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

Deep FBSDE Neural Networks for Solving Incompressible Navier-Stokes Equation and Cahn-Hilliard Equation