Overview
Neural Equivariant Multipole Operators (NEMORA) are introduced as a neural equivariant extension of the Fast Multipole Method (FMM). This architecture is designed for learning long-range tensorial representations in atomistic systems. NEMORA generalizes the FMM's analytical multipole expansion and translation operators by employing learned equivariant counterparts, operating on an adaptive spatial hierarchy. Its design couples angular degrees and forms many-body interactions across different length scales. The system retains the FMM's hierarchical organization and analytical radial factors as physical inductive biases while concurrently learning data-dependent long-range couplings. NEMORA operates with linear time and memory complexity, enabling its application to systems containing hundreds of thousands of atoms. The method can augment both symmetry-constrained and unconstrained short-range backbones.
Research Context
Equivariant graph neural networks (GNNs) serve as foundational architectures for machine-learned interatomic potentials, offering an approach to quantum-chemical accuracy with reduced computational cost. These models typically excel at describing local atomic environments. However, their performance is often constrained by finite spatial cutoffs, which truncate long-range information flow. Additionally, stacking multiple message-passing layers in GNNs can lead to issues such as over-smoothing and over-squashing. Existing approaches to extend long-range capabilities often involve prescriptive analytical propagation kernels, restrict long-range communication to scalars or degree-preserving channels, achieve only approximate equivariance, or incur super-linear computational costs. The integration of learnable long-range equivariant transport with multiscale many-body expressivity and efficient scaling for larger systems has been identified as a central challenge in this domain.
Approach
NEMORA's approach involves generalizing the FMM's analytical components—specifically its multipole expansion and translation operators—into learned equivariant counterparts. This generalization is applied within an adaptive spatial hierarchy. The method incorporates physical inductive biases by retaining the FMM's hierarchical organization and analytical radial factors. Simultaneously, NEMORA learns data-dependent long-range couplings. The operators within NEMORA are designed to couple angular degrees and facilitate many-body interactions across various length scales. The architecture is engineered to maintain linear time and memory complexity.
Findings
- NEMORA exhibits linear time and memory complexity.
- It can treat larger systems, specifically reaching hundreds of thousands of atoms.
- On non-local benchmarks, NEMORA reduces force and energy errors relative to short-range backbones.
- The reduction in force errors is by over an order of magnitude.
- The reduction in energy errors is up to three orders of magnitude.
- NEMORA's accuracy on non-local benchmarks is better than or competitive with existing long-range extensions.
- The system augments both symmetry-constrained and unconstrained short-range backbones.
Why This Matters
The ability of NEMORA to achieve linear time and memory complexity allows it to scale to systems containing hundreds of thousands of atoms. This scaling capability, combined with its significant reduction in force and energy errors on non-local benchmarks, addresses a limitation of current equivariant graph neural networks regarding long-range information flow and computational cost. By integrating learnable long-range equivariant transport with multiscale many-body expressivity, NEMORA offers an efficient and accurate method for interatomic potentials, relevant for atomistic learning applications.