AI 中文总结
研究在弱正则性假设下扩散过程、福克 - 普朗克方程和相关PF - ODE的数学关系，通过建立边际密度流的性质、研究拉格朗日问题等方法，证明流的相关特性，识别正则性条件，强调描述区别并推导稳定性估计。
AI 中文摘要
与扩散过程相关的概率流常微分方程（PF - ODE）在基于分数的生成建模中被广泛用作确定性采样器，以再现扩散的边际分布。本文在对漂移和分数的弱正则性假设下，研究扩散过程、福克 - 普朗克方程和相关PF - ODE之间的精确数学关系。首先在对扩散系数的最小假设下，建立了作为福克 - 普朗克方程解的边际密度流的存在性和唯一性。然后使用正则拉格朗日流的迪佩尔纳 - 利翁斯 - 安布罗西奥理论研究相应的拉格朗日问题，证明了流的存在性、唯一性和稳定性。还识别了扩散模型中所需正则性的充分条件，强调了欧拉描述和拉格朗日描述之间的根本区别，并构造了反例。最后推导了在学习分数近似下概率流轨迹的稳定性估计。
英文摘要
The probability-flow ordinary differential equation (PF-ODE) associated with a diffusion process is widely used in score-based generative modeling as a deterministic sampler that reproduces the marginal distributions of the diffusion. The validity of this marginal-matching property depends on the well-posedness of an ordinary differential equation whose velocity field is constructed from the score function of the diffusion. We examine the precise mathematical relation between a diffusion process, the Fokker-Planck equation and the associated PF-ODE under weak regularity assumptions on the drift and score. We establish existence and uniqueness of the marginal density flow as a solution of the Fokker--Planck equation under minimal regularity assumptions. We then study the corresponding Lagrangian problem using the DiPerna-Lions-Ambrosio theory of regular Lagrangian flows. We prove existence, uniqueness and stability of the flow, and show that it transports the initial distribution onto the diffusion marginals, under Sobolev or bounded-variation regularity of the score together with one-sided bounds on the divergence of the probability-flow velocity. We identify sufficient conditions for the required regularity in diffusion models relevant for applications. Our analysis underlines a fundamental distinction between Eulerian and Lagrangian descriptions. We construct a counterexample in which the Fokker--Planck equation has a unique density flow while the associated PF-ODE fails to admit a regular Lagrangian flow from the initial time, demonstrating that uniqueness of the density evolution does not in general imply the existence of a deterministic probability-flow representation. Finally, we derive stability estimates for probability-flow trajectories under learned score approximations. Our findings have implications for the training and deployment of score-based diffusion models.
Comments41 pages including appendix, 2 figures
