Correlational Lagrangian Schrödinger Bridge: Learning Dynamics with
  Population-Level Regularization
Yuning You Ruida Zhou Yang Shen
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
Schr\"odinger 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.