VPVnet: A Velocity-Pressure-Vorticity Neural Network Method for the Stokes’ Equations under Reduced Regularity (Communications in Computational Physics, 2022)
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 squarefunctional of the first-order velocity-pressure-vorticity (VPV) formulation ([30]) as lossfunctions. As such, only first-order derivative is required in the loss functions, hencethe 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 reducethe regularity requirement by transforming the original problem into a correspondingvariational form, while for the Stokes’ equations, the choice of approximating spacesfor the velocity and the pressure has to satisfy the LBB condition additionally. Hereby making use of the VPV formulation, lower regularity requirement is achieved withno need for considering the LBB condition. Convergence and error estimates havebeen established for the proposed method. It is worth emphasizing that the VPVnetmethod is divergence-free and pressure-robust, while classical inf-sup stable mixedfinite elements for the Stokes’ equations are not pressure-robust. Various numericalexperiments including 2D and 3D lid-driven cavity test cases are conducted to demonstrate its efficiency and accuracy.