Artificial neural network with the Levenberg-Marquardt algorithm for numerical solution of two-dimension Poisson’s equation

Abstract

This study introduces an Artificial Neural Network (ANN) framework to address the two-dimensional Poisson’s equation within a rectangular domain. It places a focus on the training process of a neural network with three layers, incorporating hidden neurons. The feedforward ANN is trained using MATLAB, which calculates weights for all neurons within the network structure. These acquired weights are subsequently applied in the trained network model to make predictions for the desired output of a specific partial differential equation. The architecture of the ANN consists of three layers: one input layer, one hidden layer, and one output layer. In this study, we specifically employ an ANN configuration with 50 hidden neurons. The training process is executed using MATLAB, utilizing the Levenberg–Marquardt algorithm (LMA) for optimization. Furthermore, the study encompasses the development of surface and contour plots that illustrate the computational solution of the partial differential equation. Additionally, error functions are graphed to assess the effectiveness of the ANN model.