Misspecified Physics Can Help and Bias Extrapolation in Hybrid Models

Hybrid models that combine physical equations with a data-driven component are expected to extrapolate more reliably than purely data-driven models as the physical component remains active beyond the training data. This study investigates whether a misspecified physical component can still provide useful extrapolative structure beyond the training data and whether its misspecification shapes the extrapolation error of the corresponding hybrid model. In a controlled reaction-diffusion system, this study compares a hybrid model with correct diffusion, hybrid models whose supplied diffusion is misspecified in its directional dependence, its discretization, its magnitude, or by omitting one process, and a purely data-driven model. All models achieve small held-out prediction errors and give similar responses to patterns well represented during training, yet their responses differ when extrapolating to patterns scarcely represented in the training data. For scarcely represented patterns, the hybrid model with correct diffusion closely follows the true response, and moderately misspecified physics gives lower errors than the purely data-driven model, while sufficiently severe misspecification loses that advantage. Hybrid models with misspecified physics carry the specific response errors of their supplied physics into extrapolation, with nearly the same magnitude and response pattern. Intervention analysis attributes the extrapolation error mainly to the misspecified physical component, as replacing the misspecified physics with the correct physics, without retraining the data-driven component, removes most of the error. A misspecified physical component can therefore provide useful extrapolative structure while also imposing a systematic extrapolation bias that follows the response error of the supplied physics.