A Multiple-Input Deep Neural Network Architecture for Solution of One-Dimensional Poisson Equation

Abstract

In this letter, we demonstrate that a multiple-input deep neural network architecture can be used for the solution of a one-dimensional (1-D) second-order boundary value problem. We investigate the solution of the 1-D Poisson equation, while using sinc- and cosine-type functions to emulate typically found electromagnetic field distributions. Network architecture, modeling of the derivative, and boundary condition criteria are implemented, and test cases are used for validation. For the considered second-order boundary value problems, we obtain $-$ 80 dB error convergence in 8.2 s, showing a successful demonstration of the method. We further investigate the effect of the number of nodes, number of layers, and learning rate on the convergence of the method.