%0 Journal Article
%T Comparison of Artificial Neural Network Architecture in Solving Ordinary Differential Equations
%A Susmita Mall
%A S. Chakraverty
%J Advances in Artificial Neural Systems
%D 2013
%I Hindawi Publishing Corporation
%R 10.1155/2013/181895
%X This paper investigates the solution of Ordinary Differential Equations (ODEs) with initial conditions using Regression Based Algorithm (RBA) and compares the results with arbitrary- and regression-based initial weights for different numbers of nodes in hidden layer. Here, we have used feed forward neural network and error back propagation method for minimizing the error function and for the modification of the parameters (weights and biases). Initial weights are taken as combination of random as well as by the proposed regression based model. We present the method for solving a variety of problems and the results are compared. Here, the number of nodes in hidden layer has been fixed according to the degree of polynomial in the regression fitting. For this, the input and output data are fitted first with various degree polynomials using regression analysis and the coefficients involved are taken as initial weights to start with the neural training. Fixing of the hidden nodes depends upon the degree of the polynomial. For the example problems, the analytical results have been compared with neural results with arbitrary and regression based weights with four, five, and six nodes in hidden layer and are found to be in good agreement. 1. Introduction Differential equations play vital role in various fields of engineering and science. The exact solution of differential equations may not be always possible [1]. So various types of well known numerical methods such as Euler, Runge-kutta, Predictor-Corrector, finite element, and finite difference methods, are used for solving these equations. Although these numerical methods provide good approximations to the solution, but these may be challenging for higher dimension problems. In recent years, many researchers tried to find new methods for solving differential equations. As such here Artificial Neural Network (ANN) based models are used to solve ordinary differential equations with initial conditions. Lee and Kang [2] first introduced a method to solve first order differential equation using Hopfield neural network models. Then, another approach by Meade and Fernandez [3, 4] has been proposed for both linear and nonlinear differential equations using -splines and feed forward neural network. Artificial neural networks based on Broyden-Fletcher-Goldfarb-Shanno (BFGS) optimization technique for solving ordinary and partial differential equations have been excellently presented by Lagaris et al. [5]. Also Lagaris et al. [6] investigated neural network methods for boundary value problems with irregular boundaries.
%U http://www.hindawi.com/journals/aans/2013/181895/