Loss Landscape Geometry and the Learning of Symmetries: Or, What Influence Functions Reveal About Robust Generalization

• URL: http://arxiv.org/abs/2601.20172v1
• Date: Wed, 28 Jan 2026 02:14:01 GMT
• Title: Loss Landscape Geometry and the Learning of Symmetries: Or, What Influence Functions Reveal About Robust Generalization
• Authors: James Amarel, Robyn Miller, Nicolas Hengartner, Benjamin Migliori, Emily Casleton, Alexei Skurikhin, Earl Lawrence, Gerd J. Kunde,
• Abstract summary: We study how neural emulators internalize physical symmetries by introducing an influence-based diagnostic.<n>This quantity probes the local geometry of the learned loss landscape.<n>We show that orbit-wise gradient coherence provides the mechanism for learning to generalize over symmetry transformations.
• Score: 0.14201057456467273
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We study how neural emulators of partial differential equation solution operators internalize physical symmetries by introducing an influence-based diagnostic that measures the propagation of parameter updates between symmetry-related states, defined as the metric-weighted overlap of loss gradients evaluated along group orbits. This quantity probes the local geometry of the learned loss landscape and goes beyond forward-pass equivariance tests by directly assessing whether learning dynamics couple physically equivalent configurations. Applying our diagnostic to autoregressive fluid flow emulators, we show that orbit-wise gradient coherence provides the mechanism for learning to generalize over symmetry transformations and indicates when training selects a symmetry compatible basin. The result is a novel technique for evaluating if surrogate models have internalized symmetry properties of the known solution operator.

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