Unbiased Physics Losses Under Model Uncertainty

Physics-informed machine learning usually penalizes violations of a chosen governing equation. However, when that equation is misspecified, minimizing the nominal residual can instead increase the error in the predicted solution. To make physics-informed machine learning effective under such misspecification, we propose compatibility training, which uses the smallest residual over a plausible family of equations. At each training step, compatibility training selects the equation in this family that best fits the current prediction and penalizes only deviations from that equation. Our theory establishes a mismatch-dependent bias floor for nominal training and characterizes the physics constraints that remain after the uncertain parameters are adjusted. We derive an exact local curvature identity that shows when the retained physics constraints and observations together identify the true solution. This curvature identity connects solution recovery to the structure and extent of the uncertainty family. Across multiple synthetic PINN benchmarks, PDEBench trajectory and operator tasks, and Darcy operator-learning experiments, compatibility training improves prediction under model mismatch, including in settings with sparse or noisy observations and field-valued uncertainty. In field-valued Darcy flow, compatibility training also reduces the true residual by more than an order of magnitude.