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

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.