The physics informed neural networks for the unsteady Stokes problems

Abstract

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.

The coupled deep neural networks for coupling of the Stokes and Darcy–Forchheimer problems

“…Additionally, some recent works have successfully solved the second-order linear elliptic equations and the high dimensional Stokes problems. [33][34][35][36] Though several excellent works have been performed in applying deep learning to solve PDEs, [37] the topic for solving complicated coupled physical problems remains to be investigated.…”

PDNNs: The parallel deep neural networks for the Navier–Stokes equations coupled with heat equation

“…Especially, this method overcomes the infeasibility and limitations of the traditional numerical methods especially for the high dimensional problems. We have successfully applied the method for simulating the three‐dimensional square cavity flow in Reference 42 and approximating the solution of the four‐dimensional Cahn–Hilliard equation in Reference 43. We present thorough numerical experiments using the parallel deep neural networks (PDNNs) to demonstrate our proposed approach. The neural network is trained by randomly sampling the space points to satisfy the PDEs residual, differential operators and boundary conditions.…”