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

2026-09-25

AI Summary

This study addresses the prohibitive computational cost of traditional mesh-based solvers for the $p$-Laplace equation in three-dimensional domains when $p$ is large. To this end, it proposes an efficient solution framework based on physics-informed neural networks (PINNs) and deep operator networks (DeepONets). Theoretically, a conditional convergence result for PINNs is established, and the universal approximation capability of DeepONets for parameterized $p$-Poisson problems is rigorously proved. Numerical experiments demonstrate that the proposed approach effectively overcomes the computational bottleneck associated with high-dimensional nonlinear problems, achieving significantly superior accuracy and efficiency compared to conventional solvers. Consequently, this work provides a new paradigm for complex three-dimensional nonlinear analysis.

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.

Problem

• infinity Laplace
• p-Laplace
• Physics-Informed Neural Networks
• Deep Operator Networks
• nonlinear analysis

Innovation

• Physics-Informed Neural Networks
• Deep Operator Networks
• p-Laplace problems
• Conditional convergence
• Universal approximation