Solving Ordinary Differential Equations Using Wavelet Neural Networks

In this paper, we present an artificial neural network (ANN) approach to approximate the solutions of ordinary differential equations (ODEs) with initial conditions. The wavelet neural networks (WNNs) with Gaussian wavelet and Mexican Hat activation functions are applied as universal approximators. The proposed methods convert the application of solving ODEs from a constrained optimization problem into an unconstrained optimization problem by satisfying the initial conditions exactly. Then, the momentum backpropagation (mBP) is employed to minimize the unsupervised error function, in which the only adjustable parameters are the weights from the hidden layer to the output layer. Initial value problems (IVPs) are solved to illustrate the applicability and accuracy of the proposed momentum backpropagation wavelet neural network (mBPWNN) methods. In comparison with the solutions of other existing ANN methods, numerical results showed that the mBPWNN methods yield a superior accuracy.