Flow matching is a powerful generative modeling framework, valued for its simplicity and strong empirical performance. However, its standard formulation treats signals on structured spaces, such as fMRI data on brain graphs, as points in Euclidean space, overlooking the rich topological features of their domains. To address this, we introduce topological flow matching, a topology-aware generalization of flow matching. We interpret flow matching as a framework for solving a degenerate Schrödinger bridge problem and inject topological information by augmenting the reference process with a Laplacian-derived drift. This principled modification captures the structure of the underlying domain while preserving the desirable properties of flow matching: a stable, simulation-free objective and deterministic sample paths. As a result, our framework serves as a drop-in replacement for standard flow matching. We demonstrate its effectiveness on diverse structured datasets, including brain fMRIs, ocean currents, seismic events, and traffic flows.

翻译：流匹配是一种强大的生成建模框架，因其简洁性和优异的经验性能而备受青睐。然而，其标准表述将结构化空间上的信号（如脑图上的fMRI数据）视为欧几里得空间中的点，忽略了其域丰富的拓扑特征。为解决此问题，我们提出拓扑流匹配，一种具有拓扑感知的流匹配泛化方法。我们将流匹配解释为求解退化薛定谔桥问题的框架，并通过向参考过程添加拉普拉斯导出的漂移项注入拓扑信息。这一原理性修改捕捉了底层域的结构，同时保留了流匹配的理想性质：稳定的无模拟目标函数和确定性样本路径。因此，我们的框架可作为标准流匹配的即插即用替代方案。我们在多种结构化数据集上验证了其有效性，包括脑fMRI、洋流、地震事件和交通流。