Artificial neural network method for solving the Navier–Stokes equations

Abstract

In this paper, a new method based on neural network is developed for obtaining the solution of the Navier–Stokes equations in an analytical function form. The solution procedure is based upon forming a trial solution consisting of two parts. The first part directly satisfies the boundary conditions and therefore, contains no adjustable parameters. The second part is constructed such that the governing equation is satisfied inside the solution domain, while the boundary conditions remain untouched. This part involves a feed-forward neural network, containing adjustable parameters (the weights), which must be determined such that the resulting approximate error function is minimized. The details of the method are discussed, and the capabilities of the method are illustrated by solving Navier–Stokes problem with different boundary conditions. The performance of the method and the accuracy of the results are evaluated by comparing with the available numerical and analytical solutions.