  It is a crucial challenge to reconstruct population dynamics using unlabeled
samples from distributions at coarse time intervals. Recent approaches such as
flow-based models or Schr\"odinger Bridge (SB) models have demonstrated
appealing performance, yet the inferred sample trajectories either fail to
account for the underlying stochasticity or are $\underline{D}$eep
$\underline{M}$omentum Multi-Marginal $\underline{S}$chr\"odinger
$\underline{B}$ridge(DMSB), a novel computational framework that learns the
smooth measure-valued spline for stochastic systems that satisfy position
marginal constraints across time. By tailoring the celebrated Bregman Iteration
and extending the Iteration Proportional Fitting to phase space, we manage to
handle high-dimensional multi-marginal trajectory inference tasks efficiently.
Our algorithm outperforms baselines significantly, as evidenced by experiments
for synthetic datasets and a real-world single-cell RNA sequence dataset.
Additionally, the proposed approach can reasonably reconstruct the evolution of
velocity distribution, from position snapshots only, when there is a ground
truth velocity that is nevertheless inaccessible.
