We consider the Schrödinger 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 modelling systems driven by the gradient of a potential energy. 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 in which non-gradient forces are at play: this is important for applications to biological systems, which naturally exist out-of-equilibrium. In the case of Gaussian marginals, we derive explicit expressions that characterise exactly the solution of both the static and dynamic Schrödinger bridge. For general marginals, we propose mvOU-OTFM, a simulation-free algorithm based on flow and score matching for learning an approximation to the Schrödinger 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：给定随机动力系统初始与最终构型的系综测量数据，并结合对动力学的先验知识，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运动参考动力学，这隐含地局限于建模由势能梯度驱动的系统。我们脱离此范式，考虑以具有一般漂移矩阵$\mathbf{A} \in \mathbb{R}^{d \times d}$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，这对应于存在非梯度力作用的非平衡系统：这对生物系统应用至关重要，因为生物系统天然处于非平衡态。对于高斯边缘分布情形，我们推导出显式表达式，精确刻画了静态与动态薛定谔桥的解。针对一般边缘分布，我们提出mvOU-OTFM——一种基于流匹配与分数匹配的免模拟算法，用于学习薛定谔桥的近似解。在基于合成与真实单细胞数据的系列问题应用中，我们证明mvOU-OTFM相比竞争方法具有更高精度，同时训练速度显著提升。