An SO(3)-equivariant reciprocal-space neural potential for long-range interactions

Lingfeng Zhang; Taoyong Cui; Dongzhan Zhou; Lei Bai; Sufei Zhang; Luca Rossi; Mao Su; Wanli Ouyang

physics.chem-phMar 19, 2026

An SO(3)-equivariant reciprocal-space neural potential for long-range interactions

Authors: Lingfeng Zhang, Taoyong Cui, Dongzhan Zhou, Lei Bai, Sufei Zhang, Luca Rossi, Mao Su, Wanli Ouyang, +1 more

Organizations: The Chinese University of Hong Kong, Hong Kong, 999077, China. · Department of Electrical and Electronic Engineering, The Hong Kong Polytechnic University, Hong Kong, 999077, China. · Shanghai Artificial Intelligence Laboratory, Shanghai, 200232, China. · Shenzhen Institute of Advanced Technology, Chinese Academy of Sciences, Shenzhen, 518055, China.

Abstract

Long-range electrostatic and polarization interactions play a central role in molecular and condensed-phase systems, yet remain fundamentally incompatible with locality-based machine-learning interatomic potentials. Although modern SO(3)-equivariant neural potentials achieve high accuracy for short-range chemistry, they cannot represent the anisotropic, slowly decaying multipolar correlations governing realistic materials, while existing long-range extensions either break SO(3) equivariance or fail to maintain energy-force consistency. Here we introduce EquiEwald, a unified neural interatomic potential that embeds an Ewald-inspired reciprocal-space formulation within an irreducible SO(3)-equivariant framework. By performing equivariant message passing in reciprocal space through learned equivariant k-space filters and an equivariant inverse transform, EquiEwald captures anisotropic, tensorial long-range correlations without sacrificing physical consistency. Across periodic and aperiodic benchmarks, EquiEwald captures long-range electrostatic behavior consistent with ab initio reference data and consistently improves energy and force accuracy, data efficiency, and long-range extrapolation. These results establish EquiEwald as a physically principled paradigm for long-range-capable machine-learning interatomic potentials.

Figures & tables

Figure 1: Overview. a , The whole model structure of EquiEwald. b , The EquiEwald long-range block takes node features and wave-vector inputs, applies an $\mathrm{SO}(3)$ linear map and gating, then decomposes degree- $\ell$ irreps into real/imag branches with $k$ -space filters driven by $\langle\mathbf{k},\mathbf{r}_{j}\rangle$ ; an inverse Fourier transform and an MLP return real-space updates per degree. Outputs across degrees are concatenated and gated, then fused with normalized local features to yield a rotationally equivariant long-range interaction update.

Panel of Figure 2 — Figure 2: Benchmark comparison between short-range and long-range models on charged molecular dimer systems. a–d Predicted versus reference energy results for polar-polar dimers. The short-range model is compared with three long-range variants: ( b ) eSCN+EwaldMP (scalar) Kosmala et al. (2023) , ( c ) eSCN+LES Cheng (2025) , and ( d ) eSCN+EquiEwald. For energy prediction, the short-range model fits only short-distance configurations and fails to generalize beyond the cutoff, while all long-range variants recover the full interaction curve, including distant test points.

Model	Test Energy (MAE)	Test Force (MAE)
eSCN ( Passaro and Zitnick, 2023 )	193.9	23.1
eSCN+EwaldMP (scalar) ( Kosmala et al., 2023 )	132.8	20.3
eSCN+EquiEwald	109.0	18.1

Table 1: Performance comparison of different long-range interaction modeling approaches on the AIMD-Chig dataset. The baseline eSCN model is compared with variants incorporating scalar EwaldMP, and the proposed EquiEwald method. Mean Absolute Errors (MAE) for energy and atomic forces are reported in meV and meV/Å , respectively.

Model	Test(Evaluation) Energy	Test(Evaluation) Force
eSCN ( Passaro and Zitnick, 2023 )	347.0	24.7
eSCN+EquiEwald	321.2	24.1
EquiformerV2 ( Liao et al., 2023 )	541.0	46.4
EquiformerV2+EquiEwald	453.0	38.4

Table 2: Mean absolute errors (MAE) of energy and force. Results are reported on the evaluation splits for OC20. Energy errors are reported in meV , and force errors are in meV/Å . Our method is highlighted in bold.

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Machine Learning Interatomic Potentials Equivariant GNNs Neural Operators Computational Materials Science

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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.

Abstract

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