  The study of high dimensional data sets often rely on their low dimensional
projections that preserve the local geometry of the original space. While
numerous methods have been developed to summarize this space as variations of
tree-like structures, they are usually non-parametric and "static" in nature.
As data may come from systems that are dynamical such as a differentiating
cell, a static, non-parametric characterization of the space may not be the
most appropriate. Here, we developed a framework, the principal flow, that is
capable of characterizing the space in a dynamical manner. The principal flow,
defined using neural ODEs, directs motion of a particle through the space,
where the trajectory of the particle resembles the principal curve of the
dataset. We illustrate that our framework can be used to characterize shapes of
various complexities, and is flexible to incorporate summaries of relaxation
dynamics.
