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.

本研究提出了使用连续标准化流（CNFs）和动态最优输运方法对种群动态进行建模，以在固定时间点观测到的种群中推断采样轨迹的方法。通过使用正则化神经 SDE 对对流扩散过程进行建模并开发了一个模型架构，可以有效地近似种群级别的动态，即使在高维数据中也可以实现，并且使用拉格朗日方法引入的先验知识可以估计具有随机行为的采样级别动态。