  Population dynamics is the study of temporal and spatial variation in the
size of populations of organisms and is a major part of population ecology. One
of the main difficulties in analyzing population dynamics is that we can only
obtain observation data with coarse time intervals from fixed-point
observations due to experimental costs or measurement constraints. Recently,
modeling population dynamics by using continuous normalizing flows (CNFs) and
dynamic optimal transport has been proposed to infer the sample trajectories
from a fixed-point observed population. While the sample behavior in CNFs is
deterministic, the actual sample in biological systems moves in an essentially
random yet directional manner. Moreover, when a sample moves from point A to
point B in dynamical systems, its trajectory typically follows the principle of
least action in which the corresponding action has the smallest possible value.
To satisfy these requirements of the sample trajectories, we formulate the
Lagrangian Schr\"odinger bridge (LSB) problem and propose to solve it
approximately by modeling the advection-diffusion process with regularized
neural SDE. We also develop a model architecture that enables faster
computation of the loss function. Experimental results show that the proposed
method can efficiently approximate the population-level dynamics even for
high-dimensional data and that using the prior knowledge introduced by the
Lagrangian enables us to estimate the sample-level dynamics with stochastic
behavior.
