  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.
