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 num