Score-based generative models have emerged as a powerful approach for sampling high-dimensional probability distributions. Despite their effectiveness, their theoretical underpinnings remain relatively underdeveloped. In this work, we study the convergence properties of deterministic samplers based on probability flow ODEs from both theoretical and numerical perspectives. Assuming access to $L^2$-accurate estimates of the score function, we prove the total variation between the target and the generated data distributions can be bounded above by $\mathcal{O}(d\sqrt{\delta})$ in the continuous time level, where $d$ denotes the data dimension and $\delta$ represents the $L^2$-score matching error. For practical implementations using a $p$-th order Runge-Kutta integrator with step size $h$, we establish error bounds of $\mathcal{O}(d(\sqrt{\delta} + (dh)^p))$ at the discrete level. Finally, we present numerical studies on problems up to $128$ dimensions to verify our theory, which indicate a better score matching error and dimension dependence.

基于分数的生成模型中，我们从理论和数值的角度研究了基于概率流ODE的确定性采样器的收敛性质，并证明了目标和生成数据分布之间的总变差可以在连续时间层面上通过d√δ（其中d表示数据维度，δ表示L2-评分匹配误差）被上界限制，并针对使用p阶Runge-Kutta积分器进行具体实现的情况，建立了离散层面上的误差界限为O(d(√δ + (dh)^p))。最后，我们进行了高达128个维度的问题的数值研究，验证了我们的理论，结果表明更好的评分匹配误差和维度依赖性。

基于分值的生成模型的概率流动常微分方程的收敛分析