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

arXiv: https://arxiv.org/abs/2610.10776 · HTML: https://mlsanity.com/paper/2610.10776/

## Key facts

- No citing papers in the map yet (data through 7 Oct 2026).
- Builds on 29 of its 131 references that are in the map.
- First posted on arXiv 7 Oct 2026.

## Builds on (most cited first, 29 of 29)

- [Hierarchical Graph Representation Learning with Differentiable Pooling](https://mlsanity.com/paper/1806.08804/) (2018)
- [Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds](https://mlsanity.com/paper/1802.08219/) (2018)
- [On the Bottleneck of Graph Neural Networks and its Practical Implications](https://mlsanity.com/paper/2006.05205/) (2020)
- [MACE: Higher Order Equivariant Message Passing Neural Networks for Fast and Accurate Force Fields](https://mlsanity.com/paper/2206.07697/) (2022)
- [Multipole Graph Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2006.09535/) (2020)
- [Equiformer: Equivariant Graph Attention Transformer for 3D Atomistic Graphs](https://mlsanity.com/paper/2206.11990/) (2022)
- [e3nn: Euclidean Neural Networks](https://mlsanity.com/paper/2207.09453/) (2022)
- [On Over-Squashing in Message Passing Neural Networks: The Impact of Width, Depth, and Topology](https://mlsanity.com/paper/2302.02941/) (2023)
- [UMA: A Family of Universal Models for Atoms](https://mlsanity.com/paper/2506.23971/) (2025)
- [Ewald-based Long-Range Message Passing for Molecular Graphs](https://mlsanity.com/paper/2303.04791/) (2023)
- [N-body Networks: a Covariant Hierarchical Neural Network Architecture for Learning Atomic Potentials](https://mlsanity.com/paper/1803.01588/) (2018)
- [Equivariant Graph Hierarchy-Based Neural Networks](https://mlsanity.com/paper/2202.10643/) (2022)
- [Erwin: A Tree-based Hierarchical Transformer for Large-scale Physical Systems](https://mlsanity.com/paper/2502.17019/) (2025)
- [Graph Neural Networks Exponentially Lose Expressive Power for Node Classification](https://mlsanity.com/paper/1905.10947/) (2019)
- [Capturing long-range interaction with reciprocal space neural network](https://mlsanity.com/paper/2211.16684/) (2022)
- [Hierarchical graph neural nets can capture long-range interactions](https://mlsanity.com/paper/2107.07432/) (2021)
- [Neural P3M: A Long-Range Interaction Modeling Enhancer for Geometric GNNs](https://mlsanity.com/paper/2409.17622/) (2024)
- [MACE-POLAR-1: A Polarisable Electrostatic Foundation Model for Molecular Chemistry](https://mlsanity.com/paper/2602.19411/) (2026)
- [A recipe for scalable attention-based MLIPs: unlocking long-range accuracy with all-to-all node attention](https://mlsanity.com/paper/2603.06567/) (2026)
- [DualEquiNet: A Dual-Space Hierarchical Equivariant Network for Large Biomolecules](https://mlsanity.com/paper/2506.19862/) (2025)
- [Linear Attention with Global Context: A Multipole Attention Mechanism for Vision and Physics](https://mlsanity.com/paper/2507.02748/) (2025)
- [Multiresolution Graph Transformers and Wavelet Positional Encoding for Learning Hierarchical Structures](https://mlsanity.com/paper/2302.08647/) (2023)
- [Fast Multipole Attention: A Scalable Multilevel Attention Mechanism for Text and Images](https://mlsanity.com/paper/2310.11960/) (2023)
- [Scalable Machine Learning Force Fields for Macromolecular Systems Through Long-Range Aware Message Passing](https://mlsanity.com/paper/2601.03774/) (2026)
- [Polarizable atomic multipoles for learning long-range electrostatics](https://mlsanity.com/paper/2605.05746/) (2026)
- [FMM-Net: neural network architecture based on the Fast Multipole Method](https://mlsanity.com/paper/2212.12899/) (2022)
- [Learning Greens Operators through Hierarchical Neural Networks Inspired by the Fast Multipole Method](https://mlsanity.com/paper/2509.20591/) (2025)
- [An SO(3)-equivariant reciprocal-space neural potential for long-range interactions](https://mlsanity.com/paper/2603.18389/) (2026)
- [Machine Learning Multiscale Interactions](https://mlsanity.com/paper/2605.25710/) (2026)

## Abstract (arXiv)

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.