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arXiv · 2609.30809

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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Tak Shing Au Yeung, Ka Chun Cheung, Hannah Potgieter, Steven J. Ruuth, Simon See. 2026-09-25. Deep-Learning Solvers and Surrogates for Infinity and p-Laplace Problems. https://arxiv.org/abs/2609.30809

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