  We present a method called Manifold Interpolating Optimal-Transport Flow
(MIOFlow) that learns stochastic, continuous population dynamics from static
snapshot samples taken at sporadic timepoints. MIOFlow combines dynamic models,
manifold learning, and optimal transport by training neural ordinary
differential equations (Neural ODE) to interpolate between static population
snapshots as penalized by optimal transport with manifold ground distance.
Further, we ensure that the flow follows the geometry by operating in the
latent space of an autoencoder that we call a geodesic autoencoder (GAE). In
GAE the latent space distance between points is regularized to match a novel
multiscale geodesic distance on the data manifold that we define. We show that
this method is superior to normalizing flows, Schr\"odinger bridges and other
generative models that are designed to flow from noise to data in terms of
interpolating between populations. Theoretically, we link these trajectories
with dynamic optimal transport. We evaluate our method on simulated data with
bifurcations and merges, as well as scRNA-seq data from embryoid body
differentiation, and acute myeloid leukemia treatment.
