Semigroup Consistency as a Diagnostic for Learned Physics Simulators
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866913162456989696 |
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| author | Shikhman, Lennon J. |
| author_facet | Shikhman, Lennon J. |
| contents | Learned physics simulators are often evaluated by one-step or short-horizon prediction error, but these metrics can miss failures in temporal composition and long-horizon rollout. For autonomous, state-complete systems, exact solution maps satisfy a semigroup law: direct evolution over $s+t$ should agree with evolution over $s$ followed by $t$. We propose normalized semigroup error as a post hoc, model-agnostic diagnostic comparing these direct and composed learned predictions. On one-dimensional heat and Burgers dynamics with time-conditioned ConvNet and FNO baselines, semigroup error is positively associated with rollout degradation, with trajectory-level Spearman correlation $ρ= 0.635$ and $95%$ CI $[0.621, 0.649]$. Semigroup regularization has mixed effects, supporting semigroup consistency primarily as an evaluation diagnostic rather than a universally beneficial training objective. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_26324 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Semigroup Consistency as a Diagnostic for Learned Physics Simulators Shikhman, Lennon J. Machine Learning Artificial Intelligence Numerical Analysis I.2.6; I.6.5; G.1.8 Learned physics simulators are often evaluated by one-step or short-horizon prediction error, but these metrics can miss failures in temporal composition and long-horizon rollout. For autonomous, state-complete systems, exact solution maps satisfy a semigroup law: direct evolution over $s+t$ should agree with evolution over $s$ followed by $t$. We propose normalized semigroup error as a post hoc, model-agnostic diagnostic comparing these direct and composed learned predictions. On one-dimensional heat and Burgers dynamics with time-conditioned ConvNet and FNO baselines, semigroup error is positively associated with rollout degradation, with trajectory-level Spearman correlation $ρ= 0.635$ and $95%$ CI $[0.621, 0.649]$. Semigroup regularization has mixed effects, supporting semigroup consistency primarily as an evaluation diagnostic rather than a universally beneficial training objective. |
| title | Semigroup Consistency as a Diagnostic for Learned Physics Simulators |
| topic | Machine Learning Artificial Intelligence Numerical Analysis I.2.6; I.6.5; G.1.8 |
| url | https://arxiv.org/abs/2605.26324 |