An Application of Neural Networks to Solve Ordinary Di fferential Equations

In this paper, we introduce a hybrid approach based on modifi ed neural networks and optimization teqnique to solve ordinary di fferential equation. Using modifi ed neural network makes that training points should be selected over the open interval (a; b) without training the network in the range of fi rst and end points. Therefore, the calculating volume involving computational error is reduced. In fact, the training points depending on the distance [a; b] selected for training neural networks are converted to similar points in the open interval (a; b) by using a new approach, then the network is trained in these similar areas. In comparison with existing similar neural networks proposed model provides solutions with high accuracy. Numerical examples with simulation results illustrate the e ectiveness of the proposed model.