A New Perspective To The Solution Of Ordinary Differential Equations Using Artificial Neural Networks

ABSTRACT

The research work in this dissertation presents a new perspective for obtaining solutions of initial value problems using Artificial Neural Networks (ANN). We discover that neural network based model for the solution of Ordinary Differential Equations (ODEs) provides a number of advantages over standard numerical methods. First, the neural network based solution is differentiable and is in closed analytic form. On the other hand most other techniques offer a discretized solution or a solution with limited differentiability. Second, the neural network based method for solving a differential equation provides a solution with very good generalization properties. In our novel approach, we consider first, second and third order homogeneous and nonhomogeneous linear ordinary differential equations, and first order nonlinear ODE. In the homogeneous case, we assume a solution in exponential form and compute a polynomial approximation using SPSS statistical package. From here we pick the unknown coefficients as the weights from input layer to hidden layer of the associated neural network trial solution. To get the weights from hidden layer to the output layer, we form algebraic equations incorporating the default sign of the differential equations. We then apply the Gaussian Radial Basis Function (GRBF) approximation model to achieve our objective. The weights obtained in this manner need not be adjusted. We proceed to develop a Neural Network algorithm using MathCAD 14 software, which enables us to slightly adjust the intrinsic biases. For first, second and third order non-homogeneous ODE, we use the forcing function with the GRBF model to compute the weights from hidden layer to the output layer. The operational neural network model is redefined to incorporate the nonlinearity seen in nonlinear differential equations. We compare exact results with the neural network results for our example ODE problems. We find the results to be in good agreement. Furthermore these compare favourably with the existing neural network methods of solution. The major advantage here is that our method reduces considerably the computational tasks involved in weight updating, while maintaining satisfactory accuracy.