  Diffusion probabilistic models (DPMs) are a key component in modern
generative models. DPM-solvers have achieved reduced latency and enhanced
quality significantly, but have posed challenges to find the exact inverse
(i.e., finding the initial noise from the given image). Here we investigate the
exact inversions for DPM-solvers and propose algorithms to perform them when
samples are generated by the first-order as well as higher-order DPM-solvers.
For each explicit denoising step in DPM-solvers, we formulated the inversions
using implicit methods such as gradient descent or forward step method to
ensure the robustness to large classifier-free guidance unlike the prior
approach using fixed-point iteration. Experimental results demonstrated that
our proposed exact inversion methods significantly reduced the error of both
image and noise reconstructions, greatly enhanced the ability to distinguish
invisible watermarks and well prevented unintended background changes
consistently during image editing. Project page:
\url{https://smhongok.github.io/inv-dpm.html}.
