Optimization of ordinary differential equations (ODE) solutions using modified recurrent neural networks

This paper introduces a hybrid approach for solving ordinary differential equations (ODE) using modified recurrent neural networks (mRNNs).The approach combines mRNNs with novel optimization techniques.Crucially, when training an mRNN, training data points should be selected from the open interval (a, b) to avoid training the network with the boundary points.This approach reduces computational errors by avoiding boundary region training.Furthermore, we propose a transformation that maps training points from a potentially broader interval [a, b] into corresponding points within the open interval (a, b), before training.This allows the network to be trained on points that are similar in the open interval, which leads to improved accuracy.The proposed model demonstrates higher accuracy compared to existing mRNN models.A numerical example and corresponding simulations demonstrate the mathematical effectiveness of this approach.