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

Yifan Zhang; Qian Tao

cs.LGAug 15, 2026

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

Authors: Yifan Zhang, Qian Tao

Organizations: School of Software Engineering, South China University of Technology Guangzhou, China

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.

Figures & tables

Figure 1: Frozen scores track residual basins and delivered endpoints (AC: Allen–Cahn; BL: Buckley–Leverett; P8: Patch-8). (a)

\Delta_{M}=100|\exp(-B_{M}/H_{M})-1|

tracks near-truth frozen-profile displacement across residual views; Table 1 reports signed results. (b) With final RBA weights frozen,

\Delta_{M}

tracks absolute parameter error over 240 fresh retrainings.

Target	Evidence	Agreement	Check
Frozen profile	Dev./locked/arch./RBA (72/36/48/240)	$r=.960/.980/.945/.982$	MAE 2.47 pp; sign 72/72; smallest CI low .918
Delivered endpoint	Fresh RBA (240)	$r=.983$ [.977,.987]; log $r=.994$	MAE 1.02 pp; sign 237/240
View alignment	Widths 32/128 (12 each)	match $r=.975/.875$ ; cross $r=.623/-.611$	MAE match 3.14/6.38; cross 10.59/14.25
Darcy vector	2-D, 2-param. (20)	full cosine .999994 [.999992,.999997]	relative error 9.50% [9.21,9.82]; full wins 20/20

Table 1: Validation summary. RBA: residual-based attention;

r

: Pearson magnitude correlation; MAE: percentage points; brackets: clustered 95% CIs. Frozen-profile entries follow development/locked/architecture/noise. The RBA endpoint test freezes final adaptive weights, and the Darcy row compares the full matrix displacement with the refined frozen-profile minimum.

PDE	Repeated-noise error (%) [95% CI]	Fisher (%)	95% coverage
Allen–Cahn	2.34 [2.21,2.47]	2.37	759/800
Buckley–Leverett	6.69 [6.31,7.08]	6.61	766/800
Burgers	17.46 [16.51,18.41]	16.73	759/800
Overall, balanced	8.83 [8.22,9.44]	8.57	2284/2400

Table 2: Matched resolution and same-data paired contrast by PDE. Panel (a) uses four layouts and 200 repeated-noise draws per PDE (2,400 fits); panel (b) uses 20 paired draws per PDE and layout (240 fits). Intervals cluster layouts within PDE–draw, and the overall rows weight PDEs equally. The repeated-noise and paired matched-forward values come from separate experiments.

Figure 2: PDE-dependent residual mechanisms and error attribution (AC: Allen–Cahn; BL: Buckley–Leverett). (a) Sensitivity shares across Gaussian residual scales. (b) Repeated-noise matched-forward error and Fisher prediction. (c) Same-data RBA-minus-forward excess with clustered 95% CIs.

Matched resolution	Small $\boldsymbol{\Delta_{M}}$	Large $\boldsymbol{\Delta_{M}}$
Tight	Inspect endpoint delivery ; model misspecification, finite budget, or nonlocal/global behavior.	Strengthen residual evidence ; inspect $B_{M},H_{M}$ .
Weak	Improve observations or revise parameter scope.	Co-design observations and residual evidence .

Table 3: Qualitative routing for scientifically material delivered error. Rows summarize matched-forward resolution; columns summarize fixed-field displacement

\Delta_{M}=100|\exp(D_{M})-1|

Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

Figure A1: Darcy numerical fidelity over 20 frozen fields (10 PCGrad and 10 PatchEvidence; seeds 10–19 for each method). The full Gauss–Newton step tracks the refined two-dimensional frozen-profile displacement; the diagonal approximation does not.

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Physics-informed neural networks (PINNs) constitute a flexible deep learning approach for solving partial differential equations (PDEs), which model phenomena ranging from heat conduction to quantum mechanical systems. Despite their flexibility, PINNs offer limited insight into how their predictions deviate from the true solution, hindering trust in their prediction quality. We propose a lightweight post-hoc method that addresses this gap by producing pointwise error estimates for PINN predictions, which offer a natural form of explanation for such models, identifying not just whether a prediction is wrong, but where and by how much. For linear partial differential equations, the error between a PINN approximation and the true solution satisfies the same differential operator as the original problem, but driven by the PINN's PDE residual as its source term. We solve this error equation numerically using finite difference methods requiring no knowledge of the true solution. Evaluated on several benchmark PDEs, our method yields accurate error maps at low computational cost, enabling targeted and interpretable validation of PINNs.

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