发表机构
University of Oxford（牛津大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文为基于分数和反射扩散模型的概率流常微分方程采样器提供理论分析，确定流存在条件，揭示稳定性与速度误差的关系，并给出设计原则，通过数值实验验证。
AI 中文摘要
概率流常微分方程（PF-ODE）被广泛用作基于分数的扩散模型的确定性采样器。其通常的合理性论证是，扩散的Fokker-Planck方程可以重写为由前向扩散的分数函数驱动的连续性方程。然而，这一恒等式并不保证由此产生的速度场生成良定义的流。我们为基于扩散和反射扩散的生成模型中此类确定性采样器的设计提供了理论见解。我们确定了正则拉格朗日PF-ODE流存在的充分条件；反向采样和可逆性需要双向散度控制。对于学习到的分数，采样器稳定性由未加权的速度误差控制，这揭示了与密度加权分数匹配的不匹配，并激励了对雅可比矩阵、散度、增长和压缩的架构控制。在流形假设下，正时间正则化证明了提前停止的PF-ODE的合理性，而常数在数据端点附近恶化；一个显式的球面例子表明，当噪声水平消失时，精确的确定性流变得奇异，尽管扩散边际分布仍然良定义。这些理论见解转化为稳定、可逆且保持约束的扩散采样器的具体设计原则。我们通过受控数值实验说明了这些设计原则的实际相关性。
英文摘要
Probability-flow ordinary differential equations (PF-ODEs) are widely used as deterministic samplers for score-based diffusion models. Their usual justification is that the Fokker--Planck equation of a diffusion can be rewritten as a continuity equation driven by the score function of the forward diffusion. This identity does not, however, guarantee that the resulting velocity field generates a well-posed flow. We provide theoretical insights into the design of such deterministic samplers for generative models based on diffusions and reflected diffusions. We identify sufficient conditions for a regular Lagrangian PF-ODE flow to exist; reverse sampling and invertibility require two-sided divergence control. For learned scores, sampler stability is controlled by an unweighted velocity error, exposing a mismatch with density-weighted score matching and motivating architectural control of Jacobians, divergence, growth, and compression. Under the manifold hypothesis, positive-time regularization justifies an early-stopped PF-ODE while constants deteriorate near the data endpoint; an explicit sphere example shows that the exact deterministic flow becomes singular as the noise level vanishes even though the diffusion marginals remain well defined. These theoretical insights translate into concrete design principles for stable, invertible, and constraint-preserving diffusion samplers. We illustrate the practical relevance of these design principles using controlled numerical experiments.
Comments40th Conference on Neural Information Processing Systems (NeurIPS 2026). Workshop: AI for Stochastic Dynamics
Journal ref40th Conference on Neural Information Processing Systems (NeurIPS 2026). Workshop: AI for Stochastic Dynamics
