Deep Learning-Based Numerical Methods for High-Dimensional Parabolic Partial Differential Equations and Backward Stochastic Differential Equations

Abstract

We study a new algorithm for solving parabolic partial differential equations (PDEs) and backward stochastic differential equations (BSDEs) in high dimension, which is based on 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 studied algorithm 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.

Keywords

PDEs / High dimension / Backward stochastic differential equations / Deep learning / Control / Feynman-Kac

Funding

U.S. Department of Energy(DOE DE-SC0009248)

U.S. Deparment of Energy(DOE DE-SC0009248)

Office of Naval Research Global(ONR N00014-13-1-0338)

National Natural Science Foundation of China(Major Program, Grant 91130005)