An overview on deep learning-based approximation methods for partial differential equations

TL;DR

This paper reviews recent deep learning-based methods for approximating high-dimensional partial differential equations, highlighting mathematical foundations, key ideas, and recent literature in this rapidly evolving research area.

Contribution

It provides an overview of mathematical results, proof ideas, and recent developments in deep learning methods for high-dimensional PDE approximation.

Findings

Deep learning methods show promise for high-dimensional PDEs

Mathematical analysis supports the effectiveness of these methods

Recent literature indicates rapid growth in this research area

Abstract

It is one of the most challenging problems in applied mathematics to approximatively solve high-dimensional partial differential equations (PDEs). Recently, several deep learning-based approximation algorithms for attacking this problem have been proposed and tested numerically on a number of examples of high-dimensional PDEs. This has given rise to a lively field of research in which deep learning-based methods and related Monte Carlo methods are applied to the approximation of high-dimensional PDEs. In this article we offer an introduction to this field of research by revisiting selected mathematical results related to deep learning approximation methods for PDEs and reviewing the main ideas of their proofs. We also provide a short overview of the recent literature in this area of research.