A derivative-fidelity failure mode in physics-informed neural networks: strengthened benchmark evidence from function-value training

A paper posted to arXiv (arXiv:2609.13171v1) formulates derivative fidelity as a failure mode of physics-informed neural networks (PINNs), showing that multilayer perceptrons trained only on function values for sin(x) and exp(x) can achieve visually accurate function approximation while producing substantially larger second-derivative errors, especially near high-curvature boundary regions. The study strengthens its hypothesis with additional tests over training-point density, activation functions, endpoint-dense evaluation, and both L2 and maximum-error diagnostics, and offers a diagnostic protocol for distinguishing value accuracy from physics-residual reliability.

arXiv:2609.13171v1 Announce Type: new Abstract: Physics-informed neural networks PINNs use automatic differentiation to impose differential-equation residuals, but good agreement in function values does not necessarily imply accurate derivatives. This paper formulates derivative fidelity as a failure mode of PINNs and tests it with one-dimensional benchmarks. Multilayer perceptrons are trained only on function values for sin x and exp x , while second derivatives obtained by automatic differentiation are evaluated separately. The hypothesis is strengthened by additional tests over training-point density, activation functions, endpoint-dense evaluation, and both L2 and maximum-error diagnostics. The results show that visually accurate function approximation can coexist with substantially larger second-derivative errors, especially near high-curvature boundary regions. The experiment provides a diagnostic protocol for distinguishing value accuracy from physics-residual reliability.