一种求解不可压缩Navier-Stokes方程和Cahn-Hilliard方程的新型深度神经网络OA

For a long time,the curse of dimensionality has severely restricted the efficiency of numerical solu-tions of high-dimensional partial differential equations(PDEs).In this paper,we extend the forward-backward stochastic neural networks(FBSNNs)constructed based on the forward-backward stochastic differ-ential equations(FBSDEs)to solve the incompressible Navier-Stokes equations.For the Cahn-Hilliard equa-tion,we derive a modified version of the equation from its widely adopted stabilized discrete scheme,which can be equivalently reformulated into a continuous parabolic system,so as to the FBSDE framework can be applied and the unknown solution of system can be approximated via neural networks.Furthermore,the pro-posed method is extended to the coupled Cahn-Hilliard-Navier-Stokes(CHNS)system.Numerical experi-ments are implemented to verify the accuracy and stability of the proposed approach.It is expected that the ob-tained results are helpful for solving the high-dimensional problem of Navier-Stokes equations and Cahn-Hilliard equations.