E(3)-Equivariant Graph Neural Networks for Data-Efficient and Accurate Interatomic Potentials (2101.03164v3)

Abstract: This work presents Neural Equivariant Interatomic Potentials (NequIP), an E(3)-equivariant neural network approach for learning interatomic potentials from ab-initio calculations for molecular dynamics simulations. While most contemporary symmetry-aware models use invariant convolutions and only act on scalars, NequIP employs E(3)-equivariant convolutions for interactions of geometric tensors, resulting in a more information-rich and faithful representation of atomic environments. The method achieves state-of-the-art accuracy on a challenging and diverse set of molecules and materials while exhibiting remarkable data efficiency. NequIP outperforms existing models with up to three orders of magnitude fewer training data, challenging the widely held belief that deep neural networks require massive training sets. The high data efficiency of the method allows for the construction of accurate potentials using high-order quantum chemical level of theory as reference and enables high-fidelity molecular dynamics simulations over long time scales.

• The paper presents NequIP, which leverages E(3)-equivariant convolutions to dramatically improve data efficiency and simulation accuracy in molecular dynamics.
• It introduces a graph-based architecture with tensor features and TFN layers that preserve physical symmetries under rotations and translations.
• Empirical results demonstrate significant reductions in energy and force errors across diverse systems, enabling high-fidelity simulations with minimal training data.

E(3)-Equivariant Graph Neural Networks for Data-Efficient and Accurate Interatomic Potentials

The paper "E(3)-Equivariant Graph Neural Networks for Data-Efficient and Accurate Interatomic Potentials" introduces Neural Equivariant Interatomic Potentials (NequIP), an advanced approach leveraging E(3)-equivariant neural networks to enhance data efficiency and accuracy in modeling interatomic potentials for molecular dynamics (MD) simulations. This addresses a significant challenge in computational chemistry and materials science where conventional machine learning interatomic potentials (ML-IPs) often suffer from substantial data requirements, limiting their usability for complex or high-fidelity simulations. NequIP stands out by significantly reducing the training data needed while improving predictive accuracy, thus providing a highly efficient alternative for MD simulations.

Equivariant Neural Networks

The core innovation lies in the use of E(3)-equivariant convolutions, which respect the symmetries of Euclidean space, including rotations, translations, and reflections. Traditional ML-IPs like SchNet or DimeNet operate on invariant features, but NequIP uses tensor features that retain equivariance properties, making internal tensor operations inherently consistent with physical transformations. This not only preserves the physical properties of molecular systems under rotations and translations but also enhances the richness of geometric information embedded in the learned representations.

Formally, a function $f: X \rightarrow Y$ is equivariant with respect to a group $G$ if applying a transformation $g \in G$ to $X$, followed by $f$, is equivalent to applying $f$ first and then transforming the result in $Y$. This property is crucial for accurate physical simulations where the output tensor (e.g., forces, dipoles) must transform appropriately under coordinate changes.

Architecture and Implementation

NequIP's architecture adopts a graph-based approach where atoms represent nodes connected by edges if they are within a specified cutoff distance $r_c$. The network utilizes tensor field network (TFN) layers, enabling direct use of geometric tensors. Each atom is associated with scalar, vector, and higher-order tensor features. The equivariant convolutions in NequIP combine invariant radial functions with spherical harmonics, ensuring the filters transform correctly under rotations and reflections.

A significant aspect of the implementation is ensuring energy conservation by deriving forces from the gradient of the potential energy predicted by the network. This guarantees that the predicted forces are consistent with gradient-derived properties essential for stable and accurate MD simulations.

Empirical Performance

The paper reports empirical results across various challenging data sets, including small organic molecules (MD-17), molecular systems at high chemical accuracy (CCSD(T)), and periodic materials such as liquid water and lithium thiophosphate (LiPS) superionic conductors. NequIP achieves state-of-the-art accuracy in these benchmarks, demonstrating superiority over existing methods like SchNet, DimeNet, and kernel-based approaches.

For instance, NequIP reduces energy and force prediction errors by significant margins across different molecular systems, even when trained on data sets as small as a few hundred configurations. In simulations of liquid water and ice (using a fraction of the data required by competitors like DeepMD), NequIP outperforms in force prediction accuracy while maintaining reasonable energy error margins, demonstrating its data efficiency and robustness.

Data Efficiency Analysis

The paper provides a detailed analysis of NequIP's data efficiency, asserting that leveraging tensor features and equivariant operations significantly enhances learning dynamics. Empirical learning curves demonstrated a shift in the log-log slope when switching from invariant to equivariant models, indicating a more substantial reduction in error rates as the training set size increases. This behavior contrasts with conventional GNN-IPs, which follow a fixed log-log slope irrespective of the method.

Implications and Future Directions

The implications of NequIP's advancements are profound for computational materials science and chemistry, enabling high-fidelity simulations of complex molecular systems with reduced computational overhead. The ability to train accurate models with minimal data opens doors to studying systems where extensive high-accuracy data would be otherwise prohibitive.

Theoretical exploration of why equivariance yields such benefits could further refine ML algorithms in this field. Moreover, understanding the many-body interactions in message-passing neural networks remains a fertile area of research.

Conclusion

NequIP exemplifies a sophisticated integration of symmetry-aware deep learning with molecular dynamics, setting a new benchmark for accuracy and data efficiency. This work paves the way for enhanced computational explorations, offering robust tools for high-fidelity simulations across diverse applications in chemistry, materials science, and beyond. Researchers can leverage NequIP to achieve computationally efficient, accurate simulations that were previously infeasible, thus accelerating advancements in these domains.