Delay Flow Matching

Flow matching (FM) based on Ordinary Differential Equations (ODEs) has achieved significant success in generative tasks. However, it faces several inherent limitations, including an inability to model trajectory intersections, capture delay dynamics, and handle transfer between heterogeneous distributions. These limitations often result in a significant mismatch between the modeled transfer process and real-world phenomena, particularly when key coupling or inherent structural information between distributions must be preserved. To address these issues, we propose Delay Flow Matching (DFM), a new FM framework based on Delay Differential Equations (DDEs). Theoretically, we show that DFM possesses universal approximation capability for continuous transfer maps. By incorporating delay terms into the vector field, DFM enables trajectory intersections and better captures delay dynamics. Moreover, by designing appropriate initial functions, DFM ensures accurate transfer between heterogeneous distributions. Consequently, our framework preserves essential coupling relationships and achieves more flexible distribution transfer strategies. We validate DFM's effectiveness across synthetic datasets, single-cell data, and image-generation tasks.