# Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations

## TL;DR

This paper introduces a novel deep learning algorithm that leverages reinforcement learning principles to efficiently solve high-dimensional parabolic PDEs and BSDEs, demonstrating accuracy in complex physics and finance models.

## Contribution

It presents a new neural network-based method that models the solution gradient as a policy, enabling effective high-dimensional PDE and BSDE solving.

## Key findings

- Efficiently solves 100-dimensional nonlinear PDEs.
- Accurately models complex physics and finance equations.
- Demonstrates scalability and effectiveness of the approach.

## Abstract

We propose a new algorithm for solving parabolic partial differential equations (PDEs) and backward stochastic differential equations (BSDEs) in high dimension, by making an analogy between the BSDE and reinforcement learning with the gradient of the solution playing the role of the policy function, and the loss function given by the error between the prescribed terminal condition and the solution of the BSDE. The policy function is then approximated by a neural network, as is done in deep reinforcement learning. Numerical results using TensorFlow illustrate the efficiency and accuracy of the proposed algorithms for several 100-dimensional nonlinear PDEs from physics and finance such as the Allen-Cahn equation, the Hamilton-Jacobi-Bellman equation, and a nonlinear pricing model for financial derivatives.