提出一种无需积分的流模型，可在流形上直接生成高质量数据。
Riemannian MeanFlow for One-Step Generation on Manifolds
- 用平行传输构建位置相关的速度场，实现流形上的平均速度建模。
- 在球面、旋转群等流形上实现单步采样，采样成本大幅降低。
- 适合需要高效生成的几何数据任务，如3D姿态生成、分子构象模拟。
流匹配可实现流形上生成模型的无仿真训练，但采样仍依赖概率流常微分方程的数值积分。本文提出黎曼均值流（Riemannian MeanFlow, RMF），将均值流扩展至流形值生成，其中速度位于随位置变化的切空间中。RMF通过平行传输定义平均速度场，并推导出连接平均速度与瞬时速度的黎曼均值流恒等式，用于内在监督。该恒等式在对数映射切空间表示下具可操作性，避免轨迹模拟和复杂几何计算。为提升优化稳定性，将RMF目标分解为两项，并采用冲突感知多任务学习缓解梯度干扰。RMF还支持通过无分类器指引实现条件生成。在球面、环面、SO(3)和SE(3)上的实验表明，其在保持生成质量的同时显著降低采样成本，实现更具竞争力的一步采样性能。
原文摘要 · Abstract (English)
Flow Matching enables simulation-free training of generative models on Riemannian manifolds, yet sampling typically still relies on numerically integrating a probability-flow ODE. We propose Riemannian MeanFlow (RMF), extending MeanFlow to manifold-valued generation where velocities lie in location-dependent tangent spaces. RMF defines an average-velocity field via parallel transport and derives a Riemannian MeanFlow identity that links average and instantaneous velocities for intrinsic supervision. We make this identity practical in a log-map tangent representation, avoiding trajectory simulation and heavy geometric computations. For stable optimization, we decompose the RMF objective into two terms and apply conflict-aware multi-task learning to mitigate gradient interference. RMF also supports conditional generation via classifier-free guidance. Experiments on spheres, tori, SO(3), and SE(3) demonstrate competitive one-step sampling with improved quality-efficiency trade-offs and substantially reduced sampling cost.
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