The physics informed neural networks for the unsteady Stokes problems

In this article, we develop the physics informed neural networks (PINNs) coupled with small sample learning for solving the transient Stokes equations. Specifically, the governing equations are encoded into the networks to construct the loss function, which involves the residual of differential equations, the initial/boundary conditions, and the residual of a handful of observations. The approximate solution was obtained by optimizing the loss function. Few sample data can rectify the network effectively and improve predictive accuracy. Moreover, the method can simultaneously solve each variable of the equations separately in a parallel framework. The information of the numerical data is compiled into the networks to enhance efficiency and accuracy in practice. Therefore, this method is a meshfree and fusion method that combined data-driven with model-driven. Inspired by the Galerkin method, the paper proves the convergence of the loss function and the capability of neural networks. Furthermore, numerical experiments are performed and discussed to demonstrate the performance of the method.