Physical Robustness under Equation Corruption

Robustness in machine learning usually means invariance, which is evaluated by perturbing an input while the correct label stays fixed. This evaluation setup fails for PDE surrogates: changing a coefficient or a boundary condition moves the physical solution, and changing the mesh or the sensor changes how the solution is measured. Physical robustness should instead be evaluated by how well the prediction tracks changes in the correct output, in both direction and magnitude. Nominal accuracy does not guarantee response fidelity: under a Burgers viscosity shift, an FNO with held-out field error below 2% still mispredicted the response by half the true response magnitude. To make physical robustness measurable and trainable, we decompose changed-target error into nominal error and sensitivity mismatch, introduce diagnostics that grade the response at held-out severities, and evaluate three training interventions on six specification-shift tasks. Our evaluation identifies effective training interventions for different perturbations, PDEs, and backbones: response alignment cut the response mismatch on public multi-viscosity Burgers data roughly sixfold and led on coefficient and boundary shifts, whereas stress augmentation cut evaluation-mesh and initial-condition error by 21.5% and 16.2% over a matched changed-label baseline. Adversarial training, a standard approach to robustness in machine learning, instead increased evaluation-mesh and initial-condition error by roughly 28%. Based on these results, we argue that a surrogate should be judged by how faithfully the prediction tracks the governing system, not by how little the prediction moves when the physics changes.