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PreprintCC BY 4.0arXiv

Deep-Learning Solvers and Surrogates for Infinity and p-Laplace Problems

  • Tak Shing Au Yeung
  • Ka Chun Cheung
  • Hannah Potgieter
  • Steven J. Ruuth
  • Simon See
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Abstract

We investigate the use of neural network solvers for infinity and p-Laplace problems, which are fundamental in nonlinear analysis and have practical applications. Our approach employs Physics-Informed Neural Networks (PINNs) and Deep Operator Networks (DeepONets) to address computational challenges associated with large p values, ranging from $2$ to $1000$, on various 2D and 3D domains. Our method offers advantages over traditional physics-based solvers, especially in three dimensions where mesh-based solvers become very costly for these problems. We also establish conditional convergence results for PINN approximations of both problems and a universal approximation result for DeepONet on the parametric p-Poisson problem. We demonstrate the effectiveness of these neural network solvers through numerical experiments and compare their performance with conventional methods.

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Cite

BibTeXRIS
BibTeX
@misc{yeung2026deeplearning,
  title = {{Deep-Learning Solvers and Surrogates for Infinity and p-Laplace Problems}},
  author = {Tak Shing Au Yeung and Ka Chun Cheung and Hannah Potgieter and Steven J. Ruuth and Simon See},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.30809},
  eprint = {2609.30809},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.30809},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Deep-Learning Solvers and Surrogates for Infinity and p-Laplace Problems
AU  - Tak Shing Au Yeung
AU  - Ka Chun Cheung
AU  - Hannah Potgieter
AU  - Steven J. Ruuth
AU  - Simon See
PY  - 2026
DA  - 2026/09/25
PB  - arXiv
DO  - 10.48550/arxiv.2609.30809
UR  - https://arxiv.org/abs/2609.30809
AB  - We investigate the use of neural network solvers for infinity and p-Laplace problems, which are fundamental in nonlinear analysis and have practical applications. Our approach employs Physics-Informed Neural Networks (PINNs) and Deep Operator Networks (DeepONets) to address computational challenges associated with large p values, ranging from $2$ to $1000$, on various 2D and 3D domains. Our method offers advantages over traditional physics-based solvers, especially in three dimensions where mesh-based solvers become very costly for these problems. We also establish conditional convergence results for PINN approximations of both problems and a universal approximation result for DeepONet on the parametric p-Poisson problem. We demonstrate the effectiveness of these neural network solvers through numerical experiments and compare their performance with conventional methods.
ER  -