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This research applies Physics-Informed Neural Networks (PINNs) and Deep Operator Networks (DeepONets) to solve challenging nonlinear partial differential equations, specifically infinity and $p$-Laplace problems, where traditional mesh-based solvers become computationally prohibitive. By establishing conditional convergence results for PINNs and universal approximation properties for DeepONets, the study demonstrates how deep learning can effectively handle high-dimensional domains and extreme parameter values ($p$ up to 1000). The work bridges theoretical machine learning guarantees with practical computational physics, offering a scalable alternative for complex nonlinear analysis tasks.
Deep-Learning Solvers and Surrogates for Infinity and p-Laplace Problems
arXiv:2609.30809v1 Announce Type: cross 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.
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.
Numerical Analysis (math.NA); Machine Learning (cs.LG)
arXiv:2609.30809 [math.NA]
(or arXiv:2609.30809v1 [math.NA] for this version)
https://doi.org/10.48550/arXiv.2609.30809 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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