Title: Topological Flow Matching
Primary area: generative models
Keywords: Flow Matching, Generative Models, Topological Deep Learning, Geometric Deep Learning, Graphs, Simplicial Complexes, Schrödinger Bridge, Optimal Transport

Abstract: 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 \emph{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 plug-and-play replacement for standard flow matching. We demonstrate its effectiveness on diverse structured datasets, including brain fMRIs, ocean currents, seismic events, and traffic flows.