Matching objectives underpin the success of modern generative models and rely on constructing conditional paths that transform a source distribution into a target distribution. Despite being a fundamental building block, conditional paths have been designed principally under the assumption of Euclidean geometry, resulting in straight interpolations. However, this can be particularly restrictive for tasks such as trajectory inference, where straight paths might lie outside the data manifold, thus failing to capture the underlying dynamics giving rise to the observed marginals. In this paper, we propose Metric Flow Matching (MFM), a novel simulation-free framework for conditional flow matching where interpolants are approximate geodesics learned by minimizing the kinetic energy of a data-induced Riemannian metric. This way, the generative model matches vector fields on the data manifold, which corresponds to lower uncertainty and more meaningful interpolations. We prescribe general metrics to instantiate MFM, independent of the task, and test it on a suite of challenging problems including LiDAR navigation, unpaired image translation, and modeling cellular dynamics. We observe that MFM outperforms the Euclidean baselines, particularly achieving SOTA on single-cell trajectory prediction.

生成模型在处理轨迹推断等任务时，传统的基于欧几里得几何的条件路径设计往往过于局限，未能很好地捕捉到底层动力学，而Metric Flow Matching（MFM）作为一种基于非模拟的条件流匹配框架，通过最小化数据诱导的黎曼度量的动能来学习近似测地线插值，实现了在数据流形上匹配矢量场，从而降低了不确定性并获得了更有意义的插值结果。该研究不局限于特定任务，并通过LiDAR导航、非配对图像翻译和建模细胞动力学等一系列具有挑战性的问题验证了MFM方法的优越性，特别在单细胞轨迹预测方面达到了SOTA水平。

数据流动匹配：实现数据流形上的平滑插值