Algorithms for Solving High Dimensional PDEs: From Nonlinear Monte Carlo to Machine Learning

TL;DR

This paper reviews recent advances in numerical algorithms for high-dimensional PDEs, highlighting stochastic and deep learning methods that potentially overcome the curse of dimensionality, with promising applications in mathematics and scientific computing.

Contribution

It provides a comprehensive review of stochastic, deep learning, and traditional methods for high-dimensional PDEs, emphasizing their potential to revolutionize computational mathematics.

Findings

Nonlinear Monte Carlo methods can solve certain high-dimensional PDEs without curse of dimensionality.

Deep learning approaches like Deep BSDE have shown promising results in high-dimensional settings.

Traditional methods are being adapted and combined with stochastic techniques for better scalability.

Abstract

In recent years, tremendous progress has been made on numerical algorithms for solving partial differential equations (PDEs) in a very high dimension, using ideas from either nonlinear (multilevel) Monte Carlo or deep learning. They are potentially free of the curse of dimensionality for many different applications and have been proven to be so in the case of some nonlinear Monte Carlo methods for nonlinear parabolic PDEs. In this paper, we review these numerical and theoretical advances. In addition to algorithms based on stochastic reformulations of the original problem, such as the multilevel Picard iteration and the Deep BSDE method, we also discuss algorithms based on the more traditional Ritz, Galerkin, and least square formulations. We hope to demonstrate to the reader that studying PDEs as well as control and variational problems in very high dimensions might very well be among the most promising new directions in mathematics and scientific computing in the near future.