Neural Lagrangian Schrödinger Bridge

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 other constraints. Recently, modeling population dynamics by using continuous normalizing flows (CNFs) and dynamic optimal transport has been proposed to infer the expected trajectory of samples from a fixed-point observed population. While the sample behavior in CNF 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 is such that the corresponding action has the smallest possible value, known as the principle of least action. To satisfy these requirements of the sample trajectories, we formulate the Lagrangian Schr\"odinger bridge (LSB) problem and propose to solve it approximately using neural SDE with regularization. We also develop a model architecture that enables faster computation. Our experiments show that our solution to the LSB problem can approximate the dynamics at the population level and that using the prior knowledge introduced by the Lagrangian enables us to estimate the trajectories of individual samples with stochastic behavior.