ML Anthology

Provably Accurate Adaptive Sampling for Collocation Points in Physics-Informed Neural Networks

Antoine Caradot, Rémi Emonet, Amaury Habrard, Abdel-Rahim Mezidi, Marc Sebban
ECML-PKDD 2025 pp. 19-37
doi:10.1007/978-3-032-06096-9_2 /ecmlpkdd/2025/caradot2025ecmlpkdd-provably/

Abstract

Despite considerable scientific advances in numerical simulation, efficiently solving PDEs remains a complex and often expensive problem. Physics-informed Neural Networks (PINN) have emerged as an efficient way to learn surrogate solvers by embedding the PDE in the loss function and minimizing its residuals using automatic differentiation at so-called collocation points. Originally uniformly sampled, the choice of the latter has been the subject of recent advances leading to adaptive sampling refinements for PINNs. In this paper, leveraging a new quadrature method for approximating definite integrals, we introduce a provably accurate sampling method for collocation points based on the Hessian of the PDE residuals. Comparative experiments conducted on a set of 1D and 2D PDEs demonstrate the benefits of our method.

PDF ECML-PKDD Semantic Scholar

Cite

Text

Caradot et al. "Provably Accurate Adaptive Sampling for Collocation Points in Physics-Informed Neural Networks." European Conference on Machine Learning and Principles and Practice of Knowledge Discovery in Databases, 2025. doi:10.1007/978-3-032-06096-9_2

Markdown

[Caradot et al. "Provably Accurate Adaptive Sampling for Collocation Points in Physics-Informed Neural Networks." European Conference on Machine Learning and Principles and Practice of Knowledge Discovery in Databases, 2025.](https://mlanthology.org/ecmlpkdd/2025/caradot2025ecmlpkdd-provably/) doi:10.1007/978-3-032-06096-9_2

BibTeX

@inproceedings{caradot2025ecmlpkdd-provably,
  title     = {{Provably Accurate Adaptive Sampling for Collocation Points in Physics-Informed Neural Networks}},
  author    = {Caradot, Antoine and Emonet, Rémi and Habrard, Amaury and Mezidi, Abdel-Rahim and Sebban, Marc},
  booktitle = {European Conference on Machine Learning and Principles and Practice of Knowledge Discovery in Databases},
  year      = {2025},
  pages     = {19-37},
  doi       = {10.1007/978-3-032-06096-9_2},
  url       = {https://mlanthology.org/ecmlpkdd/2025/caradot2025ecmlpkdd-provably/}
}