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

Linfeng Zhang, Taoyong Cui, Dongzhan Zhou, Lei Bai, Sufei Zhang, Luca Rossi, Mao Su, Wanli Ouyang, Pheng-Ann Heng

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.

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

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.
Paper Structure (10 sections, 17 equations, 2 figures, 2 tables)

This paper contains 10 sections, 17 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Results
  3. Preliminaries
  4. Overall structure of EquiEwald
  5. Experimental results
  6. Discussion
  7. Methods
  8. Model implementations
  9. Free-energy calculation
  10. Training settings

Figures (2)

  • 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.
  • 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) kosmala2023ewald, (c) eSCN+LES cheng2025latent, 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.