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 degenerate 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). Under 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.

翻译：在本文中,我们展示了一种深层次的基于学习的数值方法,用于近似高维随机部分差异方程式(SPDEs ) 。 每一步,我们的方法都依赖于预测-校正程序。更准确地说,我们将原SPDE分解成退化的SPDE和确定性的PDE。然后,在预测步骤中,我们用Euler计划解决了退化的SPDE,而在纠正步骤中,我们通过等效的后向偏差方程式(BSDE)通过深神经网络解决二级确定性PDE。根据标准假设,提出了错误估计和拟议算法的趋同率。提议的算法的效率和准确性用数字示例加以说明。