四川大学学报（自然科学版）2026，Vol.63Issue(4)：793-812,20.DOI:10.19907/j.0490-6756.250073
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
Abstract
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.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分类
数理科学引用本文复制引用
邓扬涛,贺巧琳..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[J].四川大学学报（自然科学版）,2026,63(4):793-812,20.基金项目
国家自然科学基金(12371434) （12371434）
