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 $\mathcal{O}(d^{3/4}\delta^{1/2})$ 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 $\mathcal{O}(d^{3/4}\delta^{1/2} + d\cdot(dh)^p)$ at the discrete level. Finally, we present numerical studies on problems up to 128 dimensions to verify our theory.

翻译：基于分数的生成模型已成为采样高维概率分布的一种强大方法。尽管其效果显著，但其理论基础仍相对薄弱。本研究从理论和数值两个角度，探讨了基于概率流常微分方程的确定性采样器的收敛性质。假设能够获得分数函数的 $L^2$ 精确估计，我们证明了在连续时间层面上，目标分布与生成数据分布之间的总变差上界为 $\mathcal{O}(d^{3/4}\delta^{1/2})$，其中 $d$ 表示数据维度，$\delta$ 代表 $L^2$-分数匹配误差。对于采用步长为 $h$ 的 $p$ 阶 Runge-Kutta 积分器的实际实现，我们在离散层面上建立了 $\mathcal{O}(d^{3/4}\delta^{1/2} + d\cdot(dh)^p)$ 的误差界。最后，我们在高达 128 维的问题上进行了数值研究，以验证我们的理论。