We study the geometry of conditional optimal transport (COT) and prove a dynamical formulation which generalizes the Benamou-Brenier Theorem. With these tools, we propose a simulation-free flow-based method for conditional generative modeling. Our method couples an arbitrary source distribution to a specified target distribution through a triangular COT plan. We build on the framework of flow matching to train a conditional generative model by approximating the geodesic path of measures induced by this COT plan. Our theory and methods are applicable in the infinite-dimensional setting, making them well suited for inverse problems. Empirically, we demonstrate our proposed method on two image-to-image translation tasks and an infinite-dimensional Bayesian inverse problem.

我们研究了条件最优传输的几何性质并证明了一种推广了Benamou-Brenier定理的动力学表述。利用这些工具，我们提出了一种无需模拟的基于流的条件生成建模方法。我们的方法通过三角形条件最优传输方案将任意源分布与指定目标分布相耦合。我们借鉴流匹配的框架，通过近似由该条件最优传输方案诱导的测度测地线路径来训练条件生成模型。我们的理论和方法可在无限维设置中应用，非常适用于逆问题。在实证方面，我们将我们提出的方法应用于两个图像转换任务和一个无限维贝叶斯逆问题。

基于模拟自由流动的动态条件最优传输