Learning the Simplest Neural ODE

Since the advent of the ``Neural Ordinary Differential Equation (Neural ODE)'' paper, learning ODEs with deep learning has been applied to system identification, time-series forecasting, and related areas. Exploiting the diffeomorphic nature of ODE solution maps, neural ODEs has also enabled their use in generative modeling. Despite the rich potential to incorporate various kinds of physical information, training Neural ODEs remains challenging in practice. This study demonstrates, through the simplest one-dimensional linear model, why training Neural ODEs is difficult. We then propose a new stabilization method and provide an analytical convergence analysis. The insights and techniques presented here serve as a concise tutorial for researchers beginning work on Neural ODEs.