Diagnosing Spectral Ceilings in Equivariant Neural Force Fields

TL;DR

This paper presents a spectral-injection diagnostic to measure the angular frequency preservation of equivariant neural force fields, revealing a sharp boundary in frequency recovery and providing insights into their spectral limitations.

Contribution

It introduces a novel spectral diagnostic method for analyzing equivariant neural force fields and characterizes their spectral capacity limits with theoretical and empirical validation.

Findings

The diagnostic reveals an 11.7x cliff at the predicted frequency boundary.

The boundary contrast persists across multiple trained backbones.

The spectral span theorem relates polynomial degree to spectral capacity.

Abstract

We introduce a spectral-injection diagnostic for measuring which angular frequencies a trained equivariant force-field backbone preserves: inject a controlled angular-frequency perturbation into a molecular force field, attach a lightweight Spectral Prediction Network (SPN) to the frozen backbone, and read off which frequencies are recoverable. On aspirin, a quadratic SPN attached to an L = 2 NequIP backbone recovers the boundary signal at l = 4 but collapses at l = 5: a 11.7x cliff at the predicted drL boundary, with p dropping from 0.913 to 0.078. The same boundary-vs-above contrast persists across n = 4 independently trained backbones (raw-gain delta contrast, hierarchical cluster bootstrap) and is corroborated by a denominator-free injected-residual metric (R2_inj(4) = 0.374 versus R2_inj(5) = 0.006). A finite-degree span theorem calibrates the diagnostic: for a single marked direction, degree-d polynomials of degree-L spherical-harmonic features span exactly H less than or equal to dL with multiplicity-one saturation at the boundary (scoped to single-direction degree-bounded probes, not a function-class upper bound on multi-atom MPNNs). A synthetic C5 calibration plus capacity, activation, and cross-architecture controls rule out parameter count alone as the explanation.