PIVOT: Physics-Informed Virtual Probing for Output-Space Tuning of Neural Operators

Partial differential equations underpin complex physical modeling, but costly numerical solvers limit their use in applications requiring real-time computation. Neural operators offer efficient surrogates, yet distribution shifts can compromise physical consistency and accumulate errors over long rollouts. We find that frozen operators can retain reliable local physical knowledge despite inaccurate predictions: output changes under small, physically admissible input perturbations can still closely reflect the true system’s local responses. Existing adaptation methods typically update parameters or learn auxiliary corrections, underusing this retained physical information. We introduce PIVOT, a general adaptation framework that perturbs frozen operators’ inputs to extract local physical responses for prediction correction. For each target sample, PIVOT learns physically admissible directions and queries the operator under small, opposite perturbations. Comparing paired predictions extracts response directions and magnitudes as candidates for correction. Physical consistency checks assess their reliability, while a lightweight adapter learns weights at each spatial location to combine responses into corrections. PIVOT bounds correction magnitude and progressively reduces updates that increase physical residuals, keeping the pretrained operator fully frozen. Experiments on APEBench and RealPDEBench show improved physical consistency, predictive accuracy, and long-horizon stability across diverse physical systems and distribution shifts. The complete model implementation is provided in the supplementary material.