Derivative Computation in PINNs: Automatic Differentiation, Finite Differences and Beyond
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Computer Science > Machine Learning
Title: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.
| Comments: | 22 pages, 5 figures |
| Subjects: | Machine Learning (cs.LG); Numerical Analysis (math.NA); Computational Physics (physics.comp-ph) |
| MSC classes: | 68T07, 65D25, 65M06, 65N06 |
| ACM classes: | I.2.6; G.1.4; G.1.8 |
| Cite as: | arXiv:2608.11020 [cs.LG] |
| (or arXiv:2608.11020v1 [cs.LG] for this version) | |
| https://doi.org/10.48550/arXiv.2608.11020
arXiv-issued DOI via DataCite (pending registration)
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Submission history
From: Maciej J. Mikulski [view email][v1] Tue, 11 Aug 2026 15:01:29 UTC (522 KB)
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