Graph Neural Networks for Interatomic Interactions: Advancing Stability Prediction in Drug Discovery and Materials Science

Abstract

This article explores the transformative role of Graph Neural Networks (GNNs) in predicting interatomic interactions and system stability, a critical challenge in computational chemistry and drug development. It covers the foundational principles of GNN architectures designed for molecular systems, details cutting-edge methodological advances and their applications in predicting drug-drug interactions and material properties, addresses key challenges in model stability and uncertainty quantification, and provides a comparative analysis of state-of-the-art models. Aimed at researchers and drug development professionals, this review synthesizes recent progress to guide the selection, application, and improvement of GNN-based models for robust and reliable atomic-scale simulations.

The Building Blocks: How GNNs Learn Interatomic Interactions and Molecular Representations

The accurate prediction of molecular properties and interatomic interactions is a cornerstone of modern computational chemistry and drug discovery. Graph Neural Networks (GNNs) have emerged as a powerful framework for this task, leveraging the inherent graph structure of molecules where atoms represent nodes and bonds represent edges. This paradigm allows GNNs to learn rich representations that capture complex chemical environments and interactions. The representation of molecular structures as graphs enables models to learn from large datasets and extrapolate to untrained geometries, providing remarkable predictive capabilities for properties ranging from quantum chemical energies to bioactivity and toxicity profiles [1].

This application note provides detailed protocols for representing molecular structures as graphs, with a specific focus on enabling stability prediction research through GNNs. We present standardized methodologies for graph construction, quantitative comparisons of representation schemes, and visualization techniques that facilitate model interpretability and deployment in real-world drug development pipelines.

Theoretical Foundations

Molecular Graph Theory

In molecular graph theory, a molecule is formally represented as a graph G = (V, E), where V is the set of vertices (atoms) and E is the set of edges (bonds). This representation preserves the topological connectivity of the molecule while abstracting away spatial coordinates, though geometric information can be incorporated as node and edge attributes. The graph structure naturally captures invariant molecular properties that are fundamental to chemical behavior and stability [1].

Graph neural networks operate directly on this structure through message-passing algorithms, where information is exchanged between connected nodes (atoms) across multiple layers. This enables the model to capture both local chemical environments and global molecular patterns. Recent theoretical work has shown that the message-passing mechanism in GNN interatomic potentials (GNN-IPs) allows them to capture non-local electrostatic interactions, explaining their remarkable extrapolation capability to untrained domains such as surfaces or amorphous configurations [1].

GNNs for Molecular Property Prediction

Multi-task learning (MTL) approaches have been developed to address data scarcity in molecular property prediction by leveraging correlations among related properties. However, imbalanced training datasets often degrade efficacy through negative transfer. Adaptive Checkpointing with Specialization (ACS) has been introduced as a training scheme for multi-task GNNs that mitigates detrimental inter-task interference while preserving MTL benefits [2]. ACS integrates a shared, task-agnostic backbone with task-specific trainable heads, adaptively checkpointing model parameters when negative transfer signals are detected. This approach has demonstrated practical utility in real-world scenarios, enabling accurate predictions with as few as 29 labeled samples for sustainable aviation fuel properties [2].

Application Notes

Molecular Graph Construction Protocol

Objective: Convert molecular structures into graph representations suitable for GNN processing.

Materials:

• Molecular structure files (SDF, MOL, XYZ, PDB formats)
• Python environment with RDKit or Open Babel
• NetworkX library for graph operations [3]

Procedure:

• Molecular Input: Load molecular structure using cheminformatics toolkit

• Node Creation: For each atom in the molecule:

  • Create a node with unique identifier
  • Add atom features as node attributes:
    • Atom type (element)
    • Formal charge
    • Hybridization state
    • Degree of connectivity
    • Aromaticity

• Edge Creation: For each bond in the molecule:

  • Create an edge between corresponding atom nodes
  • Add bond features as edge attributes:
    • Bond type (single, double, triple, aromatic)
    • Conjugation
    • Stereochemistry

• Graph Validation: Verify connectivity and feature consistency

• Graph Serialization: Export graph in compatible format (GraphML, DGL, PyG)

Troubleshooting:

• Aromaticity perception errors: Use Kekulization before feature extraction
• Stereochemistry preservation: Verify chiral flag in source file
• Charge representation: Check for neutralization during file conversion

Advanced Representation Schemes

Table 1: Molecular Graph Representation Schemes

Scheme	Node Features	Edge Features	Spatial Encoding	Best Use Case
Basic Graph	Element, Degree	Bond type, Conjugation	None	2D QSAR
3D-Aware Graph	Element, Hybridization	Bond type, Distance	Atomic coordinates	Conformation-dependent properties
Quantum Graph	Element, Partial charge	Bond order, Bond length	Wavefunction overlap	Reactivity prediction
Multi-Task Graph	Extended feature set	Multiple bond descriptors	Various	Low-data regimes [2]

Multi-Task Learning Implementation

Objective: Implement ACS for molecular property prediction in low-data regimes.

Background: ACS combines shared GNN backbone with task-specific heads, using adaptive checkpointing to mitigate negative transfer. Validation loss is monitored for each task, checkpointing the best backbone-head pair when a task reaches a new minimum [2].

Procedure:

• Architecture Setup:

  • Initialize shared GNN backbone (message-passing network)
  • Create task-specific MLP heads for each target property
  • Configure independent optimization paths

• Training Loop:

  • Forward pass through shared backbone
  • Task-specific processing through individual heads
  • Loss computation with masking for missing labels
  • Validation loss monitoring per task

• Checkpointing:

  • Track validation loss for each task independently
  • Save backbone-head parameters when task-specific minimum is achieved
  • Maintain best-performing specialized models for each task

• Inference:

  • Load specialized model for target property
  • Generate predictions using task-optimized parameters

Validation: Benchmark against single-task learning and conventional MTL on molecular property benchmarks (ClinTox, SIDER, Tox21) [2].

Visualization Methods

Basic Molecular Graph Diagram

Figure 1: Basic molecular graph with atom and bond types

GNN Message Passing Diagram

Figure 2: GNN message passing between atoms

ACS Training Scheme Diagram

Figure 3: ACS architecture with shared backbone and task-specific heads

Experimental Protocols

Benchmarking Protocol for Molecular Property Prediction

Objective: Evaluate GNN performance on standard molecular property prediction tasks.

Dataset Preparation:

• Source data from MoleculeNet benchmarks (ClinTox, SIDER, Tox21) [2]
• Apply Murcko scaffold splitting for realistic evaluation [2]
• Handle missing labels through loss masking [2]

Model Configuration:

• GNN architecture: Message-passing network with 4-6 layers
• Hidden dimension: 128-300 units
• Readout function: Global mean pooling or set2set
• Task heads: 2-layer MLPs with task-specific dimensions

Training Procedure:

• Optimization: Adam optimizer with learning rate 0.001
• Batch size: 32-128 depending on dataset size
• Early stopping: Based on validation performance
• Evaluation metric: ROC-AUC for classification, RMSE for regression

Validation: Compare against state-of-the-art baselines including D-MPNN and other supervised methods [2].

Ultra-Low Data Regime Protocol

Objective: Assess performance with minimal labeled data.

Procedure:

• Create artificially limited datasets (29-100 samples) [2]
• Apply ACS training scheme with adaptive checkpointing
• Compare against single-task and conventional MTL baselines
• Evaluate on real-world low-data scenario (sustainable aviation fuels)

Metrics:

• Prediction accuracy relative to experimental values
• Data efficiency: Performance vs. training set size
• Negative transfer mitigation: Task imbalance robustness [2]

Data Presentation

Table 2: Quantitative Performance Comparison on Molecular Property Benchmarks [2]

Method	ClinTox	SIDER	Tox21	Average	Parameters
STL (Single-Task)	0.823	0.635	0.758	0.739	Task-specific
MTL (Multi-Task)	0.845	0.642	0.769	0.752	Shared
MTL-GLC	0.847	0.645	0.772	0.755	Shared
ACS (Proposed)	0.876	0.649	0.775	0.767	Shared + Specialized

Table 3: Task Imbalance Analysis on ClinTox Dataset (Two Tasks) [2]

Imbalance Level	STL	MTL	MTL-GLC	ACS
Balanced (I=0)	0.823	0.845	0.847	0.876
Moderate (I=0.3)	0.801	0.832	0.838	0.861
Severe (I=0.6)	0.763	0.798	0.812	0.843
Extreme (I=0.9)	0.712	0.735	0.762	0.819

The Scientist's Toolkit

Table 4: Essential Research Reagents and Computational Tools

Tool/Reagent	Function	Application Context
RDKit	Cheminformatics toolkit	Molecular graph construction and feature calculation
NetworkX	Graph analysis and visualization	Graph manipulation and algorithm implementation [3]
PyTorch Geometric	GNN library	Implementation of graph neural network architectures
DGL-LifeSci	Domain-specific GNN tools	Pre-built models for molecular property prediction
Graphviz	Graph visualization	Generation of publication-quality diagrams [4]
ACS Training Scheme	Multi-task learning	Mitigating negative transfer in low-data regimes [2]
Message-Passing GNN	Core architecture	Learning representations from molecular graphs [1]
Adaptive Checkpointing	Training optimization	Preserving task-specific knowledge in MTL [2]

Graph Neural Networks (GNNs) have emerged as a transformative technology for modeling molecular systems, fundamentally shifting how researchers approach problems in drug discovery, materials science, and computational chemistry. The inherent graph structure of molecules—with atoms as nodes and bonds as edges—makes GNNs a natural fit for learning molecular representations. Early GNN models operated on simple graph structures with invariant features, but recent advances have incorporated geometric equivariance to account for the physical symmetries and 3D spatial relationships essential for accurate molecular property prediction.

The core learning pattern of most GNNs involves message passing, where each node aggregates feature information from its neighboring nodes to update its own representation. This process enables the network to capture complex molecular interactions and dependencies. However, traditional GNNs optimized for independent and identically distributed data often face performance degradation in real-world scenarios with out-of-distribution data, driving the development of more robust architectures that can handle distribution shifts commonly encountered in molecular systems.

Foundational GNN Architectures for Molecular Representation

Basic Architectural Framework

Most GNN architectures share a universal framework where each layer operates as a non-linear function of the form ( H^{(l+1)} = f(H^{(l)}, A) ), with ( H^{(0)} = X ) (node features) and ( H^{(L)} = Z ) (final node representations), where ( A ) represents the graph structure, typically as an adjacency matrix. The specific implementations differ primarily in how the function ( f(\cdot, \cdot) ) is designed and parameterized [5].

The Graph Convolutional Network (GCN) introduced one of the earliest and most influential propagation rules, using a layer-wise operation defined as ( f(H^{(l)}, A) = \sigma( \hat{D}^{-\frac{1}{2}}\hat{A}\hat{D}^{-\frac{1}{2}}H^{(l)}W^{(l)} ) ), where ( \hat{A} = A + I ) (adding self-loops), ( \hat{D} ) is the diagonal node degree matrix of ( \hat{A} ), and ( W^{(l)} ) is a layer-specific trainable weight matrix. This symmetric normalization ensures numerical stability while enabling effective feature propagation across graph neighborhoods [5].

Advanced Message-Passing Variants

Beyond basic graph convolutions, several specialized architectures have emerged with enhanced representational capabilities:

• Graph Attention Networks (GAT): Assign differential attention weights to neighbors during aggregation, allowing models to focus on more relevant molecular substructures [6].
• Graph Isomorphism Networks (GIN): Utilize a sum aggregator combined with multi-layer perceptrons to maximize representational capacity, theoretically as powerful as the Weisfeiler-Lehman graph isomorphism test [6].
• Message Passing Neural Networks (MPNNs): Provide a generalized framework that iteratively passes messages between neighboring nodes, with updates governed by learned neural network functions [6].

These foundational architectures excel at capturing topological relationships in molecular graphs but traditionally lack explicit mechanisms for incorporating 3D geometric information, which is crucial for accurately predicting many molecular properties.

The Critical Advancement: Geometric Equivariant GNNs

The Equivariance Principle in Molecular Systems

Geometric equivariance represents a paradigm shift in GNN design for molecular systems. While invariant models use features unchanged by transformations (e.g., bond lengths, angles), equivariant models maintain internal representations that transform consistently with input transformations. This property is essential for predicting molecular properties where directions matter, such as forces, dipole moments, and other vector-valued quantities [7] [8].

Formally, a function ( f: X \rightarrow Y ) is equivariant to a group ( G ) if for any transformation ( g \in G ), ( f(g \circ x) = g \circ f(x) ). In molecular systems, the relevant symmetry groups include SO(3) (rotations), SE(3) (rotations and translations), and E(3) (including reflections). Embedding these physical symmetries directly into network architectures—rather than applying constraints only to final outputs—has proven instrumental for achieving both data efficiency and prediction accuracy [8].

Key Equivariant Architectures and Their Innovations

Table 1: Comparison of Key Equivariant GNN Architectures for Molecular Systems

Architecture	Core Innovation	Representation Type	Key Advantages	Target Applications
E2GNN [7]	Scalar-vector dual representation	Scalar and vector features	Computational efficiency while maintaining equivariance	Interatomic potentials, force prediction
TEGNN [9]	Equivariant locally complete frames	Tensor information projection	Incorporates chemical bond constraints and higher-order tensors	Molecular dynamics prediction
NequIP [8]	Higher-order tensor representations	Spherical harmonics	State-of-the-art accuracy for complex systems	Interatomic potentials, material properties
MACE [8]	Atomic cluster expansion	Higher-order body order	High data efficiency and accuracy	Interatomic potentials
MagNet [8]	E(3)-equivariance for spins	Magnetic force vectors	Models magnetic materials with spin interactions	Magnetic force prediction

E2GNN (Efficient Equivariant GNN) addresses the computational challenges of equivariant models by employing a scalar-vector dual representation rather than relying on computationally expensive higher-order representations. The model maintains separate scalar features ( {{\bf{x}}}{i} ) and vector features ( {\overrightarrow{{\bf{x}}}}{i} ) for each node, updated through specialized geometric operations that preserve equivariance. This approach achieves significant efficiency improvements while maintaining high accuracy for interatomic potential and force predictions [7].

TEGNN (Tensor Improved Equivariant GNN) extends equivariant architectures to incorporate more sophisticated tensor information (relative position, velocity, torsion angles) while explicitly modeling chemical bonding constraints through generalized coordinates. The model employs a scalarization block that projects geometric tensors onto equivariant local frames, converting them into SO(3)-invariant scalar coefficients for message passing. This innovation allows TEGNN to leverage rich geometric information without the computational complexity of high-dimensional equivariant function embeddings [9].

Experimental Protocols and Benchmarking

Standardized Evaluation Frameworks

Robust evaluation is essential for comparing GNN architectures across molecular tasks. Standardized benchmarks have emerged using established datasets and metrics:

Table 2: Key Benchmark Datasets for Molecular GNN Evaluation

Dataset	Domain	Scale	Prediction Tasks	Key Metrics
QM9 [8] [10]	Small organic molecules	134k molecules	Quantum mechanical properties	MAE, RMSE
MD17 [9]	Molecular dynamics	3-4M configurations	Energy and forces	MAE, RMSE
ESOL [10]	Physical chemistry	1,128 molecules	Water solubility	RMSE, R²
FreeSolv [10]	Physical chemistry	642 molecules	Hydration free energy	RMSE, MAE
Lipophilicity [10]	Physical chemistry	4,200 molecules	Octanol/water distribution coefficient	RMSE, MAE

For regression tasks (energy, solubility, etc.), standard metrics include Mean Absolute Error (MAE), Root Mean Squared Error (RMSE), and correlation coefficients (R², Pearson's R). For classification tasks (toxicity, activity, etc.), common metrics include ROC-AUC, precision-recall curves (AUPRC), and F1-score [6] [10].

Implementation Protocol for Molecular Property Prediction

A standardized experimental protocol for molecular property prediction involves these key steps:

• Data Preparation and Splitting

  • Obtain molecular structures from sources like MoleculeNet or TUDatasets
  • Convert molecules to graph representations with nodes (atoms) and edges (bonds)
  • Implement dataset splitting (typically 80/10/10 train/validation/test) with scaffold splitting to assess generalization

• Model Configuration

  • For invariant GNNs (GCN, GAT, GIN): 3-6 message passing layers with hidden dimensions of 64-256
  • For equivariant GNNs (E2GNN, TEGNN): Implement geometric features with scalar-vector dimensions of 64-128
  • Use appropriate readout functions: global mean/max/sum pooling for graph-level predictions

• Training Procedure

  • Optimization: Adam optimizer with learning rate 0.001-0.0001
  • Regularization: Dropout (0.1-0.5), weight decay, and early stopping
  • Batch size: 32-128 depending on model complexity and memory constraints
  • Loss function: Mean squared error for regression, cross-entropy for classification

• Validation and Testing

  • Perform hyperparameter optimization on validation set
  • Report final performance on held-out test set with multiple random seeds
  • Conduct ablation studies to isolate contribution of architectural components

Architectural Visualizations

Molecular GNN Architecture Pathways - This diagram illustrates the parallel processing pathways for invariant and equivariant GNN architectures in molecular systems, from input structures to application outputs.

Equivariant GNN Message Passing - This workflow details the key operations in equivariant GNNs, highlighting the separate processing pathways for scalar and vector features and their interactions.

Table 3: Essential Computational Tools for Molecular GNN Research

Resource Category	Specific Tools & Libraries	Primary Function	Application Context
Deep Learning Frameworks	PyTorch, PyTorch Geometric, TensorFlow, JAX	Model implementation and training	General GNN development
Molecular Processing	RDKit, Open Babel, MDAnalysis	Molecular graph construction and featurization	Data preprocessing
Specialized GNN Libraries	DGL-LifeSci, MatterGen, e3nn	Domain-specific GNN implementations	Drug discovery, materials science
Benchmark Datasets	MoleculeNet, TUDatasets, OGB	Standardized evaluation datasets	Model benchmarking
Quantum Chemistry Data	QM9, MD17, ANI-1	High-quality reference data for training	Interatomic potential development

The evolution of GNN architectures for molecular systems has progressed from basic graph convolutional networks to sophisticated geometrically equivariant models that explicitly incorporate 3D structural information. This progression has enabled increasingly accurate predictions of molecular properties, energies, and forces at computational costs far below traditional quantum mechanical methods.

Future research directions include developing more data-efficient architectures that can learn from limited labeled data, improving out-of-distribution generalization through stable learning techniques [11], and creating more interpretable models that provide insights into molecular structure-property relationships. Additionally, the integration of large-scale pre-training approaches for molecular GNNs represents a promising avenue for developing foundation models in molecular sciences, potentially transforming the pace of discovery in drug development and materials design.

Message-passing mechanisms form the computational foundation of modern graph neural networks (GNNs) applied to molecular graphs, enabling the prediction of chemical properties, material behaviors, and bioactivities directly from structural information. In molecular contexts, where atoms naturally represent nodes and bonds represent edges, these mechanisms allow neural networks to learn from graph-structured data by iteratively exchanging information between connected entities [12]. Unlike traditional convolutional neural networks designed for grid-like data, message-passing GNNs specialize in handling the irregular connectivity patterns inherent to molecular systems, from simple organic compounds to complex crystalline structures [13] [12].

The significance of message-passing mechanisms extends beyond mere structural analysis to impactful applications in drug development and materials science. For molecular property prediction, message-passing neural networks (MPNNs) have demonstrated state-of-the-art performance in predicting quantum chemical properties, bioactivity, and physical-chemical characteristics without requiring hand-crafted feature engineering [14] [12]. This capability is particularly valuable in virtual screening and materials design, where accurate prediction of molecular behavior accelerates discovery while reducing experimental costs.

Theoretical Foundations of Message Passing

Core Mathematical Framework

Message passing in graph neural networks operates through three fundamental operations that transform node representations by aggregating information from local neighborhoods. For a molecular graph (G = (V, E)) with node features (hv) and edge features (e{vw}), the message-passing process at layer (t) can be formally described by the following equations [14] [12]:

[mv^{(t+1)} = \sum{w \in N(v)} Mt\left(hv^{(t)}, hw^{(t)}, e{vw}\right)]

[hv^{(t+1)} = Ut\left(hv^{(t)}, mv^{(t+1)}\right)]

[y = R\left({h_v^{(K)} \mid v \in G}\right)]

where:

• (M_t) is the message function that transforms neighbor information
• (U_t) is the update function that combines current node state with incoming messages
• (R) is the readout function that produces graph-level predictions
• (N(v)) denotes the neighbors of node (v)
• (K) is the total number of message-passing layers

This framework creates a powerful computational paradigm where each node progressively incorporates information from its extended neighborhood through multiple iterations, effectively capturing both local atomic environments and global molecular structure [15] [12].

Aggregation Functions and Their Properties

The choice of aggregation function significantly influences the expressive power and behavior of message-passing networks. Different functions offer distinct advantages for capturing various aspects of molecular structure:

Table 1: Comparison of Aggregation Functions in Message-Passing Networks

Aggregation Function	Mathematical Expression	Key Advantages	Molecular Applications
Sum	(mv = \sum{w \in N(v)} h_w)	Preserves complete neighborhood information	Counting specific substructures, molecular fingerprints
Mean	(m_v = \frac{1}{	N(v)	} \sum{w \in N(v)} hw)	Stable across neighborhoods of different sizes	Statistical properties, normalized features
Max	(mv = \max{w \in N(v)} h_w)	Identifies most salient features in neighborhood	Critical functional group detection
Attention-weighted	(mv = \sum{w \in N(v)} \alpha{vw} hw)	Adaptively weights neighbor importance	Complex bond interactions, protein-ligand binding

The attention mechanism, particularly implemented through multi-head attention, has demonstrated significant advantages in molecular applications by allowing the model to focus on particularly relevant atomic interactions while suppressing noise from less important connections [16] [14].

Figure 1: Message-Passing Architecture showing the flow of information from node and edge features through message functions, aggregation, and update functions to produce node and graph-level representations.

Implementation Protocols for Molecular Applications

Molecular Graph Construction

The first critical step in applying message-passing mechanisms to molecular systems is the construction of appropriate graph representations from chemical structure data:

Protocol 1: Molecular Graph Construction from SMILES

• Input Processing: Convert SMILES string to molecular structure using cheminformatics libraries (RDKit or OpenBabel)
• Node Representation: Generate atom features including:
  • Atom type (one-hot encoded)
  • Degree of connectivity
  • Formal charge
  • Hybridization state
  • Aromaticity flag
  • Atomic mass (normalized)
• Edge Representation: Generate bond features including:
  • Bond type (single, double, triple, aromatic)
  • Conjugation
  • Ring membership
  • Spatial distance (if 3D coordinates available)
• Graph Assembly: Construct adjacency matrix or adjacency list connecting atoms via bonds

For crystalline materials, additional considerations include periodic boundary conditions and longer-range interactions beyond covalent bonding, often addressed through multi-scale graph representations [16] [12].

Message-Passing Neural Network Implementation

Protocol 2: MPNN Forward Pass Implementation

• Initialization:
  • Initialize atom embeddings (hv^0) from feature vectors
  • Initialize bond embeddings (e{vw}) from bond features
  • Set number of message-passing steps (K) (typically 3-6 for molecules)

• Message-Passing Loop (for (t = 0) to (K-1)):

  • For each atom (v) in molecular graph:
    • Gather neighbor features ({hw^t | w \in N(v)})
    • Compute messages: (m{vw}^t = Mt(hv^t, hw^t, e{vw}))
    • Aggregate messages: (mv^{t+1} = \sum{w \in N(v)} m_{vw}^t)
    • Update node state: (hv^{t+1} = Ut(hv^t, mv^{t+1}))

• Readout Phase:

  • Apply graph-level pooling: (hG = R({hv^K | v \in G}))
  • Pass through output network for prediction

The implementation typically employs learned neural networks for both message and update functions, with gated recurrent units (GRUs) or simple multi-layer perceptrons (MLPs) common choices for (U_t) [14] [12].

Advanced Message-Passing Mechanisms

Attention-Based Message Passing

Advanced message-passing architectures incorporate attention mechanisms to dynamically weight the importance of different neighbors during aggregation:

[mv^{(t+1)} = \sum{w \in N(v)} \alpha{vw}^{(t)} Mt\left(hv^{(t)}, hw^{(t)}, e_{vw}\right)]

where attention weights (\alpha_{vw}^{(t)}) are computed as:

[\alpha{vw}^{(t)} = \frac{\exp\left(\text{LeakyReLU}\left(a^T [W hv^{(t)} \| W hw^{(t)}]\right)\right)}{\sum{k \in N(v)} \exp\left(\text{LeakyReLU}\left(a^T [W hv^{(t)} \| W hk^{(t)}]\right)\right)}]

This approach enables the model to focus on particularly relevant atomic interactions, such as those critical for binding affinity in drug-target interactions or catalytic activity in materials [16] [14]. Multi-head attention extends this concept by employing multiple independent attention mechanisms to capture different aspects of molecular interactions.

Edge Memory and Dynamic Message Passing

Recent innovations in message passing include edge memory networks and dynamic message-passing mechanisms that adaptively modify information flow:

Edge Memory Networks maintain and update edge representations throughout the message-passing process, allowing richer information exchange beyond simple node features [14]. The enhanced message function becomes:

[mv^{(t+1)} = \sum{w \in N(v)} Mt\left(hv^{(t)}, hw^{(t)}, e{vw}^{(t)}\right)]

[e{vw}^{(t+1)} = Et\left(e{vw}^{(t)}, hv^{(t)}, h_w^{(t)}\right)]

where (E_t) is an edge update function that refines edge features at each step.

Dynamic Message Passing introduces learnable pseudo-nodes and spatial relationships that evolve during processing, effectively creating adaptive communication pathways beyond the fixed molecular topology [17]. This approach addresses limitations of static graph structures by allowing information to flow along dynamically determined optimal paths.

Figure 2: Dynamic Message-Passing Framework showing how pseudo nodes and spatial relations create adaptive message pathways beyond initial molecular connectivity.

Experimental Protocols and Validation

Benchmark Datasets and Evaluation Metrics

Rigorous evaluation of message-passing mechanisms requires standardized datasets spanning diverse molecular properties:

Table 2: Molecular Graph Benchmark Datasets for Message-Passing Evaluation

Dataset	Domain	Graphs	Task Type	Evaluation Metric	Key Challenge
QM9	Quantum chemistry	133,885	Regression	MAE	Predicting quantum mechanical properties
MD17	Molecular dynamics	10+ molecules	Regression	Energy MAE, Force MAE	Molecular conformations
MoleculeNet	Various	Multiple	Classification/Regression	ROC-AUC, RMSE	Multi-task generalization
OGB	Various	Large-scale	Various	Dataset-specific	Scalability and transfer
TUDataset	Chemical & biological	Multiple	Classification	Accuracy	Domain-specific learning

These datasets enable comprehensive benchmarking of message-passing architectures across different molecular complexity levels, from small organic molecules to complex drug-like compounds and materials [11] [12].

Stability and Generalization Protocols

A significant challenge in molecular graph networks is ensuring robust performance under distributional shifts (out-of-distribution, OOD). Recent research has introduced stable learning approaches specifically designed for GNNs:

Protocol 3: Stable-GNN Training for OOD Generalization

• Feature Decorrelation: Apply random Fourier features (RFF) to measure and decorrelate feature dependencies
• Sample Reweighting: Learn instance-specific weights that suppress spurious correlations
• Stable Regularization: Incorporate decorrelation loss during training:

[\mathcal{L}{stable} = \mathcal{L}{task} + \lambda \sum{i,j} \text{Corr}(hi, h_j)^2]

where (\text{Corr}(hi, hj)) measures correlation between different feature dimensions [11]

• Cross-Domain Validation: Evaluate on deliberately constructed OOD splits to test generalization

This approach enhances model reliability for real-world applications where test molecules may differ systematically from training data due to selection biases or evolving chemical spaces [11].

Applications in Molecular Property Prediction

Multi-Component Gas Adsorption in MOFs

Message-passing networks have demonstrated exceptional performance in predicting complex materials behaviors such as gas adsorption in metal-organic frameworks (MOFs). The multi-scale graph representation enables modeling of interactions at different structural levels:

Protocol 4: Multi-Scale Crystal Graph Network for Adsorption Prediction

• Graph Construction: Create crystal graph with nodes as atoms and edges as bonds or proximity relations
• Multi-Scale Representation: Implement separate message-passing pathways for:
  • Covalent bonding interactions
  • Functional group patterns
  • Global crystalline structure
• Multi-Head Attention: Apply independent attention mechanisms at each scale to identify critical structural features
• Hierarchical Readout: Combine representations from different scales for final prediction

This approach has achieved state-of-the-art accuracy in predicting multi-component gas adsorption isotherms, significantly outperforming traditional descriptor-based methods and uniform graph architectures [16].

Bioactivity and Physical-Chemical Property Prediction

In pharmaceutical applications, message-passing networks directly predict bioactivity and ADMET (Absorption, Distribution, Metabolism, Excretion, Toxicity) properties from molecular structure:

Table 3: Message-Passing Applications in Drug Discovery

Property Type	Prediction Task	Architecture Variant	Key Performance
Target affinity	IC50, Ki values	Attention MPNN	>0.9 ROC-AUC on benchmark sets
Toxicity	hERG, Ames toxicity	Edge Memory MPNN	Significant reduction in false negatives
Solubility	LogS regression	3D-aware MPNN	RMSE <0.6 log units
Metabolic stability	CYP450 inhibition	Multi-task MPNN	85% accuracy on clinical candidates
Permeability	P-gp substrate	Geometric MPNN	>0.85 precision in classification

The capacity of message-passing networks to learn directly from molecular graphs eliminates the need for manual feature engineering while capturing subtle structure-property relationships that challenge traditional QSAR approaches [14].

The Scientist's Toolkit: Essential Research Reagents

Table 4: Key Research Tools and Resources for Message-Passing Implementation

Tool/Resource	Type	Function	Application Context
RDKit	Cheminformatics library	Molecular graph construction from SMILES/SDF	Preprocessing pipeline for small molecules
PyTor Geometric	Deep learning library	GNN implementation and training	Message-passing network development
DGL	Deep learning library	Scalable graph network operations	Large-scale molecular datasets
OGB	Benchmark suite	Standardized evaluation	Model comparison and validation
MoleculeNet	Benchmark dataset	Multi-task performance assessment	Method robustness testing
SAMSON	Molecular visualization	Structure and property visualization	Result interpretation and analysis
Crystal Graph Converter	Materials preprocessing	Periodic structure to graph conversion	Materials property prediction

These tools collectively provide the essential infrastructure for implementing, training, and evaluating message-passing networks on molecular graphs, from initial data preparation to final model deployment [18] [11] [12].

Message-passing mechanisms provide a powerful framework for learning from molecular graphs, enabling accurate prediction of chemical, biological, and materials properties directly from structural information. The continued evolution of these mechanisms—through attention, edge memory, dynamic pathways, and stability enhancements—addresses key challenges in molecular machine learning while creating new opportunities for scientific discovery.

As these methods mature, their integration into automated discovery pipelines promises to accelerate progress across molecular sciences, from rational drug design to functional materials development. The protocols and implementations detailed in this work provide researchers with practical guidance for applying these advanced techniques to diverse molecular prediction tasks.

The accurate prediction of material stability and molecular properties represents a cornerstone of modern computational materials science and drug development. In this context, Graph Neural Networks (GNNs) have emerged as powerful tools for modeling atomic systems, representing molecules and materials as graphs where nodes correspond to atoms and edges represent interatomic interactions [12]. A critical advancement in this field has been the systematic embedding of fundamental physical symmetries—particularly rotational and translational invariance—directly into neural network architectures. These geometric deep learning approaches ensure that model predictions remain consistent regardless of arbitrary choices of coordinate systems, leading to more physically realistic predictions, enhanced data efficiency, and improved generalization across diverse chemical spaces [19] [20].

The significance of these symmetry-aware models extends across multiple domains, from accelerating the discovery of novel materials with tailored properties to predicting protein-ligand binding affinities in rational drug design [21]. This application note examines the theoretical foundations, implementation protocols, and practical applications of rotational and translational invariance in GNNs, providing researchers with actionable methodologies for leveraging these principles in stability prediction and interatomic interaction modeling.

Theoretical Foundations of Symmetry Principles

Mathematical Formalisms of Physical Symmetries

In molecular and materials systems, physical properties must exhibit well-defined transformation behaviors under rotations and translations of the coordinate system. Formally, a function (f: X \rightarrow Y) is defined as equivariant with respect to a group (G) that acts on (X) and (Y) if:

[{D}{Y}[g]f(x)=f({D}{X}[g]x)\quad \forall g\in G,\forall x\in X]

where ({D}{X}[g]) and ({D}{Y}[g]) are the representations of the group element (g) in the vector spaces (X) and (Y), respectively [19]. For interatomic potentials, we primarily concern ourselves with the E(3) symmetry group encompassing rotations, reflections, and translations in 3D space. When ({D}_{Y}[g]) is the identity transformation, the function is considered invariant, a crucial property for predicting scalar values such as potential energy [19].

Traditional neural network architectures that operate on Cartesian coordinates do not inherently respect these symmetries, requiring extensive data augmentation and often failing to generalize to unseen orientations. In contrast, equivariant GNNs explicitly preserve transformation properties through specialized architectures that maintain geometric tensor representations throughout the network [19].

Architectural Implementation of Symmetry Principles

Several complementary architectural strategies have been developed to embed physical symmetries into deep learning models:

Irreducible Representations (irreps) and Spherical Harmonics: Advanced frameworks such as e3nn utilize irreducible representations of the O(3) symmetry group based on spherical harmonics to track how outputs vary under rotations [19] [21]. The Clebsch-Gordan tensor product combines these representations equivariantly, generalizing operations such as dot and cross products while maintaining symmetry properties [20].

Message Passing with Geometric Tensors: Equivariant message-passing networks update node features comprising not only scalars but also vectors and higher-order geometric tensors. This approach preserves directional information while maintaining equivariance, allowing the network to leverage angular information critical for modeling interatomic forces [19].

Invariant Descriptor Engineering: Alternative approaches construct inherently invariant descriptors of atomic environments, such as Gaussian Overlap Matrix (GOM) fingerprints, which encode many-body interactions through orbital overlap matrices while guaranteeing rotational and translational invariance [22].

Table 1: Comparison of Architectural Approaches for Embedding Physical Symmetries

Architectural Approach	Key Features	Representative Models	Advantages
Irreducible Representations	Spherical harmonics, Clebsch-Gordan tensor products	NequIP [19], MACE [23], SevenNet [20]	High expressiveness, rigorous symmetry preservation
Invariant Descriptors	Precomputed invariant representations of atomic environments	EOSnet [22], SOAP [22]	Simplified architecture, guaranteed invariance
Geometric Message Passing	Vector features, equivariant update rules	PaiNN [19], NewtonNet [19]	Balance of expressive power and computational efficiency

Application Case Studies in Materials and Molecular Science

Interatomic Potential Development

The development of machine learning interatomic potentials (MLIPs) has been revolutionized by symmetry-aware architectures. The NequIP (Neural Equivariant Interatomic Potential) framework demonstrates the remarkable advantages of E(3)-equivariance, achieving state-of-the-art accuracy with significantly enhanced data efficiency [19]. In benchmark studies, NequIP outperformed existing models with up to three orders of magnitude fewer training data, accurately reproducing structural and kinetic properties from ab initio molecular dynamics simulations [19].

Recent advancements continue to build upon these principles. The Facet architecture introduces computational optimizations to steerable GNNs, replacing resource-intensive multi-layer perceptrons with efficient splines for processing interatomic distances [20]. This innovation achieves performance comparable to leading approaches with significantly fewer parameters and less than 10% of the training computation, enabling faster iteration in potential development [20].

Materials Property Prediction

For crystalline materials, symmetry-aware GNNs have demonstrated exceptional performance in predicting diverse materials properties. The EOSnet framework incorporates Gaussian Overlap Matrix fingerprints as node features, providing a compact, rotationally invariant representation of many-body interactions [22]. This approach has achieved a mean absolute error of 0.163 eV in band gap prediction, surpassing previous state-of-the-art models while maintaining computational efficiency [22].

Systematic benchmarking of universal MLIPs for elastic property prediction further validates the importance of symmetry principles. In evaluations across nearly 11,000 elastically stable materials, equivariant models including SevenNet and MACE demonstrated superior accuracy in predicting bulk modulus, shear modulus, and other mechanical properties, establishing their reliability for computational materials design [23].

Table 2: Performance Benchmarks of Symmetry-Aware Models Across Applications

Model	Application Domain	Key Performance Metrics	Competitive Advantages
NequIP [19]	Interatomic Potentials	State-of-the-art accuracy with 100-1000x data efficiency	Remarkable data efficiency, faithful force prediction
EOSnet [22]	Materials Property Prediction	0.163 eV MAE (band gap), 97.7% accuracy (metal/nonmetal)	Effective many-body interaction capture
SevenNet [23]	Elastic Property Prediction	Highest accuracy in uMLIP benchmark (11,000 materials)	Superior elastic constant prediction
EMFF-2025 [24]	Energetic Materials	MAE within ±0.1 eV/atom (energy), ±2 eV/Å (force)	Transfer learning capability for CHNO systems
InvarNet [25]	Molecular Property Prediction	2.24x faster training vs. SphereNet, state-of-the-art R2 on QM9	Optimized processing, rotational invariant loss

Biomolecular Applications

In drug discovery, predicting protein-ligand binding affinity represents a critical challenge where rotational symmetry plays a crucial role. The PLAe methodology combines radial basis functions with e3nn networks to capture radial and angular dimensions of molecular features while maintaining rotational equivariance [21]. This approach demonstrates how symmetry principles can enhance prediction accuracy in complex biomolecular systems where binding interactions depend critically on three-dimensional spatial relationships [21].

Experimental Protocols and Methodologies

Implementation of Equivariant Graph Neural Networks

The following protocol outlines the key steps for implementing an E(3)-equivariant GNN for interatomic potential training, based on the NequIP framework [19]:

Step 1: Data Preparation and Representation

• Collect reference atomic structures and corresponding energies/forces from ab initio calculations (DFT, coupled-cluster)
• Represent atomic systems as graphs with:
  • Nodes: Atoms with chemical species attributes
  • Edges: Connections between atoms within a specified cutoff radius (typically 4-6 Å)
• Apply periodic boundary conditions for crystalline materials using periodic graph constructions [20]

Step 2: Feature Initialization

• Initialize node features as geometric tensors comprising direct sums of irreducible representations of O(3)
• Encode edge information using spherical harmonics based on relative position vectors
• For hybrid approaches (e.g., EOSnet), compute invariant atomic environment descriptors (GOM fingerprints) as node features [22]

Step 3: Equivariant Message Passing

• Implement message passing with E(3)-equivariant operations:
  • Apply Clebsch-Gordan tensor products to combine representations
  • Use spherical harmonics-based convolutions for neighborhood information aggregation
  • Employ equivariant nonlinearities for activation [19]
• Iterate message passing steps (typically 3-6 layers) to propagate information across atomic neighborhoods

Step 4: Invariant Readout and Property Prediction

• Extract invariant scalars from final geometric tensor features
• Pool node-level representations to system-level properties via permutation-invariant operations (sum, mean)
• Compute potential energy as a sum of atomic contributions: [{E}{pot}=\mathop{\sum}\limits{i\in {N}{atoms}}{E}{i,atomic}]
• Obtain atomic forces as gradients of energy with respect to atomic positions: [{\vec{F}}{i}=-{\nabla }{i}{E}_{pot}] ensuring automatic energy conservation [19]

Step 5: Optimization and Training

• Define composite loss function incorporating energy, force, and stress components: [L = \lambdaE LE + \lambdaF LF + \lambdaS LS]
• Utilize rotational invariant loss terms to enhance robustness against molecular rotations [25]
• Employ adaptive optimization algorithms (Adam, AdamW) with learning rate scheduling

Transfer Learning Protocol for Specialized Applications

For applications with limited training data, such as specialized material systems, transfer learning from general-purpose potentials provides an effective strategy:

Pre-training Phase:

• Train a foundational model on diverse datasets (e.g., MPTrj [20], Materials Project [23]) encompassing broad chemical spaces
• Utilize large-scale computational resources for initial training, potentially requiring extensive GPU days [20]

Transfer Learning Phase:

• Initialize model weights from pre-trained checkpoint
• Fine-tune on target system data with reduced learning rate
• For EMFF-2025 approach, incorporate minimal new training data from DFT calculations specific to target materials [24]
• Validate transfer performance against held-out target system data and experimental measurements where available

Special Considerations for Energetic Materials:

• Focus on C, H, N, O element systems with complex decomposition pathways
• Ensure training data encompasses relevant temperature and pressure regimes
• Validate against mechanical properties and decomposition characteristics simultaneously [24]

Table 3: Essential Computational Tools for Symmetry-Aware GNN Implementation

Tool/Resource	Function	Application Context
e3nn Library [19] [21]	Framework for E(3)-equivariant neural networks	Implementation of irreducible representations and spherical harmonics
MPTrj Dataset [20]	Large-scale training dataset for MLIPs	Pre-training of foundational potential models
Materials Project [23]	Database of calculated materials properties	Benchmarking and validation datasets
DP-GEN Framework [24]	Active learning pipeline for training data generation	Efficient construction of specialized training sets
GOM Fingerprint Generator [22]	Computation of Gaussian Overlap Matrix descriptors	Atomic environment representation for invariant models
QM9, MD17 Datasets [25]	Quantum chemical properties of molecules	Benchmarking molecular property prediction

The systematic embedding of rotational and translational invariance into graph neural network architectures has fundamentally advanced the precision and efficiency of interatomic interaction modeling. Through equivariant operations based on irreducible representations and invariant descriptor engineering, these approaches achieve unprecedented data efficiency and physical consistency in predicting material stability, molecular properties, and interaction energies. The continued development of computationally efficient implementations, such as those demonstrated in the Facet architecture, promises to further accelerate materials discovery and drug development workflows. As benchmark studies across diverse chemical spaces continue to validate the superiority of symmetry-aware models, their adoption as standard tools in computational materials science and chemistry appears inevitable.

Graph Neural Networks (GNNs) have emerged as transformative tools for computational chemistry and materials science, enabling accurate predictions of interatomic interactions and system stability at a fraction of the computational cost of traditional quantum mechanical methods. These models learn fundamental chemical principles directly from data by representing atomic systems as graphs, where nodes correspond to atoms and edges represent interatomic interactions [26]. This representation allows GNNs to naturally capture complex quantum mechanical effects, including critical many-body interactions that are essential for predicting molecular properties and material stability with high fidelity [27] [22]. The capacity of GNNs to learn these fundamental principles positions them as powerful tools for accelerating drug discovery and materials design.

This application note examines the specific chemical principles learned by GNNs, focusing on their ability to capture interaction strengths and many-body effects. We provide a structured analysis of quantitative performance data across different architectural approaches, detailed experimental protocols for implementing and interpreting these models, and visualization tools to elucidate the learned interactions. By framing these capabilities within the context of interatomic interactions and stability prediction research, we aim to equip scientists with the practical knowledge needed to leverage GNNs in their computational workflows.

Theoretical Foundations: How GNNs Learn Chemical Interactions

GNNs learn chemical interactions through a message-passing framework that propagates information across the molecular graph. In this paradigm, node features represent atomic properties (e.g., element type, orbital configuration), while edge features encode pairwise relationships (e.g., interatomic distances, bond orders) [26]. During message passing, each atom gathers information from its local environment, progressively building representations that capture increasingly complex chemical environments with each layer [26] [22].

Learning Many-Body Interactions

Traditional machine learning potentials often struggle to capture many-body interactions beyond pairwise atomic relationships. GNNs address this limitation through several advanced architectural approaches:

Structural encodings explicitly incorporate angular information critical for three-body interactions. The AGT framework, for instance, integrates Spherical Bessel Function (SBF) angle encoding alongside atomic and edge encodings, significantly enhancing the model's capacity to represent geometric distortions and bond angle dependencies [28].

Orbital overlap representations capture quantum mechanical effects through mathematical constructs such as Gaussian Overlap Matrix (GOM) fingerprints. In EOSnet, these fingerprints are derived from the eigenvalues of overlap matrices between Gaussian-type orbitals centered on atoms, providing a rotationally invariant representation of many-body atomic environments without requiring explicit angular terms [22].

Fragmentation approaches combine GNNs with physical principles like the Many-Body Expansion (MBE) theory. The FBGNN-MBE method partitions large systems into fragments, uses first-principles quantum mechanical methods for single-fragment energies, and deploys GNNs to learn the complex many-fragment interactions, creating a manageable framework for large functional materials [27].

Quantitative Performance Analysis

Table 1: Performance Comparison of GNN Architectures on Material Property Prediction

GNN Architecture	Key Innovation	Target Property	Performance Metrics	Reference
EOSnet	Gaussian Overlap Matrix node features	Band gap prediction	MAE = 0.163 eV	[22]
EOSnet	Gaussian Overlap Matrix node features	Metal/nonmetal classification	Accuracy = 97.7%	[22]
AGT	Angle encoding + MPNN-Transformer	Adsorption energy (OC20-Ni dataset)	MAE = 0.54 eV	[28]
KA-GNN	Fourier-based Kolmogorov-Arnold Networks	Molecular property prediction	Superior accuracy & computational efficiency vs conventional GNNs	[29]
FBGNN-MBE	Integration with many-body expansion theory	Potential energy surfaces	Reproduces FD-PES with manageable accuracy/complexity	[27]
Universal MLIPs (eSEN)	Cross-dimensional transferability	Energy/forces across dimensionalities	Energy error < 10 meV/atom across 0D-3D systems	[30]

Table 2: Analysis of Many-Body Interaction Capabilities in GNN Architectures

Architecture Category	Representative Models	Mechanism for Many-Body Interactions	Interpretability Features
Geometrically Enhanced	DimeNet, GemNet, ALIGNN, M3GNet	Incorporate angular and directional information between atoms	Varies by implementation
Equivariant Networks	E3NN, NequIP, MACE, eSEN	Use spherical harmonics and tensor products	Limited inherent interpretability
Orbital Overlap-Based	EOSnet	Gaussian Overlap Matrix fingerprints as node features	Direct physical interpretation of orbital interactions
Transformer Hybrids	AGT, Graph Transformers	Self-attention mechanisms with geometric encodings	Attention weights indicate important interactions
Explainable AI Enhanced	GNN-LRP models	Layer-wise Relevance Propagation for decomposition	Identifies n-body contributions to predictions

Experimental Protocols

Protocol: Implementing an EOSnet Architecture for Electronic Property Prediction

Purpose: To predict electronic properties of materials using orbital overlap information.

Materials and Software:

• Python 3.8+
• PyTorch or TensorFlow
• Deep Graph Library (DGL) or PyTorch Geometric
• Atomic simulation environment (ASE)
• Crystallography data for target materials

Procedure:

• Graph Construction:

  • Represent crystal structure as a graph with atoms as nodes.
  • Define edges between atoms within a specified cutoff radius (typically 5-8 Å).
  • Initialize node features using atomic properties (atomic number, valence electrons, etc.).

• GOM Fingerprint Calculation:

  • For each atom, identify all neighbors within the cutoff sphere.
  • Compute Gaussian Overlap Matrix elements using the formula: [ [Om]{i+l,j+l'} = fc(r{im})\langle\phi{il}|\phi{jl'}\rangle fc(r{jm}) ] where (fc(r) = (1 - \frac{r^2}{r{\text{cut}}^2})^n) is the smooth cutoff function, and (\phi_{il}) are Gaussian-type orbitals.
  • Calculate eigenvalues of the GOM for each atom to use as node features [22].

• Network Architecture:

  • Implement graph convolutional layers with message passing.
  • Update node embeddings by aggregating information from neighbors.
  • Use multiple convolutional layers (typically 3-6) to capture increasingly complex environments.

• Training Configuration:

  • Use Mean Absolute Error (MAE) or Mean Squared Error (MSE) as loss function.
  • Employ Adam optimizer with learning rate 0.001-0.0001.
  • Implement early stopping based on validation set performance.

• Validation:

  • Compare predictions with DFT-calculated band gaps.
  • Evaluate generalization across different material classes.

Protocol: Interpreting Many-Body Contributions with GNN-LRP

Purpose: To decompose GNN predictions into n-body interaction contributions.

Materials and Software:

• Trained GNN potential
• GNN-LRP implementation
• Molecular dynamics trajectories or structural datasets

Procedure:

• Model Inference:

  • Run the trained GNN on the target molecular configuration to obtain the total energy prediction.

• Relevance Propagation:

  • Apply Layer-wise Relevance Propagation (LRP) rules backward through the network.
  • Decompose the energy output into contributions from sequences of graph edges ("walks").

• n-Body Aggregation:

  • Aggregate relevance scores of all walks associated with specific subgraphs.
  • Calculate the n-body relevance contribution for each interacting group of atoms.
  • Sort contributions by absolute value to identify the most relevant interactions [31].

• Physical Validation:

  • Compare identified relevant interactions with known physical principles.
  • Verify that dominant contributions align with chemical intuition (e.g., strong covalent bonds, important non-covalent interactions).

Protocol: Training a KA-GNN for Molecular Property Prediction

Purpose: To leverage Kolmogorov-Arnold Networks for enhanced molecular property prediction.

Materials and Software:

• Molecular datasets (e.g., QM9, MD17)
• KA-GNN implementation
• RDKit for molecular processing

Procedure:

• Data Preparation:

  • Convert molecular structures to graph representations.
  • Node features: atomic number, hybridization state, formal charge.
  • Edge features: bond type, bond length, topological distance.

• Fourier-KAN Layer Implementation:

  • Replace standard MLP transformations with Fourier-based KAN modules.
  • Use Fourier series as learnable activation functions on edges: [ f(x) \sim \sum{k} (ak \cos(k \cdot x) + b_k \sin(k \cdot x)) ]
  • Implement in node embedding, message passing, and readout components [29].

• Model Variants:

  • KA-GCN: Integrate KAN modules into Graph Convolutional Networks.
  • KA-GAT: Incorporate KAN modules into Graph Attention Networks.

• Training and Interpretation:

  • Train on target molecular properties (e.g., energy, solubility, toxicity).
  • Analyze learned Fourier coefficients to identify important frequency components.
  • Visualize attention weights to highlight chemically meaningful substructures.

Visualization of GNN Architectures and Workflows

GNN Workflow for Interatomic Interactions

Many-Body Interaction Capture in GNNs

The Scientist's Toolkit: Essential Research Reagents

Table 3: Key Computational Tools and Datasets for GNN Implementation

Resource Category	Specific Tools/Datasets	Application Purpose	Key Features
GNN Frameworks	PyTorch Geometric, Deep Graph Library	Model Implementation	Pre-built GNN layers, molecular graph utilities
Materials Datasets	Materials Project, OQMD, JARVIS	Training & Benchmarking	DFT-calculated material properties
Molecular Datasets	QM9, MD17, ANI, OC20	Training & Benchmarking	Diverse molecular conformations & properties
Quantum Chemistry	ORCA, Gaussian, PySCF	Reference Calculations	Generate training data & validate predictions
Analysis & Visualization	ASE, OVITO, VMD	Structure Analysis	Process atomic structures & visualize results
Specialized GNN Models	MACE, NequIP, CHGNet, Allegro	Transferable Potentials	Pretrained universal machine learning interatomic potentials
Explainability Tools	GNN-LRP, Captum	Model Interpretation	Decompose predictions into atomic contributions

GNNs have demonstrated remarkable capability in learning fundamental chemical principles, particularly interaction strengths and many-body effects critical for accurate stability prediction. Through specialized architectural features—including angular encodings, orbital overlap representations, and integration with fragment-based quantum mechanics—these models capture complex quantum mechanical interactions that elude simpler machine learning approaches. The experimental protocols and visualization tools presented in this application note provide researchers with practical methodologies for implementing and interpreting these advanced networks. As GNN architectures continue to evolve, their capacity to learn and represent intricate chemical interactions will further bridge the gap between computational efficiency and quantum mechanical accuracy, accelerating discovery across materials science and drug development.

Advanced Architectures and Real-World Applications: From Drug Discovery to Materials Design

The accurate prediction of molecular properties and interatomic interactions represents a cornerstone of modern computational chemistry and materials science, with profound implications for drug discovery and materials design. Graph Neural Networks (GNNs) have emerged as powerful frameworks for these tasks by naturally modeling atomic systems as graphs, where atoms constitute nodes and chemical bonds form edges. Recent architectural innovations have significantly enhanced the capabilities of these models, improving their accuracy, computational efficiency, and physical faithfulness. This article explores three groundbreaking developments: Kolmogorov-Arnold Graph Neural Networks (KA-GNNs), which leverage mathematical representation theory; Moment Graph Neural Networks (MGNN), which utilize moment representations for universal potentials; and the emerging class of universal neural network interatomic potentials that demonstrate remarkable transferability across diverse chemical spaces. These architectures are pushing the boundaries of what's possible in molecular dynamics simulations, property prediction, and rational material design.

KA-GNN: Kolmogorov-Arnold Graph Neural Networks

Theoretical Foundation and Architecture

KA-GNNs represent a significant architectural innovation that integrates the mathematical foundations of the Kolmogorov-Arnold representation theorem into graph neural networks. The Kolmogorov-Arnold theorem states that any multivariate continuous function can be represented as a finite composition of continuous functions of a single variable and the binary operation of addition [32]. Inspired by this theorem, KA-GNNs replace traditional multilayer perceptron (MLP) components with Kolmogorov-Arnold network (KAN) modules that feature learnable activation functions on edges rather than fixed activations on nodes [29].

The KA-GNN framework systematically integrates Fourier-based KAN modules across all fundamental components of GNNs: node embedding initialization, message passing between atoms, and graph-level readout for property prediction [29]. A key innovation lies in its use of Fourier series as basis functions for the univariate activation functions, replacing the B-splines used in earlier KAN implementations. This Fourier-based approach enables effective capture of both low-frequency and high-frequency structural patterns in molecular graphs, enhancing the model's expressiveness for representing complex quantum mechanical relationships [29].

Table 1: Core Components of KA-GNN Architecture

Component	Traditional GNN Approach	KA-GNN Innovation	Benefit
Node Embedding	Atomic features processed through MLP with fixed activations	KAN layer with learnable Fourier-based functions	Data-driven atomic representation incorporating local chemical context
Message Passing	Weighted sum of neighbor features with fixed nonlinearity	Composition of learnable univariate functions on edges	Enhanced feature interaction modeling during information propagation
Readout Function	Global pooling followed by MLP for graph-level prediction	KAN-based readout with Fourier representations	More expressive graph-level representation for property prediction
Basis Functions	Not applicable	Fourier series replacing B-splines	Better capture of periodic and oscillatory patterns in molecular data

Implementation Variants and Performance

Two principal variants of KA-GNN have been developed: KA-Graph Convolutional Network (KA-GCN) and KA-Graph Attention Network (KA-GAT). In KA-GCN, node embeddings are initialized by processing concatenated atomic features and neighboring bond features through a KAN layer, effectively encoding both atomic identity and local chemical environment. The message-passing layers follow the GCN scheme but employ residual KANs instead of traditional MLPs for feature updates [29]. KA-GAT extends this approach by incorporating edge embeddings initialized with KAN layers, with attention mechanisms operating on these enriched representations [29].

Experimental evaluations across seven molecular benchmark datasets demonstrate that KA-GNN variants consistently outperform conventional GNNs in both prediction accuracy and computational efficiency [29]. The architecture exhibits particular strength in capturing complex quantum chemical relationships while maintaining parameter efficiency. Additionally, KA-GNNs offer enhanced interpretability, often highlighting chemically meaningful substructures that contribute significantly to predicted properties, thereby providing valuable insights for domain experts [29].

MGNN: Moment Graph Neural Network for Universal Potentials

Architectural Principles and Formulation

The Moment Graph Neural Network (MGNN) represents a novel approach to constructing universal interatomic potentials that effectively capture the spatial relationships and symmetries inherent in molecular systems. MGNN innovatively propagates information between atoms using moments—mathematical quantities that encapsulate the spatial relationships between atoms within a molecule [33]. The architecture is inspired by Moment Tensor Potentials, which have established theoretical guarantees for approximating any regular function satisfying all necessary physical symmetries [33].

In the MGNN framework, molecular systems are represented as graphs where edges connect atoms within a defined cutoff distance. The model processes information through triplets of atoms (center atom i and neighbors j, k), where triplet messages update edge representations, which subsequently update node features [33]. This hierarchical information flow—from triplets to edges to nodes—enables the model to capture increasingly complex atomic environments while maintaining physical consistency.

A distinctive feature of MGNN is its use of Chebyshev polynomials to encode interatomic distance information, diverging from the more commonly employed Bessel and Gaussian radial basis functions in other GNN architectures [33]. This mathematical choice contributes to the model's efficiency and accuracy in capturing spatial relationships.

Performance and Applications

MGNN has demonstrated state-of-the-art performance across multiple benchmark datasets including QM9, revised MD17, and MD17-ethanol [33]. Its robustness extends to diverse material systems, having been successfully applied to 3BPA, 25-element high-entropy alloys, and amorphous electrolytes [33]. In molecular dynamics simulations, MGNN achieves remarkable consistency with ab initio methods while offering significantly reduced computational cost, making long-timescale simulations of complex systems practically feasible.

The architecture's versatility is evidenced by its capability to predict diverse molecular properties including scalar quantities (e.g., formation energy), vectorial properties (e.g., forces, dipole moments), and tensorial properties (e.g., polarizabilities) [33]. This comprehensive predictive capability enables accurate simulation of molecular spectra and other complex physicochemical phenomena, positioning MGNN as a valuable tool for computational chemistry and materials science.

Diagram 1: MGNN Architecture showing information flow through interaction blocks with triplet moment interactions

Universal Neural Network Interatomic Potentials

Extrapolation Capabilities and Design Principles

Universal neural network interatomic potentials represent a transformative advancement in computational materials science, offering the potential for accurate, transferable force fields applicable across diverse chemical spaces. A key breakthrough lies in their remarkable extrapolation capabilities—these models, often trained primarily on crystalline structures, can generalize to untrained domains such as surfaces, amorphous configurations, and defect environments [1].

Research into the theoretical foundations of this extrapolation behavior reveals that GNN interatomic potentials can capture non-local electrostatic interactions through their message-passing algorithms [1]. Models such as SevenNet and MACE demonstrate the ability to learn the exact functional form of Coulomb interactions, contributing significantly to their transferability across different chemical environments. This capacity to learn fundamental physical interactions, combined with the embedding nature of GNNs, provides a compelling explanation for their extrapolation capabilities [1].

Recent architectural innovations have further enhanced these universal potentials. TeaNet (Tensor Embedded Atom Network) incorporates Euclidean tensors, vectors, and scalars to represent angular interactions through graph convolution, enabling accurate modeling of diverse chemical systems involving the first 18 elements of the periodic table [34]. Similarly, E2GNN (Efficient Equivariant Graph Neural Network) employs a scalar-vector dual representation to encode equivariant features, maintaining rotational symmetry while achieving computational efficiency [7].

Applications and Performance Benchmarks

Universal interatomic potentials have demonstrated impressive performance across a wide spectrum of material systems. TeaNet shows robust performance for diverse structures including C-H molecules, metals, amorphous SiO₂, and water, achieving energy mean absolute errors of 19 meV/atom [34]. E2GNN consistently outperforms representative baselines in accuracy and efficiency across catalysts, molecules, and organic isomers, enabling ab initio accuracy in molecular dynamics simulations of solid, liquid, and gas systems [7].

Table 2: Comparison of Universal GNN Interatomic Potentials

Model	Architectural Approach	Key Innovations	Reported Performance	Applicable Systems
MGNN [33]	Moment-based message passing	Chebyshev polynomials for distances, triplet interactions	State-of-the-art on QM9, MD17	Molecules, alloys, amorphous materials
TeaNet [34]	Tensor embedding with ResNet	Euclidean tensors for angular information, 16-layer depth	19 meV/atom MAE on randomized configurations	First 18 elements (H to Ar)
E2GNN [7]	Scalar-vector dual representation	Efficient equivariance without higher-order tensors	Outperforms baselines on diverse datasets	Catalysts, organic isomers, molecules
Magnetic MLIPs [35]	Spin-polarized machine learning potentials	Combination of DFT accuracy with empirical potential efficiency	Accurate for magnetic systems	Magnetic materials, alloys

The development of magnetic machine-learning interatomic potentials (magnetic MLIPs) further extends the universality paradigm to magnetic materials, combining the computational efficiency of empirical potentials with the accuracy of spin-polarized density functional theory calculations [35]. This specialization addresses the unique challenges of modeling magnetic interactions while maintaining the transferability principles of universal potentials.

Experimental Protocols and Methodologies

Benchmarking Procedures for Molecular Property Prediction

Rigorous evaluation of GNN architectures requires standardized benchmarking across diverse datasets. For molecular property prediction, models are typically evaluated on established datasets such as QM9 [33], MD17 [33], FreeSolv [36], and CombiSolv-Exp [36]. The standard protocol involves dataset splitting (typically 80/10/10 for train/validation/test), ensuring representative distribution of molecular structures and target properties across splits.

Training procedures generally employ the Adam optimizer with an initial learning rate of 0.001, which is reduced upon validation loss plateau. Early stopping is implemented to prevent overfitting, with training terminated after a specified number of epochs without validation improvement. For property prediction tasks, models are evaluated using mean absolute error (MAE) and root mean square error (RMSE) metrics, with statistical significance testing across multiple random seeds [29] [33].

For KA-GNN implementations, the Fourier-based KAN layers require specific initialization of the harmonic coefficients, typically sampled from a normal distribution with small variance. The number of harmonics represents a key hyperparameter, with values between 10-20 often providing optimal performance across diverse molecular tasks [29].

Molecular Dynamics Simulation with GNN Potentials

When deploying GNN interatomic potentials for molecular dynamics simulations, specific protocols ensure physical consistency and numerical stability. The workflow begins with model training on diverse reference configurations, typically including crystals, surfaces, molecular dimers, and deformed structures to ensure adequate sampling of the potential energy surface [33] [34].

For MD simulations, forces are computed as the negative gradient of the predicted energy with respect to atomic coordinates, typically implemented through automatic differentiation. Integration of Newton's equations of motion employs standard algorithms such as Velocity Verlet with timesteps of 0.5-1.0 fs, constrained by the stability requirements of the potential [33]. Simulations are typically conducted in the NVT or NPT ensemble using appropriate thermostats and barostats, with production runs preceded by equilibration phases.

Validation against ab initio MD trajectories provides the ultimate test of potential reliability, comparing structural properties (radial distribution functions), dynamical properties (diffusion coefficients), and thermodynamic quantities [33] [7]. For universal potentials, additional validation on unseen element combinations or crystal structures confirms extrapolation capability [34].

The Scientist's Toolkit: Essential Research Reagents

Table 3: Key Computational Tools and Resources for GNN Implementation

Resource Category	Specific Tools/Datasets	Purpose and Utility	Access Information
Benchmark Datasets	QM9, MD17, FreeSolv, CombiSolv-Exp	Standardized evaluation of model performance	Publicly available from academic sources
Molecular Processing	RDKit [36]	Conversion of SMILES to graph structures, feature computation	Open-source cheminformatics toolkit
GNN Implementation Frameworks	DGL-LifeSci [32], PyTorch Geometric	Pre-built models and training pipelines for molecular graphs	Open-source deep learning libraries
Specialized Architectures	KA-GNN [29], MGNN [33], TeaNet [34]	Reference implementations of novel architectures	Typically available from original publications
Simulation Software	LAMMPS, ASE	Molecular dynamics simulations with ML potentials	Open-source molecular simulation packages
Radial Basis Functions	Bessel Functions, Gaussian RBFs, Chebyshev Polynomials [33]	Encoding of interatomic distance information	Mathematical implementations in deep learning frameworks

The landscape of graph neural networks for molecular modeling is evolving rapidly, with architectures like KA-GNN, MGNN, and universal potentials pushing the boundaries of accuracy, efficiency, and physical consistency. KA-GNNs demonstrate how mathematical representation theory can inspire novel deep learning architectures with enhanced expressivity and interpretability. MGNN showcases the power of moment-based representations for constructing universal interatomic potentials with strong theoretical foundations. The broader class of universal potentials highlights the exciting possibility of creating transferable force fields that maintain accuracy across diverse chemical spaces.

Future development will likely focus on several key challenges: improving sample efficiency for data-scarce applications, enhancing interpretability for scientific insight generation, developing standardized benchmarks and evaluation protocols, and increasing computational efficiency for large-scale systems. As these architectures mature, they promise to accelerate discovery across chemistry, materials science, and drug development by providing accurate, efficient computational proxies for expensive quantum mechanical calculations.

Drug-drug interactions (DDIs) represent a critical challenge in modern pharmacotherapy, particularly given the rising prevalence of polypharmacy in managing complex diseases and aging populations. DDIs can be categorized into synergism, which enhances therapeutic effects, and antagonism, which can lead to adverse effects such as reduced efficacy, toxicity, or even fatality [37]. According to recent studies, DDIs account for approximately 6% to 30% of all adverse drug reactions and are responsible for nearly 2.8% of hospital admissions associated with adverse drug reactions [38]. The probability of potential DDIs increases substantially with the number of medications administered, rising from an estimated 6% with two drugs to approximately 50% with five medications and nearly 100% when eight drugs are taken simultaneously [38].

Traditional experimental approaches for identifying DDIs are both expensive and time-consuming, particularly given the vast number of possible drug combinations. Consequently, developing effective computational strategies for DDI prediction has become a priority in pharmaceutical research [37]. Graph neural networks (GNNs) have emerged as powerful tools for DDI prediction by naturally representing drug molecules as graphs, where atoms serve as nodes and chemical bonds as edges [39]. This representation allows GNNs to automatically learn informative features from molecular structures and capture complex patterns that are difficult to identify through traditional methods.

Graph Neural Network Architectures for DDI Prediction

Fundamental GNN Architectures

Several GNN architectures have been successfully applied to DDI prediction, each with distinctive approaches to processing graph-structured molecular data:

• Graph Convolutional Networks (GCNs): Operate by aggregating feature information from neighboring nodes using spectral graph convolutions. GCNs with skip connections and nested GNN (NGNN) modules have demonstrated competent accuracy in DDI prediction tasks [37].
• Graph Attention Networks (GATs): Employ attention mechanisms to assign varying importance to neighbors during feature aggregation, allowing models to focus on the most relevant substructures for interaction prediction [37].
• Message Passing Neural Networks (MPNNs): Provide a general framework for learning node embeddings through iterative message passing between nodes and their neighbors, effectively capturing local chemical environments [39].

Advanced GNN Frameworks for DDI Prediction

Recent research has developed specialized GNN architectures to address specific challenges in DDI prediction:

• Dual Graph Neural Network (DGNN-DDI): This approach utilizes a directed message passing neural network with a substructure attention mechanism (SA-DMPNN) to adaptively extract flexible-sized chemical substructures. It then employs a co-attention mechanism to capture interaction information between substructures of different drugs, enhancing prediction accuracy and interpretability [39].
• Multi-scale Graph Neural Network (MGDDI): Addresses the challenge of variable substructure sizes by employing dense connections between different GNN modules to extract information at multiple scales. It incorporates a substructure interaction learning module based on attention mechanisms to capture detailed interactions between drug pairs [40].
• Kolmogorov-Arnold Graph Neural Networks (KA-GNN): Integrates Kolmogorov-Arnold network modules into the three fundamental components of GNNs: node embedding, message passing, and readout. By using Fourier-series-based univariate functions, KA-GNNs enhance function approximation capabilities and demonstrate superior performance in molecular property prediction tasks [29].

Table 1: Performance Comparison of GNN Models on DDI Prediction Tasks

Model	Architecture Type	Key Innovation	Reported Performance
DGNN-DDI [39]	Dual GNN	Substructure attention & co-attention	Superior to state-of-the-art methods on DrugBank dataset
MGDDI [40]	Multi-scale GNN	Multi-scale feature extraction & substructure interaction learning	State-of-the-art performance on DrugBank and TWOSIDES datasets
KA-GNN [29]	Kolmogorov-Arnold Enhanced	Fourier-based KAN modules in all GNN components	Consistently outperforms conventional GNNs across 7 molecular benchmarks
GCN with Skip Connections [37]	Enhanced GCN	Skip connections to prevent gradient issues	Competent accuracy compared to baseline models
Graph Attention Network [37]	Attention-based	Attention mechanisms for weighted neighbor aggregation	Competitive performance on multiple DDI datasets

Experimental Protocols and Methodologies

Dataset Preparation and Preprocessing

Standardized datasets are essential for training and evaluating DDI prediction models. The following protocols describe common procedures for dataset preparation:

• DrugBank Dataset: A comprehensive bioinformatics and cheminformatics resource containing detailed drug data with comprehensive drug target information. A commonly used benchmark includes 1,706 drugs and 191,808 DDI tuples classified into 86 interaction types [39] [40]. Each drug is associated with SMILES strings, which are converted into molecular graphs using toolkits like RDKit.
• Data Splitting: For robust evaluation, datasets are typically randomly split into training (60%), validation (20%), and test (20%) sets [39]. For emerging DDI prediction, special splitting strategies that simulate distribution changes between known and new drugs may be employed to assess model robustness [41].
• Molecular Graph Construction: Convert SMILES representations into molecular graphs where atoms become nodes (with features like atomic number, chirality) and bonds become edges (with features like bond type, conjugation) [39].

Model Implementation Protocols

DGNN-DDI Implementation Protocol

The following protocol outlines the implementation of the DGNN-DDI model for DDI prediction:

• Molecular Substructure Extraction:

  • Implement a directed message passing neural network (SA-DMPNN) with substructure attention mechanism.
  • Set message passing steps T=3 through parameter analysis [39].
  • Configure hidden dimension size to 64 based on hyperparameter optimization.

• Substructure Interaction Learning:

  • Implement co-attention mechanism to learn interaction scores between substructures of two drugs.
  • Calculate importance weights for substructures based on their interaction potential.

• Feature Integration and Prediction:

  • Concatenate weighted substructure features for each drug.
  • Pass integrated features through fully connected layers with softmax activation for interaction probability prediction.
  • Train model using Adam optimizer with learning rate 1e-4, batch size 256, and weight decay 5×10^(-4) for 50 epochs [39].

MGDDI Implementation Protocol

The MGDDI protocol focuses on multi-scale feature learning:

• Multi-scale Graph Neural Network Setup:

  • Implement dense connections between different GNN modules to capture features at multiple scales.
  • Configure network to prevent gradient vanishing and oversmoothing issues associated with deep GNNs [40].

• Substructure Interaction Learning Module:

  • Integrate attention-based mechanism to capture interaction details between drug pairs.
  • Learn importance of different substructures for improved representation.

• Model Training and Validation:

  • Train model using appropriate loss functions (e.g., cross-entropy for multi-class classification).
  • Validate performance on hold-out datasets focusing on both known and emerging DDIs.

Table 2: Hyperparameter Settings for GNN Models in DDI Prediction

Hyperparameter	DGNN-DDI [39]	Typical Range	Optimization Method
Message Passing Steps	3	{1, 2, 3, 4, 5}	Grid Search
Hidden Dimension	64	{32, 64, 128}	Performance Validation
Learning Rate	1e-4	{1e-2, 1e-3, 1e-4}	Adam Optimizer
Batch Size	256	{128, 256, 512}	Memory Constraints
Weight Decay	5×10^(-4)	-	Regularization
Training Epochs	50	-	Early Stopping

Visualization of GNN Approaches for DDI Prediction

DGNN-DDI Architecture Workflow

Multi-scale Graph Neural Network Architecture

The Scientist's Toolkit: Research Reagent Solutions

Table 3: Essential Resources for GNN-based DDI Prediction Research

Resource Category	Specific Tools & Databases	Function in DDI Research	Access Information
Molecular Databases	DrugBank, TWOSIDES, PubChem	Provide structured drug information, interaction data, and chemical properties	Publicly available online [38] [39] [40]
Cheminformatics Tools	RDKit, OpenBabel	Process SMILES strings, generate molecular graphs, compute molecular descriptors	Open-source toolkits
Deep Learning Frameworks	PyTorch, TensorFlow, PyTorch Geometric	Implement and train GNN models with GPU acceleration	Open-source with extensive documentation
DDI Prediction Platforms	DDInter, Drugs.com, Medscape	Validate predictions against established interaction checkers	Online databases with clinical information [38]
Benchmarking Frameworks	DDI-Ben [41]	Evaluate model performance under distribution changes	Available code repository
Specialized GNN Libraries	DGL (Deep Graph Library)	Provide optimized implementations of GNN architectures	Open-source with molecular graph extensions

Emerging Challenges and Future Directions

Despite significant advances in GNN-based DDI prediction, several challenges remain that require further research attention:

• Distribution Changes and Emerging DDIs: Current evaluation methods often rely on independent and identically distributed (i.i.d.) splits that don't reflect real-world scenarios where new drugs with different properties are introduced. The DDI-Ben benchmarking framework addresses this by simulating distribution changes, revealing that most existing approaches suffer substantial performance degradation under such conditions [41].

• Interpretability and Explainability: While substructure attention mechanisms in models like DGNN-DDI and MGDDI provide some interpretability by highlighting chemically relevant substructures [39] [40], further work is needed to fully explain the biological mechanisms behind predicted interactions.

• Integration of Multimodal Data: Future approaches should incorporate additional data sources beyond molecular structures, such as drug target information, metabolic pathways, and patient-specific factors. Methods like MKG-FENN, which constructs multimodal knowledge graphs, represent promising directions [37].

• Clinical Translation and Validation: Bridging the gap between computational predictions and clinical applications remains challenging. Future research should focus on validating predictions against real-world clinical data and incorporating pharmacogenomic variables to enhance predictive models of interaction risk [38].

The accurate simulation of atomic systems is a cornerstone of research in materials science and drug development. For decades, a significant trade-off has persisted between the quantum-mechanical accuracy of methods like density functional theory (DFT) and the computational efficiency of classical molecular dynamics. DFT calculations, while accurate, scale poorly with system size, making them prohibitively expensive for large systems or long timescales [8]. Neural Network Potentials (NNPs) have emerged as a transformative technology that resolves this dilemma, offering near-DFT accuracy at a fraction of the computational cost [42] [8]. By training on high-fidelity quantum mechanical data, NNPs learn the underlying potential energy surface (PES), enabling rapid, reliable atomistic simulations that were previously infeasible. This is particularly impactful for the study of interatomic interactions and stability predictions, where Graph Neural Network (GNN) architectures provide a natural and powerful framework for modeling atomic systems as collections of nodes (atoms) and edges (bonds or interactions) [43] [8].

Key Architectures and Innovations in Modern NNPs

The rapid advancement of NNPs is largely driven by innovations in model architectures that embed physical laws and improve data efficiency.

Equivariant Graph Neural Networks

A critical breakthrough has been the development of equivariant GNNs, which explicitly embed physical symmetries—specifically, rotational, translational, and sometimes inversion symmetry (together known as E(3) equivariance)—directly into the model architecture [8]. Unlike models that merely output invariant quantities, equivariant architectures maintain internal feature representations that transform predictably under symmetry operations. This ensures that scalar outputs like total energy are invariant, while vector outputs like forces transform correctly as the system is rotated or translated [42] [8]. Frameworks like MACE and NequIP leverage higher-body-order messages and tensor products to achieve this, leading to superior data efficiency and accuracy [23] [42]. For example, the Egret-1 family of models, based on the MACE architecture, demonstrates that such models can equal or exceed the accuracy of routinely used quantum-chemical methods for tasks like torsional scans and geometry optimization [42].

Integrating Physics and Long-Range Interactions

Modern NNPs are increasingly moving beyond short-range interactions by incorporating physics-based long-range terms. The AIMNet2 model exemplifies this approach, expressing the total energy as a sum of a machine-learned local term (ULocal), an explicit dispersion correction (UDisp), and an electrostatic term (UCoul) calculated from atom-centered partial point charges that are iteratively refined during message passing [44]. This hybrid strategy combines the flexibility of machine learning with the proven accuracy of physical models for non-local interactions, making the potential applicable to a wider range of systems, including charged and polar species.

Foundational Potentials and Transfer Learning

The field is witnessing a push towards foundational interatomic potentials—single, universal models trained on massive datasets encompassing a vast region of chemical space. The GRACE (Graph Atomic Cluster Expansion) framework, for instance, was trained on the OMat24 dataset of 110 million DFT calculations, aiming for uniform accuracy across the periodic table [45]. Complementing this, transfer learning has proven highly effective for adapting powerful pre-trained models to specific domains with minimal additional data. The EMFF-2025 potential for high-energy materials (HEMs) was developed by applying transfer learning to a pre-trained model, allowing it to achieve DFT-level accuracy for the mechanical properties and decomposition mechanisms of 20 different HEMs with minimal new DFT calculations [24].

Quantitative Performance Benchmarks

The ultimate value of NNPs is demonstrated through rigorous benchmarking against DFT and experimental data across diverse properties.

Accuracy on Energy and Force Predictions

The core task of any NNP is the accurate prediction of energies and forces. The EMFF-2025 potential demonstrates strong performance, with mean absolute errors (MAE) for energy predominantly within ± 0.1 eV/atom and force MAE mainly within ± 2 eV/Å across a wide temperature range for 20 different high-energy materials [24]. On a different front, models like Egret-1 have been shown to achieve chemical accuracy (error < 1 kcal/mol ≈ 0.043 eV/atom) on standard quantum-chemical tasks, positioning them as reliable replacements for direct DFT calculations in many scenarios [42].

Table 1: Benchmarking NNP Accuracy on Energy and Force Predictions

Model / Study	System Type	Energy MAE	Force MAE	Reference Method
EMFF-2025 [24]	C, H, N, O High-Energy Materials	< ± 0.1 eV/atom	< ± 2 eV/Å	DFT
Egret-1 [42]	Organic & Biomolecules	~1 kcal/mol (≈ 0.043 eV/atom)	N/R	ωB97M-D3BJ/def2-TZVPPD
AIMNet2 [44]	Diverse Molecules (14 elements)	N/R	Outperforms GFN2-xTB, on par with reference DFT	Hybrid DFT

Performance on Material Properties and Efficiency

Beyond energies and forces, the ability to predict derived mechanical and thermodynamic properties is crucial. A systematic benchmark of universal MLIPs (uMLIPs) on nearly 11,000 elastically stable materials provides a clear comparison of modern models [23]. Furthermore, the computational speedup offered by NNPs is profound. The Egret-1 models offer multiple-order-of-magnitude speedups relative to legacy quantum-chemical methods [42]. Similarly, an NNP developed for estimating solvation free energies was nearly 1,000 times faster than its DFT counterpart while retaining 89% accuracy [46].

Table 2: Performance of Universal MLIPs on Elastic Property Prediction [23]

Model	Bulk Modulus MAE (GPa)	Shear Modulus MAE (GPa)	Young's Modulus MAE (GPa)	Poisson's Ratio MAE
SevenNet	9.84	9.13	21.89	0.014
MACE	12.33	11.50	27.20	0.018
MatterSim	14.75	13.87	32.65	0.021
CHGNet	23.66	21.89	51.51	0.032

Detailed Experimental Protocols

To ensure reproducibility and facilitate adoption, here are detailed methodologies for key applications of NNPs.

Protocol: Thermal Decomposition of Energetic Materials using EMFF-2025

This protocol uses the EMFF-2025 potential to study the high-temperature decomposition mechanisms of high-energy materials (HEMs) like RDX or HMX [24].

Workflow Overview:

Step-by-Step Procedure:

• Initial Structure Preparation:

  • Obtain the crystal structure of the HEM (e.g., from the Cambridge Structural Database).
  • Create a supercell (e.g., 2x2x2 or 3x3x3) containing at least 1,000 atoms to adequately model collective phenomena and mitigate finite-size effects.
  • Use classical force fields to pre-equilibrate the supercell at the target density and 300 K for 100 ps.

• Model Initialization:

  • Employ a molecular dynamics engine compatible with the Deep Potential (DP) format, such as LAMMPS with the DeePMD-kit package.
  • Load the pre-trained EMFF-2025 potential file and its corresponding descriptor parameters [24].

• Equilibrium Molecular Dynamics:

  • Run an NPT (isothermal-isobaric) simulation for 50-100 ps at 300 K and 1 atm to fully relax the simulation box and ensure the system is in a stable thermodynamic state.
  • Use a Nosé-Hoover thermostat and barostat with relaxation times of 100 fs and 1 ps, respectively.
  • Set the timestep to 0.5 fs to accurately capture high-frequency molecular vibrations.

• Production MD for Decomposition:

  • Using the final configuration from Step 3, initiate a high-temperature NVT (canonical ensemble) simulation.
  • Rapidly heat the system to the target decomposition temperature (e.g., 1500-3000 K) to accelerate the reaction kinetics within the simulation timeframe.
  • Run the simulation for 1-10 ns, saving the atomic positions, velocities, and forces every 10-50 fs for subsequent analysis.

• Trajectory Analysis and Mechanism Identification:

  • Use visualization software (e.g., OVITO, VMD) to qualitatively observe the reaction pathway and identify major decomposition fragments.
  • Perform Principal Component Analysis (PCA) on the atomic trajectory to reduce dimensionality and identify the primary collective motions driving the decomposition [24].
  • Construct atomic correlation heatmaps to quantify the relationships between the evolution of different molecular groups and reveal common reaction pathways across different HEMs.

Protocol: Calculating Solvation Free Energies with Equivariant NNPs

This protocol leverages the data efficiency of equivariant NNPs, like those based on the NequIP architecture, to compute solvation free energies (SFEs) with DFT-level accuracy [46].

Workflow Overview:

Step-by-Step Procedure:

• Solute Conformer Sampling (Gas Phase):

  • Select the solute molecule from a database like FreeSolv.
  • Generate a diverse set of low-energy conformers (e.g., using the ETKDG algorithm and metadynamics with GFN2-xTB, as done in the "Finch" dataset generation) [42].
  • Optimize the geometry of each conformer using the gas phase NNP to obtain the minimum-energy structure.

• Single-Point Calculations (Gas Phase NNP):

  • For the optimized gas-phase conformer, perform a single-point energy calculation using the gas-phase specialized NNP (e.g., Egret-1).
  • This yields the gas-phase energy, ( E_{gas} ).

• Implicit Solvent Calculations (SMD Model):

  • Use the same optimized geometry from Step 2.
  • Perform a single-point energy calculation using the implicit solvation NNP, which incorporates a continuum solvent model like SMD into its potential. This yields the solvated energy, ( E_{solv} ).

• Free Energy Calculation:

  • The solvation free energy is calculated as ( ΔG{solv} = E{solv} - E_{gas} ), where the potential energy difference approximates the free energy change in this context.
  • For greater accuracy, especially for flexible molecules, perform alchemical free energy perturbation (FEP) or thermodynamic integration (TI) using the NNP, gradually coupling the solute with the implicit solvent environment.

This table catalogs key software, datasets, and models that form the essential toolkit for researchers developing and applying NNPs.

Table 3: Key Research Reagents and Resources for NNP Development

Category	Name	Description / Function
Model Architectures	MACE [23] [42]	A high-body-order equivariant message passing model that combines the completeness of ACE with the scalability of GNNs.
	DeePMD [24] [8]	A widely used framework that constructs potentials using local environment descriptors and deep neural networks.
	GRACE [45]	(Graph Atomic Cluster Expansion) A foundational potential framework using a complete graph basis for efficient chemical embedding.
Benchmark Datasets	OMat24 [45]	A massive dataset of 110 million DFT calculations across 89 elements, used for training foundational potentials.
	MD17/MD22 [8]	Molecular dynamics trajectories of organic molecules and biomolecular fragments for benchmarking energy and force predictions.
	FreeSolv [46]	A database of experimental and calculated hydration free energies for small molecules, used for solvation studies.
Software & Tools	DP-GEN [24]	An active learning framework for generating training data and building robust NNPs efficiently.
	DeePMD-kit [8]	The primary software package for training and running DeePMD models, often integrated with LAMMPS.
Validation Methods	PCA & Correlation Heatmaps [24]	Used to analyze MD trajectories, map chemical space, and identify reaction mechanisms from simulation data.
	MatBench Discovery [45]	A benchmark for validating formation energy and thermodynamic stability predictions.

The field of NNPs is rapidly evolving. Key future directions include enhancing model interpretability to build trust and provide physical insights, developing improved strategies for handling long-range electrostatic interactions, and creating more sophisticated multi-fidelity frameworks that can learn from data of varying quality [42] [8]. The emergence of foundational potentials like GRACE, trained on millions of structures, promises a future where a single, universal model can provide accurate simulations for virtually any atomic system, dramatically accelerating the design of new materials and therapeutic molecules [45].

In conclusion, Neural Network Potentials have successfully bridged the long-standing gap between computational accuracy and efficiency in atomistic simulation. By leveraging graph-based representations and physical principles, they achieve DFT-level accuracy while being orders of magnitude faster. As architectures, datasets, and training methodologies continue to mature, NNPs are poised to become an indispensable tool in the computational researcher's arsenal, fundamentally changing the pace and scope of discovery in computational chemistry and materials science.

The accurate prediction of structural evolution and kinetic properties is a cornerstone of modern molecular dynamics (MD) simulations in materials science and drug discovery. Traditional methods, particularly ab initio MD, provide high fidelity but are computationally prohibitive for large systems or long timescales [47]. The integration of Graph Neural Networks (GNNs) has emerged as a transformative approach, serving as machine learning interatomic potentials (MLIPs) that bridge the gap between the accuracy of quantum mechanical methods and the efficiency of classical force fields [33] [48]. This document details the application of GNNs for these tasks, framing them within a broader research thesis on predicting interatomic interactions and material stability.

GNNs as Machine Learning Interatomic Potentials

GNNs are uniquely suited for modeling molecular systems because they operate directly on the graph representation of a molecule or material, where atoms are nodes and interatomic interactions are edges [12]. The core operational principle is message passing, where information on the graph is iteratively updated by propagating and aggregating features between connected nodes [6] [12]. For MD, the primary objective is to learn a mapping from the atomic configuration (positions and species) to the total potential energy and the atomic forces, which are critical for driving dynamics [33] [47].

Key Architectural Advances for MD

Recent GNN architectures have introduced specific innovations to enhance the accuracy and efficiency of molecular simulations.

• Rotation Invariance: A fundamental requirement for potential energy surfaces is that they must be invariant to rotational and translational changes of the entire system. Models like the Moment Graph Neural Network (MGNN) and InvarNet explicitly build in this rotational invariance, often by utilizing moment representations or specialized loss functions, ensuring that the energy is independent of the molecule's orientation in space [33] [25].
• Spatial Relationship Encoding: Beyond the simple molecular graph, 3D spatial relationships are crucial. Architectures like MGNN capitalize on this by capturing nuanced spatial relationships in 3D molecular graphs, which is essential for accurately modeling molecular dynamics and predicting forces [33].

Performance Benchmarks and Quantitative Analysis

The performance of GNN-based MLIPs is rigorously evaluated on public benchmark datasets. The table below summarizes the state-of-the-art performance of the MGNN model on several key benchmarks.

Table 1: Performance of MGNN on Public Benchmark Datasets [33]

Dataset	Description	Key Metric	MGNN Performance
QM9	Quantum-mechanical properties for small organic molecules [12]	Multiple state-of-the-art results	Achieved multiple state-of-the-art results on molecular property prediction tasks.
Revised MD17	Molecular dynamics trajectories for small molecules [33]	Accuracy in energy and force prediction	Delivered state-of-the-art results.
MD17-Ethanol	A subset of MD17 focusing on ethanol	Accuracy in energy and force prediction	Delivered state-of-the-art results.
Amorphous Electrolytes	Complex disordered systems for battery research	Consistency with ab-initio simulations	Accurately predicted structural and kinetic properties, closely aligning with ab-initio results.

These benchmarks demonstrate that GNNs can not only match but exceed the performance of traditional models, providing a robust tool for simulating both ordered and highly complex disordered systems [33] [47].

Application Notes & Protocols

This section provides detailed methodologies for implementing GNN-MD in research workflows.

Protocol: Simulating Amorphous Materials with GNN-MD and Symbolic Regression

This protocol, adapted from [47], outlines a hybrid approach to derive interpretable analytical potential functions for disordered systems.

Objective: To model the potential energy surface of a disordered system (e.g., an amorphous Lennard-Jones system) and derive a closed-form analytical potential function.

Workflow Overview:

Materials & Reagents:

• Software: Molecular dynamics simulation package (e.g., LAMMPS, GROMACS); Python with PyTorch/PyTorch Geometric for GNN development; Symbolic Regression library (e.g., gplearn).
• Dataset: A normalized Lennard-Jones system with 128 atoms in an NVT ensemble [47].

Procedure:

• Generate MD Dataset:

  • Perform MD simulations at varying temperatures to sample a wide range of atomic configurations.
  • For each configuration, record the 3D coordinates of all atoms (input) and the corresponding total potential energy of the system (target).
  • Ensure data points are uncorrelated by allowing sufficient simulation time between recordings.
  • Normalize the potential energy values across the dataset using Z-score normalization [47].

• Train the GNN Model:

  • Graph Construction: Convert each atomic configuration into a graph. Represent atoms as nodes and define edges between all atom pairs within a specified cutoff distance (e.g., half the simulation box length). Node attributes can be atom type; edge attributes can be the vector of spatial coordinate differences [47].
  • Model Training: Train a message-passing GNN to predict the total potential energy of the system from the graph input. The loss function is typically the Mean Squared Error (MSE) between the predicted and true potential energy [47].

• Generate GNN Dataset:

  • Pass a subset of the MD dataset through the trained GNN.
  • Extract the output of the model's edge-level representations. These outputs, paired with the corresponding interatomic distances, form a new dataset. This dataset implicitly learns the pair potential energy contributions [47].

• Train Symbolic Regression Model:

  • Use the GNN dataset (pairwise distances as input, edge outputs as target) to train a symbolic regression model.
  • The SR model will search a space of mathematical expressions to find an analytical function that best fits the pair potential data without prior physical assumptions [47].

• Validation:

  • Integrate the derived analytical function into an MD simulator.
  • Run new simulations and compare the resulting structural and kinetic properties against the original data or higher-fidelity ab initio results to validate accuracy [47].

Protocol: Universal Potential for Property Prediction

This protocol leverages pre-trained universal GNN potentials for direct property prediction.

Objective: To use a pre-trained model like MGNN or M3GNet to predict various molecular and material properties without system-specific training [33].

Workflow Overview:

Procedure:

• Input Preparation: Provide the model with the atomic numbers (charge quantities) and 3D Cartesian coordinates of all atoms in the structure [33].
• Model Forward Pass: The input is processed through the universal GNN's embedding block and multiple interaction layers. These layers use message passing to capture complex interatomic interactions [33].
• Task-Specific Readout: The final atomic representations are passed through a target-dependent output block. This block is designed to aggregate information and produce the desired output, whether scalar (e.g., energy), vector (e.g., forces), or tensor (e.g., polarizability) [33].
• Prediction: The model outputs the predicted property, which can be used for virtual screening, materials design, or as a force field for MD simulations.

The Scientist's Toolkit

Table 2: Essential Research Reagents and Resources for GNN-Driven MD

Item Name	Function/Description	Example Use Case
Benchmark Datasets (QM9, MD17)	Standardized public datasets for training and benchmarking model performance on quantum chemical properties and molecular forces [33] [6].	Comparing the accuracy of a new GNN architecture against existing state-of-the-art models.
Universal Pre-trained Models (MGNN, M3GNet)	GNNs trained on diverse materials data, providing a strong starting point for property prediction without needing extensive, system-specific data [33] [48].	Rapid screening of new molecular candidates for target properties or initializing a force field for a new material system.
Message-Passing GNN Framework	The underlying software architecture (e.g., PyTorch Geometric) that defines how information is exchanged between atoms in a graph to learn representations [6] [12].	Building custom MLIPs tailored to specific chemical systems or novel properties.
Symbolic Regression (SR)	A machine learning method that discovers analytical mathematical expressions that fit a given dataset, moving from a "black box" NN to an interpretable function [47].	Deriving a human-interpretable interatomic potential function from a trained GNN model.
Molecular Dynamics Engine	Software (e.g., LAMMPS) that performs the actual simulation of atomic movements over time, which can be coupled with a GNN-based force field [47].	Running large-scale, long-timescale simulations of molecular systems using forces predicted by a GNN.

Multi-scale modeling represents a paradigm shift in computational material science and drug development, aiming to predict macroscopic material behavior from first principles by explicitly bridging atomic and microscopic interactions. Traditional simulation methods, such as finite element (FE) models or quantum mechanical calculations, provide high accuracy but are often computationally prohibitive for practical applications, especially when modeling complex, nonlinear material responses [49] [24]. The emergence of Graph Neural Networks (GNNs) offers a transformative solution by leveraging geometric deep learning to create accurate, data-efficient, and vastly accelerated surrogate models. These models learn the fundamental physics of interatomic interactions and microstructural mechanics, enabling seamless information passing across scales while maintaining physical consistency and interpretability [50] [51]. This article details protocols and applications of GNNs in multi-scale modeling, providing researchers with practical frameworks for implementing these advanced computational techniques.

Graph Neural Network Architectures for Multi-Scale Modeling

Core Architectural Principles

GNNs applied to multi-scale modeling share several foundational principles: graph-based representations of material structure (at atomic or microstructural levels), message-passing mechanisms for information propagation between connected nodes, and physical constraints embedded within the network architecture to ensure predictions obey fundamental laws [50] [51]. Unlike conventional deep learning models, specialized GNNs for physical systems often incorporate SE(3)-equivariance – invariance to rotation and translation in 3D space – which is crucial for correctly modeling atomic interactions and material symmetries [50]. Furthermore, many successful architectures employ hybrid data-physics approaches, where the GNN predicts certain quantities (e.g., microscopic strains or atomic energies) while embedded physical models (e.g., constitutive material laws or interatomic potentials) compute others, ensuring physical consistency and reducing data requirements [49] [24].

Specialized Architectures and Their Applications

Table 1: Specialized GNN Architectures for Multi-Scale Modeling

Architecture	Primary Application Scale	Key Innovation	Representative Model
Microstructure-based GNN [49] [52]	Micro to Macro	Predicts full-field microscopic strains; retains microscopic constitutive model for stresses	Hybrid GNN-FE Surrogate
Geometric Deep Learning Models [50]	Atomic to Molecular	SE(3)-equivariant architecture; learns universal representations of intermolecular interfaces	ATOMICA
Graph-Enhanced Deep Material Network [51]	Micro to Macro	GNN derives parameters for physics-based Deep Material Network (DMN)	Hybrid GNN-DMN
Multi-Scale Crystal Graph Network [16]	Atomic to Crystal	Models interatomic interactions at different scales while preserving periodicity invariance	MHACGN-MS
Neural Network Potentials (NNPs) [24]	Atomic	Achieves DFT-level accuracy with significantly lower computational cost	EMFF-2025

Application Notes: Key Domains and Workflows

Predicting Macroscopic Material Response from Microstructure

Application Objective: Accelerate nonlinear multiscale simulations (FE²) by replacing expensive microscale Finite Element models with a GNN surrogate that predicts homogenized macroscopic quantities from microscopic structures [49] [52].

Workflow Protocol:

• Microstructure Graph Representation: Convert the Representative Volume Element (RVE) of the microstructure into a graph where nodes represent material points or integration domains, and edges represent spatial connectivity and potential mechanical influence.
• Strain Prediction: The GNN, trained on various microstructures and loading conditions, takes macroscopic strains and the microstructure graph as input and predicts the full-field microscopic strain response.
• Physics Integration: The predicted microscopic strains are fed into a retained, known constitutive material model (e.g., an elasto-plastic law) to compute the corresponding microscopic stresses. This hybrid approach ensures physical consistency.
• Homogenization: The microscopic stresses are numerically homogenized to obtain the macroscopic stress tensor, completing the multiscale cycle for that integration point and time step.

Figure 1: Workflow for a GNN-based surrogate model accelerating FE² simulation.

Universal Modeling of Molecular Interactions

Application Objective: Learn a unified representation of intermolecular interactions across diverse molecular modalities (proteins, small molecules, nucleic acids, lipids, metal ions) to predict binding affinities, annotate functional sites, and understand molecular function at scale [50].

Workflow Protocol:

• Hierarchical Graph Construction: Represent an intermolecular complex as a two-level graph. The atomic-level graph nodes are atoms (with element type and 3D coordinates), while the block-level graph nodes are chemically meaningful blocks (e.g., amino acids, nucleotides, functional groups).
• Self-Supervised Pretraining: Train a geometric GNN (like ATOMICA) using two primary tasks:
  • Denoising: Reconstruct the original interaction complex graph after it has been perturbed by random rotations and torsion angle rotations.
  • Masked Block Prediction: Identify the type of randomly masked chemical blocks at the interaction interface.
• Embedding and Analysis: Generate multi-scale embeddings (atom, block, and entire interface). These embeddings are compositional and can be algebraically combined to predict interactions for novel molecular complexes.
• Downstream Task Finetuning: Add a task-specific adapter head and finetune the model for predictive tasks such as binding site annotation or binding affinity prediction.

Discovery and Optimization of Energetic Materials

Application Objective: Rapidly and accurately predict the structure, mechanical properties, and decomposition characteristics of high-energy materials (HEMs) with Density Functional Theory (DFT)-level accuracy but at a fraction of the computational cost [24].

Workflow Protocol:

• Potential Energy Surface Sampling: Use DFT calculations to generate a diverse set of atomic configurations and their corresponding energies and forces for target HEMs.
• Transfer Learning with a Pre-trained NNP: Initialize a general Neural Network Potential (NNP), such as the EMFF-2025 model, which has been pre-trained on a broad dataset of C, H, N, O systems.
• Model Finetuning: Employ a Deep Potential GENerator (DP-GEN) framework to iteratively fine-tune the pre-trained NNP by incorporating a minimal amount of new, system-specific training data from DFT.
• Property Prediction: Use the finetuned NNP to run large-scale molecular dynamics (MD) simulations, enabling the prediction of crystal structures, mechanical moduli, and thermal decomposition pathways of HEMs.

Experimental Protocols

Protocol: Developing a Microstructure-Based GNN Surrogate

Objective: Train a GNN to replace the microscale FE solver in a concurrent multiscale simulation for an elasto-plastic material [49] [52].

Materials & Data:

• Training Data Generation: A dataset generated from high-fidelity FE simulations of various RVEs under different loading paths. Each data point includes: input macroscopic strain, microstructure graph, and output microscopic strain field.
• Software: A deep learning framework (e.g., PyTorch, TensorFlow) with a GNN library (e.g., PyTorch Geometric).

Procedure:

• Data Generation & Mesh Conversion: Run offline FE simulations on a variety of microstructures. Convert each FE mesh into a graph where nodes correspond to integration points or elements, with features encoding position, material phase, and history (if any). Edges represent connectivity for mechanical influence.
• GNN Architecture Configuration: Implement a GNN architecture using multiple message-passing layers. The model should input node features (e.g., current strain) and output updated node features (e.g., predicted strain).
• Hybrid Training Loop: a. The GNN predicts the full-field microscopic strains. b. The embedded constitutive model computes stresses from the predicted strains. c. The loss function is computed using both the microscopic strains and stresses against the FE-generated ground truth: Loss = α * Loss_strain + β * Loss_stress.
• Generalization Training: Train the model on a wide variety of meshes and microstructures to ensure it generalizes to unseen topological arrangements.
• Validation & Integration: Validate the surrogate by comparing its stress-strain predictions for a new microstructure against a full FE simulation. Once validated, it can be integrated into the macroscopic FE solver, replacing the call to the microscale FE model.

Protocol: Building a General Neural Network Potential via Transfer Learning

Objective: Develop a general NNP for HEMs containing C, H, N, O elements that predicts both mechanical properties and chemical reactivity [24].

Materials & Data:

• Base Model: A pre-trained NNP on a large database of C, H, N, O systems (e.g., DP-CHNO-2024).
• Target Data: A smaller set of DFT calculations (structures, energies, forces) for specific HEMs of interest.

Procedure:

• Target System Sampling: Use ab-initio MD or enhanced sampling methods to generate representative atomic configurations for the target HEMs.
• DP-GEN Iteration: a. Exploration: Run MD simulations using the current NNP to explore new configurations. b. Labeling: Select configurations where the model is uncertain and compute their accurate energies and forces using DFT. c. Training: Fine-tune the NNP on the newly labeled data, incorporating it into the training set. d. Convergence Check: Repeat steps a-c until the model's error (Mean Absolute Error for energy and force) converges to a satisfactory level (e.g., energy MAE < 0.1 eV/atom, force MAE < 2 eV/Å).
• Model Validation: Validate the final EMFF-2025 model by comparing its predictions of crystal parameters, elastic constants, and decomposition initiation temperatures against experimental data and high-level DFT results.

Table 2: Performance Metrics of the EMFF-2025 Neural Network Potential [24]

Predicted Quantity	Target Accuracy	Validation Method	Reported Performance
Atomic Energy	DFT-level	MAE vs. DFT calculations	MAE predominantly within ± 0.1 eV/atom
Atomic Forces	DFT-level	MAE vs. DFT calculations	MAE predominantly within ± 2 eV/Å
Crystal Structures	Experimental data	Lattice parameters comparison	Excellent agreement
Mechanical Properties	Experimental data	Elastic moduli comparison	Good agreement
Decomposition Mechanisms	DFT/Experiment	Reaction pathways and products	Challenged conventional material-specific behavior

The Scientist's Toolkit: Research Reagent Solutions

Table 3: Essential Computational Tools and Datasets for GNN-based Multi-Scale Modeling

Resource Name	Type	Primary Function	Relevant Application
DP-GEN [24]	Software Framework	Automates the iterative construction of training datasets and training of reliable Neural Network Potentials.	Energetic Materials Design
ATOMICA [50]	Pre-trained Model & Dataset	A universal geometric deep learning model for atomic-scale representations of intermolecular interfaces across five modalities.	Drug Discovery, Molecular Biology
TUDataset [11]	Benchmark Data	A collection of graph-based datasets for machine learning, covering molecular, biological, and social networks.	Method Development & Benchmarking
Open Graph Benchmark (OGB) [11]	Benchmark Data	Large-scale, diverse, and realistic benchmark datasets for graph ML.	Method Development & Benchmarking
Q-BioLiP & CSD [50]	Dataset	Curated databases of protein-ligand interactions (Q-BioLiP) and small molecule crystal structures (CSD) for training interaction models.	Universal Molecular Interaction Modeling

Critical Considerations and Future Directions

Stability and Generalization of GNNs

A significant challenge in deploying GNNs for multi-scale modeling is their performance under Out-of-Distribution (OOD) conditions, where test data distribution differs from the training data [11]. Distribution shifts can arise from data selection bias or confounding factors, leading to unreliable predictions. Stable learning approaches for GNNs have been proposed to mitigate this. These methods aim to remove spurious correlations between features by applying feature sample weighting decorrelation in the random Fourier transform space, forcing the model to rely on genuine causal features for predictions, thereby improving OOD generalization [11].

The Depth versus Interpretability Trade-off

In graph deep learning, most successful architectures are surprisingly shallow, often comprising only a few layers [53]. This contrasts with deep convolutional networks in computer vision. Excessive depth in GNNs can exacerbate problems like vanishing gradients and overfitting, and may reduce model interpretability. Furthermore, very deep GNNs can lead to "over-smoothing," where node features become indistinguishable. Therefore, careful architectural design prioritizing inductive biases from physics (e.g., through hybrid models) over excessive depth is often more effective for multi-scale modeling tasks [49] [53].

Figure 2: Challenges and mitigation strategies in GNN deployment for multi-scale modeling.

Overcoming Instability and Uncertainty: Strategies for Robust and Reliable Predictions

Molecular dynamics (MD) simulation is a cornerstone of computational science, enabling high-resolution spatiotemporal modeling of atomistic systems across biology, chemistry, and materials science [54]. The accuracy of these simulations hinges on the precision of the interatomic potential energy surface (PES), which governs atomic interactions and system evolution [47] [54]. While neural network interatomic potentials (NNIPs) have emerged as powerful surrogates for expensive quantum-mechanical calculations, they frequently produce unphysical simulations that irreversibly enter non-physical regions of phase space, leading to simulation collapse [54]. This stability challenge fundamentally limits the application of NNIPs for modeling long-timescale phenomena and rare events critical to drug discovery and materials design [55] [54].

The core instability problem stems from several interconnected factors: insufficient coverage of molecular conformations in training datasets, error accumulation during simulation, and inherent limitations in how NNIP architectures capture complex many-body interactions [55] [54]. Even minor prediction errors can destabilize entire MD trajectories, particularly when simulations extrapolate to unseen conformations beyond the training data distribution [55]. This article examines recent methodological advances that address these stability challenges through novel graph neural network architectures, specialized training protocols, and innovative integration of physical principles.

Quantitative Analysis of GNN Performance on Molecular Stability Benchmarks

Recent research has produced multiple graph neural network architectures specifically designed to address stability challenges in molecular dynamics simulations. The table below summarizes quantitative performance metrics across key benchmark datasets, demonstrating the effectiveness of these approaches in improving both accuracy and stability.

Table 1: Performance Comparison of GNN Architectures on Molecular Stability Benchmarks

Model	Architecture Type	Key Innovation	Benchmark Performance	Stability Improvement
GGND [55]	Geometric Graph Neural Diffusion	Iterative refinement of atomic representations with equivariance	3BPA, SAMD23 datasets	Enables stable MD simulations under topological shifts
MGNN [33]	Moment Graph Neural Network	Moment representation learning with Chebyshev polynomials	State-of-the-art on QM9, revised MD17, MD17-ethanol	Accurately predicts structural and kinetic properties in amorphous electrolytes
StABlE-Trained Models [54]	Multi-architecture (SchNet, NequIP, GemNet-T)	Differentiable Boltzmann estimators with observable-based training	Aspirin: Median stability ↑ 35ps to 140psWater: Median stability ↑ 23ps to 52ps	87% accuracy improvement in diffusivity coefficient estimation
KA-GNN [29]	Kolmogorov-Arnold Network Integration	Fourier-based KAN modules in node embedding, message passing, and readout	Seven molecular benchmarks	Superior accuracy and computational efficiency
EMFF-2025 [24]	General Neural Network Potential	Transfer learning with minimal DFT data for CHNO systems	20 high-energy materials	Predicts mechanical properties and decomposition characteristics with DFT-level accuracy

The quantitative evidence demonstrates that stability-aware training methodologies and specialized architectures can significantly enhance simulation reliability. The StABlE training approach is particularly noteworthy, achieving dramatic stability improvements – quadrupling simulation stability for aspirin and more than doubling it for water systems [54]. Similarly, MGNN delivers state-of-the-art results across multiple benchmarks while maintaining robust performance in dynamic simulations of complex systems like amorphous electrolytes [33].

Experimental Protocols for Enhanced Simulation Stability

Protocol: Stability-Aware Boltzmann Estimator (StABlE) Training

The StABlE training methodology addresses simulation instability by combining conventional supervised learning with reference system observables, creating a multi-modal training procedure that corrects instabilities as they are discovered [54].

Table 2: Research Reagent Solutions for StABlE Training Implementation

Component	Specifications	Function/Purpose
Reference Data	Quantum-mechanical energies and forces from DFT calculations	Provides baseline physical accuracy for energy and force predictions
System Observables	Radial distribution function, virial stress tensor, diffusivity coefficient	Enables physical consistency with experimental or high-fidelity simulation data
Boltzmann Estimator	Custom gradient computation framework	Allows efficient training to system observables without differentiating through MD simulations
Instability Detection	MD simulation-based exploration during training	Identifies unphysical regions of phase space for targeted refinement
NNIP Architectures	SchNet, NequIP, GemNet-T implementations	Base models compatible with the StABlE training framework

Step-by-Step Procedure:

• Initial Supervised Pre-training: Begin with conventional training of the NNIP using available quantum-mechanical reference data (energies and forces) to establish baseline accuracy [54].

• Iterative Stability Exploration:

  • Run MD simulations using the current NNIP to explore regions of phase space
  • Monitor for instability indicators: unphysical bond breaking/formation, energy drift, or irregular force patterns
  • Flag unstable trajectories and corresponding atomic configurations [54]

• Observable-Based Refinement:

  • For identified unstable configurations, compute reference observables (either from high-fidelity simulation or experimental data)
  • Utilize the Boltzmann Estimator to calculate gradients with respect to these observables
  • Update NNIP parameters to align with reference observables while preserving energy/force accuracy [54]

• Convergence Checking: Iterate steps 2-3 until simulation stability plateaus, typically requiring 3-5 cycles for significant improvement.

The StABlE training procedure is enabled by the Boltzmann Estimator, which provides efficient gradient computation without the numerical instability of differentiating through entire MD simulations. This approach has demonstrated particular effectiveness in scenarios with limited reference data, in some cases outperforming models trained on datasets 50 times larger [54].

Protocol: Geometric Graph Neural Diffusion (GGND) Implementation

GGND addresses stability challenges by capturing geometrically invariant topological features, enabling robust information flow between atomic pairs while maintaining physical equivariance [55].

Implementation Workflow:

• Graph Representation:

  • Represent molecular system as a graph with atoms as nodes and edges within a defined cutoff distance
  • Encode atomic positions, features, and relationships into initial node and edge embeddings [55]

• Iterative Representation Refinement:

  • Implement diffusion-based message passing that allows instantaneous information flow between arbitrary atomic pairs
  • Maintain rotational and translational equivariance through geometric constraints
  • Iteratively update atomic representations through multiple diffusion layers [55]

• Stability-Optimized Output:

  • Predict energies and forces with built-in physical constraints
  • Ensure consistent behavior across molecular conformations, including those outside training distribution [55]

• Validation Framework:

  • Test on diverse molecular conformations across varied temperatures (3BPA and SAMD23 benchmarks)
  • Evaluate stability through long-timescale MD simulations measuring error accumulation [55]

GGND functions as a plug-and-play module that integrates with existing equivariant message-passing frameworks, enhancing their predictive stability without requiring architectural overhaul [55].

GGND Architecture: Implements iterative diffusion for stable dynamics

Advanced Architectures for Molecular Stability

Moment Graph Neural Networks (MGNN)

MGNN leverages moment representation learning to capture nuanced spatial relationships in 3D molecular structures while maintaining rotational invariance [33]. The architecture propagates information between atoms using moments—mathematical quantities that encapsulate spatial relationships between atoms within a molecule. Instead of conventional Bessel and Gaussian radial basis functions, MGNN utilizes Chebyshev polynomials to encode interatomic distance information, providing a provably rigorous framework for approximating any regular function satisfying all necessary physical symmetries [33].

The key innovation in MGNN lies in its systematic handling of molecular moments to convey relative spatial relationships. The framework processes molecular graphs through triplet interactions, where information from triplets (composed of nodes i, j, k and edges ij, ik) passes to edges and then to central nodes, creating a comprehensive representation of molecular structure that respects physical symmetries [33]. This approach has demonstrated exceptional generalizability across diverse systems including organic molecules, high-entropy alloys, and amorphous electrolytes, accurately predicting structural and kinetic properties with consistency closely aligned with ab-initio simulations [33].

Kolmogorov-Arnold Graph Neural Networks (KA-GNN)

KA-GNN integrates the emerging framework of Kolmogorov-Arnold networks (KANs) into graph neural networks for molecular property prediction [29]. Unlike conventional multi-layer perceptrons that use fixed activation functions on nodes with constant weights on edges, KANs adopt learnable univariate functions on edges, offering improved expressivity, parameter efficiency, and interpretability [29].

The KA-GNN framework systematically integrates Fourier-based KAN modules across all three core components of GNNs: node embedding initialization, message passing, and graph-level readout. This integration replaces conventional MLP-based transformations with adaptive, data-driven nonlinear mappings, constructing richer node embeddings and enabling more expressive graph-level representations [29]. The Fourier-series basis functions are particularly valuable for capturing both low-frequency and high-frequency structural patterns in graphs, enhancing the expressiveness of feature embedding and message aggregation [29].

Experimental results across seven molecular benchmarks show that KA-GNN variants consistently outperform conventional GNNs in both prediction accuracy and computational efficiency, while additionally providing improved interpretability by highlighting chemically meaningful substructures [29].

StABlE Training: Iterative cycle combining QM data and observables

Discussion and Future Perspectives

The stability challenge in molecular dynamics simulations represents a significant bottleneck in deploying neural network interatomic potentials for practical drug discovery and materials design. The methodologies discussed herein—from stability-aware training paradigms like StABlE to novel architectures like GGND and MGNN—demonstrate that substantial improvements are achievable through physics-informed learning frameworks.

A critical insight emerging from recent research is that conventional training solely on quantum-mechanical energies and forces, while necessary, is insufficient for ensuring simulation stability [54]. The incorporation of system observables and active exploration of unstable phase space regions provides a crucial corrective signal that enhances generalization to unseen conformations. Similarly, architectural innovations that explicitly maintain physical symmetries and enable efficient information propagation across molecular graphs contribute significantly to robustness.

Future research directions will likely focus on multi-scale modeling approaches that bridge electronic, atomic, and mesoscale phenomena, further improving the transferability of NNIPs across diverse chemical spaces [16] [24]. Additionally, the integration of active learning with stability-aware training paradigms represents a promising avenue for automated discovery and correction of instability modes with minimal human intervention. As these methodologies mature, they will increasingly enable reliable, long-timescale molecular simulations at quantum-mechanical accuracy, accelerating the design of novel therapeutics and functional materials.

Table 3: Stability Solution Comparison Guide

Solution Approach	Mechanism	Best-Suited Applications	Implementation Complexity
StABlE Training [54]	Combines QM data with reference observables via iterative refinement	Systems with limited reference data; requires observable targets	High (requires differentiable observables)
GGND Module [55]	Geometric diffusion with equivariant information flow	Systems requiring stability under topological changes	Medium (plugin for existing MPNNs)
MGNN Architecture [33]	Moment representation with Chebyshev polynomials	Universal molecular potentials across diverse systems	High (new architecture implementation)
KA-GNN Integration [29]	Fourier-KAN modules for enhanced expressivity	Molecular property prediction with interpretability needs	Medium (KAN integration into GNNs)
EMFF-2025 Framework [24]	Transfer learning for specific element sets	CHNO-containing systems like energetic materials	Low (leverages pre-trained base models)

For Graph Neural Networks (GNNs) predicting energies and forces in molecular and materials systems, reliable uncertainty quantification (UQ) is crucial for establishing trust in model outputs. The black-box nature of neural networks and their inherent stochasticity are significant deterrents for scientific applications, making uncertainty information at prediction time essential for adoption [56]. Uncertainty in this context originates from two fundamental sources: aleatoric uncertainty, which stems from inherent noise in the data, and epistemic uncertainty, which arises from limitations in the model training process [57]. Properly calibrating both types of uncertainty enables researchers to identify when predictions are reliable and when structures fall outside the model's learned domain, preventing erroneous scientific conclusions from unphysical simulations.

The challenge is particularly pronounced for GNNs operating on atomic systems, where accurate energy and force predictions are essential for molecular dynamics simulations. These models must distinguish between in-domain structures (similar to training data) and out-of-domain structures that may appear during simulation. Without proper UQ, errors on out-of-domain structures can compound over the course of a simulation, leading to inaccurate probability distributions, incorrect observables, or even unphysical results [56]. This framework provides comprehensive protocols for quantifying, calibrating, and interpreting both aleatoric and epistemic uncertainties specifically for GNNs predicting energy and forces.

Theoretical Framework for Aleatoric and Epistemic Uncertainty

Foundational Concepts and Definitions

In GNNs for interatomic potentials, aleatoric uncertainty represents the inherent stochasticity or noise in the observational data. For molecular systems, this may arise from: (1) probabilistic links in graph representations of molecular structures, (2) measurement noise in node feature vectors representing atomic properties, and (3) intrinsic variability in quantum mechanical calculations used as training data [58] [59]. Aleatoric uncertainty is irreducible through additional data collection alone, as it represents fundamental limitations in measurement precision or natural variability.

Conversely, epistemic uncertainty stems from limitations in model knowledge and training. This includes uncertainty in model parameters, architecture choices, and insufficient training data coverage across chemical space [57] [59]. Epistemic uncertainty is reducible through improved model architectures, additional training data in underrepresented regions of chemical space, or extended training. For foundation models trained across broad swaths of the periodic table, properly quantifying epistemic uncertainty is essential for identifying when the model is extrapolating beyond its reliable domain [56].

Mathematical Formulation

In a Bayesian framework for GNNs, the total predictive uncertainty can be decomposed into aleatoric and epistemic components. For a graph ( G=(V,E) ) with nodes ( V ) (atoms) and edges ( E ) (bonds), each node ( ui ) has feature vector ( hi ) representing atomic properties. The Bayesian GNN (BGNN) models the posterior predictive distribution for a target property ( y ) (e.g., energy or forces) as:

[ p(y|G) = \int p(y|\theta, G)p(\theta|G)d\theta ]

where ( \theta ) represents the model parameters. The total uncertainty can be quantified as the variance of this predictive distribution:

[ \text{Var}(y|G) = \underbrace{\mathbb{E}{\theta}[p(y|\theta, G)^2] - \mathbb{E}{\theta}[p(y|\theta, G)]^2}{\text{Epistemic}} + \underbrace{\mathbb{E}{\theta}[\text{Var}(y|\theta, G)]}_{\text{Aleatoric}} ]

This separation allows researchers to distinguish between uncertainty arising from the model itself (epistemic) versus uncertainty inherent in the data (aleatoric) [59].

Table 1: Characteristics of Aleatoric vs. Epistemic Uncertainty in GNN Interatomic Predictions

Characteristic	Aleatoric Uncertainty	Epistemic Uncertainty
Origin	Data noise and inherent stochasticity	Model limitations and insufficient training data
Reducibility	Irreducible with more data	Reducible with improved models or more data
Quantification Methods	Quantile regression, Assumed Density Filtering	Model ensembling, Monte Carlo dropout
Dependence on System Size	Often increases with chemical complexity	Higher in underrepresented regions of chemical space
Typical Manifestation in MD	Consistent across similar structures	Spikes when encountering novel atomic environments

Methodological Approaches for Uncertainty Quantification

Ensemble-Based Methods for Epistemic Uncertainty

Ensemble methods provide a powerful approach for quantifying epistemic uncertainty by training multiple models and measuring their prediction disagreement. For foundation models with high computational training costs, readout ensembling offers an efficient alternative where only the final readout layers are fine-tuned for each ensemble member [56].

Protocol 3.1.1: Readout Ensembling for Foundation Models

• Initialization: Start with a pre-trained foundation model (e.g., MACE-MP-0 [56]).
• Ensemble Generation: For each of ( N ) ensemble members (typically 5-10 models):
  • Freeze all model parameters except the final readout layers
  • Fine-tune on different bootstrap samples (typically 80% of available data)
  • Use different random initializations for readout layers
• Uncertainty Calculation: For a new structure, calculate epistemic uncertainty as:
  • Standard deviation across ensemble predictions: ( \sigma{\text{epistemic}} = \sqrt{\frac{1}{N}\sum{i=1}^{N}(y_i - \bar{y})^2} )
  • Confidence intervals using Student's t-distribution

This approach significantly reduces computational costs compared to full-model ensembling while maintaining the learned representations of the foundation model [56].

Bayesian Methods for Aleatoric Uncertainty

Bayesian approaches provide a principled framework for propagating aleatoric uncertainty through all layers of a GNN. The Assumed Density Filtering (ADF) method efficiently approximates how uncertainty from probabilistic inputs propagates through the network:

Protocol 3.2.1: Aleatoric Uncertainty Propagation with ADF

• Input Uncertainty Specification:

  • Define probability distributions for node features: ( hi \sim \mathcal{N}(\mui, \sigma_i^2) )
  • Specify probabilistic links with weights ( p_{ij} \in [0,1] ) representing connection certainty [59]

• Forward Pass with Moment Matching:

  • For each layer, compute the first and second moments (mean and variance) of the output distribution
  • Approximate non-linear activations using Taylor expansions or analytical solutions

• Output Distribution:

  • Obtain predictive distribution at final layer with both mean and variance
  • Aleatoric uncertainty quantified as variance of this distribution

This approach systematically propagates input uncertainty through the node embedding and feedforward modules of GNNs, providing a complete picture of how data uncertainty affects final predictions [59].

Quantile Regression for Aleatoric Uncertainty

Quantile regression provides a non-parametric approach to capturing aleatoric uncertainty by predicting conditional quantiles rather than point estimates:

Protocol 3.3.1: Quantile Regression Implementation

• Architecture Modification:

  • Modify the network to have dual readout layers with separate loss functions
  • One readout penalized asymmetrically to target upper quantile (e.g., 95th percentile)
  • Second readout penalized to target lower quantile (e.g., 5th percentile)

• Asymmetric Loss Function:

  • For quantile ( \tau ), use loss: ( L_\tau(y, \hat{y}) = \begin{cases} \tau \cdot |y - \hat{y}| & \text{if } y > \hat{y} \ (1-\tau) \cdot |y - \hat{y}| & \text{otherwise} \end{cases} )
  • Upper quantile readout uses ( \tau = 0.95 )
  • Lower quantile readout uses ( \tau = 0.05 )

• Uncertainty Quantification:

  • Calculate 90% confidence interval as difference between upper and lower quantile predictions
  • Use this interval width as measure of aleatoric uncertainty [56]

This approach directly captures the variability in the training data distribution without requiring distributional assumptions.

Diagram 1: Workflow for uncertainty quantification and calibration in GNN interatomic predictions. The process begins with uncertain inputs, propagates uncertainty through the model, separates uncertainty types, and produces calibrated outputs with reliability scores.

Experimental Protocols and Implementation

Benchmarking and Evaluation Framework

Comprehensive evaluation of UQ methods requires specialized benchmarks that test performance under diverse conditions. The Tartarus benchmark provides a suite of molecular design tasks that simulate real-world challenges in materials science, pharmaceuticals, and chemical reactions [60].

Protocol 4.1.1: UQ Evaluation on Tartarus Benchmark

• Dataset Preparation:

  • Select diverse tasks from Tartarus covering organic emitters, protein ligands, and reaction substrates
  • Include both single-objective (7 tasks) and multi-objective (2 tasks) optimization problems
  • Utilize computational methods including conformer sampling, DFT calculations, and docking simulations

• Model Training:

  • Implement D-MPNN (Directed Message Passing Neural Network) architecture using Chemprop
  • Integrate UQ via probabilistic improvement optimization (PIO) for guiding optimization
  • Compare uncertainty-aware vs. uncertainty-agnostic approaches

• Evaluation Metrics:

  • Optimization success rate across chemical space
  • Ability to satisfy multiple competing objectives simultaneously
  • Reliability of uncertainty estimates via calibration curves

This benchmarking approach systematically evaluates whether UQ integration enables effective optimization across broad, open-ended chemical spaces [60].

Black-Box UQ for Pre-Trained Models

For scenarios where model architecture cannot be modified, black-box UQ methods provide uncertainty estimates without requiring changes to the underlying model:

Protocol 4.2.1: Ensemble-Based Black-Box UQ

• Ensemble Construction:

  • Create ensemble of 25 identical architecture models ("experts")
  • Train each expert on bootstrap samples (80% of training data)
  • Maintain original model architecture and training procedure

• Uncertainty Signal Generation:

  • Calculate uncertainty as disagreement between ensemble predictions
  • For forces, measure spread of force predictions across ensemble
  • Apply scaling to account for heteroscedasticity in data

• Out-of-Distribution Detection:

  • Establish empirical threshold for UQ signal indicating OOD behavior
  • Flag predictions with UQ signal above threshold for further investigation
  • Use UQ signal correlation with model convergence for training optimization [57]

This approach is particularly valuable for sealed models or when working with foundation models where architectural changes are impractical.

Table 2: Comparison of UQ Methods for GNN Interatomic Potentials

Method	Uncertainty Type Captured	Computational Cost	Implementation Complexity	Best Use Cases
Readout Ensembling	Primarily epistemic, some aleatoric	Moderate	Medium	Foundation model fine-tuning
Full Model Ensembling	Both epistemic and aleatoric	High	Low	Small to medium datasets
Monte Carlo Dropout	Epistemic	Low	Medium	Models with dropout layers
Quantile Regression	Aleatoric	Low	High	Data with heteroscedastic noise
Assumed Density Filtering	Aleatoric	Medium	High	Probabilistic graph inputs
Black-Box Ensembling	Epistemic	High	Low	Pre-trained or sealed models

Research Reagent Solutions

Table 3: Essential Software Tools and Datasets for UQ in GNN Interatomic Predictions

Resource	Type	Primary Function	Application in UQ
Chemprop	Software Framework	Directed Message Passing Neural Networks	Implements D-MPNN with UQ capabilities for molecular property prediction [60]
Tartarus Benchmark	Dataset Suite	Molecular design tasks with diverse objectives	Evaluating UQ method performance across chemical space [60]
GuacaMol Benchmark	Dataset Suite	Drug discovery optimization tasks	Testing multi-objective optimization with UQ [60]
MACE-MP-0	Foundation Model	Pretrained NNP across periodic table	Base for readout ensembling and transfer learning [56]
MPtrj Dataset	Training Data	1.6M materials from Materials Project	Training and testing foundation model UQ [56]
Bootstrap Sampling	Statistical Method	Creating diverse training subsets	Generating ensemble diversity for epistemic UQ [57]

Advanced Applications and Case Studies

Multi-Objective Molecular Optimization

Uncertainty quantification proves particularly advantageous in multi-objective molecular optimization tasks, where balancing competing objectives is challenging:

Protocol 6.1.1: UQ-Enhanced Multi-Objective Optimization

• Probabilistic Improvement Optimization (PIO):

  • Quantify likelihood that candidate molecules exceed predefined property thresholds
  • Use UQ to reduce selection of molecules outside model's reliable range
  • Promote candidates with superior properties while maintaining reliability

• Implementation:

  • Apply PIO method within genetic algorithm optimization framework
  • Use UQ to guide exploration-exploitation trade-off in chemical space
  • Focus on satisfying multiple property constraints simultaneously rather than extreme values

• Validation:

  • Compare against uncertainty-agnostic approaches on benchmark tasks
  • Evaluate success rate in finding molecules satisfying all objectives
  • Assess chemical diversity of optimized candidates [60]

This approach demonstrates substantially improved optimization success in most cases, particularly for balancing competing objectives in practical molecular design scenarios.

Long-Range Interaction Modeling

For GNNs capturing long-range interactions in materials systems, explicit UQ is essential for identifying unreliable predictions in complex electrostatic environments:

Protocol 6.2.1: UQ for Polarizable Force Fields

• Architecture Integration:

  • Combine equivariant message-passing neural networks with explicit polarizable long-range potential
  • Implement polarizable charge equilibration (PQEq) scheme optimizing electrostatic interaction energies
  • Capture polarization effects under external electric fields

• Uncertainty Quantification:

  • Apply readout ensembling to foundation model with polarizable interactions
  • Quantify uncertainty in charge transfer and polarization predictions
  • Identify regions of chemical space with poor electrostatic behavior prediction

• Application:

  • Mechanical properties prediction of materials
  • Ionic diffusivity in solid-state electrolytes
  • Ferroelectric phase transitions
  • Reactive dynamics at electrode-electrolyte interfaces [61]

This approach enables accurate uncertainty estimation for systems where long-range interactions dominate material behavior, extending the applicability of GNN interatomic potentials.

Diagram 2: GNN architecture for uncertainty quantification in energy and force predictions. The system processes atomic structures through message-passing networks, then separates into dedicated pathways for aleatoric and epistemic uncertainty before producing calibrated predictions with reliability scores.

Performance Metrics and Validation

Quantitative Assessment of UQ Quality

Rigorous validation of UQ methods requires specialized metrics beyond traditional accuracy measures:

Protocol 7.1.1: UQ Calibration Assessment

• Calibration Curves:

  • Plot observed error versus predicted uncertainty across confidence intervals
  • Assess whether 90% confidence intervals contain approximately 90% of observations
  • Identify underconfident or overconfident uncertainty estimates

• Sharpness Evaluation:

  • Measure the spread of predictive distributions
  • Prefer methods that provide tight confidence intervals while maintaining calibration
  • Balance between informativeness and reliability

• Downstream Task Performance:

  • Evaluate optimization success rates in molecular design benchmarks
  • Assess active learning efficiency in reducing prediction error with minimal data
  • Measure simulation reliability in molecular dynamics applications [60] [56]

Comparative Performance Analysis

Table 4: Performance Comparison of UQ Methods on Standard Benchmarks

UQ Method	MAE (meV/e⁻)	Calibration Error	OOD Detection AUC	Multi-Objective Success Rate
Readout Ensembling	0.721	0.15	0.89	78%
Quantile Regression	0.890	0.08	0.76	72%
MC Dropout	0.951	0.22	0.82	65%
Full Ensembling	0.685	0.12	0.91	81%
Black-Box Ensembling	0.743	0.18	0.87	75%

Data adapted from performance evaluations on Tartarus and MPtrj benchmarks [60] [56]. Metrics represent relative performance across methods, with lower values preferred for MAE and Calibration Error, and higher values preferred for OOD Detection AUC and Success Rate.

These results demonstrate that readout ensembling provides favorable balance between computational efficiency and UQ quality, while quantile regression offers superior calibration for aleatoric uncertainty. The choice of method depends on specific application requirements and computational constraints.

The accuracy of machine learning force fields (MLFFs) has traditionally been benchmarked against quantum-mechanical calculations of energy and forces. However, even models with excellent energy fidelity can produce unstable molecular dynamics (MD) simulations that drift into non-physical states, limiting their utility for predicting experimentally relevant observables [62] [63]. This article details the application of Stability-Aware Training with Differentiable Boltzmann Estimators (StABlE), a novel paradigm that directly addresses this challenge. By integrating reference observables and a differentiable path from MD simulations back to model parameters, StABlE Training produces more robust and data-efficient MLFFs, a critical advancement for research in drug development and materials science [62] [63] [64].

Core Protocol: StABlE Training Methodology

StABlE Training introduces a multi-modal procedure that supplements traditional energy and forces supervision with direct supervision from system-level observables. The protocol corrects instabilities without requiring additional costly ab-initio calculations [62] [63].

Principle and Workflow

The method iteratively seeks out and corrects unstable regions of the potential energy surface (PES) by running short, parallel MD simulations and comparing simulation-derived observables against reference data.

Table 1: Key Concepts in StABlE Training

Concept	Description	Function in Protocol
Differentiable Boltzmann Estimator	A generalization of implicit differentiation techniques for stochastic algorithms [62].	Enables end-to-end automatic differentiation through MD simulations, allowing gradient flow from observables back to force field parameters.
Multi-Modal Supervision	Joint training using both reference quantum-mechanical data (energy/forces) and system observables [62] [63].	Ensures the model learns both local quantum-mechanical accuracy and global, long-timescale simulation stability.
Stability-Aware Loss	A composite loss function combining energy/forces loss and an observables-based loss [62].	Directly penalizes model parameters that lead to simulations producing unphysical observables.
Reference Observables	Experimentally measurable or highly accurate computationally derived system properties (e.g., radial distribution functions) [62].	Provides a physical constraint on simulation trajectories, grounding the model in real-world behavior.

The following diagram illustrates the iterative StABlE Training workflow, which combines traditional supervision with a novel stability-correction cycle via the Boltzmann Estimator.

Step-by-Step Experimental Protocol

This protocol is designed for a single active learning iteration within the broader StABlE framework.

• Step 1: Initial Model Preparation

  • Begin with a pre-trained MLFF model using any modern architecture (e.g., NequIP, MACE, SchNet). The model should have a baseline level of accuracy on energy and force predictions from a standard quantum-mechanical dataset [62].

• Step 2: Parallel MD Simulation and Trajectory Analysis

  • Procedure: Launch multiple independent, short MD simulations (e.g., 10-100 ps) in parallel using the current MLFF. The initial structures should be drawn from a diverse set of relevant thermodynamic states [62].
  • Stability Check: Monitor simulations for instabilities, indicated by abnormal energy drift, bond breaking, or system collapse. Flag unstable trajectories for correction [62] [63].
  • Observables Calculation: From the stable portions of the trajectories, compute one or more target observables, ( O_{MD} ). Common choices include Radial Distribution Functions (RDFs) or diffusion coefficients [62].

• Step 3: Differentiable Boltzmann Estimation and Loss Computation

  • Core Operation: Pass the calculated observables ( O_{MD} ) through the Differentiable Boltzmann Estimator. This module allows the calculation of gradients of the observables with respect to the MLFF parameters, ( \theta ) [62].
  • Loss Function Calculation: Compute the total loss, ( \mathcal{L}{total} ): ( \mathcal{L}{total} = \lambda{QM}\mathcal{L}{QM}(E{pred}, F{pred}, E{DFT}, F{DFT}) + \lambda{Obs}\mathcal{L}{Obs}(O{MD}, O{Ref}) ) where ( \mathcal{L}{QM} ) is the standard quantum-mechanical loss (e.g., MAE), ( \mathcal{L}{Obs} ) is the observables loss (e.g., MSE), and ( \lambda ) are weighting coefficients [62].

• Step 4: Model Update and Iteration

  • Backpropagation: Compute gradients ( \frac{\partial \mathcal{L}{total}}{\partial \theta} ) using standard backpropagation for ( \mathcal{L}{QM} ) and via the Boltzmann Estimator for ( \mathcal{L}_{Obs} ) [62].
  • Parameter Update: Update the model parameters ( \theta ) using a stochastic gradient descent algorithm.
  • Iteration: Repeat from Step 2 until simulation stability and observable convergence are achieved. The number of required iterations is system-dependent [62].

Applications and Experimental Validation

The StABlE training framework has been empirically validated across diverse molecular systems, demonstrating broad applicability.

Table 2: Quantitative Performance of StABlE-Trained Models

System Class	Key Metric	StABlE Performance	Baseline MLFF Performance	Reference/Note
Organic Molecules	Simulation Stability	Significant improvement in stability and data efficiency [62].	Produces unstable simulations [62].	Stability gain not replicable by merely reducing timestep [62].
Tetrapeptides	Stability & Observables	High agreement with reference observables [62].	Poor agreement with reference observables [62].	Demonstrates utility for biologically relevant systems [62].
Condensed Phase	Stability & Observables	Significant improvement in stability and data efficiency [62].	Produces unstable simulations [62].	Validated for complex, periodic systems [62].
Energetic Materials (CHNO)	Energy/Force MAE	MAE within ~0.1 eV/atom and ~2 eV/Å [24].	Pre-trained model showed significant deviations [24].	EMFF-2025 model, showcasing transfer learning [24].

Case Study: Protocol for Organic Molecule and Tetrapeptide Stability

• Objective: To assess and improve the stability of an MLFF simulating small organic molecules and tetrapeptides in explicit solvent.
• Experimental Setup:
  • MLFF Architecture: A graph neural network-based potential like NequIP or a message-passing network [62].
  • Reference Data: DFT-level energies/forces and reference RDFs from high-level theory or experiment.
  • Simulation Details: Multiple ~100 ps NVT simulations initiated from different solvent configurations.
• StABlE-specific Protocol:
  • The target observable ( O_{Ref} ) is the O-H RDF of the solvent (e.g., water).
  • Instability is defined as a deviation of the simulation RDF (( O{MD} )) from ( O{Ref} ) exceeding a predefined threshold.
  • The Boltzmann Estimator is configured for the specific RDF calculation, enabling gradient computation for this structural observable.
• Validation: Success is measured by the ability to run multi-nanosecond simulations without drift and achieve a near-perfect match between simulated and reference RDFs [62].

The Scientist's Toolkit: Research Reagent Solutions

Table 3: Essential Components for Implementing StABlE Training

Item	Category	Function in Protocol	Implementation Note
Differentiable Boltzmann Estimator	Algorithm	Core enabler; allows gradient propagation through stochastic MD simulations [62].	Provided in the official StABlE codebase [65].
Modern MLFF Architecture	Software/Model	Base model for learning the potential energy surface (e.g., NequIP, MACE, SchNet) [62].	Must be compatible with automatic differentiation frameworks (PyTorch, JAX).
Reference Quantum-Mechanical Data	Dataset	Provides ( \mathcal{L}_{QM} ) supervision for energies and forces [62].	Can be a small, targeted dataset. StABlE improves data efficiency [62].
Reference Observables (( O_{Ref} ))	Dataset	Provides ( \mathcal{L}_{Obs} ) supervision for simulation stability and accuracy [62].	Can be experimental data or derived from high-fidelity ab-initio MD.
Molecular Dynamics Engine	Software	To run parallel simulations for stability probing (e.g., HOOMD-blue, LAMMPS, OpenMM).	Must be integrated into the training loop, often via an interface like JAX-MD.
StABlE Training Codebase	Software	Reference implementation of the training procedure [65].	Available at: https://github.com/ASK-Berkeley/StABlE-Training [65].

Integration with Broader Research Context

The StABlE paradigm aligns with several key trends in machine learning for science. It is a form of semi-empirical force field development, integrating physical first principles with empirical data to overcome the limitations of purely physical models [62] [63]. This approach is particularly powerful in the context of foundation models for chemistry, where a base model, pre-trained on vast datasets (e.g., GNoME's 89 million structures), can be rapidly fine-tuned for specific systems with limited data using StABlE's observable-guided training [66]. Furthermore, the drive for stability is central to other large-scale discovery efforts. For instance, the GNoME project used active learning to discover millions of stable crystals, underscoring that stability is both a primary goal and a key constraint in materials and molecular modeling [67]. StABlE Training provides a practical methodology to enforce this constraint directly during the training of force fields.

The application of Graph Neural Networks (GNNs) to the prediction of interatomic interactions represents a frontier in computational materials science and drug discovery. A significant challenge in this domain is that generating high-quality, large-scale training data from quantum mechanical calculations or experimental assays is often prohibitively expensive and time-consuming [68] [69]. Consequently, researchers are increasingly turning to data-efficient learning strategies to develop robust models where data is scarce. This application note details two such powerful strategies—transfer learning and stable learning for small data—and provides structured protocols for their implementation in the context of stability prediction and interatomic potential modeling.

The core challenge in data-scarce regimes is to build models that generalize well beyond their limited training sets, particularly to out-of-distribution (OOD) data. The strategies outlined below address this by leveraging pre-existing knowledge and by explicitly designing models to be invariant to spurious correlations.

Table 1: Comparison of Data Efficiency Strategies for GNNs in Scientific Applications.

Strategy	Core Principle	Key Advantage	Exemplary Model/ Framework	Reported Performance
Transfer Learning	Leverages knowledge (e.g., atomic descriptors) from a model pre-trained on a large, general dataset to a new, specific task with limited data [68] [24].	Drastically reduces required training data and time for new systems; enables fast adaptation of universal potentials [68].	franken [68], EMFF-2025 [24], ThermoMPNN [70]	Training time reduced from tens of hours to minutes; accurate potentials trained with tens of structures [68].
Stable Learning (for Small Data)	Uses feature sample weighting and decorrelation in Random Fourier Feature space to eliminate spurious correlations, forcing the model to rely on genuine causal features [11].	Improves model robustness and prediction stability on unseen test distributions (OOD generalization) [11].	Stable-GNN (S-GNN) [11]	Surpasses state-of-the-art GNNs; reduces prediction bias in OOD settings [11].
Multi-Task & Joint Learning	Simultaneously trains a single model on multiple related tasks (e.g., multiple toxicity endpoints), allowing it to learn more generalized representations [71].	Improves predictive accuracy and data efficiency by sharing information across tasks; effective for small-scale data [71].	JLGCN-MTT [71]	AUC improved by over 10% in 11 of 12 toxicity tasks with small sample sizes [71].

The following workflow diagram illustrates the sequential process for applying the franken transfer learning framework to develop a machine learning interatomic potential (MLIP).

Figure 1: The franken framework workflow for transfer learning of interatomic potentials.

Application Protocols

Protocol A: Transfer Learning for Interatomic Potentials usingfranken

This protocol describes how to adapt a general-purpose, pre-trained GNN to a new chemical system for molecular dynamics simulations with minimal data.

Research Reagent Solutions

Table 2: Essential Tools and Datasets for Transfer Learning with franken.

Item Name	Function / Description	Example / Source
Pre-trained GNN Potentials	Provides a source of rich, transferable atomic descriptors.	MACE-MP-0, CHGNet [68] [72]
Quantum Mechanical Reference Data	Small, system-specific dataset for fine-tuning. Used as ground truth.	DFT calculations for target system [68]
franken Framework	Open-source software implementing the transfer learning and RFF pipeline.	https://franken.readthedocs.io [68]
Random Fourier Features (RFF)	A scalable approximation of kernel methods that enables fast linear regression on top of GNN descriptors [68].	Integrated within franken

Step-by-Step Procedure

• Descriptor Extraction:

  • Input: A pre-trained GNN model (e.g., MACE-MP-0) and your small dataset of atomic structures for the target system.
  • Action: Pass each atomic configuration through the pre-trained GNN. Instead of using the final output (energy/forces), extract the intermediate atomic representations or descriptors from the network's message-passing layers [68] [72].
  • Output: A set of high-dimensional, physically meaningful descriptors for each local atomic environment.

• Feature Mapping with Random Fourier Features:

  • Input: The extracted atomic descriptors.
  • Action: Project these descriptors into a randomized finite-dimensional feature space using RFF. This step efficiently approximates a kernel function, transforming the nonlinear regression problem into a linear one [68].
  • Output: A transformed, fixed-dimensional feature vector for each atom suitable for fast linear regression.

• Lightweight Regression:

  • Input: The RFF-mapped features and their corresponding target properties (energy, forces) from the reference quantum mechanical data.
  • Action: Train a linear regressor (e.g., ridge regression) to predict the total energy as a sum of local atomic contributions. Forces can be learned alongside the energy using automatic differentiation [68] [72].
  • Output: A fine-tuned, accurate, and fast interatomic potential for the target system.

• Validation and Simulation:

  • Validate the potential on a held-out set of configurations by comparing predicted energies and forces to DFT reference data.
  • Deploy the validated potential in molecular dynamics simulation packages (e.g., LAMMPS) to run stable simulations [68] [24].

Protocol B: Stable-GNN (S-GNN) for Robust Prediction with Small Data

This protocol is designed to improve the generalization and OOD performance of GNNs when training data is limited, by de-correlating features.

The Stable-GNN framework incorporates a sample re-weighting mechanism to de-correlate features in the random Fourier feature space, ensuring the model bases its predictions on genuine causal features.

Figure 2: The Stable-GNN (S-GNN) training workflow for OOD generalization.

Step-by-Step Procedure

• Base Representation Learning:

  • Input: The small training dataset of graphs (e.g., molecular graphs or crystal graphs).
  • Action: Process the input graphs through a standard GNN (e.g., GCN, GAT) to generate initial node embeddings [11].

• Random Fourier Feature Mapping:

  • Input: The learned node embeddings from the GNN.
  • Action: Map these embeddings into the RFF space. This explicit mapping allows for efficient computation of feature independence with linear complexity, O(nD) [11].

• Sample Weighting for Decorrelation:

  • Input: The RFF-mapped features of the training samples.
  • Action: Compute instance-specific weights for each training sample. The optimization goal is to find a set of weights such that in the reweighted data distribution, all features become independent [11]. This is achieved by minimizing the Frobenius norm of the cross-covariance matrix in the RFF space.
  • Output: A set of sample weights, ωᵢ.

• Weighted Model Training:

  • Input: The original training data and the computed sample weights.
  • Action: Retrain or fine-tune the base GNN model using a weighted loss function. The sample weights ωᵢ modulate the importance of each instance, forcing the model to focus on learning from causal features rather than spurious correlations [11].
  • Output: A Stable-GNN (S-GNN) model with improved robustness and reduced prediction bias on unseen data distributions.

The integration of transfer learning and stable learning paradigms presents a powerful and essential toolkit for advancing research on GNNs for interatomic interactions and stability prediction. By following the protocols outlined in this document, researchers and developers can construct accurate, robust, and data-efficient models, thereby accelerating the discovery and design of new materials and therapeutic compounds.

The application of Graph Neural Networks (GNNs) to predict the stability of materials based on interatomic interactions represents a frontier in computational materials science. Models such as GNoME (Graph Networks for Materials Exploration) have demonstrated remarkable capability in discovering novel stable crystals by modeling materials at the atomic level, where atoms and their bonds are represented as graphs with nodes denoting individual atoms and edges capturing interatomic interactions [43]. However, the "black-box" nature of these sophisticated models has raised significant concerns within the scientific community regarding the rationality and legitimacy of their decision-making processes, ultimately limiting their trusted application in critical domains like drug discovery and materials design [73].

Explainable AI (XAI) for GNNs addresses this critical challenge by making the internal mechanisms of these models transparent and elucidating the mapping relationships between inputs and outputs. In the context of interatomic stability prediction, explainability is not merely about understanding which atoms or bonds contributed to a stability prediction, but about uncovering the fundamental physical principles that govern material stability. This transparency is crucial for establishing system trust, ensuring controllability, identifying and correcting model errors, avoiding ethical issues, and meeting regulatory requirements in pharmaceutical and materials development [73]. By providing interpretable insights into GNN predictions, researchers can transform these models from black-box predictors into collaborative tools for scientific discovery.

Current XAI Methodologies for GNNs

Taxonomy of Explainability Approaches

Multiple specialized approaches have been developed to interpret GNN predictions, each with distinct mechanisms and advantages for interatomic interaction analysis. The table below summarizes the primary categories of GNN explainability methods:

Table 1: Taxonomy of GNN Explainability Methods

Category	Representative Methods	Core Mechanism	Advantages	Limitations for Stability Prediction
Gradient/Feature-Based	SA, Guided BP, CAM, Grad-CAM [73]	Customized backpropagation and hidden-layer feature integration	Direct connection to model parameters	Localized explanations; High computational cost with dimensionality
Perturbation-Based	GNNExplainer, PGExplainer, ZORRO, GraphMask [73]	Analyzing output changes from input variations	Model-agnostic; Intuitive methodology	Limited perturbation ranges; May miss extreme behaviors
Surrogate-Based	GraphLime, RelEx, CXPlain, PGM-Explainer [73]	Approximating complex models with interpretable local substitutes	Flexibility in explanation format	Potential approximation errors; Context understanding limitations
Decomposition-Based	LRP, Excitation BP, GNN-LRP [73]	Propagating predictions backward through layers	Theoretical grounding; Layer-wise relevance	Computational complexity; Generalization challenges
Subgraph-Based	Key Subgraph Retrieval [73]	Identifying critical connected substructures	Captures functional groups; High accuracy	Dependency on quality of training data

Quantitative Performance Comparison

Recent benchmarking studies have evaluated these approaches across multiple datasets relevant to molecular and materials analysis. The performance metrics below provide guidance for selecting appropriate XAI methods for interatomic stability prediction:

Table 2: Performance Comparison of XAI Methods on Molecular Datasets

Method	BA3 Dataset Accuracy (%)	Mutagenicity Dataset Accuracy (%)	Benzene Dataset Accuracy (%)	Computational Efficiency
SA	85.30	70.50	72.80	Medium
Grad-CAM	86.70	72.30	74.20	Medium
GNNExplainer	92.50	78.90	81.60	Low
CXPlain	94.80	80.10	83.70	Low
PGM	91.20	76.40	79.50	Low
Key Subgraph Retrieval	99.25	82.40	85.90	High

The key subgraph retrieval method demonstrates superior performance, achieving 99.25% accuracy on the BA3 dataset and 82.40% on the Mutagenicity dataset [73]. This approach is particularly relevant for interatomic stability prediction as it identifies connected substructures (similar to functional groups in molecules) that predominantly influence model decisions, offering both high accuracy and computational efficiency by leveraging pre-trained GNN embeddings without requiring model retraining.

Application to Interatomic Interactions Stability Prediction

GNoME Framework and Interpretability Needs

The GNoME (Graph Networks for Materials Exploration) framework exemplifies the cutting-edge application of GNNs to materials stability prediction. This system leverages GNNs to model materials at the atomic level, representing atoms as nodes and interatomic interactions as edges in a graph structure [43]. Descriptors of elemental properties are embedded in the node features, and the GNNs learn to predict energetic properties of molecules through message passing along these edges [43]. The framework employs active learning in conjunction with Density Functional Theory (DFT) calculations to iteratively expand its knowledge, alternating between using GNNs to screen candidate materials and refining predictions with computational chemistry methods.

Despite its impressive performance in discovering novel stable materials, the GNoME framework faces interpretability challenges that limit scientific insight. While it can accurately predict material stability, the specific atomic configurations, bonding patterns, and physical principles driving these predictions remain largely opaque. This limitation hinders the ability of researchers to extract fundamental knowledge about stability rules that could guide rational materials design beyond the screening capabilities of the model itself.

Specialized XAI Protocol for Stability Prediction

The following protocol provides a structured methodology for interpreting GNN-based stability predictions, with specific adaptations for interatomic interactions analysis:

Diagram 1: XAI workflow for interatomic stability prediction

Protocol 1: Interpretable Stability Prediction for Crystalline Materials

Objective: To explain GNN-based stability predictions for crystalline materials by identifying critical atomic substructures and interactions.

Materials and Inputs:

• Crystallographic Information Files (CIF) or atomic coordinate data
• Pre-trained GNoME model or equivalent GNN for stability prediction
• Computational resources for DFT validation (optional but recommended)

Procedure:

• Graph Representation Construction

  • Convert crystal structure to graph representation with atoms as nodes and bonds as edges
  • Node features: atomic number, electronegativity, atomic radius, valence electron count
  • Edge features: bond length, bond type, calculated bond order

• GNN Inference with Attention Mechanisms

  • Utilize GAT-based architectures with built-in attention mechanisms
  • Record attention weights across all message-passing layers
  • Generate latent node and edge embeddings for analysis

• Subgraph Importance Analysis

  • Apply key subgraph retrieval method to identify critical atomic motifs
  • Calculate Euclidean distances between node embeddings in latent space
  • Cluster connected atoms with similar importance scores
  • Rank identified substructures by contribution to stability prediction

• Cross-validation with Physical Principles

  • Compare identified critical substructures with known crystal chemistry principles
  • Validate against DFT calculations for representative examples
  • Correlate with experimental stability data when available

Interpretation Guidelines:

• High-attention edges typically indicate critical bonding interactions affecting stability
• Atomic clusters with cohesive importance scores suggest functional subunits
• Discrepancies between model attention and theoretical expectations may indicate model errors or novel physical insights

Advanced XAI Techniques and Failure Diagnosis

Robustness-Interpretability Trade-offs

Recent research has revealed critical tensions between model robustness and interpretability in GNNs. A comprehensive benchmark study evaluating six GNN architectures across five datasets demonstrated that defense mechanisms against adversarial attacks significantly impact interpretability metrics [74]. The study employed four key interpretability metrics—Fidelity, Stability, Consistency, and Sparsity—to evaluate how robustness-enhancing techniques affect explanation quality.

For interatomic stability prediction, these findings highlight a crucial consideration: methods to make GNNs more robust to noisy or incomplete crystallographic data may inadvertently reduce the faithfulness of explanations. This trade-off necessitates careful evaluation of application requirements—whether the priority is absolute predictive accuracy or scientifically plausible explanations that can guide materials design.

Failure Diagnosis Protocol

When GNN stability predictions contradict established knowledge or experimental evidence, systematic diagnosis is essential to identify failure root causes:

Diagram 2: Failure diagnosis protocol for GNN stability prediction

Protocol 2: Failure Diagnosis for Stability Prediction Models

Objective: To systematically identify root causes of erroneous stability predictions in GNN models and implement appropriate corrections.

Diagnostic Procedure:

• Explanation Fidelity Assessment

  • Calculate fidelity metric by removing important features identified by XAI and measuring prediction change
  • Evaluate stability by measuring explanation consistency across similar atomic environments
  • Assess sparsity to determine if explanations are overly diffuse

• Data Artifact Detection

  • Analyze training data for distributional shifts or annotation errors
  • Identify potential spurious correlations in training data (e.g., specific space groups overrepresented in stable crystals)
  • Check for dataset imbalance that might bias predictions

• Architectural Limitation Evaluation

  • Test for oversmoothing in deep GNN architectures by monitoring node differentiation across layers
  • Evaluate expressive power limitations using synthetic benchmarks
  • Assess message passing effectiveness for long-range interactions in crystals

Remediation Strategies:

• For oversmoothing: Implement architectures like GraphCON that mitigate this issue through oscillatory dynamics [75]
• For limited expressive power: Employ higher-order GNNs or subgraph-based approaches
• For data limitations: Incorporate active learning with DFT calculations to strategically expand training data

Table 3: Essential Research Reagents and Computational Tools for GNN XAI

Category	Specific Tools/Methods	Application Context	Key Functionality
GNN Architectures	GCN, GAT, GraphSAGE, GIN [74]	Baseline models for materials stability prediction	Graph representation learning with varying expressive power
XAI Libraries	GNNExplainer, PGExplainer, GraphLIME [73]	Post-hoc interpretation of trained models	Generate feature importance scores and visual explanations
Subgraph Analysis	Key Subgraph Retrieval [73]	Identification of critical atomic motifs	Euclidean distance-based retrieval of important connected components
Robustness Evaluation	Fidelity, Stability, Consistency, Sparsity metrics [74]	Assessment of explanation quality	Quantify different aspects of explanation reliability
Materials Datasets	Materials Project, OQMD, ICSD	Training and benchmarking stability models	Curated crystal structures with stability annotations
Validation Tools	DFT codes (VASP, Quantum ESPRESSO)	Physical validation of model predictions	First-principles calculation of formation energies

Emerging Frontiers and Future Directions

The field of XAI for GNNs in interatomic interactions prediction continues to evolve rapidly. Several promising directions are emerging:

Causal Interpretation Methods: Beyond correlational explanations, new approaches like Causal Intervention Graph Neural Networks (CIGNN) are being developed to distinguish causal relationships from spurious correlations in atomic graphs [76]. This is particularly important for stability prediction where underlying physical mechanisms are of primary interest.

Integrated Physical Constraints: Future methods will more deeply incorporate physical constraints and invariances directly into explanation frameworks, ensuring that interpretations respect known thermodynamic and quantum mechanical principles.

Uncertainty-Aware Explanations: Developing explanations that convey not only importance but also uncertainty in interpretations will be crucial for high-stakes applications in materials and drug design.

As GNNs continue to advance in predicting interatomic interactions and material stability, the parallel development of robust explainability methods will ensure these models serve as true scientific tools that not only predict but also illuminate the fundamental principles governing material stability and reactivity.

Benchmarking Performance: Validating and Comparing GNN Models for Stability Prediction

Within the rapidly evolving field of graph neural network (GNN) research for molecular and materials science, standardized benchmark datasets are indispensable for developing, evaluating, and comparing new machine learning models. These benchmarks provide consistent, high-quality data derived from quantum mechanical calculations, enabling rigorous testing of a model's ability to predict interatomic interactions and material stability. The QM9, MD17, and Materials Project datasets represent three pillars in this domain, each addressing a unique aspect of computational prediction. QM9 focuses on the chemical space of small organic molecules, MD17 provides insights into molecular dynamics, and the Materials Project offers a vast repository of inorganic crystal structures and their properties. Their widespread adoption has been a key driver in the progress of GNNs, which have emerged as powerful tools because they operate directly on the natural graph representation of atomic structures [12]. This application note details these critical datasets and their associated experimental protocols to guide researchers in leveraging them effectively.

Dataset Specifications and Quantitative Comparison

The following tables summarize the core characteristics of the QM9, MD17, and Materials Project datasets, providing researchers with key quantitative data for experimental planning.

Table 1: Core Dataset Specifications

Dataset	Primary Content	# of Entries	# of Heavy Atoms	Key Elements	Key Properties
QM9 [77] [78]	Small organic molecules	133,885 (core); ~134k (extended)	Up to 9 (CONF)	C, N, O, F	Geometric, energetic, electronic, & thermodynamic properties
MD17 [79] [33]	Molecular dynamics trajectories	Multiple small molecules	Varies	C, H, O (in benchmarks)	Energies & molecular forces
Materials Project [79]	Bulk inorganic crystals	> -	-	Full periodic table	Formation energy, Band gap, Elastic tensors, etc.
QCDGE [78]	Small organic molecules (w/ excited states)	443,106	Up to 10 (CNOF)	C, N, O, F	27 ground- & excited-state properties

Table 2: Dataset Properties and Applications in GNN Research

Dataset	Quantum Chemistry Level	Primary Use Case in GNNs	Notable GNN Benchmarks
QM9 [77] [78]	B3LYP/6-31G(2df,p)	Universal prediction of molecular properties	MGNN [33], InvarNet [25]
Revised MD17 [33]	Higher-level reference calculations	Molecular force field & dynamics learning	MGNN (SOTA on ethanol) [33]
Materials Project [79]	DFT (various codes)	Prediction of solid-state material properties	CGNN [80]
QCDGE [78]	B3LYP/6-31G* & ωB97X-D/6-31G*	Prediction of excited-state properties	-

Dataset-Specific Experimental Protocols

Protocol for GNN Training and Evaluation on QM9

The QM9 dataset serves as a standard for evaluating a model's comprehensive ability to predict a wide array of quantum mechanical properties.

Workflow Overview:

Detailed Methodology:

• Data Acquisition and Partitioning: Download the QM9 dataset. Standard practice involves a random 80/10/10 split for training, validation, and testing to evaluate model interpolation capability. For a more challenging assessment of generalizability, scaffold splitting based on molecular substructures can be used [81].
• Feature Encoding: Initialize node (atom) features. These typically include atomic number, chirality, formal charge, and hybridization state. Edge (bond) features are initialized with bond type, conjugation, and stereochemistry [81]. Models like MGNN use the atomic number and 3D Cartesian coordinates directly, generating more complex geometric features internally [33].
• Model Training:
  • Architecture: Implement a GNN such as an Message Passing Neural Network (MPNN) [12] or a more recent invariant architecture like MGNN [33] or InvarNet [25].
  • Input: Feed the molecular graph (atoms as nodes, bonds as edges) into the network.
  • Targets: The model simultaneously predicts 12+ target properties (e.g., internal energy U, enthalpy H, free energy G, heat capacity Cv, etc.) for each molecule.
  • Loss Function: The standard loss function is the Mean Absolute Error (MAE) between the predicted and DFT-calculated values. For the QM9 dataset, this is typically averaged across all properties or reported for key ones. The total loss is: Loss = Σ |y_pred - y_true| / N.
  • Optimization: Use a standard optimizer like Adam to minimize the loss via backpropagation.
• Validation and Testing: The model's performance is evaluated on the held-out validation and test sets. The primary metric is MAE (in eV or meV for energies). A lower MAE indicates a model better at approximating the underlying DFT potential energy surface.

Protocol for Molecular Dynamics with MD17

The MD17 dataset benchmarks a model's capability to learn atomic forces, which are crucial for stable and accurate molecular dynamics simulations.

Workflow Overview:

Detailed Methodology:

• Data Preparation: The dataset consists of molecular dynamics trajectories, meaning multiple conformations of the same molecule. Each data point includes the molecular geometry, total potential energy, and the force vector on each atom.
• Model and Loss Function: The key difference from QM9 training is the loss function. To accurately capture forces, the loss must include both energy and force components. A common formulation is: Loss = λ₁ * MAE(Energy_pred, Energy_true) + λ₂ * MAE(Forces_pred, Forces_true) where λ₁ and λ₂ are weighting coefficients. Forces are calculated as the negative gradient of the predicted energy with respect to the atomic coordinates: F = -∇_R E_pred. This requires the GNN to be differentiable with respect to its spatial inputs [33] [25].
• Training and Benchmarking: Models are trained on a subset of the trajectory and evaluated on unseen frames. Performance is reported as the Root Mean Square Error (RMSE) for energies (meV) and forces (meV/Å). State-of-the-art models like MGNN have achieved remarkably low force errors on the revised MD17 and MD17-ethanol benchmarks [33].

Protocol for Material Property Prediction with Materials Project

The Materials Project provides a large-scale database for predicting properties of periodic crystal structures, which is essential for materials design.

Detailed Methodology:

• Graph Representation of Crystals: The primary challenge is converting a periodic crystal into a graph. This is typically done by creating nodes for each atom in the unit cell and connecting edges between atoms that are within a predefined cutoff radius, considering periodic boundary conditions.
• Property Prediction: Once the graph is constructed, GNNs similar to those used for molecules can be applied. The readout function aggregates atom-level representations into a crystal-level embedding, which is then used for regression or classification of material properties such as formation energy, band gap, and elastic moduli [12] [79]. Models like CGNN have been benchmarked on data derived from the Materials Project [80].
• Evaluation: Models are evaluated using standard metrics like MAE for formation energy (eV/atom) and band gap (eV). Dataset splitting can be random or based on crystal structure to assess generalization to novel material classes.

The Scientist's Toolkit: Essential Research Reagents

Table 3: Key Computational Tools and Datasets for GNN Research

Resource Name	Type	Primary Function	Relevance to GNNs
OGB [81] [82]	Software Package	Data loaders & evaluators	Provides standardized data loaders and evaluation scripts for benchmark datasets like ogbg-molhiv.
RDKit [81]	Cheminformatics Library	Molecule processing & featurization	Converts SMILES strings to molecular graphs and generates atom/bond features for model input.
SchNetPack [79]	Neural Network Library	Pre-built GNN models	Offers implementations of models like SchNet for fast prototyping on QM9 and MD17.
PyTorch Geometric / DGL [81]	Graph Learning Frameworks	GNN model construction	The primary deep learning frameworks used to build and train custom GNN architectures.
Coulomb Matrix [77]	Molecular Representation	Pre-defined input feature	An early, fixed representation for molecules; provides a baseline for GNN performance.
Message Passing Framework [12] [33]	Neural Network Paradigm	Learning on graphs	The conceptual and mathematical foundation for most modern GNNs in chemistry.

In molecular dynamics (MD) simulations, the machine learning community has traditionally prioritized low force mean absolute error (MAE) as the primary metric for evaluating interatomic potentials. However, emerging research consistently demonstrates that low force MAE does not guarantee stable, physically realistic MD trajectories [83]. This document outlines application notes and protocols for assessing Graph Neural Network Interatomic Potentials (GNN-IPs) using metrics beyond accuracy, focusing on molecular dynamics stability and adherence to fundamental chemical principles. These protocols are essential for researchers developing reliable force fields for drug discovery and materials science applications where simulation stability directly impacts predictive validity.

Core Stability Metrics and Quantitative Benchmarks

While force MAE provides a basic measure of regression accuracy, it fails to capture the dynamic error accumulation that leads to simulation failure. The table below summarizes key quantitative metrics for comprehensive GNN-IP evaluation.

Table 1: Key Performance Metrics for GNN-IP Stability Assessment

Metric Category	Specific Metric	Target Value / Observation	Significance
Simulation Integrity	Trajectory Lifetime / Stability Time	Pre-trained models sustain trajectories ~3x longer than models trained from scratch [83]	Measures duration before unphysical events (e.g., bond breakage) occur.
Structural Fidelity	Pair-Distance Distribution Function [83]	Close alignment with reference ab initio MD results	Validates the preservation of molecular geometry and conformation over time.
Extrapolation Performance	Force MAE on Far-from-Equilibrium Structures [84]	Lower errors indicate better generalization	Assesses model robustness on high-energy, non-equilibrium configurations.
Physical Soundness	PES Realism under Extreme Hydrostatic Pressure [84]	Absence of unphysical energy wells or spikes	Ensures the model behaves correctly under non-ambient conditions.

The critical limitation of force MAE was demonstrated in a study comparing GemNet-T models trained on the MD17 dataset with and without pre-training on the large, diverse OC20 dataset. While both approaches achieved similarly low force MAEs (~5 meV/Å), the model trained from scratch failed catastrophically in MD simulations, while the pre-trained model maintained stable trajectories three times longer [83]. This underscores that stability must be measured directly.

Experimental Protocols for Assessing MD Stability

Protocol: Molecular Dynamics Stability and Trajectory Lifetime Assessment

This protocol evaluates the stability of GNN-IPs in production-level MD simulations.

I. Research Reagent Solutions

Table 2: Essential Materials for Stability Experiments

Item	Function / Description	Example
Benchmark Dataset	Provides diverse, high-quality reference data for training and testing.	MD17 dataset (ab initio energies/forces for small organic molecules) [85].
Pre-training Dataset	Large, chemically diverse dataset to initialize model weights for improved generalization.	OC20 dataset [83].
Simulation Software	Platform to run MD simulations using the trained GNN-IP.	ASE, LAMMPS, SchNetPack.
Validation Software	Tools for analyzing simulation outputs and structural properties.	In-house scripts for calculating pair-distance distribution functions.

II. Methodology

• Model Preparation: Train two instances of your GNN-IP architecture (e.g., GemNet-T).
  • Model A: Train from scratch on the target dataset (e.g., 10k samples from MD17).
  • Model B: Pre-train on a large, diverse dataset (e.g., OC20), then fine-tune on the same target dataset as Model A [83].
• Simulation Setup:
  • System: Initialize a molecular configuration (e.g., Aspirin from MD17).
  • Thermostat: Employ a Nosé–Hoover thermostat to maintain a target temperature (e.g., 500 K) [83].
  • Integrator: Use the velocity Verlet algorithm with a time step of 0.5 fs [83].
  • Duration: Run for a sufficient number of steps (e.g., 600,000 steps for 300 ps).
• Data Collection: Periodically save atomic positions and velocities for analysis.

III. Analysis and Evaluation

• Trajectory Visualization: Visually inspect the simulation trajectory for unphysical events, such as bond breaking, atom clustering, or evaporation.
• Quantitative Stability Metric: Record the simulation time step at which the first unphysical event occurs. Compare the "stable lifetime" between Model A and Model B.
• Structural Validation: Compute the pair-distance distribution function, ( h(r) ), for the generated trajectory and compare it against a reference ab initio MD trajectory [83]. Significant deviations indicate a loss of structural fidelity.

Protocol: Evaluating Extrapolation to Unseen Conformations

This protocol tests a model's ability to adhere to chemical principles when presented with molecular conformations outside its training distribution.

I. Research Reagent Solutions

• Stability-Centric Architectures: Use models designed for stability, such as Geometric Graph Neural Diffusion (GGND), which iteratively refines atomic representations to capture geometrically invariant features [55].
• Diverse Benchmark Datasets: Employ datasets containing off-equilibrium structures, such as MP-ALOE, which includes high-energy configurations and large magnitude forces [84].

II. Methodology

• Training: Train a model on a dataset containing primarily near-equilibrium structures (e.g., the standard MD17 split).
• Testing:
  • Near-Equilibrium Test: Evaluate force MAE on a held-out test set from the same distribution.
  • Far-from-Equilibrium Test: Evaluate force MAE on a dedicated benchmark of off-equilibrium structures (e.g., from MP-ALOE or via active learning sampling) [84].
  • Stability under Topological Shifts: Test on benchmarks like 3BPA and SAMD23 that encompass diverse conformations across varied temperatures, inducing significant topological shifts [55].
• Static Deformation: Apply extreme hydrostatic pressures to equilibrium structures and evaluate the physical soundness of the predicted Potential Energy Surface (PES) for the absence of unphysical features [84].

III. Analysis and Evaluation

• Compare the relative increase in force MAE between near-equilibrium and far-from-equilibrium tests. A smaller performance gap indicates better extrapolation capability.
• For static deformation, visualize the PES. A physically realistic PES should be smooth and lack severe non-physical oscillations or discontinuities.

Theoretical Underpinnings: Mechanisms of Stability and Extrapolation

Understanding why GNN-IPs succeed or fail is crucial for designing better models. A key theoretical insight is that the message-passing algorithm in GNNs enables them to learn non-local electrostatic interactions, which is a primary factor in their ability to extrapolate to untrained geometric domains, such as surfaces or amorphous configurations [1]. Universal GNN-IPs like SevenNet have been shown to accurately infer Coulomb interactions in untrained domains, though they may struggle with non-local forces from the kinetic energy term [1].

Frameworks like Geometric Graph Neural Diffusion (GGND) enhance stability by enabling instantaneous information flow between arbitrary atomic pairs while maintaining rotational equivariance. This iterative refinement of atomic representations helps capture long-range interactions and geometrically invariant topological features, thereby alleviating error accumulation in long-time-scale MD simulations [55].

Universal Machine Learning Interatomic Potentials (uMLIPs) represent a transformative advancement in computational materials science, enabling high-throughput atomistic simulations at near-density functional theory (DFT) accuracy but at a fraction of the computational cost. These models serve as foundational tools for predicting material properties, stability, and dynamics across diverse chemical spaces. Within the broader context of graph neural network research for interatomic interactions and stability prediction, this analysis provides a structured comparison of three prominent uMLIP architectures: CHGNet, MACE, and SevenNet. We examine their architectural principles, performance across key material property benchmarks, and provide detailed protocols for their practical application in materials discovery workflows.

The performance and applicability of uMLIPs are fundamentally guided by their underlying architectural choices. The table below summarizes the core architectural characteristics of CHGNet, MACE, and SevenNet.

Table 1: Architectural Comparison of CHGNet, MACE, and SevenNet

Feature	CHGNet	MACE	SevenNet
Core Architectural Principle	Charge-informed graph neural network [86] [87]	Higher-order equivariant message passing [88] [20]	Steerable equivariant GNN [89] [20]
Key Innovation/Differentiator	Explicit inclusion of magnetic moments to infer atomic charge states [86]	Higher-body order messages using symmetric tensor products [88]	Efficient nonlinear gates for node updates; compute-intensive MLPs for interatomic distances [20]
Symmetry Handling	E(3)-invariance (Rotation, translation, permutation) [87]	E(3)-equivariance [88]	E(3)-equivariance [20]
Representative Pretrained Model	CHGNet (Published 2023) [86]	MACE-MP-0 (Published 2024) [89] [90]	SevenNet-0 (Published 2025) [89]
Training Dataset	Materials Project Trajectory (MPtrj, ~1.5M structures) [86] [87]	Materials Project Trajectory (MPtrj) [89] [90]	Not explicitly stated; likely large-scale materials database

A critical differentiator for CHGNet is its explicit use of magnetic moments as a proxy for atomic charge states, allowing it to model the coupling between electronic states and ionic rearrangements, which is crucial for capturing the chemistry of transition metal ions [86] [87]. In contrast, MACE and SevenNet prioritize geometric equivariance through more sophisticated, higher-order representations. MACE employs a high-body order message-passing framework, which enhances its data efficiency and accuracy [88], whereas SevenNet's design focuses on computational efficiency, for instance, by using nonlinear gates instead of more expensive tensor product operations for mixing node information [20].

Performance Benchmarking

Quantitative benchmarking is essential for guiding model selection for specific research tasks. The following tables consolidate performance data from recent large-scale assessments.

Accuracy on Core Properties

Table 2: Benchmarking Performance on Energy, Force, and Phonon Predictions

Model	Energy MAE (meV/atom)	Force MAE (meV/Å)	Phonon Spectrum Accuracy	Notable Strengths/Weaknesses
CHGNet	30 [86]	77 [86]	Substantial inaccuracies reported [89]	Strengths: Charge-informed MD; good for geometry relaxation [89] [86].Weaknesses: Higher energy error; lower phonon accuracy [89].
MACE (MACE-MP-0)	~22 (estimated from leaderboard) [89]	Information missing	High accuracy [89]	Strengths: High overall accuracy and data efficiency [89] [88].Weaknesses: Computationally intensive training [20].
SevenNet (SevenNet-0)	Information missing	Information missing	High accuracy [89]	Strengths: High phonon and elastic property accuracy [89] [91].Weaknesses: High computational cost for training (90+ days on A100) [20].

Application-Specific Performance

Table 3: Performance on Downstream Applications and Tasks

Model	Geometry Relaxation Failure Rate	Elastic Property Prediction	Demonstrated Applications
CHGNet	0.09% (Lowest) [89]	Less effective overall [91]	Phase transformations, Li-ion battery materials [86] [92]
MACE (MACE-MP-0)	~0.2% (Medium) [89]	Balances accuracy with efficiency [91]	General-purpose simulations [88]
SevenNet (SevenNet-0)	~0.2% (Medium) [89]	Highest accuracy [91]	Not explicitly stated in results

The geometry relaxation failure rate is a critical metric for practical high-throughput screening. CHGNet demonstrates exceptional reliability in this task, failing to converge for only 0.09% of structures in a benchmark of ~10,000 materials, which is attributed to its robust force predictions near equilibrium [89]. For predicting mechanical properties, SevenNet achieves the highest accuracy in elastic property prediction, while MACE offers a favorable balance between accuracy and computational efficiency [91].

Experimental Protocols and Workflows

General uMLIP Application Workflow

The following diagram illustrates a standardized workflow for employing uMLIPs in crystal structure prediction and screening, integrating steps from high-throughput computational studies [92].

Protocol for Phonon Property Calculation

Phonon spectra are critical for assessing dynamical stability and thermal properties. The following protocol is adapted from benchmarks evaluating uMLIPs on phonon calculations [89] [90].

• Initial Structure Sourcing: Obtain the crystal structure of interest from a database like the Materials Project [89] or generate it via a structure prediction algorithm [92].
• Geometry Relaxation:
  • Tool: Use the uMLIP's built-in relaxation function or an external calculator interfaced with a package like ASE or VASP.
  • Parameters: Relax the structure until the forces on all atoms are below a threshold of 0.005 eV/Å and the stress tensor components are below 0.1 GPa. Monitor for convergence failures, which are more common with models like eqV2-M and ORB [89].
• Force Calculation and Phonon Dispersion:
  • Method: Employ the finite-displacement method as implemented in packages like Phonopy.
  • Supercell: Construct a supercell of sufficient size (often 2x2x2 or 3x3x3 for simple unit cells) to capture long-range interactions.
  • Displacements: Calculate forces for a set of atomic displacements (default 0.01 Å is typically sufficient) using the uMLIP as the force calculator.
  • Post-Processing: Phonopy uses the force sets to compute the dynamical matrix and diagonalize it to obtain phonon frequencies and eigenvectors across the Brillouin zone.
• Validation: Compare the uMLIP-predicted phonon band structure with DFT-PBE reference data where available. Be aware that the choice of exchange-correlation functional (PBE vs. PBEsol) introduces variability comparable to the error of some uMLIPs [89].

Protocol for Elastic Constant Calculation

Elastic constants describe the response of a material to external strain and are key for mechanical property assessment [91].

• Pre-Relaxation: Fully relax the crystal structure using the uMLIP (as in Section 4.2, Step 2) to ensure it is at a local energy minimum.
• Strain Application:
  • A set of independent finite strains (both tensile and shear) is applied to the relaxed unit cell. The number and type of strains depend on the crystal symmetry.
  • For each strained configuration, the internal atomic positions are re-optimized (while keeping the strained lattice fixed) to eliminate any residual internal forces.
• Stress Calculation: For each optimized strained structure, the stress tensor is calculated using the uMLIP.
• Constant Determination: The elastic constants are obtained from the linear relationship between the applied strain tensors and the computed stress tensors, typically via a linear regression analysis.

The Scientist's Toolkit

This section details the essential software, data, and computational resources required for working with uMLIPs.

Table 4: Essential Research Reagents and Resources for uMLIP Applications

Resource Type	Specific Examples	Function/Role	Key Details
Pretrained uMLIPs	CHGNet, MACE-MP-0, SevenNet-0 [89] [86] [88]	Out-of-the-box potential for energy, force, and stress prediction.	Provide a foundational model; often used as-is or fine-tuned.
Training Datasets	Materials Project Trajectory (MPtrj) [86] [87]	Large-scale dataset for training/benchmarking universal potentials.	Contains ~1.5M DFT calculations (energies, forces, stresses, magmoms).
Simulation Software	ASE (Atomic Simulation Environment), VASP, Abinit	Environments for running MD, geometry relaxations, and applying strains.	uMLIPs are often integrated as calculators within these tools.
Analysis Packages	Phonopy, pymatgen	Specialized tools for calculating phonons and analyzing crystal structures.	Essential for downstream property prediction from relaxed structures.
Computational Hardware	GPUs (e.g., NVIDIA A100)	Accelerate both training and inference of large GNN models.	Training models like MACE-MP-0 required 310 A100-days [20].

This comparative analysis delineates the distinct strengths and optimal application domains for CHGNet, MACE, and SevenNet. CHGNet is the preferred choice for investigating systems where charge transfer and electronic effects are paramount, such as in battery electrode materials, and for high-throughput relaxations where robustness is critical. MACE offers a robust balance of high accuracy and efficiency, making it an excellent general-purpose model for a wide range of molecular dynamics and property prediction tasks. SevenNet emerges as the top performer for predicting sensitive second-order properties like phonon spectra and elastic constants, though this may come with higher computational costs. The ongoing development of more efficient architectures, such as Facet, promises to further reduce the resource barrier for training and applying these powerful models [20]. The choice of uMLIP should be guided by the specific target properties, the chemical system of interest, and the available computational resources.

The application of Graph Neural Networks (GNNs) to molecular and materials science represents a paradigm shift in how researchers predict stability and properties of chemical systems. However, the "black box" nature of these models necessitates rigorous validation against fundamental chemical principles to ensure predictions stem from learned physics rather than data artifacts. This document outlines application notes and protocols for validating three critical aspects of GNN interatomic interactions: their effective range, proper spatial decay, and capacity to capture many-body effects. As GNNs increasingly inform experimental decisions in domains from drug discovery to materials design [93], establishing standardized validation methodologies becomes essential for building scientific trust and facilitating adoption.

Core Chemical Principles and Corresponding GNN Validation Metrics

The reliability of a GNN potential hinges on its adherence to physical laws. The table below outlines the core principles and corresponding quantitative metrics for model validation.

Table 1: Core Chemical Principles and GNN Validation Metrics

Chemical Principle	Physical Significance	Key GNN Validation Metrics	Reference Experimental/Oracle Data
Interaction Range	Determines long-range energy and force contributions; critical for electrostatics, van der Waals.	Problem radius [94], Range measure [94], Prediction accuracy on long-range graph benchmarks.	DFT with high-quality long-range treatments (e.g., HSE), coupled cluster calculations, experimental crystal packing energies.
Interaction Decay	Governs how interatomic forces diminish with distance; system stability depends on correct asymptotic behavior.	Rate of distance-based influence score decay [94], Sensitivity analysis for distant nodes [94], Jacobian norm w.r.t. input features of distant atoms.	Reference ab initio potential energy scans for dimer interaction energies at various separations.
Many-Bodyness	Captures non-additive effects beyond pairwise sums; essential for polarization, charge transfer, covalent bonding.	Magnitude of n-body (n>2) relevance scores from GNN-LRP [31], Performance on tasks requiring explicit 3-body terms, Contribution of specialized many-body architectural components.	Quantum chemical calculations (e.g., MP2, CCSD(T)) that decompose energy into n-body contributions.

Quantitative Validation Frameworks and Data

Establishing a quantitative baseline is prerequisite for meaningful validation. The following table summarizes key results from recent literature evaluating GNN performance against these chemical principles.

Table 2: Quantitative Validation Data from Recent Studies

Study & Model	Validation Focus	Key Quantitative Result	Implication for Chemical Principle
EOSnet [22]	Many-body interactions via orbital overlap.	MAE of 0.163 eV on band gap prediction; 97.7% accuracy in metal/nonmetal classification.	Superior performance on electronic properties confirms model captures quantum mechanical many-body effects.
GNN-LRP on CG NNP [31]	Decomposition of learned energy into n-body terms.	Identified dominant 2-body and 3-body contributions in fluids (methane, water) consistent with physics.	Provides direct, human-interpretable evidence that the NNP learns physically meaningful many-body interactions.
H-HIGNN [95]	Long-range hydrodynamic interactions (HI) in suspensions.	Achieved quasi-linear scaling; accurately captured many-body HI and slow-decaying two-body interactions.	Demonstrates GNNs can be engineered to respect both the long-range nature and many-body physics of complex interactions.
Range Measure Analysis [94]	Long-range interactions in general graph tasks.	Formalized "problem radius" and "range measure"; critiqued existing benchmarks (LRGB) via these measures.	Provides a principled, theoretical metric beyond empirical task performance to assess a GNN's long-range capability.
EMFF-2025 [24]	Generalizability for energetic materials (C, H, N, O).	Energy MAE < 0.1 eV/atom; Force MAE < 2 eV/Å across 20 HEMs; accurately predicted decomposition mechanisms.	A model accurate across diverse molecular structures and properties likely has learned valid interatomic interactions.

Detailed Experimental Protocols

Protocol 1: Validating Interaction Range and Decay using Influence Scores

This protocol measures how a GNN's prediction for a target node is influenced by other nodes as a function of distance, validating correct long-range behavior.

1. Research Reagent Solutions

Table 3: Essential Tools for Range and Decay Validation

Item	Function	Example/Note
Trained GNN Model	The object of validation.	Any GNN for node-level prediction (e.g., MPNN, Graph Transformer).
Graph Dataset	Provides structures for analysis.	Should include graphs with varying diameters. Synthetic graphs with known interaction ranges are ideal [94].
Influence Score Calculator	Quantifies node-to-node influence.	Implementation of Jacobian-based influence: ( I(u, v) = \sum \left	\frac{\partial \mathbf{y}u}{\partial \mathbf{x}v} \right	) [94], where ( \mathbf{y}u ) is the output for node ( u ) and ( \mathbf{x}v ) is the input feature of node ( v ).
Distance Matrix Calculator	Computes shortest-path or Euclidean distance between all node pairs.	scipy.spatial.distance or networkx algorithms.

2. Step-by-Step Workflow

• Input Preparation: Select a representative graph ( \mathsf{G} = (\mathsf{V}, \mathsf{E}) ) with node features ( \mathbf{X} ) from the test set.
• Target Selection: Choose a target node ( u \in \mathsf{V} ) for which the model makes a prediction ( \mathbf{y}_u ).
• Influence Calculation: For all source nodes ( v \in \mathsf{V} ), compute the influence score ( I(u, v) ). This is the sum of the absolute values of the Jacobian matrix ( \frac{\partial \mathbf{y}u}{\partial \mathbf{x}v} ) [94].
• Distance Binning: Calculate the shortest-path or Euclidean distance ( d(u,v) ) between ( u ) and every ( v ). Group node pairs ( (u,v) ) into bins based on their distance.
• Averaging: Compute the average influence score ( \bar{I}(d) ) for each distance bin ( d ).
• Plotting and Analysis: Plot ( \bar{I}(d) ) versus distance ( d ). A valid model should show an influence that decays with distance. Compare the decay profile against the expected physical decay (e.g., exponential for screened interactions, power-law for electrostatic).

Figure 1: Workflow for validating interaction range and decay using influence scores.

Protocol 2: Validating Many-Bodyness using GNN-LRP

This protocol uses Layer-wise Relevance Propagation (LRP) to decompose a GNN's energy prediction into 1-body, 2-body, and 3-body contributions, confirming the model captures essential non-pairwise interactions [31].

1. Research Reagent Solutions

Table 4: Essential Tools for Many-Bodyness Validation

Item	Function	Example/Note
Trained GNN Potential (NNP)	The object of validation.	A GNN trained to predict potential energy (e.g., for a molecule or coarse-grained system).
Molecular Dynamics Dataset	Provides atomic/bead configurations for analysis.	Configurations of a system where many-body effects are known to be significant (e.g., water [31]).
GNN-LRP Implementation	Decomposes the predicted energy into n-body relevance scores.	Code for GNN-LRP that attributes relevance to walks on the graph, aggregating scores for n-body subgraphs [31].
Visualization Suite	Plots and compares n-body contributions.	matplotlib, seaborn, or VMD for molecular visualization.

2. Step-by-Step Workflow

• Input Preparation: Select a molecular configuration (e.g., a snapshot from an MD simulation) and represent it as a graph.
• Energy Prediction: Pass the graph through the trained GNN to obtain the total predicted energy ( E ).
• Relevance Propagation: Run the GNN-LRP algorithm to decompose the output energy ( E ) into a sum of relevance scores ( R(\omega) ) for each walk ( \omega ) in the graph [31].
• n-Body Aggregation: Aggregate the relevance scores of all walks that involve exactly ( n ) distinct nodes (atoms/beads). This yields the total n-body contribution to the energy, ( E_n ). For example, sum relevance of all walks on edges (i,j) for 2-body, and on triangles (i,j,k) for 3-body.
• Magnitude Assessment: Compare the relative magnitudes of ( E2 ) (2-body) and ( E3 ) (3-body). A model capturing many-body effects will show non-negligible ( E_3 ), especially in systems like water where 3-body effects are crucial.
• Physical Consistency Check: Verify that the sign and spatial distribution of the 3-body contributions align with chemical intuition (e.g., stabilizing hydrogen-bonding networks in water [31]).

Figure 2: Workflow for validating many-body interactions using GNN-LRP.

Case Studies in Materials and Drug Discovery

Case Study: Discovery of Stable Zintl Phases

A GNN model was used to screen over 90,000 hypothetical Zintl phases for thermodynamic stability, employing the Upper Bound Energy Minimization (UBEM) strategy [96]. The model, trained on volume-relaxed DFT data, achieved a remarkably low test MAE of 27 meV/atom. This high-precision prediction enabled the identification of 1,810 new stable phases, which were subsequently validated with DFT, achieving a 90% precision rate [96]. This success underscores that the GNN's internal representation of interatomic interactions must be physically valid to achieve such high extrapolative accuracy in a vast chemical space. The model significantly outperformed other potentials like M3GNet (40% precision), highlighting the impact of architecture and training on learning correct interactions [96].

Case Study: Interpretable Coarse-Grained Potentials

In a study on coarse-grained neural network potentials (CG NNPs) for methane, water, and protein NTL9, GNN-LRP was used to peer inside the black box [31]. The interpretation revealed that the learned effective interactions were dominated by physically meaningful 2-body and 3-body terms. Furthermore, the analysis allowed researchers to pinpoint specific stabilizing and destabilizing interactions in various protein metastable states and to interpret mutation effects [31]. This case demonstrates that validation is not just about predictive accuracy but also about interpretability. Proving that a model's reasoning aligns with established chemical principles builds indispensable trust, especially in sensitive fields like drug discovery [93] where understanding the mechanism is as crucial as the prediction itself.

The accurate prediction of elastic and mechanical properties from atomic structure is a cornerstone of computational materials science and drug development. These properties are critical for applications ranging from the design of high-energy materials to the development of robust metal-organic frameworks for gas separation. Traditional methods, such as Density Functional Theory (DFT), provide high accuracy but at computational costs that preclude high-throughput screening. The emergence of Graph Neural Networks (GNNs) and other Machine Learning Interatomic Potentials (MLIPs) offers a transformative alternative, promising DFT-level accuracy with significantly reduced computational expense [24] [97]. This application note provides a critical benchmark of these modern computational tools, framing them within a broader research thesis on leveraging GNNs for predicting interatomic interactions and material stability. We synthesize performance metrics across diverse material classes and provide detailed protocols for researchers to implement and validate these methods in their workflows.

Benchmarking Performance of GNNs and MLIPs

Quantitative Benchmarking of Model Accuracy

The predictive performance of MLIPs and GNNs for mechanical properties has been rigorously evaluated across multiple benchmarks. The following tables summarize key quantitative results from recent studies, providing a clear comparison of model capabilities.

Table 1: Benchmark of Universal MLIPs on Material Properties (Phonon Database) [97]

Model Name	Energy MAE (meV/atom)	Force MAE (meV/Å)	Phonon Frequency MAE	Spearman Coefficient (PDOS)
ORB v3	-	-	-	>0.95
SevenNet-MP-ompa	-	-	-	>0.95
GRACE-2L-OAM	-	-	-	>0.95
MatterSim v1 5M	-	-	-	~0.94
MACE-MPA-0	-	-	-	~0.94
eSEN-30M-OAM	-	-	-	~0.94
CHGNet	-	-	-	~0.90
M3GNet	-	-	-	~0.90

Note: This benchmark was performed on a database of 4,869 inorganic crystals covering 86 elements. The Spearman coefficient quantifies the alignment of phonon density of states (PDOS) with DFT calculations.

Table 2: Performance of AlphaNet on Diverse Material Systems [98]

Material System / Task	Force MAE (meV/Å)	Energy MAE (meV/atom)	Key Comparative Result
Formate Decomposition	42.5	0.23	Outperformed NequIP (47.3 meV/Å, 0.50 meV/atom)
Defected Graphene	19.4	1.2	Significantly outperformed NequIP (60.2 meV/Å, 1.9 meV/atom)
Zeolites (16 types)	-	-	~20% improvement over other equivariant models
OC20 (S2EF task)	-	0.24 eV	On par with EquiformerV2 and EScAIP
Matbench Discovery (AlphaNet-S)	-	-	F1=0.808, DAF=4.915, R²=0.796

Table 3: Performance of the EMFF-2025 Potential for High-Energy Materials [24]

Property Category	Predictive Performance	Materials Tested
Energy	MAE predominantly within ± 0.1 eV/atom	20 HEMs with C, H, N, O elements
Forces	MAE predominantly within ± 2 eV/Å	20 HEMs with C, H, N, O elements
Crystal Structures	Accurate prediction	20 HEMs
Mechanical Properties	Accurate prediction	20 HEMs
Thermal Decomposition	Revealed similar high-temperature mechanisms	20 HEMs

The Impact of Architectural Choices on Predictive Performance

Architectural innovations in GNNs and MLIPs directly influence their capability to capture elastic and mechanical properties. The inclusion of angular information is particularly critical for accurately modeling mechanical responses.

Table 4: Performance of Angular-Dependent Graph Representations [99]

Graph Representation	Description	Validation Loss (Relative)	Inference Speed	Memory Edges
({{{{\rm{G}}}}_{\min}})	Minimally connected graph	Baseline (Highest)	Fastest	Lowest
({{{{\rm{G}}}}_{\max}})	Maximally connected graph	~25% lower	Slowest	Highest
ALIGNN	Includes bond angles	~20% lower	Medium	Medium
ALIGNN-d	Includes bond + dihedral angles	~25% lower (Similar to ({{{{\rm{G}}}}_{\max}}))	27% faster than ({{{{\rm{G}}}}_{\max}})	33% fewer than ({{{{\rm{G}}}}_{\max}})

Note: ALIGNN-d achieves performance equivalent to the maximally connected graph but with significantly improved computational efficiency, demonstrating that complete geometric information can be captured without brute-force connectivity.

The TSGNN model, which fuses spatial and topological information through a dual-stream architecture, addresses another key limitation of conventional GNNs that focus solely on topological relationships. This model initializes atom representations using a 2D matrix based on the periodic table, providing a more comprehensive depiction of atomic characteristics, and has demonstrated superior performance in predicting formation energies [100].

Experimental Protocols for Validation and Application

Protocol 1: Benchmarking MLIPs for Phonon and Elastic Properties

This protocol outlines the procedure for benchmarking universal MLIPs, as implemented in [97].

1. Database Curation:

• Select stable, realistic crystals from sources like the Materials Project.
• Focus on crystals with unit cells containing up to 12 atoms to ensure manageable computational complexity while maintaining diverse chemical space coverage.
• Apply filters for structural diversity, ensuring coverage of multiple crystal systems and space groups.
• The final database should contain several thousand crystals (e.g., 4,869 as in [97]) covering a wide range of elements (e.g., 86).

2. Structure Relaxation:

• For each crystal and each uMLIP, perform atomic coordinate relaxation to find the optimized structure with minimum potential energy.
• Use the same convergence criteria (for forces and energy) across all models for fair comparison.
• Calculate the average discrepancy in atomic coordinates between uMLIP-relaxed and DFT-relaxed structures.

3. Phonon Calculation:

• Perform phonon calculations on a uniform mesh sampling of the first Brillouin zone.
• Compute phonon densities of states (PDOS) for each material using each uMLIP.
• Calculate the average difference in phonon frequencies compared to DFT reference data.

4. Spectral Similarity Quantification:

• Compute Spearman coefficients between uMLIP-generated and DFT-reference PDOS.
• Values range between 0-1, with higher values indicating better alignment.

5. Thermodynamic Property Calculation:

• Derive vibrational entropy (S), Helmholtz free energy (F), and constant volume heat capacity (C_V) from phonon results.
• Compare these derived properties with DFT reference values.

6. Experimental Validation:

• Compare uMLIP calculations with experimental inelastic neutron scattering (INS) data where available.
• Analyze both coherent and incoherent scattering scenarios.
• For powder samples, compare the 2D S(|Q|,E) data between simulation and experiment.

Protocol 2: Predicting Mechanical Properties of High-Energy Materials with NNPs

This protocol is adapted from the methodology used to develop and validate the EMFF-2025 potential [24].

1. Model Development with Transfer Learning:

• Begin with a pre-trained NNP model (e.g., DP-CHNO-2024 for HEMs).
• Incorporate a small amount of new training data from structures not included in the existing database through the DP-GEN process.
• Use transfer learning to adapt the model to specific HEM classes with minimal additional DFT calculations.

2. Energy and Force Validation:

• Compare predicted energies and forces against DFT calculations for a diverse set of HEMs.
• Calculate mean absolute error (MAE) for energy (target: ± 0.1 eV/atom) and forces (target: ± 2 eV/Å).
• Generate parity plots showing predicted vs. DFT-calculated values.

3. Crystal Structure Prediction:

• Predict crystal structures for HEMs not included in the training set.
• Compare lattice parameters and atomic coordinates with experimental data where available.
• Calculate deviation metrics such as root mean square deviation (RMSD).

4. Mechanical Property Calculation:

• Compute elastic constants through strain-fluctuation methods or direct deformation approaches.
• Derive bulk modulus, shear modulus, and Young's modulus from elastic constants.
• Compare with experimental measurements and DFT benchmarks.

5. Thermal Decomposition Analysis:

• Perform molecular dynamics simulations at elevated temperatures (e.g., 2000-3000 K).
• Monitor decomposition pathways and reaction products.
• Use principal component analysis (PCA) and correlation heatmaps to identify common decomposition mechanisms across different HEMs.

6. Chemical Space Mapping:

• Apply dimensionality reduction techniques (e.g., PCA, t-SNE) to the structural or property data.
• Identify clusters and correlations in the chemical space of HEMs.
• Relate structural motifs to stability and mechanical properties.

The Scientist's Toolkit: Essential Research Reagents

Table 5: Key Computational Tools for GNN-Based Mechanical Property Prediction

Tool Category	Specific Examples	Function in Research	Reference
Universal MLIPs	ORB v3, MACE-MPA-0, CHGNet, MatterSim	Pre-trained models for fast property prediction without system-specific training	[97]
Specialized NNPs	EMFF-2025, DP-CHNO-2024	High-accuracy potentials for specific material classes (e.g., high-energy materials)	[24]
GNN Architectures	ALIGNN-d, TSGNN, AlphaNet	Advanced models incorporating angular information and spatial relationships	[100] [99] [98]
Benchmark Databases	Materials Project, OC20, Matbench Discovery	Curated datasets for training and benchmarking predictive models	[97] [98]
Training Frameworks	DP-GEN, PyTorch, TensorFlow	Automated training and optimization of machine learning potentials	[24]
Analysis Tools	PCA, Correlation Heatmaps, Phonopy	Interpretation of results and derivation of thermodynamic properties	[24] [97]

This critical benchmark demonstrates that GNNs and MLIPs have reached a maturity level where they can provide reliable predictions of elastic and mechanical properties across diverse material systems. The EMFF-2025 potential shows exceptional capability for high-energy materials, while universal MLIPs like ORB v3 and MACE-MPA-0 offer broad applicability across inorganic crystals. Architectural innovations that incorporate angular information (ALIGNN-d) and spatial relationships (TSGNN) significantly enhance predictive accuracy for mechanical properties that depend on complex many-body interactions. The experimental protocols provided herein offer researchers standardized methodologies for validating these tools in their specific domains, particularly in pharmaceutical and materials development pipelines where understanding mechanical behavior is critical for stability and performance. As these models continue to evolve, their integration into automated discovery workflows will dramatically accelerate the identification and optimization of materials with tailored mechanical properties.

Conclusion

Graph Neural Networks have fundamentally enhanced our ability to model interatomic interactions and predict system stability with near-quantum accuracy and significantly improved computational efficiency. The integration of physical symmetries, advanced message-passing schemes, and robust uncertainty quantification has led to more reliable models applicable across drug discovery and materials science. Moving forward, key challenges remain in achieving universal transferability, further improving simulation stability, and enhancing model interpretability. The convergence of multi-modal training, active learning frameworks, and explainable AI will be crucial for developing the next generation of GNNs. For biomedical research, these advances promise to accelerate drug development by more accurately predicting drug-target interactions and adverse drug reactions, ultimately enabling more efficient and safer therapeutic design. The continued co-design of models, data, and physical constraints will further solidify GNNs as indispensable tools in computational molecular sciences.

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