Eric Fock's arXiv test catches wrong PDE models behind low RMSE

Eric Fock (arXiv:2608.16925) has released a new statistical instrument that tackles a silent failure mode in hybrid physics-AI models: when a neural network or PDE-parameter estimator gets the operator wrong but still produces low error on training data. The instrument reads from a single fit—no oracle needed—whether the postulated operator is misspecified, separating that failure from merely unidentifiable parameters. On a self-adjoint parabolic inverse problem, the information-matrix statistic shows median 0.19 under correct specification, with a rejection rate of 0.033 (below the pre-registered 0.10 ceiling). Under two misspecifications, the statistic jumps to 224 and 85, firing in every replicate.

Crucially, Fock shows why this matters: a misspecified estimator's in-domain RMSE is 2.7×10⁻², below observation noise for σ≥0.05, while the coefficient is wrong by 29.7% at zero noise and 31.2% at the loudest—standard accuracy checks are blind. The failure isn't architectural; one-parameter fits, bare parameters, and MLPs with 49 and 241 params all converge to the same pseudo-true, matched in closed form to 0.07%, whereas physics-informed networks converge elsewhere. The paper also reports a pre-registered negative where a neural estimator loses to Tikhonov-regularized inversion, and where the guarantee holds but a trained network violates it. This is a much-needed diagnostic for scientific machine learning.