Graph Neural Networks (GNNs), especially message-passing neural networks (MPNNs), have emerged as powerful architectures for learning on graphs in diverse applications. However, MPNNs face challenges when modeling non-local interactions in systems such as large conjugated molecules, metals, or amorphous materials. Although Spectral GNNs and traditional neural networks such as recurrent neural networks and transformers mitigate these challenges, they often lack extensivity, adaptability, generalizability, computational efficiency, or fail to capture detailed structural relationships or symmetries in the data. To address these concerns, we introduce Matrix Function Neural Networks (MFNs), a novel architecture that parameterizes non-local interactions through analytic matrix equivariant functions. Employing resolvent expansions offers a straightforward implementation and the potential for linear scaling with system size. The MFN architecture achieves state-of-the-art performance in standard graph benchmarks, such as the ZINC and TU datasets, and is able to capture intricate non-local interactions in quantum systems. The code and the datasets will be made public.

Reductive Lie Groups, such as the orthogonal groups, the Lorentz group, or the unitary groups, play essential roles across scientific fields as diverse as high energy physics, quantum mechanics, quantum chromodynamics, molecular dynamics, computer vision, and imaging. In this paper, we present a general Equivariant Neural Network architecture capable of respecting the symmetries of the finite-dimensional representations of any reductive Lie Group. Our approach generalizes the successful ACE and MACE architectures for atomistic point clouds to any data equivariant to a reductive Lie group action. We also introduce the lie-nn software library, which provides all the necessary tools to develop and implement such general G-equivariant neural networks. It implements routines for the reduction of generic tensor products of representations into irreducible representations, making it easy to apply our architecture to a wide range of problems and groups. The generality and performance of our approach are demonstrated by applying it to the tasks of top quark decay tagging (Lorentz group) and shape recognition (orthogonal group).

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.