  A key appeal of the recently proposed Neural Ordinary Differential Equation
(ODE) framework is that it seems to provide a continuous-time extension of
discrete residual neural networks. As we show herein, though, trained Neural
ODE models actually depend on the specific numerical method used during
training. If the trained model is supposed to be a flow generated from an ODE,
it should be possible to choose another numerical solver with equal or smaller
numerical error without loss of performance. We observe that if training relies
on a solver with overly coarse discretization, then testing with another solver
of equal or smaller numerical error results in a sharp drop in accuracy. In
such cases, the combination of vector field and numerical method cannot be
interpreted as a flow generated from an ODE, which arguably poses a fatal
breakdown of the Neural ODE concept. We observe, however, that there exists a
critical step size beyond which the training yields a valid ODE vector field.
We propose a method that monitors the behavior of the ODE solver during
training to adapt its step size, aiming to ensure a valid ODE without
unnecessarily increasing computational cost. We verify this adaptation
algorithm on a common bench mark dataset as well as a synthetic dataset.
