Solving Coupled Systems of Differential Equations Using the Length Factor Artificial Neural Network Method

The length factor artificial neural network method for solving differential equations has previously been shown to successfully solve boundary value problems involving partial differential equations. This manuscript extends the method to solve coupled systems of partial differential equations, including accurate approximation of local Nusselt numbers in boundary layers and solving the Navier-Stokes equations for the entry length problem. With strengths including an explicit and continuous approximate solution, elimination of meshing concerns and simple implementation for nonlinear differential equations, this method is emerging as a viable alternative to traditional numerical techniques such as the finite element method. INTRODUCTION Many problems in science and engineering involve differential equations (DEs) sufficiently complicated to require numerical techniques for approximating their solutions. Traditionally, the finite difference [1], finite element [2], and boundary element [3] methods have been employed to numerically solve DEs. Although powerful and widespread, these tools do have drawbacks including problematic discretization of the problem domain and complications in solving nonlinear DEs. Artificial neural networks (ANNs) have emerged as an alternative method for numerical solution of DEs [4-12]. Methods using ANNs generally avoid the drawbacks of traditional numerical techniques. For example, domain discretization often involves simple square grids where no special treatment is necessary for nonlinear DEs. ANN methods begin with a trial approximate solution (TAS) continuous over the problem domain whose value is influenced by a number of ANN parameters initialized to random values. Those parameters are then optimized to most closely approximate a solution to the given DE equation. While most ANN methods follow this general strategy, they vary greatly in implementation. The length factor ANN method featured here was developed to be simple in order to improve accessibility to those less familiar with ANNs. The hallmark of this method is that the boundary conditions (BCs) associated with the DE are automatically satisfied during all stages of training the ANN, including during initialization of network parameters with random values. Such an approach removes the BC constraint, allowing a simpler and more straightforward optimization stage compared with other ANN methods. The length factor method for solving DEs has already been shown to successfully solve partial differential equations (PDEs) in two and three dimensions [8]. The main contribution of this manuscript is to expand the method to solve coupled systems of PDEs including the two-dimensional steady Navier-Stokes equations. The method is first demonstrated with a toy problem on an irregularly shaped domain, it produces an approximation for the local Nusselt number in the Blasius boundary layer, and it models the entrance region of flow between two infinite flat plates. FORM OF THE TRIAL APPROXIMATE SOLUTION Consider the two-dimensional, second-order differential equation defined by