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

Understand

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.