Wrong Operator or Blind Design? A Reference-Free Diagnostic for Physics-Informed Coefficient Learning

Eric Fock

cs.LGAug 4, 2026

Wrong Operator or Blind Design? A Reference-Free Diagnostic for Physics-Informed Coefficient Learning

Authors: Eric Fock

Organizations: PIMENT Laboratory, Universit´e de La R´eunion, Le Tampon 97430, La R´eunion, France

Abstract

Physics-informed neural networks and hybrid models infer PDE coefficients from noisy data. When a trained network returns one, no standard check says whether to trust it. We show what those checks report when the operator is wrong: one sensor aggregating several diffusion sources. On one parabolic benchmark at $2\%$ noise, the in-domain error is $1.4$ times the noise while the identified diffusivity settles $30\%$ off. Every least-squares minimiser reaches that value, which drifts $27\%$ across windows; the network, whose objective is composite, settles $1.3\%$ away. The checks stay as silent when the design is blind to a rate of a richer operator, though the remedies are opposite. We develop a reference-free diagnostic, read in the physical parameter, not the weights, without retraining the network: an information-matrix test on the residuals, a heterogeneity statistic across window refits, and a Fisher-rank statistic on the design at the rates the single fit postulates. On the analytic head the specification test holds its pre-registered ceiling and rejects every misspecified replicate of both benchmark configurations, with a notch against a missing reaction term. The rank statistic is exactly zero only where the design is blind; a wrong operator confined to that mode leaves the specification test mute, and the rank statistic says so before any fit. The window reading exceeds its ceiling by one seed in thirty. A network frozen at its minimum returns the same verdicts; one stopped short rejects as a wrong operator would.

Figures & tables

	well-spec.	single-missp.	double-missp.
modes	$[1,3,5]$	$[1,3,5]$	$[1,3,5,7]$
amplitudes $c_{n}$	$[1,0.3,0.1]$	idem	$[1,0.3,0.1,0.05]$
rates $\bar{\alpha}_{n}$	$0.010$ (all)	$\{0.010,0.005,0.020\}$	$\{0.010,0.006,0.018,0.014\}$
config hash	32cf4a5526cb	7935b218c150	0d7c972548f5
postulated operator	$\partial_{t}u=\alpha\,\partial_{xx}u$ , one scalar $\alpha$

TABLE I: The three generating configurations.

Panel of Figure 2 — Fig. 1: The estimators, grouped by the objective they minimize. (a) Least-squares class ( Hyb_scalar , MLP_scalar , MLP_scalar_big , Hyb_modal , Hyb_field , cf , cf2 ). The postulated form is solved analytically and the network, when there is one, emits the coefficient. (b) Composite class ( PINN , PINN_small , PINN+corrector ). Here a network emits the state and the coefficient enters via the residual channel of ( 3 ). The plateau of Section VI-B is shared within a class and differs across classes.

Panel of Figure 3 — Fig. 2: The diagnostic on one fit ( $I_{N}$ of ( 5 ), $T_{5}$ of ( 6 ), $Q$ of Section VII ; any estimator of Fig. 1 ). Reading rule at the pre-registered thresholds: $I_{N}$ mute and $T_{5}\geq 1$ , none flagged on this design. $I_{N}$ fires and $T_{5}\geq 1$ , wrong operator, change the physics; $I_{N}$ mute and $T_{5}<1$ , design blind to the named rate, change the experiment; on such a design the silence says nothing about the operator (Section VII ). Both, pre-registered and not run (certified cells in Section VII ). For a trained network, see Section VIII-C .

Estimator	well-spec.	single-missp.	double-missp.
Hyb_scalar †	$2.2\times 10^{-8}$	$0.297$	$0.223$
cf (curve fit, $\alpha$ )	$1.3\times 10^{-7}$	$0.297$	$0.223$
cf2 ( $\alpha+c_{n}$ )	$3.6\times 10^{-8}$	$0.365$	$0.268$
Hyb_modal (correct)	$3.3\times 10^{-6}$	$1.5\times 10^{-6}$	$2.7\times 10^{-3}$
Hyb_field	$8.8\times 10^{-6}$	$0.419$	$0.477$
PINN	$1.9\times 10^{-3}$	$0.303$	$0.226$

TABLE II: Identification error

\alpha_{\mathrm{err}}

at zero noise (median,

n=30

), against the nominal

\alpha_{0}

Estimator	floor	$/$ bound	plateau	separation
Hyb_scalar, cf	$0.60\%$	$1.44$	$29.8\%$	$49$
cf2	$0.65\%$	$1.55$	$36.5\%$	$56$
Hyb_field	$1.21\%$	$2.88$	$42.0\%$	$35$
PINN	$0.70\%$	$1.67$	$30.6\%$	$44$
PINN_small	$12.95\%$	$30.9$	$37.4\%$	$2.9$

TABLE III: Precision floor under correct specification against the Cramér–Rao bound, and its separation from the single-misspecification plateau. Median over

n=30

\sigma=0.02

; efficient-estimator median

0.42\%

Least-squares surrogate (analytic field)
Regime	median $I_{N}$	rejection (in)	(OOS)	threshold
well-specified	$0.193$	$0.033$	$0.10$	$2.41$
single-missp.	$223.9$	$1.00$	$1.00$	$2.66$
double-missp.	$85.0$	$1.00$	$1.00$	$2.59$
Trained network, frozen before testing
well-specified	$0.220$	$0.067$	$0.133$	$2.41$

TABLE IV: Information-matrix statistic on a single fit,

n=30

, at the detection design (

\sigma=0.02

, plug-in scale, per-seed

\hat{\alpha}

). Threshold: bootstrap

95

th percentile per regime;

\chi^{2}(1)

reference

3.84

. The last three rows report the same reading when the deployed field is a trained network frozen before testing. Both misspecified network cells were run under a separate pre-registration, in sample only.

Estimator	params	$\hat{\alpha}$ (median)	bootstrap CI 95
cf (curve fit)	$1$	$\equiv$ Hyb_scalar	–
Hyb_scalar (bare)	$1$	$0.0070284$	$[0.0070178,0.0070402]$
MLP_scalar	$49$	$0.0070296$	$[0.0070190,0.0070408]$
MLP_scalar_big	$241$	$0.0070296$	$[0.0070190,0.0070408]$
PINN (composite)	$8578$	$0.0069391$	$[0.0069141,0.0069567]$

TABLE V: Family invariance within the least-squares class, and the PINN’s departure. Single misspecification,

\sigma=0

n=30

, in sample. This cell has its own training budget (Table S14 of the supplementary material). Its PINN value is not the same run as the PINN column of Table II .

Fig. 3: Bias–variance decomposition of the identification error across noise levels and regimes. (a) Under correct specification the extrapolation MSE is pure variance (

\propto\sigma^{2}

). A noise-invariant bias plateau dominates under misspecification. (b) At

\sigma=0.02

the ratio

B^{2}/V

(filled: bias 2 , hatched: variance) separates the regimes by three orders of magnitude. (c) The variance rises with slope

1.6

–

2.0

in all regimes. Only the well-specified slope (

1.99\pm 0.01

) meets the pre-specified band

2\pm 0.15

. The misspecified ones (

1.70\pm 0.12

1.61\pm 0.14

) fall below it, a shortfall reported, not explained.

\delta

: misspecification amplitude, true rates

\alpha_{0}(1,\,1-\delta,\,1+2\delta)

;

\delta=0.5

is Table I ’s single misspecification.

Fig. 4: Empirical plateau versus White’s pseudo-true

\alpha^{*}

(closed-form objective, no network). The median identification settles on

\alpha^{*}

to within

0.07\%

across the three regimes.

Fig. 5: Identification error against noise. Scalar PINN, corrector baseline (PINN plus learned discrepancy [ 26 ] ), scalar

\chi

-head and correctly structured modal head. Median over

n=30

paired seeds. The modal head dominates the corrector up to

\sigma\approx 0.3

under Holm correction; beyond, the gap is no longer certified at

n=30

Design	$I_{N}$ rejection	median $I_{N}$	$T_{5}$ (rank)
correct, $c_{5}=0.1$ (pivot)	$0.033$	$0.193$	$2159$
correct, $c_{5}=10^{-3}$	$0.067$	$0.387$	$0.216$
correct, $c_{5}=0$ (non-identifiable)	$0.067$	$0.384$	$0.000$
wrong operator, $c_{5}=0.1$	$1.00$	$223.9$	$2378$
wrong op. (mode $5$ ), $c_{5}=10^{-3}$	$0.067$	$0.389$	$0.216$

TABLE VI: Discrimination on one fit: two readings, two axes.

n=30

, detection design, pre-registered thresholds reused (ceiling

0.10

;

T_{5}<1

). The first row is the pivot, bit-identical to the well-specified row of Table IV . Rows descend along the identifiability axis; the last two rows are wrong-operator designs, the last one with the wrong rate on the unseen mode only (supplement, Section S6).

T_{5}

at the rate of the single fit (

0.0100

;

0.0070

in the fourth row, where the true rate

0.020

would give

1237

Fig. 6: The observation-window axis. (a) Same generating field, four disjoint quarters of $[0,1]$ (their union is the detection design, same noise). Median identified rate of the scalar operator over $n=30$ seeds (dots), $\pm$ one Fisher standard error (band). Analytic pseudo-true value per window (crosses, computed before the run). Invariant by identity under a correct operator, drifting by $27\%$ of $\alpha_{0}$ under aggregated rates. (b) Median identified rate over sliding windows $(t_{0},\Delta T)$ on $[0,2]$ under single misspecification. The correct operator reads $10.00$ in every cell ( $9.97$ – $10.02$ , same windows and seeds). Hatched cells: $T_{5}<1$ at the generating rates of the field, an oracle reading. The deployable reading at the postulated rate stays $\geq 286$ everywhere (Section VIII-B ).

claim	threshold	measured
network beats oracle-tuned Tikhonov at field recovery	error ratio $\leq 0.70$	$4.03$ , $2.45$ , $1.79$ worse at $\sigma=0.01,0.02,0.05$
field head collapses into a flat anisotropic valley	$\kappa\geq 10$	$\kappa=2.27$ ( $2.08$ – $2.69$ ; $2.57$ under correct specification; floor $2$ )
modal field is “free” flexibility	fits no better than constants	fits better
three-channel network, correct operator: $I_{N}$ specific	rejection $\leq 0.10$	$7/30$ , $30/200$ ; $5/30$ , $6/30$ , $22/30$ after training changes
per-quarter network, correct operator: $Q$ specific (surrogate’s threshold)	rejection $\leq 0.10$	$8/30$
surrogate, correct operator: $Q$ specific	rejection $\leq 0.10$	$4/30$ ; $11/200$ (extension)

TABLE VII: Pre-registered claims rejected by their own threshold;

n=30

unless stated.

Explore similar work

Learning Physics from an Imperfect Ancestor
Sep 21, 2026S. Mohammad Mousavi, Teeratorn Kadeethum, Nikolaos Bouklas +1Parametric Physics-Informed Neural Network Partial Differential Equations
Sibley School of Mechanical and Aerospace Engineering, Cornell University · AI Lab, Siemens Energy · Department of Civil and Systems Engineering, Johns Hopkins University
Silent Failures in Physics-Informed Neural Networks: Parameter Poisoning and the Limits of Loss-Based Validation
Jun 23, 2026David McShannon, Nicholas DietrichParametric Physics-Informed Neural Network Partial Differential Equations
On the training of physics-informed neural operators for solving parametric partial differential equations
Jun 4, 2026Nanxi Chen, Chuanjie Cui, Airong Chen +2Parametric Physics-Informed Neural Network Physics-Informed Learning
College of Civil Engineering, Tongji University, Shanghai 200092 · Department of Engineering Science, University of Oxford, Oxford OX1 3PJ · Institute for Foundations of Data Science, Yale University, New Haven, CT 06520

Sep 21, 2026, cs.LGJ/K move · Enter open · S save

Learning Physics from an Imperfect Ancestor

S. Mohammad Mousavi, Teeratorn Kadeethum, Nikolaos Bouklas, Somdatta Goswami

Sibley School of Mechanical and Aerospace Engineering, Cornell University · AI Lab, Siemens Energy · Department of Civil and Systems Engineering, Johns Hopkins University

Parametric Physics-Informed Neural Network Partial Differential Equations Basin

Open paper AskPDF arXiv

Neural operators evaluate parametric partial differential equations cheaply but degrade sharply outside their training distribution. Physics-informed neural networks avoid dependence on labeled data, yet their optimization can be basin-fragile: when the governing residual admits multiple solutions, a PINN trained from scratch may converge to a physically incorrect state despite achieving a small residual. We show that these failure modes can be addressed jointly: an imperfect NO provides the structural prior needed to place a PINN in the correct solution basin, while the PDE residual refines the solution beyond the operator's accuracy. We introduce a three-stage framework that freezes the spatial basis of a physics-informed NO, extrapolates its solution branch to an out-of-distribution parameter using a polynomial continuation prior, and distills the resulting field into a fresh PINN. The NO need not be accurate at the target; it transfers solution-branch information, while PDE residual minimization in the PINN governs convergence. We evaluate the framework on three nonlinear PDEs: 1D viscous Burgers, 2D steady Allen-Cahn near a pitchfork bifurcation, and 2D steady lid-driven cavity flow. For Allen-Cahn, where the trivial solution satisfies the PDE residual exactly, a standard PINN collapses to the trivial zero branch, whereas distillation from the crude extrapolated operator recovers the non-trivial branch that matches the finite-difference reference. For the lid-driven cavity, extrapolating to a Reynolds number of Re = 3200 accelerates convergence to the correct physical state, achieving competitive accuracy using fewer parameters and optimization steps than recent literature baselines. These results establish a simple principle: an NO need not accurately predict the solution to be useful; it only needs to identify the correct basin from which PINN optimization can recover it.

Abstract

Explore similar work

Learning Physics from an Imperfect Ancestor

Silent Failures in Physics-Informed Neural Networks: Parameter Poisoning and the Limits of Loss-Based Validation

On the training of physics-informed neural operators for solving parametric partial differential equations