Loss Landscape Geometry and the Learning of Symmetries: Or, What Influence Functions Reveal About Robust Generalization
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| Autores principales: | , , , , , , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866914285767098368 |
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| author | Amarel, James Miller, Robyn Hengartner, Nicolas Migliori, Benjamin Casleton, Emily Skurikhin, Alexei Lawrence, Earl Kunde, Gerd J. |
| author_facet | Amarel, James Miller, Robyn Hengartner, Nicolas Migliori, Benjamin Casleton, Emily Skurikhin, Alexei Lawrence, Earl Kunde, Gerd J. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_20172 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Loss Landscape Geometry and the Learning of Symmetries: Or, What Influence Functions Reveal About Robust Generalization Amarel, James Miller, Robyn Hengartner, Nicolas Migliori, Benjamin Casleton, Emily Skurikhin, Alexei Lawrence, Earl Kunde, Gerd J. Machine Learning Computational Physics 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. |
| title | Loss Landscape Geometry and the Learning of Symmetries: Or, What Influence Functions Reveal About Robust Generalization |
| topic | Machine Learning Computational Physics |
| url | https://arxiv.org/abs/2601.20172 |