A Deep Neural Network Surrogate for High-Dimensional Random Partial Differential Equations

TL;DR

This paper introduces a deep neural network-based method to efficiently approximate solutions to high-dimensional random PDEs, overcoming the curse of dimensionality with a mesh-free, flexible framework.

Contribution

The authors develop a novel deep learning framework that approximates high-dimensional random PDEs using residual networks, enabling mesh-free solutions on irregular domains.

Findings

Achieves accurate solutions comparable to Monte Carlo finite element methods.

Handles irregular domains without mesh generation.

Demonstrates effectiveness on diffusion and heat conduction problems.

Abstract

Developing efficient numerical algorithms for the solution of high dimensional random Partial Differential Equations (PDEs) has been a challenging task due to the well-known curse of dimensionality. We present a new solution framework for these problems based on a deep learning approach. Specifically, the random PDE is approximated by a feed-forward fully-connected deep residual network, with either strong or weak enforcement of initial and boundary constraints. The framework is mesh-free, and can handle irregular computational domains. Parameters of the approximating deep neural network are determined iteratively using variants of the Stochastic Gradient Descent (SGD) algorithm. The satisfactory accuracy of the proposed frameworks is numerically demonstrated on diffusion and heat conduction problems, in comparison with the converged Monte Carlo-based finite element results.