Semigroup Consistency as a Diagnostic for Learned Physics Simulators
For researchers developing learned physics simulators, this provides a post-hoc diagnostic to detect long-horizon failures that short-horizon metrics miss.
Learned physics simulators are often evaluated by short-horizon prediction error, which can miss failures in long-horizon rollouts. The authors propose normalized semigroup error as a diagnostic, showing it correlates with rollout degradation (Spearman ρ=0.635) on heat and Burgers dynamics.
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.