MACE: Higher Order Equivariant Message Passing Neural Networks for Fast and Accurate Force Fields

TL;DR

MACE introduces higher-order, equivariant message passing in MPNNs to overcome two-body limitations in force-field models. By expanding messages up to four-body correlations through efficient tensor contractions, the method achieves state-of-the-art accuracy with only two message-passing iterations, enabling fast, highly parallelizable training and inference. Empirical results on rMD17, 3BPA, and AcAc demonstrate strong accuracy, data efficiency, and extrapolation capabilities, with substantial speed advantages over prior equivariant MPNNs. The work highlights the importance of body-order in learning local environments and suggests broad applicability to larger systems and condensed phases while maintaining computational practicality.

Abstract

Creating fast and accurate force fields is a long-standing challenge in computational chemistry and materials science. Recently, several equivariant message passing neural networks (MPNNs) have been shown to outperform models built using other approaches in terms of accuracy. However, most MPNNs suffer from high computational cost and poor scalability. We propose that these limitations arise because MPNNs only pass two-body messages leading to a direct relationship between the number of layers and the expressivity of the network. In this work, we introduce MACE, a new equivariant MPNN model that uses higher body order messages. In particular, we show that using four-body messages reduces the required number of message passing iterations to just two, resulting in a fast and highly parallelizable model, reaching or exceeding state-of-the-art accuracy on the rMD17, 3BPA, and AcAc benchmark tasks. We also demonstrate that using higher order messages leads to an improved steepness of the learning curves.

Figures (4)

• Figure 1: Energy and force errors of BOTNet, NequIP, and MACE ($L = 2$) on the 3BPA dataset at different temperatures as a function of the number of layers.
• Figure 2: Learning curve of force errors (MAE in eV / Å) for aspirin from the rMD17 dataset for different models. Left: Two layers of invariant ($L=0$) MACE with increasing body order $\nu \in \{1,2,3\}$. Center: Two layers of MACE with $\nu = 3$ and increasing equivariance $L \in \{0,1,2\}$. Right: Two layers of MACE with $\nu = 1$ and increasing equivariance $L \in \{0,1,2\}$. In each case the slope ($s$) is indicated.
• Figure 3: Energy predictions on three cuts through the potential energy surface of the 3-(benzyloxy)pyridin-2-amine (3BPA) molecule by BOTNet, NequIP, and MACE ($L=2$). The ground-truth energy (DFT) is shown in black. For each cut, the curves have been shifted vertically so that the lowest ground-truth energy is zero.
• Figure 4: Left: energy predictions for a dihedral slice of the DFT potential energy surface of acetylacetone. Right: energy predictions for the proton transfer in acetylacetone. Error bars indicate one standard deviation computed over three runs. The histograms show the distribution of the training data along the relevant coordinate.