Beyond Field Accuracy: Two-Axis Diagnosis of Inverse-PINN Parameter Error

Abstract:Inverse physics-informed neural networks (PINNs) can reconstruct a field accurately while returning an incorrect physical parameter. We introduce a two-axis post-training diagnosis that separates finite-sample resolution under a specified observation-and-estimation protocol from the signed parameter preference encoded by the final learned field and residual metric. The first axis repeatedly fits noisy observations with a matched forward estimator. At known synthetic truth, the second freezes the field and residual view and computes a local score displacement toward a nearby residual-profile minimum. Endpoint consistency then tests whether joint training delivers that preference under the same final view. Across three synthetic one-dimensional, scalar-parameter PDEs, matched-forward mean absolute relative error ranges from 2.34 percent to 17.46 percent. The displacement tracks frozen-profile minima across locked seeds, architectures, and fresh-noise retraining (r from .945 to .982), and it tracks delivered signed log-error in 240 fresh-noise RBA runs (r = .994; 237/240 correct directions). A coupled two-parameter Darcy check validates the full matrix calculation. The axes are complementary diagnostic coordinates, not additive error components or a deployable oracle-free estimator. Together, they route follow-up work toward observations, residual evidence, or endpoint delivery.

Preprint with supplementary appendix. 26 pages, 3 figures

Machine Learning (cs.LG); Numerical Analysis (math.NA)

arXiv:2608.15373 [cs.LG]

(or arXiv:2608.15373v1 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2608.15373 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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