Artificial neural network based numerical solution of ordinary differential equations

Abstract

In this investigation we introduced the method for solving Ordinary Differential Equations (ODEs) using artificial neural network. The feed forward neural network of the unsupervised type has been used to get the approximation of the given ODEs up to the required accuracy without direct use of the optimization techniques. The problem is formulated in such a manner that it satisfies the initial/boundary conditions by its construction. The trail solution of the ODE is the sum of two terms. The first term satisfies the initial or boundary conditions, while the second one is the feed forward neural output produced by n number of inputs and h number of hidden sigmoid units. The error gradient has been reduced by applying general learning method to get the desired output.The results have been verified for different problems and the convergence of Artificial Neural Network (ANN) output has been checked for arbitrary points. It may be noted that the interpolation is also possible through this process. The advantage of neuron processor is that the output can be produced to any arbitrary accuracy, while the targets or exact results are unknown or hard to find out.