Riemannian Neural Hamiltonian Flows: Geodesic Symplectic Transport and Interpretability

Riemannian manifolds are powerful models for problems with non-Euclidean structure. Recent work in geometric machine learning has focused on training deep neural networks to learn Riemannian metrics from high-dimensional data. However, many geometric algorithms require repeated evaluations of the exponential and logarithmic maps, whose high computational cost restricts the utility of learned Riemannian manifolds in practice. We propose to address this cost by training a conditional invertible neural network, called a flow chart, to approximate normal coordinates around points on a given Riemannian manifold. With this approach, the exponential and logarithmic maps can be approximated by fast Euclidean operations in the model's learned coordinate space. Our main contribution is an efficient loss function for training flow charts that is nonnegative and, under suitable domain assumptions, is zero precisely when the chart is normal. Across sixteen synthetic and data-derived geometries, flow charts are competitive with a specialized neural geodesic interpolator for logarithmic map estimation, while also being able to estimate the exponential map. When pretrained, flow charts provide a median speedup of 60-70x over numerical baselines at comparable logarithmic map estimation error, and their learned coordinates improve the accuracy of a numerical geodesic solver.

Scalable approaches to learning neural dynamical systems have placed diffusion, flow matching, and few-step generators in flow maps at the forefront of generative modeling. Such state-of-the-art models have limited access to explicit model densities despite evolving from classical generative modeling approaches that leveraged tractable likelihoods, e.g., Normalizing Flows and Autoregressive models. In this paper, we combine these threads of few-step generation and tractable sample likelihoods and introduce Normalizing Flow Maps (NFM), a class of exactly invertible pushforward maps that generalizes normalizing flows and flow maps. More precisely, NFMs learn the integrator of an underlying dynamical system, but crucially, they provide efficient exact likelihood evaluation by construction. To enable scalable training of NFMs, we derive two complementary axes of consistency inherent to NFMs that yield (self) distillation objectives: (1) pointwise matching, which unlocks $\ell_2$-based flow-map style regression losses, and (2) distributional matching, which matches pushforward distributions to the marginals of a prescribed forward process. On conditional image generation on CIFAR-10 and ImageNet-64, NFMs achieve state-of-the-art sample quality among parameter-matched normalizing flows, while supporting likelihood-based RLHF. For equilibrium sampling in Boltzmann Generators, NFMs outperform prior state-of-the-art normalizing flows on both single-system and transferable peptide benchmarks.

Many scientific datasets, such as molecular conformational ensembles or single-cell tissue measurements, are naturally modeled as meta-distributions: distributions over probability measures on non-Euclidean domains. Existing generative methods largely assume Euclidean geometry and fail to capture this structure. We introduce Riemannian Wasserstein Entropic Flow Matching (RWEFM), a generative framework on the Wasserstein space $\mathcal{P}_2(\mathcal{M})$ of a Riemannian manifold $(\mathcal{M},g)$. RWEFM is trained by regressing a neural vector field onto Riemannian optimal transport velocities, using McCann displacement interpolations as conditional paths. We confirm theoretically that this construction leads to a valid flow matching approach on $\mathcal{P}_2(\mathcal{M})$ and introduce the Riemannian Entropic Map, a GPU-efficient approximation of the optimal transport map on manifolds. Our experiments show that by respecting the intrinsic geometry of the data, RWEFM can generate whole single-cell samples in hyperspherical latent spaces and protein conformational ensembles on the torus. As RWEFM requires only a geodesic distance and a projection operator, it is not restricted to manifolds with closed-form geometry, which we demonstrate by generating distributions on a general triangulated mesh.