Abstract
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.
Highlights
Differential equations play vital role in various fields of engineering and science
We propose a method for solving ordinary differential equations using feed forward neural network as a basic approximation element and error back propagation algorithm [24, 25] by fixing hidden nodes as per the required accuracy
This paper presents a new approach to solve ordinary differential equations by using regression based artificial neural network model
Summary
Differential equations play vital role in various fields of engineering and science. The exact solution of differential equations may not be always possible [1]. 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. Numerical solution of elliptic partial differential equation using radial basis function neural networks has been presented by Jianyu et al [15]. Shirvany et al [16] proposed multilayer perceptron and radial basis function (RBF) neural networks with a new unsupervised training method for numerical solution of partial differential equations. We propose a method for solving ordinary differential equations using feed forward neural network as a basic approximation element and error back propagation algorithm [24, 25] by fixing hidden nodes as per the required accuracy.
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