Multilayer perceptron neural networks with novel unsupervised training method for numerical solution of the partial differential equations

Abstract

In this paper by using MultiLayer Perceptron and Radial Basis Function (RBF) neural networks, a novel method for solving both kinds of differential equation, ordinary and partial differential equation, is presented. From the differential equation and its boundary conditions, the energy function of the network is prepared which is used in the unsupervised training method to update the network parameters. This method was implemented to solve the nonlinear Schrodinger equation in hydrogen atom and triangle-shaped quantum well. Comparison of this method results with analytical solution and two well-known numerical methods, Runge–kutta and finite element, shows the efficiency of Neural Networks with high accuracy, fast convergence and low use of memory for solving the differential equations.