🤖 AI Summary
Existing metrics, such as relative L² error, often fail to comprehensively assess the numerical plausibility of learned PDE simulators. To address this limitation, this work proposes the first architecture-agnostic, post-hoc diagnostic framework that systematically audits a model’s behavior as an approximate evolution operator through structural indicators—including semigroup consistency, energy dynamics, and response to perturbations. Requiring only reference trajectories, predictions, equation metadata, and simulation configurations, the framework uniformly evaluates diverse architectures such as FNOs, DeepONets, U-Nets, and ResNets. Experiments across five canonical PDE benchmarks demonstrate that even when L² errors are low, structural metrics can exhibit significant degradation, thereby underscoring the necessity and efficacy of the proposed approach.
📝 Abstract
Learned PDE simulators are increasingly used as low-cost replacements for expensive numerical solvers, but standard relative $L^2$ error does not determine whether a learned model behaves as a coherent numerical time propagator. This paper presents a diagnostic software suite for auditing learned PDE simulators as approximate evolution operators. The suite provides architecture-independent, post hoc diagnostics for relative state error, semigroup consistency, finite-difference generator discrepancy, energy behavior, integral balance, admissibility constraints, perturbation response, and scaling-law consistency. The software is designed around a minimal contract: reference trajectories, a learned propagator or saved predictions, equation metadata, and a diagnostic configuration specifying which structures are meaningful for the problem under study. We validate the suite on five benchmark PDE tasks: two-dimensional incompressible Navier-Stokes, shallow-water dynamics, active matter, three-dimensional compressible Navier-Stokes, and three-dimensional magnetohydrodynamics, using FNO, DeepONet, U-Net, and ResNet-style surrogate models together with controlled underfit and oversmoothed variants. The validation study shows that relative $L^2$ error can remain moderate, or even improve, while structural diagnostics deteriorate substantially. The package therefore supports software-level auditing of learned PDE simulators by reporting an interpretable diagnostic panel rather than collapsing model behavior into a single state-error score.