Response Geometry for Decisions with Neural PDE Surrogates

Neural PDE surrogates can fit familiar solution fields while responding incorrectly to changes in physical parameters. We study whether this missing response information improves three decisions: choosing a forward predictor, choosing a surrogate for parameter recovery, and repairing an existing predictor. Response geometry compares parameter derivatives of model and reference fields. An exact pathwise error identity connects these derivatives to extrapolation; sensor projection explains local inverse sensitivity, and the rollout chain rule explains why one-step derivative fit can miss accumulated error. We evaluate these connected uses on the one-dimensional viscous Burgers equation, the two-dimensional steady Darcy flow equation with heterogeneous permeability, and the two-dimensional forced incompressible Navier-Stokes equations in vorticity form, separating ID-only probes, shift-calibrated scores, and target-assisted comparisons. A prespecified response selector reduces early-time Navier-Stokes OOD relative field error from 0.9934 to 0.3377, while its Burgers counterpart ranks population risk better but selects worse. Completed repair tests reduce Navier-Stokes field error from 0.6284 to 0.4899 using rollout-plus-response supervision, with a corrected paired-test gain and additional compute disclosed. The same repair worsens inverse parameter recovery. Burgers shows smaller path-repair gains; Darcy's reliable small gain is a direct-field control. These findings make response geometry useful as a task-specific decision variable: its information can improve selection and prediction, but field accuracy, inverse recovery, and long-time physical fidelity require separate tests.