NEMORA: Neural Equivariant Multipole Operators for Long-Range Atomistic Learning

2026-10-07

AI Summary

This study addresses the loss of long-range information in graph neural networks due to truncation, as well as the low computational efficiency and lack of strict equivariance in existing extension methods. To overcome these limitations, this work proposes NEMORA, a framework that for the first time neuralizes the fast multipole method (FMM) into a learnable, strictly equivariant operator. By integrating adaptive spatial hierarchies with an equivariant propagation mechanism, NEMORA achieves linear-complexity modeling of long-range interatomic potentials while preserving physical inductive biases. Experimental results demonstrate that NEMORA scales to systems comprising hundreds of thousands of atoms, reducing force prediction errors by one order of magnitude and energy errors by up to three orders of magnitude, thereby significantly outperforming existing state-of-the-art methods in accuracy.

Abstract

Equivariant graph neural networks have emerged as foundational architectures for machine-learned interatomic potentials, approaching quantum-chemical accuracy at a fraction of the computational cost. These models describe local atomic environments accurately, but finite spatial cutoffs truncate long-range information flow, and stacking message-passing layers can lead to over-smoothing and over-squashing. Existing long-range extensions either prescribe a fixed analytical propagation kernel, restrict long-range communication to scalars or degree-preserving channels, are only approximately equivariant, or incur super-linear computational cost. Combining learnable long-range equivariant transport with multiscale many-body expressivity and efficient scaling for larger systems remains a central challenge. We introduce Neural Equivariant Multipole Operators (NEMORA), a neural equivariant extension of the Fast Multipole Method (FMM) for learning long-range tensorial representations. NEMORA generalizes the FMM's analytical multipole expansion and translation operators to learned equivariant counterparts on an adaptive spatial hierarchy. Its operators couple angular degrees and form many-body interactions across length scales, retaining the FMM's hierarchical organization and analytical radial factors as physical inductive biases while learning data-dependent long-range couplings. NEMORA evaluates in linear time and memory complexity, allowing it to treat larger systems than other long-range methods reaching hundreds of thousands of atoms, and it augments both symmetry-constrained and unconstrained short-range backbones. On non-local benchmarks, it reduces force and energy errors relative to the short-range backbones by over an order of magnitude and up to three orders of magnitude, respectively, which is better than or competitive with existing long-range extensions in accuracy.

Problem

• equivariant graph neural networks
• long-range interactions
• interatomic potentials
• over-smoothing
• computational scaling

Innovation

• Equivariant Graph Neural Networks
• Fast Multipole Method
• Long-range Interactions
• Multipole Operators
• Linear Complexity