A Deep Learning Neural Network Framework for Solving Singular Nonlinear Ordinary Differential Equations

This paper presents a deep learning feedforward neural network framework as a numerical tool to approximate the solutions to singular nonlinear ordinary differential equations arising in physiology. Artificial neural network (ANN) models are best suited for optimization problems. By casting the solution finding algorithm into a weighted-residual type minimization procedure, the ANN models are shown to be capable of approximating the solution to varieties of equations: ordinary differential equations, partial differential equations and integral equations. In this paper, we utilize a mean-square type loss function in an unsupervised learning network—without actually requiring exact or approximate solution for training—to find the approximate solution to varieties singular differential equations. The total loss function includes the strong form differential equation along with initial/boundary condition terms. During the training phase, our network is trained using a linear interpolation of the small data set, then it adjusts itself to approximate the curvature so that the total loss value is small. The fine-tuning of network’s hyperparameters, which are the weights and bias, is done through the backpropogation step. The efficiency of the proposed method is demonstrated by approximating the solutions to several singular differential equations. We report that the relative error in all the problems are of the order of 10 - 4 to 10 - 7 . The entire computational framework is developed using Python programming platform along with Tensorflow and Keras libraries. Our study has several advantages: first, the entire neural network architecture is easy to implement; second, there is no need for exact solution during the training phase; third, only few data points and a linear interpolation are needed for training and then the network adjusts itself to match the curvature. Furthermore, our study presents a comprehensive comparative examination to settle several issues regarding the choice of various parameters in the network. © 2023, The Author(s), under exclusive licence to Springer Nature India Private Limited.