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

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, or incur super-linear computational cost. Combining learnable long-range equivariant transport with multiscale many-body expressivity and efficient scaling 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 is linear in time and memory, allowing it to treat larger systems than other long-range methods, and it augments both symmetry-constrained and unconstrained short-range backbones. On non-local benchmarks, it reduces energy and force errors relative to the short-range backbones by up to three orders of magnitude, and it is competitive with or better than existing long-range extensions in accuracy.