Case Study: Ensemble Learning with SHAP Interpretation

Experimental Setup and Results

In a recent study on student dropout prediction (a proxy for complex classification in scientific domains), researchers implemented an ensemble framework (EASE-Predict) combining five machine learning algorithms (Random Forest, Gradient Boosting, Extra Trees, Logistic Regression, and SVM) with voting and stacking ensemble models [51]. The dataset comprised 4,424 instances with 36 features. SHAP analysis revealed that second-semester curricular units completion accounted for 60% of prediction influence, followed by tuition payment status (35%) and scholarship availability (12%) [51].

Table 3: Performance Comparison of Ensemble vs. Individual Models

Model Type	Accuracy	AUC (Dropout)	AUC (Graduate)	Performance Variance (σ)
Ensemble (EASE-Predict)	77.4%	0.913	0.930	0.014
Best Individual Model (Random Forest)	73.1%	0.904	0.927	0.0189
Improvement	+4.3%	+0.009	+0.003	-0.0049

Statistical significance of the ensemble's superior performance was confirmed using McNemar's test (p < 0.05) [51]. This demonstrates how ensemble methods enhance predictive performance while SHAP provides actionable insights into the driving features.

SHAP Visualization Workflow

The Scientist's Toolkit: Essential Research Reagents

Table 4: Essential Tools for SHAP Analysis in Scientific Research

Tool/Software	Function	Application Context
SHAP Python Library	Core computation of Shapley values	Model-agnostic and model-specific explanations
ACV Library	Advanced Shapley values with correlated features	Scenarios with high feature interdependence
Scikit-learn	Implementation of base ML models	Building ensemble learners
XGBoost/LightGBM	Gradient boosting frameworks	High-performance base learners for ensembles
Matplotlib/Seaborn	Custom visualization of results	Publication-quality figures
Jupyter Notebooks	Interactive analysis environment	Exploratory model interpretation
InterpretML	Explainable Boosting Machines	Baseline interpretable models for validation

SHAP provides an essential bridge between complex ensemble models and scientific interpretability in thermodynamic stability research. By leveraging game-theoretically optimal Shapley values, researchers can move beyond black-box predictions to gain actionable insights into the fundamental drivers of material behavior. The integration of SHAP with ensemble learning creates a powerful framework that combines state-of-the-art predictive performance with meaningful scientific explanations, enabling more informed decision-making in drug development and materials science. As ensemble methods continue to evolve in sophistication, corresponding advances in explainable AI approaches like SHAP will be crucial for ensuring these technologies yield not just predictions, but understanding.

Optimizing Hyperparameters and Training for Robust Ensemble Performance

The pursuit of novel materials with specific properties is a significant challenge in materials science, compounded by the vastness of compositional space. Accurately predicting thermodynamic stability is a crucial first step, as it can efficiently winnow out compounds that are difficult to synthesize, thereby accelerating materials development [1]. Traditional methods for determining stability, such as density functional theory (DFT) calculations,, while accurate, are computationally intensive and inefficient for exploring new compounds [1].

Machine learning (ML) offers a promising alternative, enabling rapid and cost-effective predictions of compound stability [1]. However, models built on a single hypothesis or a specific piece of domain knowledge can introduce significant inductive bias, limiting their accuracy and generalizability [1]. Ensemble methods, which combine multiple models, have emerged as a powerful technique to mitigate these limitations. By leveraging the strengths and diversity of several base learners, ensemble models can achieve superior performance and robustness. This document provides detailed application notes and protocols for optimizing such ensembles, with a specific focus on predicting thermodynamic stability from electron configuration data within a drug development context, where new solid forms of active pharmaceutical ingredients (APIs) are critical.

Ensemble Framework and Base-Layer Models

The core of a robust ensemble lies in the strategic combination of diverse models that leverage different assumptions or domains of knowledge. This diversity helps ensure that the weaknesses of one model are compensated by the strengths of another.

The Stacked Generalization Framework

The Electron Configuration models with Stacked Generalization (ECSG) framework is a potent architecture for this purpose [1]. It operates on two levels:

• Base-Level: Several distinct models are trained on the same dataset.
• Meta-Level: A super-learner model is trained to optimally combine the predictions of the base-level models, thereby producing the final output.

This approach amalgamates models rooted in distinct domains of knowledge, effectively mitigating inductive biases and harnessing a synergy that enhances overall performance [1]. Experimental results have demonstrated that such a framework can achieve an Area Under the Curve (AUC) score of 0.988 in predicting compound stability and exhibits exceptional sample efficiency, requiring only one-seventh of the data used by existing models to achieve the same performance [1].

Recommended Base-Level Models

For research focused on electron configuration and thermodynamic stability, the ECSG framework integrates three complementary models. The selection is based on incorporating domain knowledge from different scales.

• Electron Configuration Convolutional Neural Network (ECCNN): This model uses electron configuration (EC) as its fundamental input. The EC delineates the distribution of electrons within an atom, encompassing energy levels and the electron count at each level. It is an intrinsic atomic property that introduces less inductive bias compared to manually crafted features and is conventionally used as input for first-principles calculations [1].

  • Architecture: The input is a matrix (e.g., 118×168×8) encoded from the EC of materials. This is processed through two convolutional layers (e.g., 64 filters of size 5×5), followed by batch normalization, max pooling, and fully connected layers for prediction [1].

• Roost (Representations from Ordered Or Unordered STructure): This model conceptualizes the chemical formula as a complete graph of elements. It employs graph neural networks with an attention mechanism to learn the relationships and message-passing processes among atoms, effectively capturing interatomic interactions critical for thermodynamic stability [1].

• Magpie (Materials Property Generator): This model emphasizes statistical features derived from various elemental properties, such as atomic number, mass, and radius. It calculates statistics like mean, mean absolute deviation, range, minimum, maximum, and mode across the elements in a compound. The model is typically trained using gradient-boosted regression trees (e.g., XGBoost) [1].

The following diagram illustrates the flow of data and models within this ensemble framework.

Data Preparation and Feature Engineering Protocol

Robust ensemble performance is contingent on high-quality, well-preprocessed data.

• Primary Data: For training and validation, leverage large materials databases such as the Materials Project (MP) or the Open Quantum Materials Database (OQMD), which provide extensive data on formation energies and computed stability [1].
• Composition-Based Input: The models discussed are composition-based, meaning they use only the chemical formula as input. This is advantageous for exploring new materials where structural information is unknown a priori [1].
• Electron Configuration Encoding: For the ECCNN model, raw composition data must be transformed into an electron configuration matrix. This involves representing the electron occupancy for each element in the compound across different atomic orbitals and energy levels, forming a structured matrix input [1].

Data Preprocessing Workflow

The following protocol ensures data is clean, consistent, and model-ready.

• Categorical Feature Encoding:

  • Application: If the dataset includes categorical variables (e.g., solvent types in solubility studies, space group symbols), use One-Hot Encoding.
  • Protocol: Convert each category into a separate binary feature (0 or 1). This ensures the model treats each category as an independent attribute without imposing ordinal bias [52].

• Data Normalization:

  • Application: Scale all numerical features to a consistent range. This is critical for models sensitive to input scale and improves training stability.
  • Protocol: Apply Min-Max Scaling to transform features into a [0, 1] range using the formula: ( X{\text{scaled}} = \frac{X - X{\text{min}}}{X{\text{max}} - X{\text{min}}} ) [52].
  • Alternative: The Z-score normalization method can also be employed to standardize the data distribution by subtracting the mean and dividing by the standard deviation [53].

• Outlier Detection:

  • Application: Identify and remove anomalous data points that could negatively impact model training.
  • Protocol: Use the Elliptic Envelope technique, which assumes a multivariate normal distribution and defines an elliptical region covering the core data points (e.g., 95% or 99%). Observations falling outside this boundary are considered outliers and removed [52].

Hyperparameter Optimization and Training Procedures

Hyperparameter tuning is essential for maximizing the performance of each base model and the meta-learner.

Optimization Strategies

• Metaheuristic Algorithms: These are effective for complex, non-convex optimization landscapes. For instance, the Pelican Optimization Algorithm (POA) and Binary Grey Wolf Optimization (BGWO) have been successfully used for hyperparameter tuning and feature selection, respectively [53]. The Stochastic Fractal Search (SFS) algorithm is another powerful option for fine-tuning model parameters to enhance prediction accuracy [52].
• Ensemble-Based Hyperparameter Determination: A novel approach involves the automated determination of hyperparameters using statistical estimators constructed from an ensemble of models. This requires parallel training of hundreds of models sampling the hyperparameter space and is highly effective for stabilizing model uncertainty estimates [54].

Base Model Training and Tuning

The table below summarizes key hyperparameters and optimization techniques for the recommended base models.

Table 1: Base Model Hyperparameter Optimization Guide

Model	Key Hyperparameters	Recommended Optimization Technique	Performance Insight
ECCNN	Number/filter size of convolutional layers, pooling strategy, dense layer units.	Pelican Optimization Algorithm (POA) [53] or Stochastic Fractal Search (SFS) [52].	Achieved AUC of 0.988 for stability prediction within the ECSG framework [1].
Roost	Graph neural network architecture, attention mechanism parameters, learning rate.	Ensemble-based hyperparameter determination via parallel training [54].	Effectively captures complex interatomic interactions [1].
Magpie (XGBoost)	Number of trees, max depth, learning rate, subsample ratio.	Bayesian Optimization or Advanced Propensity Score Modelling [55].	Provides robust baseline via statistical features of elemental properties [1].

Meta-Learner Training Protocol

• Generate Predictions: Use the trained base-level models (ECCNN, Roost, Magpie) to generate prediction scores on the validation set.
• Create Meta-Dataset: Construct a new dataset where the features are the prediction outputs from the base models, and the target is the true label (e.g., stable/unstable).
• Train Super-Learner: Train a meta-learner on this new dataset. A simple linear model or Logistic Regression often works well, but more complex models like XGBoost can also be used. Hyperparameter tuning for the meta-learner is also crucial and can be performed using the techniques mentioned above.

The following diagram outlines the logical sequence of the hyperparameter optimization process for the ensemble.

Performance Evaluation and Validation

A rigorous evaluation is necessary to validate the ensemble's predictive power and robustness.

• Key Metrics:

  • Area Under the Curve (AUC): A key metric for binary classification tasks like stability prediction. The ECSG framework achieved an AUC of 0.988 [1].
  • R-squared (R²): For regression tasks (e.g., predicting formation energy). Well-optimized models can achieve R² values >0.99 on test data [52].
  • Mean Squared Error (MSE)/Root MSE (RMSE): Measure prediction error. For instance, a Bayesian Neural Network achieved an MSE of 3.07×10⁻⁸ in a solubility prediction task [52].
  • Mean Absolute Percentage Error (MAPE): Provides a relative measure of error. The Neural Oblivious Decision Ensemble (NODE) model achieved a MAPE of 0.1835 in pharmaceutical solubility analysis [52].

• Validation with First-Principles Calculations:

  • Protocol: The ultimate validation for predicted novel stable compounds involves Density Functional Theory (DFT) calculations. Compute the formation energy of the candidate compound and determine its energy above the convex hull (ΔHd) to verify thermodynamic stability [1] [56]. Validation results from such calculations have confirmed the remarkable accuracy of ensemble methods in correctly identifying stable compounds [1].

The Scientist's Toolkit: Research Reagent Solutions

Table 2: Essential Resources for Ensemble-Driven Materials Discovery

Item / Resource	Function / Application
Materials Project (MP) / OQMD Databases	Provide large, curated datasets of computed material properties for training and validation [1].
VASP (Vienna Ab initio Simulation Package)	Software for performing DFT calculations to validate model predictions and generate new training data [56].
Moment Tensor Potential (MTP)	A class of machine-learning interatomic potentials for fast, accurate energy and force predictions, useful for data generation [56].
Stochastic Fractal Search (SFS) / Pelican Optimization Algorithm (POA)	Metaheuristic algorithms for hyperparameter optimization [52] [53].
XGBoost	A scalable tree-boosting system, ideal for the Magpie featurization and as a potential meta-learner [1].
PyTorch/TensorFlow	Deep learning frameworks for implementing and training complex models like ECCNN and Roost [1].
Elliptic Envelope (Scikit-learn)	Tool for outlier detection in the data preprocessing stage [52].

Proving the Paradigm: Benchmarking Performance and Validating Discoveries

In computational materials science and drug discovery, accurately predicting properties like thermodynamic stability is paramount for accelerating the development of new compounds and therapies. The Area Under the Receiver Operating Characteristic Curve (AUC-ROC) has emerged as a critical evaluation metric, particularly for classification tasks involving imbalanced data, where the outcome of interest—such as a stable compound or an active drug candidate—is rare. This Application Note provides a structured framework for benchmarking machine learning models using AUC scores and complementary error metrics, with a specific focus on ensemble methods within electron configuration and thermodynamic stability research. We present standardized protocols, quantitative benchmarks, and visualization tools to enable researchers to conduct robust, reproducible model evaluations.

Quantitative Performance Benchmarks

The following tables consolidate key quantitative findings from recent literature to establish performance benchmarks for model comparison.

Table 1: Comparative AUC Performance of Stability Prediction Models

Model / Framework	Dataset	AUC Score	Key Application Context
ECSG (Ensemble) [1]	JARVIS	0.988	Predicting thermodynamic stability of inorganic compounds
AUC-Maximizing Ensemble [57]	Multiple (Simulations)	~20-30% AUC risk reduction vs. baselines	General binary classification; outperforms non-AUC maximizing methods
Super Learner with AUC Metalearning [58]	Imbalanced biomedical data	Outperforms top base algorithm	Biomedical classification with increasing class imbalance

Table 2: Key Error Metrics for Comprehensive Model Evaluation

Metric	Formula / Principle	Interpretation & Use Case
Accuracy [59]	(True Positives + True Negatives) / Total Predictions	Overall correctness; can be misleading for imbalanced data.
Precision [60]	True Positives / (True Positives + False Positives)	Measures false positive cost; critical when false positives are costly (e.g., drug candidate prioritization).
Recall (Sensitivity) [60]	True Positives / (True Positives + False Negatives)	Measures false negative cost; vital for rare event detection (e.g., stable material identification).
F1-Score [60]	2 * (Precision * Recall) / (Precision + Recall)	Harmonic mean of precision and recall; balances the two concerns.
PR-AUC [61]	Area under the Precision-Recall curve	Preferred over ROC-AUC for highly imbalanced datasets; focuses on minority class performance.

Experimental Protocols for Model Benchmarking

Protocol 1: Implementing an AUC-Maximizing Super Learner Ensemble

This protocol details the metalearning approach for constructing an ensemble that directly optimizes the AUC, based on the Super Learner algorithm [58].

• Base Learner Library Construction: Assemble a diverse set of L base learning algorithms, {ψ₁,..., ψₗ}. Diversity is key and can include:

  • Different algorithm classes (e.g., Random Forest, Support Vector Machine, Neural Networks).
  • The same algorithm with different hyperparameter settings (e.g., multiple Random Forests with different mtry values).
  • Models incorporating different domain knowledge (e.g., Magpie, Roost, ECCNN) [1].

• Level-One Data Generation via Cross-Validation:

  • Partition the training set into V cross-validation folds (typically V=5 or 10).
  • For each base learner ψₗ, perform V-fold cross-validation. This generates an n-dimensional vector of cross-validated predicted values, where n is the number of observations in the training set.
  • Combine these vectors into an n × L matrix Z (the "level-one" data).

• Metalearning for AUC Maximization:

  • The metalearning algorithm Φ is applied to the level-one data Z and the true outcome vector Y.
  • The objective is to find the weight vector α = (α₁,..., αₗ) that minimizes the rank loss (1 - AUC). This is a nonlinear optimization problem.
  • In practice, use an AUC-maximizing metalearner (e.g., as implemented in the SuperLearner R package) to solve for α.

• Final Ensemble Training:

  • Train each base learner ψₗ on the entire training set to obtain {ψ̂₁,..., ψ̂ₗ}.
  • The final Super Learner prediction for a new input X is the metalearner combination: f(X) = Σ αₗ ψ̂ₗ(X).

Protocol 2: Threshold-Independent AUC Evaluation

This protocol outlines the standard procedure for calculating and interpreting the AUC-ROC, a threshold-independent metric [61].

• Probability Score Generation: For a given model (base or ensemble), obtain the predicted probability (or score) for the positive class (e.g., "stable") on a test set.

• ROC Curve Construction:

  • Vary the classification threshold from 0 to 1.
  • For each threshold, calculate the True Positive Rate (TPR/Sensitivity) and False Positive Rate (FPR/1-Specificity).
  • Plot TPR (y-axis) against FPR (x-axis). The resulting curve is the ROC curve.

• AUC Calculation:

  • Calculate the Area Under this ROC Curve (AUC).
  • The AUC can be computed numerically using the trapezoidal rule or libraries like scikit-learn's roc_auc_score [61].
  • Interpretation:
    • AUC = 1.0: Perfect class separation.
    • AUC = 0.5: Performance equivalent to random guessing.
    • AUC < 0.5: Performance worse than random guessing.
    • AUC 0.7-0.8: Reasonable for baseline models.
    • AUC > 0.9: High performance, often required for high-stakes applications like fraud or medical diagnosis [61].

Protocol 3: Evaluation under Class Imbalance using PR-AUC

For imbalanced datasets (e.g., rare stable materials), the Precision-Recall AUC (PR-AUC) is often more informative than ROC-AUC [61].

• Scenario Identification: Switch to PR-AUC when the positive class frequency falls below roughly 10% [61].

• PR Curve Construction:

  • Vary the classification threshold from 0 to 1.
  • For each threshold, calculate Precision (Positive Predictive Value) and Recall (Sensitivity).
  • Plot Precision (y-axis) against Recall (x-axis).

• PR-AUC Calculation:

  • Calculate the Area Under the Precision-Recall Curve.
  • Implementation can be done using sklearn.metrics.precision_recall_curve and auc functions [61].

Workflow Visualization

The following diagram illustrates the logical workflow for benchmarking ensemble models using AUC-maximizing strategies, as detailed in the experimental protocols.

The Scientist's Toolkit: Research Reagent Solutions

Table 3: Essential Computational Tools for Ensemble AUC Benchmarking

Tool / Resource	Type	Function in Protocol	Example/Reference
SuperLearner R Package	Software Library	Implements the Super Learner ensemble algorithm with various metalearning options, including AUC maximization.	[58]
Scikit-learn (Python)	Software Library	Provides production-ready functions for metric calculation (e.g., roc_auc_score, precision_recall_curve).	[61]
AUC-Maximizing Metalearner	Algorithm	The core algorithm used in the metalearning step to directly optimize the ensemble weights for AUC.	[58] [57]
Domain-Specific Base Models	Model Architecture	Base learners that incorporate distinct domain knowledge (e.g., electron configuration, atomic graphs, elemental properties).	ECCNN, Roost, Magpie [1]
Structured Materials Database	Data Resource	Provides labeled data (stable/unstable compounds) for training and benchmarking.	JARVIS, Materials Project (MP) [1]

The discovery of new functional materials, such as those for optoelectronics, thermoelectrics, or catalysis, is often gated by the challenge of confirming their thermodynamic stability. A compound that is not thermodynamically stable may be difficult or impossible to synthesize. The integration of ensemble machine learning (ML) models with first-principles calculations has created a powerful pipeline for accelerating this discovery process. Ensemble ML, particularly models based on electron configurations like the Electron Configuration models with Stacked Generalization (ECSG), can rapidly screen vast compositional spaces to identify promising candidate materials [1]. However, these predictions require rigorous validation. This protocol details the application of first-principles calculations, primarily based on Density Functional Theory (DFT), to confirm the structural, electronic, and thermodynamic stability of compounds identified by ensemble ML models, forming the critical experimental bridge in a computational materials discovery workflow.

The validation process is a multi-stage sequence that begins with the output from an ensemble ML screening. The diagram below outlines the complete workflow from initial candidate selection to final stability confirmation.

Core Computational Protocols

Protocol for Structural Optimization and Stability

Objective: To determine the most stable crystal structure and its lattice parameters for a given composition, confirming its structural integrity.

Detailed Methodology:

• Software Selection: Employ a robust DFT package. Common choices include:
  • WIEN2k: Uses the full-potential linearized augmented plane wave (FP-LAPW) method, known for high accuracy [62].
  • VASP: Utilizes the projector-augmented wave (PAW) method, offering a good balance of accuracy and efficiency [56].
  • CASTEP: An alternative plane-wave pseudopotential code [63].
• Initial Structure Modeling: For completely novel compounds with no known structure, use crystal structure prediction algorithms or adopt a known prototype structure (e.g., α-NaFeO₂-type for thallium rare-earth selenides [62]).
• Exchange-Correlation Functional: Select an appropriate functional. The Perdew-Burke-Ernzerhof (PBE) Generalized Gradient Approximation (GGA) is a common starting point [63] [56]. For more accurate band gaps, meta-GGA functionals like TB-mBJ are recommended [62].
• Geometry Optimization: Use an algorithm like BFGS to relax the atomic positions and lattice parameters until the following convergence criteria are met [63]:
  • Total energy change per atom: < 1.0 × 10⁻⁶ eV
  • Hellmann-Feynman force on each atom: < 0.03 eV/Å
  • Stress on the unit cell: < 0.05 GPa

Key Outputs:

• Equilibrium lattice parameters (a, b, c, α, β, γ).
• Equilibrium volume (V₀) and bulk modulus.
• Total energy of the compound in its ground-state structure.

Protocol for Electronic Structure Analysis

Objective: To characterize the electronic properties of the compound, which are crucial for determining its potential applications (e.g., as a semiconductor or metal).

Detailed Methodology:

• Self-Consistent Field (SCF) Calculation: Perform a high-precision SCF calculation on the optimized structure using a dense k-point mesh for Brillouin zone integration (e.g., 5 × 5 × 8 for hexagonal systems [63]).
• Band Structure and Density of States (DOS):
  • Calculate the electronic band structure along high-symmetry paths in the Brillouin zone.
  • Calculate the total and partial density of states (PDOS) to identify contributions from specific atomic orbitals.
• Functional Selection for Accuracy: As demonstrated in the study of Tl(Nd/Gd/Tb)Se₂, different functionals (LDA, PBE-GGA, TB-mBJ) can predict varying electronic behaviors (e.g., half-metallic vs. semiconducting) [62]. The TB-mBJ functional is widely regarded as more reliable for bandgap estimation due to its better agreement with experimental data [62].

Key Outputs:

• Electronic band structure plot.
• Density of States (DOS) and Partial DOS (PDOS) profiles.
• Value of the electronic band gap (if any).

Protocol for Thermodynamic Stability (Convex Hull Construction)

Objective: To definitively confirm the thermodynamic stability of the compound with respect to decomposition into other phases in its chemical space.

Detailed Methodology:

• Formation Energy Calculation: Calculate the formation energy (ΔH(f)) of the compound using the formula: *Δ*H*(f) = E(total) - Σ*n*(i)E(i) where *E*(total) is the total energy of the compound, and n(i) and *E*(i) are the number of atoms and reference energy of element i, respectively.
• Identify Competing Phases: Compile a list of all known compounds in the relevant chemical system (e.g., A-B-C) from databases like the Materials Project (MP) or Open Quantum Materials Database (OQMD) [1] [64].
• Calculate Competitor Energies: Perform DFT calculations to obtain the formation energies of all competing phases, or retrieve them from validated databases.
• Construct the Convex Hull: Plot the formation energies of all compounds as a function of composition. The convex hull is the set of points with the lowest formation energy at any given composition.
• Stability Assessment: A compound is considered thermodynamically stable if its formation energy lies on the convex hull. If its energy lies above the hull, the energy difference (ΔH(_d), decomposition energy) indicates its instability [1] [64].

Key Outputs:

• Formation energy (ΔH(_f)) of the target compound.
• Convex hull diagram for the chemical system.
• Decomposition energy (ΔH(_d)), if applicable.

Data Presentation and Analysis

The following tables summarize key quantitative data and computational parameters from cited studies and this protocol.

Table 1: Electronic Property Predictions for TlRESe₂ Compounds Using Different DFT Functionals [62]

Compound	LDA/PBEsol/WC	PBE-GGA	TB-mBJ
TlNdSe₂	Half-metallic	Semiconducting	Semiconducting
TlGdSe₂	Semiconducting	Semiconducting	Semiconducting
TlTbSe₂	Half-metallic (spin-down)	Half-metallic (spin-down)	Semiconducting

Table 2: Key Parameters for DFT Validation Calculations

Calculation Step	Key Parameter	Typical Value / Setting	Purpose
Structural Optimization	Energy Convergence	< 1.0 × 10⁻⁶ eV/atom [63]	Ensure ground state is reached
	Force Convergence	< 0.03 eV/Å [63]	Ensure atomic forces are minimized
	k-point Mesh	System-dependent (e.g., 5x5x8 [63])	Sample Brillouin zone accurately
Electronic Structure	Cut-off Energy	500 eV [63] / 520 eV [56]	Basis set completeness for plane waves
	Functional	PBE-GGA, TB-mBJ [62]	Describe electron exchange & correlation

The Scientist's Toolkit: Essential Research Reagents

This section details the essential computational "reagents" and tools required to execute the validation protocols described above.

Table 3: Key Research Reagent Solutions for First-Principles Validation

Item Name	Function / Description	Example Packages
DFT Software Package	Core engine for performing electronic structure calculations and determining total energies, forces, and electronic properties.	WIEN2k [62], VASP [56], CASTEP [63], Quantum ESPRESSO
Exchange-Correlation Functional	A critical approximation to describe quantum mechanical interactions between electrons; choice impacts accuracy of results.	PBE-GGA [63] [56], TB-mBJ [62], LDA
Pseudopotential / PAW Dataset	Represents the core electrons and nucleus, allowing use of a plane-wave basis set for valence electrons; improves computational efficiency.	PAW Potentials [56], Ultrasoft Pseudopotentials [63]
Materials Database	Source of crystal structures and reference data for competing phases, essential for convex hull construction.	Materials Project (MP) [1] [64], Open Quantum Materials Database (OQMD) [64]
Machine Learning Framework	For the initial high-throughput screening of candidate materials based on composition and electron configuration.	ECSG Framework [1]

Visualization of the DFT Validation Logic

The decision process for validating a novel compound's stability, following the protocols in Section 3, is summarized in the logic diagram below.

In ensemble machine learning for scientific discovery, the choice of featurization method—how molecular or material structures are translated into numerical descriptors—is a critical determinant of model performance and interpretability. This analysis examines two prominent strategies: featurization based on fundamental electron configuration (EC) and the use of hand-crafted custom descriptors. Electron configuration describes the distribution of electrons in atomic orbitals, providing a first-principles representation of atoms within a compound [65]. In contrast, custom descriptors often encompass a suite of engineered features derived from domain knowledge, such as statistical aggregates of elemental properties [1]. Within the context of predicting thermodynamic stability—a key property for materials design and drug development—this document provides a detailed comparison of these approaches, supported by quantitative data and executable protocols for the research community.

Theoretical Background & Key Concepts

Electron Configuration (EC) as a Feature

The electron configuration of an element delineates the arrangement of its electrons within atomic orbitals (e.g., 1s², 2s², 2p⁴) [65]. When used for machine learning, this intrinsic atomic property is encoded to represent a material's composition. The underlying hypothesis is that the EC fundamentally governs an atom's chemical behavior, including its bonding and stability, thereby serving as a feature with low inductive bias. Recent research has successfully leveraged EC convolutions to predict the thermodynamic stability of inorganic compounds with high accuracy [1].

Custom Descriptors

Custom descriptors are human-engineered features grounded in specific domain knowledge. In materials science and chemistry, a common set of custom descriptors is the Magpie feature set. Magpie generates statistical summaries (mean, range, mode, etc.) of a wide array of elemental properties (e.g., atomic number, mass, radius, electronegativity) for a given compound [1]. Other custom approaches may involve graph-based representations that model a chemical formula as a network of atoms to capture interatomic interactions [1]. The quality and completeness of the domain knowledge used to create these descriptors directly influence model performance.

The Ensemble Approach and Stacked Generalization

Ensemble methods combine multiple machine learning models to achieve superior predictive performance and robustness compared to any single model. Stacked Generalization (SG) is an advanced ensemble technique where the predictions of several base-level models (e.g., models trained on EC, Magpie, and graph-based features) are used as input features to train a meta-learner [1]. This framework allows the super learner to synergistically integrate the strengths of diverse featurization strategies, mitigating the individual biases inherent in each approach [1].

Quantitative Comparative Analysis

The following tables summarize the core characteristics and performance metrics of the two featurization methods, drawing from research on thermodynamic stability prediction.

Table 1: Characteristics of Featurization Methods

Feature	Electron Configuration (EC)	Custom Descriptors (e.g., Magpie)
Theoretical Basis	First-principles quantum mechanics [65]	Empirical domain knowledge & heuristics [1]
Information Scale	Atomic/Electronic structure	Atomic & interatomic properties [1]
Primary Advantage	Low inductive bias; intrinsic atomic property [1]	Direct encoding of known, relevant properties [1]
Primary Challenge	May require complex encoding for ML	Potential for large inductive bias if domain knowledge is incomplete [1]
Representative Model	ECCNN (Electron Configuration CNN) [1]	Magpie (statistical features with XGBoost) [1]

Table 2: Performance in Thermodynamic Stability Prediction

Metric	ECCNN (EC-based) [1]	Magpie (Custom Descriptors) [1]	ECSG (Ensemble) [1]
AUC (Area Under the Curve)	Reported as part of ensemble	Reported as part of ensemble	0.988
Sample Efficiency	High (Achieves target performance with ~1/7 of the data required by other models) [1]	Lower than ECCNN	High (Inherits efficiency from base models like ECCNN)
Key Strength	Captures fundamental electronic structure	Provides statistically aggregated elemental trends	Mitigates individual model bias; leverages synergy

Experimental Protocols

Protocol A: Training an Electron Configuration CNN (ECCNN) Model

This protocol details the process for developing a convolutional neural network using electron configuration features for stability prediction.

I. Research Reagent Solutions

Item	Function/Specification
JARVIS/MP/OQMD Database	Source of labeled data (formation energies, stability labels) [1].
EC Encoder	Custom script to convert material composition into a 118 (elements) x 168 (features) x 8 (channels) tensor representing electron configurations [1].
Deep Learning Framework	TensorFlow or PyTorch for building and training the CNN.
High-Performance Computing (HPC) Cluster	For handling intensive CNN training computations.

II. Step-by-Step Procedure

• Data Acquisition and Preprocessing:
  • Download the dataset of inorganic compounds and their corresponding decomposition energies (ΔH_d) or stable/unstable labels from a database like the Materials Project (MP) or JARVIS [1].
  • Clean the data, handling missing values and ensuring a balanced representation if possible.
• Feature Engineering (EC Encoding):
  • For each element in the periodic table, compute its electron configuration.
  • For a given chemical formula, encode the overall material's electron configuration into a 3D tensor (118 x 168 x 8) as described in the source literature [1]. This serves as the input matrix for the CNN.
• Model Architecture and Training (ECCNN):
  • Input Layer: Accepts the (118, 168, 8) EC tensor.
  • Convolutional Layers: Implement two convolutional layers, each with 64 filters of size 5x5. Use a ReLU activation function.
  • Pooling and Normalization: After the second convolution, apply Batch Normalization (BN) followed by a 2x2 Max Pooling operation.
  • Fully Connected Layers: Flatten the output and connect to one or more dense layers for the final regression (ΔH_d) or classification (stable/unstable) output.
  • Compilation and Training: Compile the model with an appropriate optimizer (e.g., Adam) and loss function (e.g., Mean Squared Error for regression). Train the model on the prepared dataset, using a separate validation set to monitor for overfitting.
• Model Validation:
  • Evaluate the trained ECCNN model on a held-out test set. Report standard metrics such as AUC, accuracy, and Mean Absolute Error (MAE) for regression tasks.

Protocol B: Building a Custom Descriptor Model with Magpie and XGBoost

This protocol outlines the construction of a model using the Magpie descriptor set.

I. Research Reagent Solutions

Item	Function/Specification
pymatgen Library	Python library for materials analysis, often used to generate Magpie descriptors.
XGBoost Library	Scalable and optimized library for gradient boosting machines.
scikit-learn	Used for data splitting, preprocessing, and model evaluation.

II. Step-by-Step Procedure

• Data Acquisition: Follow the same data sourcing procedure as in Protocol A.
• Feature Engineering (Magpie Descriptors):
  • Using a library like pymatgen, generate the Magpie feature vector for each compound in your dataset. This will create a vector of statistical features (mean, deviation, range, etc.) derived from a list of elemental properties [1].
• Data Preparation for ML:
  • Split the dataset into training and test sets (e.g., 80/20 split). It is critical to hold out a test set that the model never sees during training to obtain an unbiased performance estimate [66].
  • Standardize the feature values (e.g., scale to zero mean and unit variance) using the StandardScaler from scikit-learn.
• Model Training (XGBoost):
  • Train an XGBoost regressor or classifier on the training data. Initially, use default hyperparameters to establish a baseline performance [66].
• Model Validation:
  • Use the trained model to make predictions on the test set. Calculate performance metrics (e.g., R² score for regression) and visualize the results, for instance, by plotting predicted versus true values to identify any systematic errors [66].

Protocol C: Implementing a Stacked Generalization Ensemble (ECSG)

This protocol describes integrating multiple featurization methods into a super learner.

I. Research Reagent Solutions

• All items listed in Protocols A and B.
• A meta-learning algorithm (e.g., Linear Regression, Logistic Regression, or another XGBoost model) for the final predictor.

II. Step-by-Step Procedure

• Train Base-Level Models: Independently train at least three distinct base models on the same training set. As per the ECSG framework, these should include:
  • ECCNN: The electron configuration-based model from Protocol A.
  • Roost: A graph-based model representing interatomic interactions [1].
  • Magpie+XGBoost: The custom descriptor model from Protocol B [1].
• Generate Base-Level Predictions: Use each trained base model to make predictions on a validation set (or use k-fold cross-validation on the training set to generate out-of-fold predictions). These predictions form the new feature set for the meta-learner.
• Train the Meta-Learner: Train a relatively simple model (the meta-learner) using the base models' predictions as input features and the original target values as the output.
• Final Evaluation and Inference: To make a prediction for a new compound, pass it through all base models to get their predictions, then feed these predictions into the trained meta-learner to obtain the final, ensemble prediction.

Workflow and Relationship Visualization

The following diagram illustrates the logical workflow and integration of the different featurization methods within the stacked generalization ensemble, as implemented in Protocol C.

Diagram 1: ECSG Ensemble Architecture. The workflow shows how material composition is processed by three distinct base models, each using a different featurization strategy. Their predictions are combined into a meta-feature vector used to train a meta-learner, which produces the final, refined stability prediction [1].

The comparative analysis reveals that electron configuration featurization and custom descriptors offer complementary strengths. The ECCNN approach, rooted in fundamental physics, demonstrates remarkable sample efficiency, achieving high accuracy with significantly less data [1]. This makes it particularly valuable for exploring new chemical spaces where data is scarce. In contrast, custom descriptors like Magpie provide a robust, interpretable framework built on well-established elemental trends.

For the critical task of predicting thermodynamic stability—a gateway property for discovering new materials and optimizing molecular compounds—the ensemble approach (ECSG) proves superior. By integrating models based on electron configuration, graph networks, and custom descriptors, the ensemble effectively mitigates the inductive biases of any single method, resulting in state-of-the-art predictive accuracy (AUC = 0.988) [1]. This synergistic framework provides a powerful and efficient strategy for accelerating the discovery of stable compounds, from novel inorganic materials to potential drug candidates, by intelligently navigating vast compositional spaces.

Accurately predicting thermodynamic stability is a cornerstone in the design of novel functional materials, from structural alloys to next-generation optoelectronic compounds. This application note presents a dual case study validation within a broader thesis on ensemble machine learning electron configuration thermodynamic stability research. We examine the application of advanced computational protocols to two distinct material classes: the binary Ti-N system, relevant for hard coatings and aerospace applications, and lead-free perovskites (LFPs), which are emerging as sustainable alternatives in photovoltaics. By comparing and contrasting the methodologies, performance metrics, and specific challenges associated with predicting stability in these systems, this note provides a validated framework and practical tools for researchers and scientists engaged in computational materials discovery and drug development where molecular stability is paramount.

The following tables consolidate key quantitative findings from the validated case studies, highlighting the performance of different predictive models and the resulting material properties.

Table 1: Performance Metrics of Stability Prediction Models in Ti-N and Perovskite Systems

Material System	Prediction Method	Key Performance Metric	Value	Reference
Ti-N System	Moment Tensor Potential (MTP)	Formation Energy RMSE (Training)	2.1 meV/atom	[56]
Ti-N System	Moment Tensor Potential (MTP)	Formation Energy RMSE (Testing)	6.8 meV/atom	[56]
Ti-N System	Moment Tensor Potential (MTP)	Max. Deviation from Convex Hull (0K)	10 meV/atom	[56]
Complex Concentrated Alloys (CCAs)	Histogram Gradient Boosting Classifier	Phase Prediction Accuracy (Thermodynamic)	85.0%	[67]
Complex Concentrated Alloys (CCAs)	Gradient Boosting Classifier	Phase Prediction Accuracy (Composition)	82.3%	[67]
General Compositional Models	Density Functional Theory (DFT)	Mean Absolute Deviation of ΔHf	~0.1 eV/atom	[68]
General Compositional Models	Machine-Learned Formation Energies	Stability Prediction Error (ΔHd)	~0.1 eV/atom	[68]

Table 2: Computed Properties of Validated Lead-Free Perovskite Compounds

Material Composition	Band Gap (eV)	Elastic Constant / Mechanical Property	Stability Assessment	Reference
K₂AgSbBr₆ (Pristine)	0.554 (PBE)	Bulk Modulus: 24.34 GPa	Dynamically & mechanically stable	[69]
K₂CuSbBr₆ (Cu⁺ Doped)	0.444 (PBE)	Bulk Modulus: 24.93 GPa	Dynamically & mechanically stable	[69]
K₂AgBiBr₆ (Bi³⁺ Doped)	1.547 (PBE)	Bulk Modulus: 21.81 GPa	Dynamically & mechanically stable	[69]
Mg₃BiI₃	0.867 (HSE06)	Ductile	Mechanically stable	[70]
Mg₃BiBr₃	1.626 (HSE06)	Brittle	Mechanically stable	[70]
Mg₃NF₃	6.789 (HSE06)	Brittle	Mechanically stable	[70]
RbSnF₃ (Pristine)	1.748 (PBE)	N/A	Stable cubic phase	[71]
RbSnF₃ (In Doped)	1.192 (PBE)	N/A	Stable cubic phase	[71]

Experimental & Computational Protocols

Protocol 1: Interatomic Potential Development for Ti-N Systems

This protocol outlines the procedure for developing and validating a Moment Tensor Potential (MTP) for predicting the thermodynamic stability and mechanical properties of a binary system like Ti-N [56].

• Objective: To create a machine-learned interatomic potential that reliably predicts formation energies and elastic constants across various stoichiometries (e.g., Ti₂N, Ti₃N₂, TiN) beyond the limitations of traditional potentials.
• Step-by-Step Workflow:
  • DFT Dataset Generation:
    • Perform first-principles calculations using software like VASP.
    • Employ the PBE-GGA exchange-correlation functional and the Projector Augmented-Wave (PAW) method [56].
    • Use a high plane-wave kinetic energy cutoff (e.g., 520 eV) and a Monkhorst-Pack k-point mesh with a resolution finer than (2π⋅0.03 Å⁻¹) for Brillouin zone sampling [56].
    • Relax structures of known and predicted Ti-N compounds (Ti, Ti₂N, Ti₃N₂, TiN, etc.) until total energies converge to within 10⁻⁹ eV/atom [56].
    • Compute elastic constants for select phases to serve as benchmarks.
  • Training Dataset Curation:
    • Carefully select a training set that encompasses the structural diversity of the Ti-N system, including similar and dissimilar phases, to ensure the potential's robustness and transferability [56].
  • MTP Parameterization:
    • Configure the MTP with a maximum tensor level (levmax) of 22 and a radial cut-off distance between 2.0 and 7.0 Å to balance computational efficiency and descriptive power [56].
    • Utilize 421 basis functions and 2621 moment tensor descriptors to represent the atomic environments [56].
  • Validation and Testing:
    • Validate the trained MTP by comparing its predictions of formation energy against a held-out testing dataset from DFT.
    • Calculate the energy above the convex hull for various compositions to assess thermodynamic stability. A maximum deviation of 10 meV/atom from the hull is considered acceptable [56].
    • Compare MTP-predicted elastic constants against DFT benchmarks to validate mechanical property predictions [56].

Protocol 2: Stability Screening for Lead-Free Perovskites

This protocol describes a combined DFT and machine learning approach for high-throughput stability screening of lead-free perovskite compositions.

• Objective: To efficiently identify stable, synthesizable lead-free perovskite compositions with target electronic properties for optoelectronic applications.
• Step-by-Step Workflow:
  • Initial Compositional Screening:
    • Calculate the Goldschmidt tolerance factor (τ) for candidate compositions ABX₃ or A₂BB'X₆. Compositions with τ between ~0.78 and 1.10 are more likely to form stable perovskite structures [71].
  • First-Principles Stability Assessment:
    • Perform structural optimization using DFT codes such as Quantum ESPRESSO or CASTEP.
    • Use the GGA-PBE functional for initial relaxations. For more accurate electronic properties, employ hybrid functionals like HSE06 [69] [70].
    • Compute the formation energy ΔHf of the compound from its constituent elements.
    • Construct the compositional convex hull by calculating ΔHf for all competing phases in the chemical space. The decomposition enthalpy ΔHd is the key metric, defined as the energy difference between the compound and the convex hull. Compounds with ΔHd ≤ 0 are thermodynamically stable [68].
  • Dynamic and Mechanical Stability Checks:
    • Perform phonon dispersion calculations using Density Functional Perturbation Theory (DFPT) to ensure no imaginary frequencies (dynamic stability) [70].
    • Calculate the elastic tensor. Verify that the Born-Huang criteria for mechanical stability are satisfied for the crystal structure [69] [70].
  • Machine Learning for Accelerated Discovery:
    • For large-scale screening, train machine learning models on existing DFT databases.
    • Use compositional or structural features. Note that purely compositional models may predict formation energy well but can perform poorly on stability (ΔHd) due to the subtle energy differences involved [68].
    • Ensemble methods like the Histogram Gradient Boosting Classifier have shown high accuracy (>84%) for phase prediction in complex alloys and can be adapted for perovskites [67].

Workflow Visualization

The following diagram illustrates the integrated computational workflow for material stability prediction, as applied in the featured case studies.

Integrated Workflow for Stability Prediction

The Scientist's Toolkit: Research Reagent Solutions

This section details essential computational tools and their functions in stability prediction workflows.

Table 3: Essential Computational Tools for Stability Prediction

Tool / Resource	Type	Primary Function in Stability Research
VASP	Software Package	First-principles DFT calculations for energy, force, and electronic structure analysis [56].
Quantum ESPRESSO	Software Package	An open-source suite for DFT simulations, using plane-wave basis sets and pseudopotentials [70] [71].
Moment Tensor Potential (MTP)	Machine Learning Interatomic Potential	A fast and accurate ML-based potential for molecular dynamics and property prediction [56].
Histogram Gradient Boosting Classifier	Machine Learning Algorithm	An ensemble learning model effective for classifying material phases from compositional or thermodynamic data [67].
HSE06 Functional	Computational Method	A hybrid exchange-correlation functional in DFT that provides more accurate electronic band gaps than GGA-PBE [69] [70].
Materials Project Database	Online Database	A repository of DFT-calculated data for known and predicted compounds, used for training ML models and constructing convex hulls [68].

Within the field of materials science and drug development, accurately predicting thermodynamic stability is a fundamental challenge. Traditional approaches, particularly Density Functional Theory (DFT), have served as a computational cornerstone for determining electronic structures and energies of molecules and solids [8]. However, the substantial computational cost of DFT, which scales with system size and desired accuracy, poses a significant bottleneck for high-throughput screening [1]. This application note details the profound speed and resource advantages offered by modern ensemble machine learning (ML) models over pure DFT, specifically within the context of research focused on predicting thermodynamic stability from electron configuration.

Computational Burden of Traditional DFT

Fundamental Principles and Scaling

Density Functional Theory bypasses the need to solve the complex many-electron wavefunction by using the electron density as the fundamental variable, as established by the Hohenberg-Kohn theorems [72] [8]. The widely used Kohn-Sham scheme introduces a system of non-interacting electrons whose density matches that of the real system. The total energy is expressed as:

[ E[\rho] = Ts[\rho] + V{\text{ext}}[\rho] + J[\rho] + E_{\text{xc}}[\rho] ]

Here, ( E_{\text{xc}}[\rho] ) is the exchange-correlation functional, which encapsulates all non-trivial many-body effects. The accuracy of a DFT calculation critically depends on the approximation used for this unknown functional [72].

The computational cost of a DFT calculation is dominated by:

• Numerical Integration: The exchange-correlation functional must be evaluated numerically on a grid of points in space. The chosen grid size (e.g., UltraFine) dramatically impacts both accuracy and computational time [73].
• Basis Set Size: Expanding the Kohn-Sham orbitals in a basis set (e.g., def2-SVP, cc-pVTZ) is necessary. Larger basis sets provide better accuracy but exponentially increase the number of integrals to compute [74] [75].
• System Size: The cost typically scales as O(N³) with the number of electrons, making calculations for large systems (e.g., >100 atoms) or high-throughput searches across thousands of compounds prohibitively expensive [1].

The Functional Ladder and its Cost

The search for more accurate functionals has led to "Jacob's Ladder," a hierarchy of approximations climbing from local to more non-local descriptions [72] [74].

Table 1: Hierarchy of DFT Functionals and Associated Computational Cost

Functional Rung	Description	Example Functionals	Typical Relative Cost
LDA	Local Density Approximation; simplest form.	SVWN5	Low (Baseline)
GGA	Adds dependence on density gradient.	PBE, BLYP, BP86	Low - Medium
meta-GGA	Adds dependence on kinetic energy density.	TPSS, SCAN, M06-L	Medium
Hybrid	Mixes in a portion of exact Hartree-Fock exchange.	B3LYP, PBE0	High
Double-Hybrid	Includes a perturbative correlation correction.	B2PLYP	Very High

While climbing this ladder often improves accuracy, it comes at a significant and sometimes dramatic increase in computational cost. For instance, hybrid functionals like B3LYP require the construction of the exact exchange matrix, which scales poorly with system size [72] [75].

The Machine Learning Paradigm Shift

Core Concept and Workflow

Machine learning offers a paradigm shift from first-principles calculation to data-driven prediction. The core idea is to train statistical models on existing datasets of computed (e.g., DFT) or experimental properties. Once trained, these models can predict properties for new, unseen compounds almost instantaneously.

The following workflow diagram illustrates the complementary relationship between DFT and ML, and the integrated process of a sophisticated ensemble ML approach:

Figure 1: Integrated Workflow of DFT and Ensemble ML for Stability Prediction. The diagram shows how DFT-generated data trains ML models, which then provide fast predictions for new compounds.

Quantitative Advantages of Ensemble ML

The transition from pure DFT to ML is justified by staggering improvements in computational efficiency. Recent research demonstrates that an ensemble ML framework based on stacked generalization can achieve an Area Under the Curve (AUC) score of 0.988 in predicting compound stability. Most notably, this model required only one-seventh of the data used by existing models to achieve equivalent performance, highlighting its exceptional sample efficiency [1] [76].

Table 2: Quantitative Comparison of Computational Efficiency: Pure DFT vs. Ensemble ML

Metric	Pure DFT	Ensemble ML (ECSG)	Advantage
Time per Prediction	Minutes to Hours	Seconds	>100x Faster
Data Efficiency	Requires full SCF calculation per system	Achieves target accuracy with 1/7 the data [1]	~7x More Efficient
Scaling with System Size	O(N³) or worse	Near O(1) after training	Drastic improvement for high-throughput screening
Primary Resource Cost	CPU/GPU cycles per calculation	Initial data generation & model training	Shift from recurring to one-time cost

This efficiency enables the rapid screening of vast compositional spaces, a task that is economically unfeasible with pure DFT alone [1].

Detailed Protocols

Protocol 1: High-Throughput Stability Screening via Ensemble ML

This protocol uses the ECSG (Electron Configuration with Stacked Generalization) framework to efficiently predict the thermodynamic stability of inorganic compounds [1].

1. Data Curation and Input Encoding

• Source: Obtain training data from established materials databases (e.g., Materials Project, OQMD) containing computed formation energies ((\Delta H_d)) and convex hull stability data [1].
• Feature Engineering: Encode the chemical composition of each compound using multiple feature sets to reduce inductive bias:
  • ECCNN Input: Create a 118 (elements) × 168 × 8 matrix representing the electron configuration (EC) of the constituent atoms [1].
  • Magpie Features: Calculate statistical features (mean, deviation, range) for elemental properties like atomic number, radius, and electronegativity [1].
  • Roost Graph: Represent the chemical formula as a complete graph to model interatomic interactions [1].

2. Base-Model Training

• ECCNN: Implement a Convolutional Neural Network with two convolutional layers (64 filters, 5×5), batch normalization, max pooling, and fully connected layers. Train using the EC matrix as input [1].
• Magpie Model: Train a Gradient-Boosted Regression Tree (XGBoost) model using the statistical feature set [1].
• Roost Model: Train a Graph Neural Network with an attention mechanism on the graph representation of the chemical formulas [1].

3. Stacked Generalization (Super Learner)

• Use the predictions of the three base models (ECCNN, Magpie, Roost) as input features for a meta-learner model [1].
• Train this meta-model on the true stability labels to learn the optimal way to combine the base predictions, resulting in the final ECSG model [1].

4. Validation and Deployment

• Validate: Assess the final model on a held-out test set, targeting performance metrics like AUC >0.98 [1].
• Deploy: Use the trained ECSG model to predict the stability of new, unexplored compositions by simply inputting their elemental formula and electron configuration data [1].

Protocol 2: Benchmarking ML Predictions with Targeted DFT

This protocol validates ML predictions and obtains highly accurate energetics for a shortlist of promising candidates, balancing speed with reliability.

1. Generate Candidate List

• Apply Protocol 1 to screen a vast compositional space (e.g., thousands of compounds) for predicted stability.
• Select a shortlist (e.g., tens of compounds) of the most promising candidates based on ML-predicted stability and other desired properties [1] [77].

2. DFT Calculation Setup

• Software: Use a quantum chemistry package like Gaussian, ORCA, or a plane-wave code for solid-state systems.
• Functional & Basis Set:
  • For initial geometry optimizations, use a robust pure functional like BP86 or M06-L with a def2-SVP basis set for a good balance of speed and accuracy [74] [75].
  • For final single-point energies, use a hybrid functional like PBE0 or ωB97X-D with a larger triple-zeta basis set (e.g., def2-TZVP) and include empirical dispersion corrections (D3) for improved energetics [73] [74] [75].
• Numerical Grid: Use a fine integration grid (e.g., UltraFine in Gaussian) to ensure numerical accuracy [73].

3. Thermodynamic Stability Assessment

• For each candidate, compute its total energy.
• Construct the convex hull for the relevant phase diagram using data from the Materials Project or by computing energies of all competing phases.
• Calculate the decomposition energy ((\Delta H_d)) to definitively determine thermodynamic stability [1].

The Scientist's Toolkit: Essential Research Reagents

This section details key computational "reagents" required to implement the protocols described above.

Table 3: Essential Tools and Datasets for Ensemble ML and DFT-Based Stability Research

Item Name	Type / Category	Function and Critical Notes
Materials Project (MP)	Database	Primary source for training data; provides computed structural and energetic properties for over 150,000 inorganic compounds [1].
ECCNN Feature Set	Software/Descriptor	Encodes the electron configuration of elements into a matrix format, providing intrinsic atomic-level information to the ML model [1].
Magpie	Software/Descriptor	Generates statistical features from elemental properties, providing composition-level domain knowledge to the ML ensemble [1].
Roost	Software/Descriptor	Represents a chemical formula as a graph, allowing the model to learn from interatomic interactions via a message-passing neural network [1].
Gaussian 16/09	Software Suite	Industry-standard software for molecular DFT calculations; offers a wide array of functionals, basis sets, and dispersion corrections [73].
def2-SVP / def2-TZVP	Basis Set	Family of efficient, modern Gaussian-type orbital basis sets. def2-SVP is recommended for optimizations, def2-TZVP for accurate energies [74] [75].
D3 Dispersion Correction	Software/Algorithm	Adds empirical van der Waals corrections to standard DFT functionals, crucial for accurate intermolecular and intramolecular dispersion interactions [73] [75].
UltraFine Integration Grid	Computational Parameter	A predefined numerical grid (e.g., in Gaussian) for evaluating the exchange-correlation functional; essential for production-level accuracy and recommended as a default [73].

The integration of ensemble machine learning models for high-throughput screening, complemented by targeted DFT validation, represents a transformative advancement in computational materials science and drug development. This synergistic approach leverages the exceptional speed and data efficiency of ML—which can perform predictions in seconds and achieve high accuracy with significantly less data—while retaining the quantitative precision of DFT for final validation. This powerful combination dramatically accelerates the discovery of new thermodynamically stable compounds and materials, enabling researchers to navigate vast chemical spaces with unprecedented efficiency and reliability.

Conclusion

The integration of ensemble machine learning with electron configuration data represents a significant leap forward in predicting thermodynamic stability. This approach successfully addresses key challenges such as model bias, data scarcity, and computational cost, as evidenced by its high accuracy and remarkable sample efficiency. The ECSG framework, which synergizes models based on electron configuration, atomic properties, and interatomic interactions, provides a robust and generalizable tool. For biomedical and clinical research, this methodology opens new avenues for the rapid in-silico screening of stable drug-like molecules and novel inorganic compounds with desired properties, potentially slashing years from development timelines. Future work should focus on expanding these models to dynamically disordered systems, integrating kinetic stability predictions, and creating user-friendly platforms to make this powerful technology accessible to a broader range of scientists in drug discovery and materials design.

References

• 1. Predicting thermodynamic stability of inorganic compounds ... [https://www.nature.com/articles/s41467-024-55525-y]
• 2. Quantitative and scalable approaches for measuring protein ... [https://pmc.ncbi.nlm.nih.gov/articles/PMC9337709/]
• 3. UChicago scientists discover thermodynamics-defying ... [https://news.uchicago.edu/story/uchicago-scientists-discover-thermodynamics-defying-materials-could-revolutionize-evs]
• 4. Thermodynamic stability and electronic structure properties ... [https://www.sciencedirect.com/science/article/abs/pii/S016943322100129X]
• 5. The thermodynamic pharma challenge: A cross-cutting ... [https://www.sciencedirect.com/science/article/pii/S0928098725001770]
• 6. Stability Toolkit for the Appraisal of Bio/Pharmaceuticals ... [https://www.mdpi.com/2673-4532/6/3/25]
• 7. Computational Modeling of Protein Stability: Quantitative ... [https://www.sciencedirect.com/science/article/pii/S0969212620301222]
• 8. Density functional theory - Wikipedia [https://en.wikipedia.org/wiki/Density_functional_theory]
• 9. quantum chemistry - When and Why does Density Functional Theory (DFT) fail? - Matter Modeling Stack Exchange [https://mattermodeling.stackexchange.com/questions/526/when-and-why-does-density-functional-theory-dft-fail]
• 10. Stability Oracle: a structure-based graph-transformer ... [https://pmc.ncbi.nlm.nih.gov/articles/PMC11266546/]
• 11. Electronic Configuration — ADF 2025.1 documentation [https://www.scm.com/doc/ADF/Input/Electronic_Configuration.html]
• 12. Electron configuration-based neural network model to predict ... [https://pubs.rsc.org/en/content/articlehtml/2020/ra/d0ra05873d]
• 13. Electron Configuration - Chemistry LibreTexts [https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Supplemental_Modules_(Physical_and_Theoretical_Chemistry)/Quantum_Mechanics/10%3A_Multi-electron_Atoms/Electron_Configuration]
• 14. Ensemble machine learning for predictive accuracy and ... [https://www.sciencedirect.com/science/article/abs/pii/S0013468624017183]
• 15. Ensemble Learning The 2025 Ultimate Guide Smarter AI Model [https://w3schools.cloud/master-ensemble-learning-the-2025-ultimate-guide/]
• 16. Master Ensemble Methods in Machine Learning for 2025 [https://bigfootdigitalmarketingllc.com/bigfootagencymarketingnews/choosing-the-best-ensemble-method-bagging-boosting-or-stacking-for-2025]
• 17. Subsampled Ensemble Can Improve Generalization Tail ... [https://openreview.net/forum?id=VJn4UKnEYS]
• 18. Bias recognition and mitigation strategies in artificial ... [https://www.nature.com/articles/s41746-025-01503-7]
• 19. Stacked Generalization: An Introduction to Super Learning [https://pmc.ncbi.nlm.nih.gov/articles/PMC6089257/]
• 20. An interpretable stacking ensemble learning framework ... [https://www.frontiersin.org/journals/pharmacology/articles/10.3389/fphar.2022.975855/full]
• 21. Development and validation of machine learning models ... [https://bmcpsychiatry.biomedcentral.com/articles/10.1186/s12888-025-06549-1]
• 22. Lead-free perovskites for next-generation applications [https://pubs.rsc.org/en/content/articlehtml/2025/ma/d5ma00681c]
• 23. HaoZou-csu/ECSG [https://github.com/Haozou-csu/ECSG]
• 24. A chemically inspired convolutional neural network using ... [https://pubs.rsc.org/en/content/articlelanding/2023/ta/d3ta01767b]
• 25. Predicting thermodynamic stability of inorganic compounds ... [https://pmc.ncbi.nlm.nih.gov/articles/PMC11696921/]
• 26. 王建新 [https://faculty.csu.edu.cn/wangjianxin1/en/lwcg/106091/content/60479.htm]
• 27. Wafer-scale integration of two-dimensional perovskite ... [https://www.nature.com/articles/s41467-024-52840-2]
• 28. Band gap predictions of double perovskite oxides using ... [https://www.nature.com/articles/s43246-023-00373-4]
• 29. A review on research progress of double perovskite oxides for ... [https://pubs.rsc.org/en/content/articlehtml/2025/lf/d4lf00395k]
• 30. Direct band gap prediction of single and double perovskite ... [https://www.sciencedirect.com/science/article/abs/pii/S0925838825036631]
• 31. Electronic, structure, mechanical, optical, and ... [https://www.sciencedirect.com/science/article/abs/pii/S0022369724000945]
• 32. Universal machine learning aided synthesis approach ... [https://pmc.ncbi.nlm.nih.gov/articles/PMC10761762/]
• 33. Machine learning analysis of molecular dynamics ... [https://www.nature.com/articles/s41598-025-11392-1]
• 34. Machine Learning Decoding of Electronic Thresholds in ... [https://www.sciencedirect.com/science/article/abs/pii/S0304389425029814]
• 35. A universal machine learning model for the electronic ... [https://arxiv.org/html/2508.17418v1]
• 36. AI-powered prediction of critical properties and boiling points [https://jcheminf.biomedcentral.com/articles/10.1186/s13321-025-01062-9]
• 37. New machine-learning application to help researchers ... [https://news.mit.edu/2025/chemxploreml-app-helps-predict-chemical-properties-0724]
• 38. Active Learning in Machine Learning: Guide & Strategies ... [https://encord.com/blog/active-learning-machine-learning-guide/]
• 39. A comprehensive benchmark of active learning strategies ... [https://www.nature.com/articles/s41598-025-24613-4]
• 40. Enhancing composition - based materials property prediction by... [https://arxiv.org/html/2511.03371v1]
• 41. Prediction and Construction of Energetic Materials Based ... [https://pmc.ncbi.nlm.nih.gov/articles/PMC9821915/]
• 42. Small data machine learning in materials science [https://www.nature.com/articles/s41524-023-01000-z]
• 43. Targeted nano-energetic material exploration through ... [https://www.sciencedirect.com/science/article/pii/S2666647224000629]
• 44. Artificial intelligence approaches for materials-by-design of ... [https://arxiv.org/abs/2211.08179]
• 45. 18 SHAP – Interpretable Machine Learning [https://christophm.github.io/interpretable-ml-book/shap.html]
• 46. AI in Drug Discovery: The Role of SHAP in Pharmacology [https://www.reprocell.com/blog/biopta/ai-in-drug-discovery-the-role-of-shap-in-pharmacology]
• 47. 17 Shapley Values – Interpretable Machine Learning [https://christophm.github.io/interpretable-ml-book/shapley.html]
• 48. Practical guide to SHAP analysis: Explaining supervised ... [https://pmc.ncbi.nlm.nih.gov/articles/PMC11513550/]
• 49. The right way to compute your Shapley Values [https://towardsdatascience.com/the-right-way-to-compute-your-shapley-values-cfea30509254/]
• 50. Predicting learning achievement using ensemble learning ... [https://journals.plos.org/plosone/article?id=10.1371/journal.pone.0312124]
• 51. Student Dropout Prediction Using Ensemble Learning with ... [https://www.icck.org/article/abs/jsspa.2025.321501]
• 52. Development of several machine learning based models ... [https://www.nature.com/articles/s41598-025-13090-4]
• 53. An ensemble of deep representation learning with ... [https://www.nature.com/articles/s41598-025-15005-9]
• 54. Hyperparameter Optimisation in Deep Learning from ... [https://indico.in2p3.fr/event/33627/contributions/154604/]
• 55. Real-World Data and Causal Machine Learning to Enhance ... [https://pmc.ncbi.nlm.nih.gov/articles/PMC12579681/]
• 56. Prediction of Mechanical Properties and Thermodynamic ... [https://arxiv.org/html/2507.18873v1]
• 57. Ensemble Learning for AUC Maximization via Surrogate Loss [https://openreview.net/forum?id=kbxjkoF42x]
• 58. AUC-Maximizing Ensembles through Metalearning - PMC [https://pmc.ncbi.nlm.nih.gov/articles/PMC4912128/]
• 59. How to Check the Accuracy of Your Machine Learning ... [https://www.deepchecks.com/how-to-check-the-accuracy-of-your-machine-learning-model/]
• 60. 12 Important Model Evaluation Metrics for Machine ... [https://www.analyticsvidhya.com/blog/2019/08/11-important-model-evaluation-error-metrics/]
• 61. How the AUC Score Prevents AI Model Failures [https://galileo.ai/blog/auc-score]
• 62. First-principles calculations to investigate structural stability ... [https://www.sciencedirect.com/science/article/abs/pii/S3050471625000353]
• 63. First-principles study of the ground-state properties of ternary ... [https://pubs.rsc.org/en/content/articlehtml/2025/ma/d5ma00091b]
• 64. Review of computational approaches to predict the thermodynamic ... [https://link.springer.com/article/10.1007/s10853-022-06915-4]
• 65. Electronic Structure of Atoms (Electron Configurations) (3.4) [https://louis.pressbooks.pub/chemistry1/chapter/electronic-structure-of-atoms-electron-configurations/]
• 66. Machine Learning 101: How to train your first QSAR model [https://optibrium.com/knowledge-base/machine-learning-101-how-to-train-your-first-qsar-model-blog/]
• 67. Applicability of machine learning approaches for predicting ... [https://www.sciencedirect.com/science/article/pii/S294982282500766X]
• 68. A critical examination of compound stability predictions ... [https://www.nature.com/articles/s41524-020-00362-y]
• 69. Electronic and optical properties of lead-free K₂AgSbBr₆ ... [https://www.nature.com/articles/s41598-025-24417-6]
• 70. Newly predicted halide perovskites Mg 3 AB 3 (A = N, Bi; B = F ... [https://pubs.rsc.org/en/content/articlehtml/2025/ra/d4ra09093d]
• 71. Re-designing lead-free halide perovskite RbSnF₃ [https://link.springer.com/article/10.1007/s11082-025-08327-5]
• 72. The "Charlotte's Web" of Density-Functional Theory (DFT) [https://rowansci.com/publications/the-charlottes-web-of-dft]
• 73. Density Functional (DFT) Methods [https://gaussian.com/dft/]
• 74. DFT Functional Selection Criteria [https://chemistry.stackexchange.com/questions/27302/dft-functional-selection-criteria]
• 75. Best‐Practice DFT Protocols for Basic Molecular ... [https://pmc.ncbi.nlm.nih.gov/articles/PMC9826355/]
• 76. Predicting thermodynamic stability of inorganic compounds ... [https://pubmed.ncbi.nlm.nih.gov/39747064/]
• 77. Machine Learning Prediction of Thermodynamic Stability ... [https://chemrxiv.org/engage/chemrxiv/article-details/660b0e1ce9ebbb4db924b22d]