Artificial neural network with the Levenberg-Marquardtalgorithm for numerical solution of two-dimension Poisson’sequation

Abstract

This study introduces an Artificial Neural Network (ANN) framework to address the two-dimensional Poisson’sequation within a rectangular domain. It places a focus on the training process of a neural network with threelayers, incorporating hidden neurons. The feedforward ANN is trained using MATLAB, which calculates weights forall neurons within the network structure. These acquired weights are subsequently applied in the trained networkmodel to make predictions for the desired output of a specific partial differential equation. The architecture of theANN consists of three layers: one input layer, one hidden layer, and one output layer. In this study, we specificallyemploy an ANN configuration with 50 hidden neurons. The training process is executed using MATLAB, utilizing theLevenberg–Marquardt algorithm (LMA) for optimization. Furthermore, the study encompasses the development ofsurface 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.