The Design Space of E(3)-Equivariant Atom-Centered Interatomic Potentials

TL;DR

The paper introduces Multi-ACE, a unifying framework that connects Atomic Cluster Expansion with Euclidean-equivarient neural potentials, enabling systematic exploration of design choices in E(3)-equivariant interatomic potentials. It presents BOTNet, a body-ordered, interpretable variant of NequIP that preserves core equivariant tensor operations while simplifying nonlinearities and radial bases. Through extensive ablation studies on in- and out-of-domain accuracy and extrapolation, the work identifies critical factors such as normalization, channel coupling, and element embedding that govern performance and generalization. The results demonstrate that BOTNet achieves state-of-the-art or competitive accuracy across diverse benchmarks (rMD17, 3BPA, acetylacetone) and exhibits robust extrapolation, highlighting the utility of the Multi-ACE design space for rapid experimentation and future model development.

Abstract

The rapid progress of machine learning interatomic potentials over the past couple of years produced a number of new architectures. Particularly notable among these are the Atomic Cluster Expansion (ACE), which unified many of the earlier ideas around atom density-based descriptors, and Neural Equivariant Interatomic Potentials (NequIP), a message passing neural network with equivariant features that showed state of the art accuracy. In this work, we construct a mathematical framework that unifies these models: ACE is generalised so that it can be recast as one layer of a multi-layer architecture. From another point of view, the linearised version of NequIP is understood as a particular sparsification of a much larger polynomial model. Our framework also provides a practical tool for systematically probing different choices in the unified design space. We demonstrate this by an ablation study of NequIP via a set of experiments looking at in- and out-of-domain accuracy and smooth extrapolation very far from the training data, and shed some light on which design choices are critical for achieving high accuracy. Finally, we present BOTNet (Body-Ordered-Tensor-Network), a much-simplified version of NequIP, which has an interpretable architecture and maintains accuracy on benchmark datasets.

Figures (10)

• Figure 1: Illustration of the construction of high body-order ACE features. First, a neighborhood graph is constructed with each node being labeled by its state. Then, the one-particle basis is computed for each edge. After that, a pooling operation is performed to create permutation invariant $A$-functions of semi-local environments. To construct higher body-order features, the product basis is formed by taking tensor-products of all coupled indices of the $A$-functions. Finally, to create equivariant messages, the $\bm{B}$ basis is formed by first specifying the required equivariance and evaluating the corresponding symmetrisation integral. We illustrate here the case of invariant $\bm{B}$ basis.
• Figure 2: Block structure of weight matrices for an equivariant linear operation. As only linear combinations of features of the same representations (here $l_{0},l_{1},l_{2}$) are allowed to interact, the weight matrix is block diagonal.
• Figure 3: Comparison of the clusters formed by two iterations of message passing with cutoff $r_\text{cut}$ at each iteration on the left and the clusters formed by ACE with cutoff $2 r_\text{cut}$ on the right. In principle, both methods incorporate information from a distance of up to $2 r_\text{cut}$, but in the case of the MPNN, only atoms that can be reached through a chain of closer intermediates contribute.
• Figure 4: Decomposition of the total energy predicted by BOTNet for the H-transfer pathway in Acetylacetone. $E_{0}$ corresponds to the "1-body" atomic energies. The contributions from $E_{1}$ to $E_{4}$ represent energies of increasing body order (2-body to 5-body, respectively). The curves are shifted by the energy of the last configuration in the transfer path, which is annotated in red above each plot. All energies are in eV.
• Figure 5: Illustration of the architectures of NequIP (a) and BOTNet (b). Panel (c) contains illustrations of components the architectures have in common.