aiqbench analysis
This research highlights a critical limitation in physics-informed neural networks (PINNs) where accurate function approximation can mask significant errors in derivative calculations, particularly near high-curvature boundaries. By establishing a diagnostic protocol that separates value accuracy from residual reliability, the study offers essential guidance for improving the robustness of AI models solving differential equations. This work directly advances the field by addressing a fundamental failure mode in how machine learning handles physics-based constraints.
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.
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.
Machine Learning (cs.LG)
arXiv:2609.13171 [cs.LG]
(or arXiv:2609.13171v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2609.13171 Focus to learn more arXiv-issued DOI via DataCite
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