Convergence Analysis of Probability Flow ODE for Score-based Generative
  Models
Daniel Zhengyu Huang Jiaoyang Huang Zhengjiang Lin
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.