The dynamic Schr\"odinger bridge problem provides an appealing setting for solving optimal transport problems by learning non-linear diffusion processes using efficient iterative solvers. Recent works have demonstrated state-of-the-art results (eg. in modelling single-cell embryo RNA sequences or sampling from complex posteriors) but are limited to learning bridges with only initial and terminal constraints. Our work extends this paradigm by proposing the Iterative Smoothing Bridge (ISB). We integrate Bayesian filtering and optimal control into learning the diffusion process, enabling constrained stochastic processes governed by sparse observations at intermediate stages and terminal constraints. We assess the effectiveness of our method on synthetic and real-world data and show that the ISB generalises well to high-dimensional data, is computationally efficient, and provides accurate estimates of the marginals at intermediate and terminal times.

翻译：动态薛定谔桥问题通过利用高效的迭代求解器学习非线性扩散过程，为解决最优输运问题提供了富有吸引力的框架。近期研究已在多项任务中取得前沿成果（例如建模单细胞胚胎RNA序列或从复杂后验分布中采样），但这些方法仅局限于学习具有初始和终端约束的桥。本研究通过提出迭代平滑桥（ISB）扩展了该范式。我们将贝叶斯滤波与最优控制融入扩散过程的学习，使得受稀疏中间观测及终端约束的随机过程得以约束。我们在合成数据与真实数据上验证了方法的有效性，结果表明ISB能良好泛化至高维数据，计算效率高，并能精确估计中间时刻与终端时刻的边缘分布。