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

A novel deep neural network for solving incompressible Navier-Stokes equation and Cahn-Hilliard equation

摘要

长期以来，维数灾难严重制约了高维偏微分方程（partial differential equation，PDE）数值求解方法的计算效率。本文将基于正倒向随机微分方程（forward-backward stochastic neural network，FBSDE）构建的正倒向随机神经网络（forward-backward stochastic differential equation，FBSNN）方法推广至不可压缩 Navier-Stokes方程的求解。针对 Cahn-Hilliard 方程，本文从其广泛采用的稳定化离散格式出发，推导出该方程的一种修正形式，并将其等价地重构为连续抛物型系统，从而使其能够纳入 FBSDE 框架之中，并借助神经网络对系统的未知解进行逼近。进一步地，本文将所提出的方法拓展至耦合的 Cahn-Hilliard-Navier-Stokes（CHNS）系统。数值实验结果验证了所提方法的精度与稳定性。本文所得结果有望为 Navier-Stokes 方程和 Cahn-Hilliard 方程相关高维问题的数值求解提供有益参考。

Abstract

For a long time， the curse of dimensionality has severely restricted the efficiency of numerical solutions 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 differential equations （FBSDEs） to solve the incompressible Navier-Stokes equations.For the Cahn-Hilliard equation， 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 proposed method is extended to the coupled Cahn-Hilliard-Navier-Stokes （CHNS） system.Numerical experiments are implemented to verify the accuracy and stability of the proposed approach.It is expected that the obtained results are helpful for solving the high-dimensional problem of Navier-Stokes equations and Cahn-Hilliard equations.

关键词

倒向随机微分方程 / 神经网络 / Navier-Stokes / Cahn-Hilliard方程

Key words

forward-backward stochastic differential equation; neural network; Navier-Stokes equation; Cahn-Hilliard equation

基金资助

国家自然科学基金(12371434)