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A key appeal of the recently proposed Neural Ordinary Differential Equation(ODE) framework is that it seems to provide a continuous-time extension ofdiscrete residual neural networks. As we show herein, though, trained NeuralODE models actually depend on the specific numerical method used duringtraining. 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 smallernumerical error without loss of performance. We observe that if training relieson a solver with overly coarse discretization, then testing with another solverof equal or smaller numerical error results in a sharp drop in accuracy. Insuch cases, the combination of vector field and numerical method cannot beinterpreted as a flow generated from an ODE, which arguably poses a fatalbreakdown of the Neural ODE concept. We observe, however, that there exists acritical step size beyond which the training yields a valid ODE vector field.We propose a method that monitors the behavior of the ODE solver duringtraining to adapt its step size, aiming to ensure a valid ODE withoutunnecessarily increasing computational cost. We verify this adaptationalgorithm on a common bench mark dataset as well as a synthetic dataset.

ResNet After All? Neural ODEs and Their Numerical Solution