Flow Matching on General Manifolds via Pulling Back Geodesic Convex Latent Manifolds

Generative models that construct conditional paths transforming a source distribution into a target distribution have seen substantial success in modeling scientific data. However, traditional methods typically assume Euclidean geometry and fail to adhere to data manifolds, falling short on tasks such as trajectory inference. Recent flow matching methods on general manifolds often relies on approximating geodesics without closed forms, suffers from limited theoretical and implementation grounding, or trades generation quality to learn velocity fields. As a result, we propose GCL-FM, a novel simulation-free framework that enables interpolation on unknown data manifolds by explicitly enforcing a $\textbf{G}$eodesic $\textbf{C}$onvex $\textbf{L}$atent subamnifold for $\textbf{F}$low \textbf{M}atching. Our theoretical analysis shows that GCL-FM possesses desirable guarantees for modeling manifold transport dynamics, including robustness to data noise. We demonstrate that GCL-FM achieves superior performance on trajectory inference for multiple biological and computational fluid dynamics datasets, while requiring significantly less computational cost compared to prior methods, providing a principled and efficient route to high-fidelity manifold-aware conditional generation for real-world transport dynamics in high dimensions.