NearestPDE: Structured Backward Error for Ambiguity-Aware PDE Diagnosis

We introduce NearestPDE, a post-hoc framework for diagnosing computed partial differential equation (PDE) fields through structured backward error. It ranks admissible coefficient, source, boundary, and geometry changes using physically scaled residual sensitivities, retaining indistinguishable explanations in a set. We establish conditions for unique recovery and limits of point diagnosis under observational overlap. Split-conformal calibration provides finite-sample marginal coverage under exchangeability within the declared dictionary. A disjoint residual grid selects a corrective re-solve without test-time reference solutions. Controlled tests recover all 1,980 identifiable cases and flag all 240 collinear controls. On nonlinear finite-difference fields, top-1 accuracy averages 100% across three noise levels, versus 78% for matched orthogonal matching pursuit (OMP). Contextual screening recovers three-cause supports with up to 1,023 localized atoms; OMP remains stronger in several transfer settings. Independent re-solves reduce forward error in 1,820/1,824 one-dimensional transfer cases. The framework separates ranked explanations, calibrated uncertainty, and testable model corrections within the declared perturbation dictionary.