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

James Amarel, Robyn Miller, Nicolas Hengartner, Benjamin Migliori, Emily Casleton, Alexei Skurikhin, Earl Lawrence, Gerd J. Kunde

Track: Tiny Paper Track (Page limit: 3-5 pages)

Keywords: Influence function, symmetry learning, fluid flow

Abstract:

We introduce a diagnostic for symmetry learning in PDE surrogates: the influence function computed across symmetry-
related states. On compressible Euler flows, our diagnostic reveals that a UNet exhibits partial but unstable influence across
square group actions and translations, whereas a ViT reaches lower prediction error yet shows largely orthogonal updates
across orbits. This exposes an optimization-symmetry tradeoff: stronger inductive biases promote data efficiency but can
couple updates rigidly; flexible architectures optimize easily but ignore physical structure. Our diagnostic offers a reproducible test for whether training dynamics propagate information across symmetry orbits, a necessary ingredient for robust generalization in scientific machine learning.

Submission Number: 26