  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.
