This work employs Physics-Informed Neural Networks and Deep Operator Networks to address computational challenges associated with large $p$ values, ranging from $2 to $1000$, on various 2D and 3D domains and establishes conditional convergence results for PINN approximations of both problems.
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.
A robust training framework for PINNs solving nonlinear partial differential equations, together with a rigorous error analysis, is proposed and extended to a parametric setting in which the exponent, the source term, and the boundary condition may all vary with the parameters.
A physics-informed random feature method for countering part of the spectral bias which PINN-based solvers are facing for a certain class of PDEs, and develops a rigorous approximation error analysis and derive high-probability error bounds on the H^1$ norm.
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MIT News · Artificial Intelligence· news.mit.eduOct 13, 2026
Machine-learning networks can learn without detailed instructions from a designer. Those that do may offer insight into how the brain reorganizes itself during learning.
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