Derivative Computation in PINNs: Automatic Differentiation, Finite Differences and Beyond

Abstract:We systematically investigate finite-difference (FD) derivative computation in Physics-Informed Neural Networks (PINNs) as an alternative to automatic differentiation (AD). On three benchmark PDEs we show that, with a properly calibrated step size, FD matches AD in accuracy on every problem while running faster across the full tested batch-size range and using substantially less GPU memory, and that a stochastic variant we propose outperforms AD on a stationary problem. We further show that for neural architectures with inter-sample dependencies (e.g. BatchNorm, self-attention) the standard PyTorch autograd idiom is silently incorrect; the correct per-sample alternative is computationally infeasible at PINN-relevant batch sizes, while FD provides a forward-only approximation that is empirically an order of magnitude closer to the true per-sample derivative.

Machine Learning (cs.LG); Numerical Analysis (math.NA); Computational Physics (physics.comp-ph)

I.2.6; G.1.4; G.1.8

arXiv:2608.11020 [cs.LG]

(or arXiv:2608.11020v2 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2608.11020 Focus to learn more arXiv-issued DOI via DataCite

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