  It is increasingly common to encounter data from dynamic processes captured
by static cross-sectional measurements over time, particularly in biomedical
settings. Recent attempts to model individual trajectories from this data use
optimal transport to create pairwise matchings between time points. However,
these methods cannot model continuous dynamics and non-linear paths that
entities can take in these systems. To address this issue, we establish a link
between continuous normalizing flows and dynamic optimal transport, that allows
us to model the expected paths of points over time. Continuous normalizing
flows are generally under constrained, as they are allowed to take an arbitrary
path from the source to the target distribution. We present TrajectoryNet,
which controls the continuous paths taken between distributions to produce
dynamic optimal transport. We show how this is particularly applicable for
studying cellular dynamics in data from single-cell RNA sequencing (scRNA-seq)
technologies, and that TrajectoryNet improves upon recently proposed static
optimal transport-based models that can be used for interpolating cellular
distributions.
