We consider the Schr\"odinger bridge problem which, given ensemble measurements of the initial and final configurations of a stochastic dynamical system and some prior knowledge on the dynamics, aims to reconstruct the "most likely" evolution of the system compatible with the data. Most existing literature assume Brownian reference dynamics and are implicitly limited to potential-driven dynamics. We depart from this regime and consider reference processes described by a multivariate Ornstein-Uhlenbeck process with generic drift matrix $\mathbf{A} \in \mathbb{R}^{d \times d}$. When $\mathbf{A}$ is asymmetric, this corresponds to a non-equilibrium system with non-conservative forces at play: this is important for applications to biological systems, which are naturally exist out-of-equilibrium. In the case of Gaussian marginals, we derive explicit expressions that characterise the solution of both the static and dynamic Schr\"odinger bridge. For general marginals, we propose mvOU-OTFM, a simulation-free algorithm based on flow and score matching for learning the Schr\"odinger bridge. In application to a range of problems based on synthetic and real single cell data, we demonstrate that mvOU-OTFM achieves higher accuracy compared to competing methods, whilst being significantly faster to train.

翻译：我们考虑薛定谔桥问题，该问题在给定随机动力系统初始与最终配置的系综测量值以及关于动力学的某些先验知识的前提下，旨在重建与数据兼容的系统“最可能”演化路径。现有文献大多假设布朗参考动力学，并隐含地局限于势驱动的动力学。我们脱离这一范畴，考虑由具有一般漂移矩阵 $\mathbf{A} \in \mathbb{R}^{d \times d}$ 的多元奥恩斯坦-乌伦贝克过程描述的参考过程。当 $\mathbf{A}$ 非对称时，这对应于存在非保守力作用的非平衡系统：这对于生物系统应用至关重要，因为生物系统天然存在于非平衡状态。在高斯边缘分布情形下，我们推导出刻画静态与动态薛定谔桥解的显式表达式。对于一般边缘分布，我们提出 mvOU-OTFM——一种基于流匹配与分数匹配的免模拟算法，用于学习薛定谔桥。在基于合成与真实单细胞数据的一系列问题应用中，我们证明 mvOU-OTFM 相较于竞争方法实现了更高的精度，同时训练速度显著更快。