Forward-Backward Stochastic Neural Networks: Deep Learning of High-dimensional Partial Differential Equations

TL;DR

This paper introduces a scalable deep learning method for solving high-dimensional partial differential equations by approximating solutions with neural networks trained via stochastic differential equations, overcoming traditional grid-based limitations.

Contribution

The authors develop a novel deep neural network approach that leverages forward-backward stochastic differential equations to efficiently solve high-dimensional PDEs without discretization.

Findings

Successfully applied to 100-dimensional Black-Scholes-Barenblatt equations

Achieved accurate solutions for Hamilton-Jacobi-Bellman equations in high dimensions

Demonstrated scalability and effectiveness of the method

Abstract

Classical numerical methods for solving partial differential equations suffer from the curse dimensionality mainly due to their reliance on meticulously generated spatio-temporal grids. Inspired by modern deep learning based techniques for solving forward and inverse problems associated with partial differential equations, we circumvent the tyranny of numerical discretization by devising an algorithm that is scalable to high-dimensions. In particular, we approximate the unknown solution by a deep neural network which essentially enables us to benefit from the merits of automatic differentiation. To train the aforementioned neural network we leverage the well-known connection between high-dimensional partial differential equations and forward-backward stochastic differential equations. In fact, independent realizations of a standard Brownian motion will act as training data. We test the effectiveness of our approach for a couple of benchmark problems spanning a number of scientific domains including Black-Scholes-Barenblatt and Hamilton-Jacobi-Bellman equations, both in 100-dimensions.