Neural Network Based Variational Methods for Solving Quadratic Porous Medium Equations in High Dimensions

TL;DR

This paper introduces neural network-based variational methods to solve high-dimensional quadratic porous medium equations, exploring strong and weak formulations and demonstrating their effectiveness through numerical tests.

Contribution

It presents novel neural network approaches for high-dimensional QPME, including strong and weak variational formulations with convergence guarantees.

Findings

Neural network methods effectively solve high-dimensional QPME.

Strong formulation guarantees convergence in $L^1$ sense.

Weak formulations provide alternative solution representations.

Abstract

In this paper, we propose and study neural network based methods for solutions of high-dimensional quadratic porous medium equation (QPME). Three variational formulations of this nonlinear PDE are presented: a strong formulation and two weak formulations. For the strong formulation, the solution is directly parameterized with a neural network and optimized by minimizing the PDE residual. It can be proved that the convergence of the optimization problem guarantees the convergence of the approximate solution in the $L^1$ sense. The weak formulations are derived following Brenier, Y., 2020, which characterizes the very weak solutions of QPME. Specifically speaking, the solutions are represented with intermediate functions who are parameterized with neural networks and are trained to optimize the weak formulations. Extensive numerical tests are further carried out to investigate the pros and cons of each formulation in low and high dimensions. This is an initial exploration made along the line of solving high-dimensional nonlinear PDEs with neural network based methods, which we hope can provide some useful experience for future investigations.