Correlational Lagrangian Schrödinger Bridge: Learning Dynamics with Population-Level Regularization
This work addresses a problem in fields like cytodynamics and fluid mechanics where limited and heterogeneous data hinder accurate dynamics modeling, offering a novel approach with potential applications in biological systems.
The paper tackles the challenge of modeling system dynamics from cross-sectional samples with heterogeneous particle behaviors, introducing the Correlational Lagrangian Schrödinger Bridge (CLSB) framework that uses population-level regularization to improve generalizability, demonstrating its superiority in single-cell sequencing data for tasks like simulating cell development and predicting drug responses.
Accurate modeling of system dynamics holds intriguing potential in broad scientific fields including cytodynamics and fluid mechanics. This task often presents significant challenges when (i) observations are limited to cross-sectional samples (where individual trajectories are inaccessible for learning), and moreover, (ii) the behaviors of individual particles are heterogeneous (especially in biological systems due to biodiversity). To address them, we introduce a novel framework dubbed correlational Lagrangian Schrödinger bridge (CLSB), aiming to seek for the evolution "bridging" among cross-sectional observations, while regularized for the minimal population "cost". In contrast to prior methods relying on \textit{individual}-level regularizers for all particles \textit{homogeneously} (e.g. restraining individual motions), CLSB operates at the population level admitting the heterogeneity nature, resulting in a more generalizable modeling in practice. To this end, our contributions include (1) a new class of population regularizers capturing the temporal variations in multivariate relations, with the tractable formulation derived, (2) three domain-informed instantiations based on genetic co-expression stability, and (3) an integration of population regularizers into data-driven generative models as constrained optimization, and a numerical solution, with further extension to conditional generative models. Empirically, we demonstrate the superiority of CLSB in single-cell sequencing data analyses such as simulating cell development over time and predicting cellular responses to drugs of varied doses.