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Deep FBSDE Neural Networks for Solving Incompressible Navier-Stokes Equation and Cahn-Hilliard Equation (2401.03427v2)

Published 7 Jan 2024 in math.NA, cs.NA, and physics.flu-dyn

Abstract: 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.

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