Symmetry Diagnostics for Equivariant Molecular Force Fields: When a Bulk Metric Manufactures a Gauge Artifact

Abstract

Equivariant graph neural networks encode the E(3) symmetries that molecular force fields must obey, yet they are often trained against geometric objectives whose own symmetry is rarely examined. We develop a symmetry-sensitive diagnostic protocol that combines rigid-transform invariance and equivariance metrics, a parity analysis of graph-level targets, an auxiliary-head predictability measure, and torsion and mirror perturbation tests. It separates three effects that bulk error metrics conflate. Applied to reduced GotenNet models on rMD17, it yields a clear empirical picture. The architecture preserves rotation, translation, and reflection to numerical precision (∼10−6 relative force error), so the auxiliary head leaves equivariance intact. The geometric targets, however, differ in parity: cosτ and |cosτ| are parity-even, while sinτ and the scalar triple product are parity-odd and near-zero-mean. The contrast between cosτ and |cosτ| reflects torsional-moment scaling, not parity. Across three-seed paired runs, the auxiliary effect on force mean absolute error (MAE) is dominated by seed noise. A seed-consistent molecule-dependent effect in raw energy MAE disappears when each model’s predicted energies are shifted by a constant, even though forces and conformer energy differences remain unchanged, showing how the bulk metric manufactures a gauge artifact. We offer preliminary design principles for symmetry-preserving auxiliary objectives rather than a new force-field benchmark.