Simulation-Free Schrödinger Bridges on Riemannian Manifolds

Schrödinger bridges provide a principled framework for stochastic transport between distributions. Extending simulation-free Schrödinger bridge learning to Riemannian manifolds is challenging because neither Brownian transition kernels nor conditional bridge distributions are generally available in closed form. We introduce SSR-FM, a simulation-free framework for approximating Schrödinger bridges on Riemannian manifolds through flow and score matching. Using a small-noise heat-kernel expansion, SSR-FM constructs a geometry-aware approximation to the endpoint transport cost and obtains the corresponding entropic coupling by Sinkhorn. Conditional bridge marginals are then approximated by Gaussian fluctuations around minimizing geodesics, yielding tractable flow and score targets without simulating diffusion paths during training. Under regularity assumptions, we show that the reconstructed SDE reproduces the marginal distributions of the resulting approximate probability path. Across benchmarks with analytically specified and data-estimated geometries, SSR-FM preserves manifold structure while achieving competitive hidden-time distribution matching among the existing methods.