Physics-informed structural diagnostics of model–data agreement beyond scalar metrics

Abstract Physics-informed machine-learning models increasingly incorporate physical laws and constraints to improve data efficiency and predictive robustness; yet their validation remains dominated by pooled scalar accuracy metrics that are largely insensitive to violations of the underlying governing relationships. Here we introduce a physics-informed validation framework, the Agreement–Entropy Map (AEM), which diagnoses model–data agreement by distinguishing structural incompatibility from conditional stochastic dispersion, rather than by defining a scalar metric or additive error decomposition. Conditioned on a physically motivated linearization of the governing relation and evaluated on matched comparison domains, AEM combines regression geometry with an information-theoretic dispersion measure based on a Gaussian plug-in entropy of residuals, without requiring distributional modeling or inferential assumptions. The framework applies uniformly to experiment–experiment and model–experiment comparisons and is agnostic to model class, architecture, and training procedure. Using thermodynamic systems as a canonical physics-governed testbed, we show that AEM reveals structural bias, variance-driven artefacts, and ensemble effects that remain undetected by conventional scalar validation metrics. By identifying when stochastic interpretation is admissible under a shared physical structure, AEM provides a general and interpretable validation principle for physics-informed machine learning, particularly in regimes involving limited, heterogeneous, or damaged data.