Neural networks solve high p-Laplace equations in 3D efficiently
Deep-Learning Solvers and Surrogates for Infinity and p-Laplace Problems
Machine Learning
Summary
Some math problems used in physics and engineering are very hard to solve, especially when a certain parameter called p is large. The authors show that special types of neural networks can solve these problems more easily than traditional methods, especially in three dimensions where classic techniques get very slow. They also prove some math guarantees about how well these neural networks can approximate the solutions. Their experiments show these neural network solvers work well and can save computing time.
What this means in practice
- •For computational engineers: Accelerate solving nonlinear PDEs with high p values in 3D domains using neural networks without costly meshes.
- •For computer graphics developers: Use neural network solvers to efficiently approximate complex physical phenomena modeled by p-Laplace equations in visual simulations.
Authors
Tak Shing Au Yeung, Ka Chun Cheung, Hannah Potgieter, Steven J. Ruuth, Simon See
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.