# Correlational Lagrangian Schr"odinger Bridge: Learning Dynamics with Population-Level Regularization

arXiv: https://arxiv.org/abs/2402.10227 · HTML: https://mlsanity.com/paper/2402.10227/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 51 references that are in the map.
- First posted on arXiv 4 Feb 2024.

## Builds on (most cited first, 14 of 14)

- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Graph Attention Networks](https://mlsanity.com/paper/1710.10903/) (2017)
- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling](https://mlsanity.com/paper/2106.01357/) (2021)
- [Scalable Gradients for Stochastic Differential Equations](https://mlsanity.com/paper/2001.01328/) (2020)
- [How to train your neural ODE: the world of Jacobian and kinetic regularization](https://mlsanity.com/paper/2002.02798/) (2020)
- [Neural Stochastic Differential Equations: Deep Latent Gaussian Models in the Diffusion Limit](https://mlsanity.com/paper/1905.09883/) (2019)
- [TrajectoryNet: A Dynamic Optimal Transport Network for Modeling Cellular Dynamics](https://mlsanity.com/paper/2002.04461/) (2020)
- [Riemannian Continuous Normalizing Flows](https://mlsanity.com/paper/2006.10605/) (2020)
- [OT-Flow: Fast and Accurate Continuous Normalizing Flows via Optimal Transport](https://mlsanity.com/paper/2006.00104/) (2020)
- [Action Matching: Learning Stochastic Dynamics from Samples](https://mlsanity.com/paper/2210.06662/) (2022)
- [Scalable Unbalanced Optimal Transport using Generative Adversarial Networks](https://mlsanity.com/paper/1810.11447/) (2018)
- [Deep Momentum Multi-Marginal Schr"odinger Bridge](https://mlsanity.com/paper/2303.01751/) (2023)

## Built on by (most cited first, 2 of 2)

- [Modeling Cell Dynamics and Interactions with Unbalanced Mean Field Schr"odinger Bridge](https://mlsanity.com/paper/2505.11197/) (2025)
- [Variational Regularized Unbalanced Optimal Transport: Single Network, Least Action](https://mlsanity.com/paper/2505.11823/) (2025)

## Abstract (arXiv)

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 textitindividual-level regularizers for all particles textithomogeneously (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.