Understanding the continuous evolution of populations from discrete temporal snapshots is a critical research challenge, particularly in fields like developmental biology and systems medicine where longitudinal tracking of individual entities is often impossible. Such trajectory inference is vital for unraveling the mechanisms of dynamic processes. While Schr\"odinger Bridge (SB) offer a potent framework, their traditional application to pairwise time points can be insufficient for systems defined by multiple intermediate snapshots. This paper introduces Multi-Marginal Schr\"odinger Bridge Matching (MSBM), a novel algorithm specifically designed for the multi-marginal SB problem. MSBM extends iterative Markovian fitting (IMF) to effectively handle multiple marginal constraints. This technique ensures robust enforcement of all intermediate marginals while preserving the continuity of the learned global dynamics across the entire trajectory. Empirical validations on synthetic data and real-world single-cell RNA sequencing datasets demonstrate the competitive or superior performance of MSBM in capturing complex trajectories and respecting intermediate distributions, all with notable computational efficiency.

翻译：理解从离散时间快照中推断种群的连续演化是一个关键的研究挑战，尤其在发育生物学和系统医学等领域，其中对个体实体进行纵向追踪通常是不可能的。这种轨迹推断对于揭示动态过程的机制至关重要。虽然薛定谔桥（SB）提供了一个强大的框架，但其传统上应用于成对时间点的方法，对于由多个中间快照定义的系统可能不够充分。本文介绍了多边际薛定谔桥匹配（MSBM），这是一种专为多边际SB问题设计的新算法。MSBM将迭代马尔可夫拟合（IMF）扩展到能有效处理多个边际约束。该技术确保了对所有中间边际的鲁棒性约束，同时保持了所学得的全局动力学在整个轨迹上的连续性。在合成数据和真实世界单细胞RNA测序数据集上的实证验证表明，MSBM在捕捉复杂轨迹和尊重中间分布方面具有竞争力或更优的性能，且均表现出显著的计算效率。