  We propose Riemannian Flow Matching (RFM), a simple yet powerful framework
for training continuous normalizing flows on manifolds. Existing methods for
generative modeling on manifolds either require expensive simulation, are
inherently unable to scale to high dimensions, or use approximations for
limiting quantities that result in biased training objectives. Riemannian Flow
Matching bypasses these limitations and offers several advantages over previous
approaches: it is simulation-free on simple geometries, does not require
divergence computation, and computes its target vector field in closed-form.
The key ingredient behind RFM is the construction of a relatively simple
premetric for defining target vector fields, which encompasses the existing
Euclidean case. To extend to general geometries, we rely on the use of spectral
decompositions to efficiently compute premetrics on the fly. Our method
achieves state-of-the-art performance on many real-world non-Euclidean
datasets, and we demonstrate tractable training on general geometries,
including triangular meshes with highly non-trivial curvature and boundaries.
