发表机构
CEA, DAM, DIF（法国原子能委员会）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出黎曼神经哈密顿流，结合流形固定动能、学习势与测地积分器，实现可解释的生成建模，并在多种空间验证其性能与可解释性。
AI 中文摘要
哈密顿归一化流因其相空间映射可逆且保体积而成为有吸引力的生成模型，但大多数神经构造是在欧几里得空间中制定的。我们引入了黎曼神经哈密顿流，它结合了黎曼流形的固定动能、一个学习的标量势和一个显式的测地跳跃式积分器。我们的分析解释了如何使学习到的哈密顿量变得可解释。每个可归一化的势都定义了一个隐式分布，位置边际最初沿着该分布与基分布之间的相对得分加速。匹配势是可解释的特化，其中隐式分布即目标分布。在各向同性高斯情形下，该机制对应于相空间旋转。局部调和分析将此结果推广到流形上一般目标的每个模态周围。学习势与匹配势之间的差距是基分布的残余记忆与模型偏差之和，当位置基分布已转移到动量时，这两个势一致。这在前者比目标更宽泛时可以实现。在欧几里得、双曲和球面空间上的数值实验表明，与黎曼连续归一化流相比，样本质量和数值成本具有竞争力，并确认了学习势的可解释性。
英文摘要
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.
