Flow matching (FM) has emerged as a powerful framework for learning dynamic transport maps between two empirical distributions. However, less explored is the setting with intermediate observed marginals that can help constrain the flows between the endpoints. This "multimarginal" regime is central to modeling temporal evolution in dynamical systems in many scientific domains that can sample sequential distributions. We tackle this problem with a novel approach that leverages the connection between FM and dynamic optimal transport (OT), softly steering the flow towards the intermediate marginals through potential terms in the dynamic OT action. By extending the conditional FM learning target to incorporate these potentials, we derive an efficient, simulation-free algorithm for multimarginal FM that offers considerable flexibility in the spatiotemporal dynamics of the learned flows. We demonstrate state-of-the-art performance and training efficiency of OT-potential FM (OTP-FM) on diverse single-cell RNA sequencing, oceanographic, and meteorological datasets. Our code is available at https://github.com/Bexorg-Inc/OTP-FM.

翻译：流匹配（Flow Matching, FM）已成为学习两个经验分布间动态输运映射的强大框架。然而，针对可帮助约束端点间流动的中间观测边际情形的研究仍较少。这一“多边际”范式对于许多科学领域中可采样时序分布的动态系统时间演化建模至关重要。我们提出一种新方法解决该问题，通过利用FM与动态最优输运（Optimal Transport, OT）之间的联系，在动态OT作用量中引入势项以软性引导流趋向中间边际。通过扩展条件FM学习目标以纳入这些势，我们推导出一种高效、无模拟的多边际FM算法，为所学习流的时空动力学提供了显著灵活性。我们在多种单细胞RNA测序、海洋学及气象数据集上展示了OT势流匹配（OTP-FM）的最优性能与训练效率。我们的代码详见https://github.com/Bexorg-Inc/OTP-FM。