  In this paper, we propose a deep learning-based method, deep Euler method
(DEM) to solve ordinary differential equations. DEM significantly improves the
accuracy of the Euler method by approximating the local truncation error with
deep neural networks which could obtain a high precision solution with a large
step size. The deep neural network in DEM is mesh-free during training and
shows good generalization in unmeasured regions. DEM could be easily combined
with other schemes of numerical methods, such as Runge-Kutta method to obtain
better solutions. Furthermore, the error bound and stability of DEM is
discussed.
