\section{Conclusion}\label{sec:conclusion}


In this work, we have provided the first polynomial-time guarantees for the probability flow ODE implementation of SGMs with corrector steps and exhibited improved dimension dependence of the ODE framework over prior results for DDPMs (\textit{i.e.,} the SDE framework).
Our analysis raises questions relevant for practice, of which we list a few.
\begin{itemize}
    \item Although we need the corrector steps for our proof, are they in fact necessary for the algorithm to work efficiently in practice?
    \item Is it possible to obtain even better dimension dependence, perhaps using higher-order solvers and stronger smoothness assumptions? 
    \item Can we obtain improved dimension dependence even in the non-smooth setting, compared to the result of~\cite{CheLeeLiu23ImprovedSGM}?
\end{itemize}

\edit{We also list several limitations of our work, namely:
\begin{itemize}
    \item Our analysis only covers the probability flow ODE corresponding to the OU forward process. We leave the study of more general dynamics for future study.
    \item Our guarantees require the score function to be learned to $L^2$ accuracy $\widetilde O(\varepsilon/\sqrt L)$, which is more stringent than the prior works~\cite{Chenetal23diffmodels, CheLeeLiu23ImprovedSGM, leelutan23sgmgeneral} and may be an technical artefact of our proof.
    \item We have not validated our theoretical findings with large-scale experiments. In particular, it is still unclear whether flow-based methods can outperform the standard DDPM algorithm in practical, high dimensional settings.
\end{itemize}
}

\edit{
\section{Acknowledgments}
We want to thank Yin Tat Lee for his valuable comments, shaping the direction of this research project in its early stages. SC was supported by NSF Award 2103300 for part of this work.}