Correcting Trained PINNs from Residuals: Operator-Kernel Inference of the Error Field

A trained physics-informed neural network (PINN) retains computable PDE residuals and boundary and initial-condition defects that contain information about its remaining solution error. We use these defects to reconstruct a global correction while keeping the network fixed. The governing equation turns them into operator observations of the unknown error, exactly for linear PDEs and through a frozen linearization for nonlinear PDEs. Applying the observation operators to a kernel produces a finite reconstruction system whose hyperparameters are fitted from the defects. Regularized linear solves then recover the correction without observing the true error or retraining the PINN. Our analysis characterizes the conditional RKHS estimator and separates finite-data reconstruction from nonlinear linearization. Experiments across canonical PDEs, held-out physical instances and irregular geometry show reductions in solution error. On a nonlinear variable-coefficient problem with directly specified forcing, the complete procedure substantially improves the trained field. For this problem, it also achieves lower error than neural error-field fitting and continued training given larger compute allowances. Budget comparisons reveal problem-dependent tradeoffs, while resolved Newton references identify reconstruction as the larger discrepancy on the studied Burgers cases. These results connect observable equation defects to a usable correction of an existing neural field.