  Ordinary Differential Equations are generally too complex to be solved
analytically. Approximations thereof can be obtained by general purpose
numerical methods. However, even though accurate schemes have been developed,
they remain computationally expensive: In this paper, we resort to the theory
of modified equations in order to obtain ''on the fly'' cheap numerical
approximations. The recipe consists in approximating, prior to that, the
modified field associated to the modified equation by neural networks.
Elementary convergence results are then established and the efficiency of the
technique is demonstrated on experiments.
