Detecting and Discriminating Operator Misspecification in Hybrid PDE-Parameter Learning: a Reference-Free Instrument, with Discrimination Bounded In Sample

Abstract

We build an instrument that reads, from a single fit and with no oracle, whether the operator a hybrid PDE-parameter estimator postulates is wrong-and separates that from a merely unidentifiable parameter. On one self-adjoint parabolic inverse problem, an information-matrix statistic with plug-in scale and per-seed parameter has median 0.19 under correct specification, rejection rate $0.033$ against a pre-registered ceiling of $0.10$, and rises to $224$ and $85$ under two misspecifications, firing in every replicate. On a correctly specified but non-identifiable design it stays mute-$0.050$ at $n=200$, Clopper-Pearson $[0.024,0.090]$-while a rank statistic collapses to zero at a pre-registered boundary $c_5^{\ast }=2.15\times 10^{-3}.$ Two readings of one fit therefore separate the two failures across the three designs a deployable test reaches. That separation is the contribution; detection alone is a crowded flank. In sample it is a bound, out of sample a direction. It is needed because the usual accuracy check is blind: the misspecified estimator's in-domain RMSE is $2.7\times 10^{-2}$, below the observation noise for $\sigma \ge 0.05,$ while the coefficient is wrong by $29.7\%$ at zero noise, $31.2\%$ at the loudest. Nor is the failure architectural: a one-parameter curve fit, a bare parameter and multilayer perceptrons of $49$ and $241$ parameters converge to the same pseudo-true, matched in closed form to $0.07\%,$ whereas a physics-informed network, with its composite objective, converges to a disjoint one. We report where the instrument is blind, a pre-registered negative where a neural estimator loses to Tikhonov-regularized inversion at recovery, and the hypothesis under which its guarantee holds but a trained network violates it.

14 pages, 8 figures. Supplementary material (5 pp.) included as an ancillary file