ESFM: Reconstructing Unbalanced Dynamics via Reduction and Extended-Space Flow Matching

Biological population dynamics are often unbalanced, involving simultaneous changes in cellular states and population abundance. Existing unbalanced flow-matching methods typically rely on cone geometry-based loss formulations. While mathematically consistent, these formulations often induce biologically unfavorable state evolutions. We propose Extended Space Flow Matching (ESFM), which lifts unbalanced dynamics into conservative transport in an extended space of state and log mass. This jointly models state transitions and abundance changes as a single conservative process, enabling prescribed conditional paths to be integrated with compatible endpoint couplings and simulation-free flow matching objectives. For linear interpolation of state and log mass, ESFM admits an analytical reduction to a tractable optimal transport formulation. On synthetic and real single-cell snapshot datasets, ESFM accurately predicts both state and abundance dynamics. Extended space formulations thus offer a flexible, biologically preferred framework for unbalanced flow matching.