A predictor-corrector deep learning algorithm for high dimensional stochastic partial differential equations

In this paper, we present a deep learning-based numerical method for approximating high dimensional stochastic partial differential equations (SPDEs). At each time step, our method relies on a predictor-corrector procedure. More precisely, we decompose the original SPDE into a de- generate SPDE and a deterministic PDE. Then in the prediction step, we solve the degenerate SPDE with the Euler scheme, while in the correction step we solve the second-order deterministic PDE by deep neural networks via its equivalent backward stochastic differential equation (BSDE). Un- der standard assumptions, error estimates and the rate of convergence of the proposed algorithm are presented. The efficiency and accuracy of the proposed algorithm are illustrated by numerical examples.