Solving high-dimensional partial differential equations using deep learning

TL;DR

This paper presents a deep learning method for solving high-dimensional PDEs by reformulating them with backward stochastic differential equations and approximating solutions with neural networks, effectively overcoming the curse of dimensionality.

Contribution

It introduces a novel deep learning algorithm that efficiently solves general high-dimensional parabolic PDEs using neural networks and stochastic reformulation, advancing computational capabilities.

Findings

Effective in high dimensions for nonlinear PDEs

Accurate and computationally efficient

Applicable to diverse fields like finance and physics

Abstract

Developing algorithms for solving high-dimensional partial differential equations (PDEs) has been an exceedingly difficult task for a long time, due to the notoriously difficult problem known as the "curse of dimensionality". This paper introduces a deep learning-based approach that can handle general high-dimensional parabolic PDEs. To this end, the PDEs are reformulated using backward stochastic differential equations and the gradient of the unknown solution is approximated by neural networks, very much in the spirit of deep reinforcement learning with the gradient acting as the policy function. Numerical results on examples including the nonlinear Black-Scholes equation, the Hamilton-Jacobi-Bellman equation, and the Allen-Cahn equation suggest that the proposed algorithm is quite effective in high dimensions, in terms of both accuracy and cost. This opens up new possibilities in economics, finance, operational research, and physics, by considering all participating agents, assets, resources, or particles together at the same time, instead of making ad hoc assumptions on their inter-relationships.