A feed-forward neural network framework for the solution of linear and nonlinear ordinary differential equations

Differential equations (DEs) play a key role in modeling the physical, chemical, and biological processes. Virtually every phenomenon occurring in nature can be described by an equation with dependent variable(s), more than one in some instances, and an independent variable(s). Only in some special circumstances, the DEs are tractable to certain well-known analytical techniques. In other instances, the solutions are approximated by prominent collocation methods such as finite elements, finite differences, finite volume techniques etc. Unfortunately, for certain Boundary Value Problems (BVPs) devising a proper approximation method is both challenging and difficult. In this work, we study a feed-forward multilayer perceptron Artificial Neural Network (ANN) framework to approximate the solution to varieties of two-point BVPs. The trail solution is the sum of two parts: the first part satisfies the boundary conditions while the second approximates the solution which is obtained by the inputs to the number of hidden layers. The mean-squared loss function is minimized by a class of gradient descent optimization methods to obtain the desired output. Our plan is to employ the proposed deep-learning architecture to a variety of DEs and report the convergence of the ANN output.