Deep FBSDE Neural Networks for Solving Incompressible Navier-Stokes Equation and Cahn-Hilliard Equation
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866911925566177280 |
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| author | Deng, Yangtao He, Qiaolin |
| author_facet | Deng, Yangtao He, Qiaolin |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_03427 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Deep FBSDE Neural Networks for Solving Incompressible Navier-Stokes Equation and Cahn-Hilliard Equation Deng, Yangtao He, Qiaolin Numerical Analysis Fluid Dynamics 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. |
| title | Deep FBSDE Neural Networks for Solving Incompressible Navier-Stokes Equation and Cahn-Hilliard Equation |
| topic | Numerical Analysis Fluid Dynamics |
| url | https://arxiv.org/abs/2401.03427 |