Deep-Learning Solvers and Surrogates for Infinity and p-Laplace Problems
math.NA, cs.LG, cs.NA
Submitted: 2026-09-25
Updated: 2026-09-25
Code: https://github.com/hannah-potgieter/infinity-and-p-Laplace-solvers
License: http://creativecommons.org/licenses/by/4.0/
The gist: We investigate the use of neural network solvers for infinity and p-Laplace problems, which are fundamental in nonlinear analysis and have practical applications.
Terminology
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.
Sources
- Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations
- Fourier Neural Operator for Parametric Partial Differential Equations
- Neural Operator: Graph Kernel Network for Partial Differential Equations
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