PDECheck: A Diagnostic Software Suite for Auditing Learned PDE Simulators

Abstract

PDECheck audits learned PDE simulators through callable propagators or saved predictions. It evaluates relative $L^2$ error, temporal composition, finite-difference generators, and equation-specific physical residuals, retaining horizons and missing prerequisites. Analytic and manufactured defects verify the kernels. A heat-equation sweep quantifies generator sensitivity to timestep, frequency, grid, and precision. A three-equation, five-seed FNO study compares supervised, PINO-style, and composition-regularized training. An accuracy-matched advection comparison has state errors below 0.3% and a semigroup-error ratio of 1.48 (95% interval 1.34-1.62). The composition penalty lowers semigroup error but increases composed-state and longer-rollout prediction error. A time-reparameterization check produces identical low-error endpoints with different compositions. Applications to five external equation families exercise the readers and physical contracts on coarse grids. The release includes reproducibility artifacts and measured CPU/CUDA costs.

arXiv comment42 pages. Revised and expanded version; PDECheck software release v0.3.3. Submitted to Computer Physics Communications. Code available at https://github.com/lennonshikhman/diagnostics_for_physics