Error estimation using neural network technique for solving ordinary differential equations

In this paper, we present a numerical method to solve ordinary differential equations (ODEs) by using neural network techniques in a deferred correction method framework. Similar to the deferred or error correction techniques, a provisional solution of the ODE is preferentially calculated by any lower-order scheme to satisfy given initial conditions, and the corresponding error is investigated by fully connected neural networks and structured to obtain sufficient magnitude of the error. Numerical examples are illustrated to demonstrate the efficiency of the proposed scheme.