Neural Lagrangian Schr\"odinger Bridge: Diffusion Modeling for Population Dynamics

TL;DR

This paper introduces a novel neural Lagrangian Schr"odinger bridge approach using neural SDEs to model population dynamics, capturing stochastic sample trajectories more accurately than deterministic models, especially in high-dimensional settings.

Contribution

It formulates the Lagrangian Schr"odinger bridge problem and develops a neural SDE-based method with architecture improvements for efficient population trajectory inference.

Findings

Efficiently approximates population dynamics in high-dimensional data.

Captures stochastic sample-level trajectories using prior knowledge.

Outperforms existing methods in modeling population movement.

Abstract

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.