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Volume 31, Issue 3
VPVnet: A Velocity-Pressure-Vorticity Neural Network Method for the Stokes’ Equations under Reduced Regularity

Yujie Liu & Chao Yang

Commun. Comput. Phys., 31 (2022), pp. 739-770.

Published online: 2022-03

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  • Abstract

We present VPVnet, a deep neural network method for the Stokes’ equations under reduced regularity. Different with recently proposed deep learning methods [40,51] which are based on the original form of PDEs, VPVnet uses the least square functional of the first-order velocity-pressure-vorticity (VPV) formulation ([30]) as loss functions. As such, only first-order derivative is required in the loss functions, hence the method is applicable to a much larger class of problems, e.g. problems with nonsmooth solutions. Despite that several methods have been proposed recently to reduce the regularity requirement by transforming the original problem into a corresponding variational form, while for the Stokes’ equations, the choice of approximating spaces for the velocity and the pressure has to satisfy the LBB condition additionally. Here by making use of the VPV formulation, lower regularity requirement is achieved with no need for considering the LBB condition. Convergence and error estimates have been established for the proposed method. It is worth emphasizing that the VPVnet method is divergence-free and pressure-robust, while classical inf-sup stable mixed finite elements for the Stokes’ equations are not pressure-robust. Various numerical experiments including 2D and 3D lid-driven cavity test cases are conducted to demonstrate its efficiency and accuracy.

  • AMS Subject Headings

76D07, 76M25, 65N12, 65N15, 35B45

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COPYRIGHT: © Global Science Press

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@Article{CiCP-31-739, author = {Liu , Yujie and Yang , Chao}, title = {VPVnet: A Velocity-Pressure-Vorticity Neural Network Method for the Stokes’ Equations under Reduced Regularity}, journal = {Communications in Computational Physics}, year = {2022}, volume = {31}, number = {3}, pages = {739--770}, abstract = {

We present VPVnet, a deep neural network method for the Stokes’ equations under reduced regularity. Different with recently proposed deep learning methods [40,51] which are based on the original form of PDEs, VPVnet uses the least square functional of the first-order velocity-pressure-vorticity (VPV) formulation ([30]) as loss functions. As such, only first-order derivative is required in the loss functions, hence the method is applicable to a much larger class of problems, e.g. problems with nonsmooth solutions. Despite that several methods have been proposed recently to reduce the regularity requirement by transforming the original problem into a corresponding variational form, while for the Stokes’ equations, the choice of approximating spaces for the velocity and the pressure has to satisfy the LBB condition additionally. Here by making use of the VPV formulation, lower regularity requirement is achieved with no need for considering the LBB condition. Convergence and error estimates have been established for the proposed method. It is worth emphasizing that the VPVnet method is divergence-free and pressure-robust, while classical inf-sup stable mixed finite elements for the Stokes’ equations are not pressure-robust. Various numerical experiments including 2D and 3D lid-driven cavity test cases are conducted to demonstrate its efficiency and accuracy.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2021-0104}, url = {http://global-sci.org/intro/article_detail/cicp/20297.html} }
TY - JOUR T1 - VPVnet: A Velocity-Pressure-Vorticity Neural Network Method for the Stokes’ Equations under Reduced Regularity AU - Liu , Yujie AU - Yang , Chao JO - Communications in Computational Physics VL - 3 SP - 739 EP - 770 PY - 2022 DA - 2022/03 SN - 31 DO - http://doi.org/10.4208/cicp.OA-2021-0104 UR - https://global-sci.org/intro/article_detail/cicp/20297.html KW - Stokes’ equations, deep neural network method, first-order velocity-pressure-vorticity formulation. AB -

We present VPVnet, a deep neural network method for the Stokes’ equations under reduced regularity. Different with recently proposed deep learning methods [40,51] which are based on the original form of PDEs, VPVnet uses the least square functional of the first-order velocity-pressure-vorticity (VPV) formulation ([30]) as loss functions. As such, only first-order derivative is required in the loss functions, hence the method is applicable to a much larger class of problems, e.g. problems with nonsmooth solutions. Despite that several methods have been proposed recently to reduce the regularity requirement by transforming the original problem into a corresponding variational form, while for the Stokes’ equations, the choice of approximating spaces for the velocity and the pressure has to satisfy the LBB condition additionally. Here by making use of the VPV formulation, lower regularity requirement is achieved with no need for considering the LBB condition. Convergence and error estimates have been established for the proposed method. It is worth emphasizing that the VPVnet method is divergence-free and pressure-robust, while classical inf-sup stable mixed finite elements for the Stokes’ equations are not pressure-robust. Various numerical experiments including 2D and 3D lid-driven cavity test cases are conducted to demonstrate its efficiency and accuracy.

Yujie Liu & Chao Yang. (2022). VPVnet: A Velocity-Pressure-Vorticity Neural Network Method for the Stokes’ Equations under Reduced Regularity. Communications in Computational Physics. 31 (3). 739-770. doi:10.4208/cicp.OA-2021-0104
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