Creating fast and accurate force fields is a long-standing challenge in computational chemistry and materials science. Recently, Equivariant Message Passing Neural Networks (MPNNs) have emerged as a powerful tool for building machine learning interatomic potentials, outperforming other approaches in terms of accuracy. However, they suffer from high computational cost and poor scalability. Moreover, most MPNNs only pass two-body messages leading to an intricate relationship between the number of layers and the expressivity of the features. This work introduces MACE, a new equivariant MPNN model that uses higher order messages, and demonstrates that this leads to an improved learning law. We show that by using four-body messages, the required number of message passing iterations reduces to just one, resulting in a fast and highly parallelizable model, reaching or exceeding state of the art accuracy on the rMD17 and 3BPA benchmark tasks. Our implementation is available at https://github.com/ACEsuit/mace.

Creating fast and accurate force fields is a long-standing challenge in computational chemistry and materials science. Recently, Equivariant Message Passing Neural Networks (MPNNs) have emerged as a powerful tool for building machine learning interatomic potentials, outperforming other approaches in terms of accuracy. However, they suffer from high computational cost and poor scalability. Moreover, most MPNNs only pass two-body messages leading to an intricate relationship between the number of layers and the expressivity of the features. This work introduces MACE, a new equivariant MPNN model that uses higher order messages, and demonstrates that this leads to an improved learning law. We show that by using four-body messages, the required number of message passing iterations reduces to just one, resulting in a fast and highly parallelizable model, reaching or exceeding state of the art accuracy on the rMD17 and 3BPA benchmark tasks.

This work proposes a geometric insight into equivariant message passing on Riemannian manifolds. As previously proposed, numerical features on Riemannian manifolds are represented as coordinate-independent feature fields on the manifold. To any coordinate-independent feature field on a manifold comes attached an equivariant embedding of the principal bundle to the space of numerical features. We argue that the metric this embedding induces on the numerical feature space should optimally preserve the principal bundle's original metric. This optimality criterion leads to the minimization of a twisted form of the Polyakov action with respect to the graph of this embedding, yielding an equivariant diffusion process on the associated vector bundle. We obtain a message passing scheme on the manifold by discretizing the diffusion equation flow for a fixed time step. We propose a higher-order equivariant diffusion process equivalent to diffusion on the cartesian product of the base manifold. The discretization of the higher-order diffusion process on a graph yields a new general class of equivariant GNN, generalizing the ACE and MACE formalism to data on Riemannian manifolds.