We compare the discretize-optimize (Disc-Opt) and optimize-discretize (Opt-Disc) approaches for time-series regression and continuous normalizing flows (CNFs) using neural ODEs. Neural ODEs are ordinary differential equations (ODEs) with neural network components. Training a neural ODE is an optimal control problem where the weights are the controls and the hidden features are the states. Every training iteration involves solving an ODE forward and another backward in time, which can require large amounts of computation, time, and memory. Comparing the Opt-Disc and Disc-Opt approaches in image classification tasks, Gholami et al. (2019) suggest that Disc-Opt is preferable due to the guaranteed accuracy of gradients. In this paper, we extend the comparison to neural ODEs for time-series regression and CNFs. Unlike in classification, meaningful models in these tasks must also satisfy additional requirements beyond accurate final-time output, e.g., the invertibility of the CNF. Through our numerical experiments, we demonstrate that with careful numerical treatment, Disc-Opt methods can achieve similar performance as Opt-Disc at inference with drastically reduced training costs. Disc-Opt reduced costs in six out of seven separate problems with training time reduction ranging from 39% to 97%, and in one case, Disc-Opt reduced training from nine days to less than one day.
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