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

摘要

We propose a new algorithm for solving parabolic partial differentialequations (PDEs) and backward stochastic differential equations (BSDEs) in highdimension, by making an analogy between the BSDE and reinforcement learningwith the gradient of the solution playing the role of the policy function, andthe loss function given by the error between the prescribed terminal conditionand the solution of the BSDE. The policy function is then approximated by aneural network, as is done in deep reinforcement learning. Numerical resultsusing TensorFlow illustrate the efficiency and accuracy of the proposedalgorithms for several 100-dimensional nonlinear PDEs from physics and financesuch as the Allen-Cahn equation, the Hamilton-Jacobi-Bellman equation, and anonlinear pricing model for financial derivatives.