Efficient algorithms for solving high-dimensional partial differential equations (PDEs) has been an exceedingly difficult task for a long time, due to the curse of dimensionality. We extend the forward-backward stochastic neural networks (FBSNNs) which depends on forward-backward stochastic differential equation (FBSDE) to solve incompressible Navier-Stokes equation. For Cahn-Hilliard equation, we derive a modified Cahn-Hilliard equation from a widely used stabilized scheme for original Cahn-Hilliard equation. This equation can be written as a continuous parabolic system, where FBSDE can be applied and the unknown solution is approximated by neural network. Also our method is successfully developed to Cahn-Hilliard-Navier-Stokes (CHNS) equation. The accuracy and stability of our methods are shown in many numerical experiments, specially in high dimension.

翻译：长期以来，由于维数灾难，求解高维偏微分方程的高效算法一直是一项极其困难的任务。本文推广了基于正向-反向随机微分方程的向前-向后随机神经网络，用于求解不可压缩Navier-Stokes方程。针对Cahn-Hilliard方程，我们从广泛使用的原始Cahn-Hilliard方程稳定化格式中推导出修正的Cahn-Hilliard方程。该方程可表述为连续抛物系统，从而可应用FBSDE方法，并通过神经网络逼近未知解。本方法亦成功拓展至Cahn-Hilliard-Navier-Stokes耦合方程组。大量数值实验（特别是高维情形）验证了所提方法的精度与稳定性。