  Learning dynamics governed by differential equations is crucial for
predicting and controlling the systems in science and engineering. Neural
Ordinary Differential Equation (NODE), a deep learning model integrated with
differential equations, is popular in learning dynamics recently due to its
robustness to irregular samples and its flexibility to high-dimensional input.
However, the training of NODE is sensitive to the precision of the numerical
solver, which makes the convergence of NODE unstable, especially for
ill-conditioned dynamical systems. In this paper, to reduce the reliance on the
numerical solver, we propose to enhance the supervised signal in the training
of NODE. Specifically, we pre-train a neural differential operator (NDO) to
output an estimation of the derivatives to serve as an additional supervised
signal. The NDO is pre-trained on a class of basis functions and learns the
mapping between the trajectory samples of these functions to their derivatives.
To leverage both the trajectory signal and the estimated derivatives from NDO,
we propose an algorithm called NDO-NODE, in which the loss function contains
two terms: the fitness on the true trajectory samples and the fitness on the
estimated derivatives that are outputted by the pre-trained NDO. Experiments on
various kinds of dynamics show that our proposed NDO-NODE can consistently
improve the forecasting accuracy with one pre-trained NDO. Especially for the
stiff ODEs, we observe that NDO-NODE can capture the transitions in the
dynamics more accurately compared with other regularization methods.
