VPVnet: a velocity-pressure-vorticity neural network method for the Stokes' equations under reduced regularity

TL;DR

VPVnet introduces a neural network approach for Stokes' equations that requires only first-order derivatives, enabling solutions for problems with low regularity and avoiding the LBB condition.

Contribution

The paper proposes VPVnet, a neural network method based on the VPV formulation that handles low-regularity Stokes' problems without LBB constraints and ensures divergence-free, pressure-robust solutions.

Findings

Achieves convergence and error estimates for the method.

Demonstrates efficiency and accuracy through 2D and 3D tests.

Applicable to problems with non-smooth solutions.

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 non-smooth 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.