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

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.

Related

• Derivative Computation in PINNs: Automatic Differentiation, Finite Differences and Beyond
• Multi-Fidelity Physics-Informed Neural Networks with Bayesian Uncertainty Quantification and Adaptive Residual Learning for Efficient Solution of Parametric Partial Differential Equations
• Automatic Differentiation from Scratch: How PyTorch Computes Gradients in Physics-Informed Neural Networks

Source: arXiv cs.LG | 2026-09-15