Riemannian Neural Hamiltonian Flows: Geodesic Symplectic Transport and Interpretability

Hamiltonian normalizing flows are attractive generative models because their phase-space maps are invertible and volume preserving, but most neural constructions are formulated in Euclidean space. We introduce Riemannian Neural Hamiltonian Flows, which combine the fixed kinetic energy of a Riemannian manifold, a learned scalar potential, and an explicit geodesic leapfrog integrator. Our analysis explains how the learned Hamiltonian can be made interpretable. Every normalizable potential defines an implicit profile, and the position marginal initially accelerates along the relative score between that profile and the base. The matched potential is the interpretable specialization for which the implicit profile is the target. In the isotropic Gaussian case, the mechanism corresponds to a phase-space rotation. A local harmonic analysis extends this result around each mode of a general target on a manifold. The gap between the learned and the matched potential is the sum of a residual memory of the base and a bias of the model, and the two potentials agree when the position base has been transferred to the momentum. This can be achieved when the former is broader than the target. Numerical experiments on Euclidean, hyperbolic, and spherical spaces show competitive sample quality and numerical cost against a Riemannian continuous normalizing flow, and confirm the interpretability of the learned potential.