Sequential Structure-Sensitive Residual Diagnostics for PDE Inverse Problems

Residual norms are often used to decide whether a fitted scientific model is adequate. In smoothing inverse problems, however, systematic model error can be hidden in observation space: residuals remain small even when parameters, predictions, or quantities of interest are biased. We study a sequential diagnostic that looks for spatial structure in residuals rather than only their magnitude. It combines evidence from a bank of residual-pattern experts in an e-process and rejects a fixed fitted model once the accumulated evidence crosses a threshold. For independent monitoring data, this gives anytime-valid type-I error control. Across three inverse problems (elliptic diffusion, Stokes flow, and an ice-stream inversion) the e-process detects misspecified fits from a fraction of the observations and identifies the residual patterns carrying the evidence. Practical residual-RMS checks accept the same fits despite materially wrong quantities of interest, while structure-sensitive batch tests detect them only after seeing the full record. The method offers a lightweight verification layer for scientific workflows, including AI-enabled pipelines, in which fitted models may be produced faster than experts can inspect them.