行走于分数流形：学习数据流形上的连续时间生成动力学
Walking the Score Manifold: Continuous-time Generative Dynamics on Learned Data Manifolds
- DFKI Kaiserslautern（凯泽斯劳滕德国人工智能研究中心）
- RPTU Kaiserslautern（凯泽斯劳滕-兰道工业大学）
- Imperial College London（帝国理工学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出在分数模型学习的流形上学习连续时间向量场，实现任意时间戳生成与时间超分辨率，并通过横向稳定性目标增强鲁棒性，在视频、PDE和分子动力学数据上验证了有效性。
AI中文摘要：
对时间相关数据的生成建模通常在离散时间网格上进行，这限制了监督仅能作用于训练数据中观测到的时间戳。相反，我们将生成过程视为在学习到的数据流形上的连续时间演化。为此，我们利用预训练的基于分数的模型作为几何先验，并学习一个向量场，该向量场使数据沿分数诱导的插值路径演化。由于这些动力学遵循尊重分数模型所学几何的转移，它们支持在任意时间戳进行生成，并能实现超越训练数据离散化的时间超分辨率。此外，这种几何表述使我们能够通过回归目标实现无模拟的向量场训练。为了提高长时程展开的鲁棒性，我们引入了一个促进路径相对横向指数稳定性的目标。虽然该目标源于稳定性理论，但它具有实际解释，即作为横向于插值路径的去噪分数匹配。进一步，我们将该框架扩展到概率设置，以建模合理未来轨迹上的分布。我们在自然视频和科学动力学数据上展示了该方法，包括时间超分辨率、基于偏微分方程的时空场以及分子动力学。我们的结果表明，基于分数的先验为学习随机连续时间生成动力学提供了坚实基础。
英文摘要：
Generative modeling of time-dependent data is typically formulated on a discrete temporal grid, restricting supervision to the observed timestamps in the training data. We instead frame generation as continuous-time evolution on a learned data manifold. To this end, we leverage pretrained score-based models as geometric priors and learn a vector field that evolves data along score-induced interpolation paths. Because these dynamics follow transitions that respect the geometry learned by the score model, they support generation at arbitrary timestamps and temporal super-resolution beyond the discretization of the training data. Moreover, this geometric formulation allows us to train the vector field simulation-free through a regression objective. To improve long-horizon rollout robustness, we introduce an objective that promotes path-relative transverse exponential stability. While motivated by stability theory, it admits a practical interpretation as denoising score matching transverse to the interpolation path. Further, we extend the framework to a probabilistic setting that models a distribution over plausible future trajectories. We demonstrate the method on natural video and scientific dynamical data, including temporal super-resolution, PDE-based spatiotemporal fields, and molecular dynamics. Our results show that score-based priors provide a strong foundation for learning stochastic continuous-time generative dynamics.
