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This research introduces a diagnostic tool to evaluate how neural emulators of partial differential equations internalize physical symmetries by analyzing the geometry of their loss landscapes. By measuring gradient propagation along group orbits, the study reveals that orbit-wise coherence is the mechanism enabling generalization over symmetry transformations in fluid flow models. This work advances the theoretical understanding of machine learning dynamics in scientific computing, specifically regarding how models learn invariant properties without explicit architectural constraints.
Loss Landscape Geometry of Partial Differential Equation Emulators: Or, Symmetry Learning via Gradient Alignment
arXiv:2601.20172v2 Announce Type: replace 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.
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.
Machine Learning (cs.LG); Computational Physics (physics.comp-ph)
arXiv:2601.20172 [cs.LG]
(or arXiv:2601.20172v2 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2601.20172 Focus to learn more arXiv-issued DOI via DataCite
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