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

arXiv:2608.16925v1 Announce Type: new 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^*=2.15imes10^{-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 2.7imes 10^{-2}, below the observation noise for sigmageq 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.

Source: arXiv cs.LG | 2026-08-19

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