# Wrong Operator or Blind Design? A Post-Training Diagnostic for Physics-Informed Coefficient Learning

arXiv: https://arxiv.org/abs/2608.16925 · HTML: https://mlsanity.com/paper/2608.16925/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 18 of its 48 references that are in the map.
- First posted on arXiv 4 Aug 2026.

## Builds on (most cited first, 18 of 18)

- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [Generalized Out-of-Distribution Detection: A Survey](https://mlsanity.com/paper/2110.11334/) (2021)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [Universal Differential Equations for Scientific Machine Learning](https://mlsanity.com/paper/2001.04385/) (2020)
- [BayesFlow: Learning complex stochastic models with invertible neural networks](https://mlsanity.com/paper/2003.06281/) (2020)
- [Deep Learning is Singular, and That's Good](https://mlsanity.com/paper/2010.11560/) (2020)
- [Detecting Model Misspecification in Amortized Bayesian Inference with Neural Networks](https://mlsanity.com/paper/2112.08866/) (2021)
- [Correcting model misspecification in physics-informed neural networks (PINNs)](https://mlsanity.com/paper/2310.10776/) (2023)
- [Enhancing Explainability of Neural Networks through Architecture Constraints](https://mlsanity.com/paper/1901.03838/) (2019)
- [Physics-informed neural networks for solving forward and inverse problems in complex beam systems](https://mlsanity.com/paper/2303.01055/) (2023)
- [On structural and practical identifiability](https://mlsanity.com/paper/2102.05100/) (2021)
- [Understanding and Mitigating Extrapolation Failures in Physics-Informed Neural Networks](https://mlsanity.com/paper/2306.09478/) (2023)
- [The Deep Promotion Time Cure Model](https://mlsanity.com/paper/2305.11575/) (2023)
- [A cautionary tale of model misspecification and identifiability](https://mlsanity.com/paper/2507.04894/) (2025)
- [PDEInvBench: A Comprehensive Dataset and Design Space Exploration of Neural Networks for PDE Inverse Problems](https://mlsanity.com/paper/2605.25353/) (2026)
- [Sequential Structure-Sensitive Residual Diagnostics for PDE Inverse Problems](https://mlsanity.com/paper/2607.02101/) (2026)
- [The Dynamics of Discoverability: How Trajectories and Priors Shape Equation Recovery](https://mlsanity.com/paper/2607.18490/) (2026)

## Abstract (arXiv)

Physics-informed and hybrid models integrate data and physics, and read learned coefficients as physical. Yet a small prediction error guarantees neither that the postulated operator is right, nor that the design identifies the coefficient, nor that training reached its minimum. On one parabolic benchmark where a sensor aggregates several diffusion rates and the truth is known, a correct operator recovers the coefficient to machine precision. A wrong one leaves it 30% off while the in-domain error is 1.36 times the noise, and drifts 27% across windows. At 7% noise the error falls below the noise; the coefficient stays 30% off. Without the truth, a post-training coefficient diagnostic questions the trained model with three classical statistics, without oracle; the first two need no refit. Does it explain the data? Can the design resolve the named physics? Does the coefficient depend on the window? The specification test keeps its pre-registered false-alarm ceiling and rejects every replicate of both wrong operators tested; a network frozen at its minimum returns the same verdicts. Where the design is blind, a wrong operator confined to the unseen rate leaves the test mute; the design reading flags it beforehand. A network stopped short of its minimum mimics a wrong operator; an analytic second fit separates the two. The test has a notch against a missing reaction term, miscalibrates under correlated noise and loses power beyond 7% noise. Each warning names its remedy (operator, experiment or training); silence certifies nothing.