Horizontal Flow Matching for Single-Cell Spatiotemporal Dynamics

Learning the spatiotemporal dynamics of cells from spatial transcriptomics requires connecting snapshots of different cells acquired in different coordinate frames. A rotation or translation of a section can appear as cell movement and distort the inferred correspondence. We introduce horizontal flow matching, which learns changes in tissue organization, molecular state, and cell abundance up to shared rigid motion. The dynamics follow a continuity-reaction equation whose action assigns no cost to shared rigid motion, and the resulting horizontal projector removes this component from spatial prediction errors during learning and from predicted velocities during generation. An equivalent endpoint problem jointly estimates correspondence and relative pose in Wasserstein-Fisher-Rao geometry and yields analytic paths whose velocity and growth serve as regression targets. Regressing these targets recovers the horizontal dynamics under ideal population conditions. Our model, HORIZON, uses an equivariant transformer to predict velocity and growth from the current tissue, so that a tissue can be generated continuously from a single initial snapshot. On a simulated tissue observed in perturbed coordinate frames, alignment improves reconstruction and projection reduces rigid drift. On four biological time courses, HORIZON attains lower reconstruction error than state-of-the-art methods across spatial, molecular, and abundance metrics, with consistent gains when interpolating held-out snapshots.