Learning high dimensional stochastic partial differential equations in a predictor-corrector framework

In this paper, we propose a deep learning-based numerical method for approximating high dimensional stochastic partial differential equations in a predictor-corrector framework. At each time step, the original equation is decomposed into a degenerate stochastic partial differential equation to serve as the prediction step, and a second-order deterministic partial differential equation to serve as the correction step. The solution of the degenerate stochastic partial differential equation is approximated by the Euler method, and the solution of the partial differential equation is approximated by neural networks via the equivalent backward stochastic differential equation. Under standard assumptions, the rate of convergence and the error estimates in terms of the approximation error of neural networks are provided. The efficiency and accuracy of the proposed algorithm are illustrated by the numerical results.