  Denoising Diffusion Probabilistic Models (DDPMs) can generate high-quality
samples such as image and audio samples. However, DDPMs require hundreds to
thousands of iterations to produce final samples. Several prior works have
successfully accelerated DDPMs through adjusting the variance schedule (e.g.,
Improved Denoising Diffusion Probabilistic Models) or the denoising equation
(e.g., Denoising Diffusion Implicit Models (DDIMs)). However, these
acceleration methods cannot maintain the quality of samples and even introduce
new noise at a high speedup rate, which limit their practicability. To
accelerate the inference process while keeping the sample quality, we provide a
fresh perspective that DDPMs should be treated as solving differential
equations on manifolds. Under such a perspective, we propose pseudo numerical
methods for diffusion models (PNDMs). Specifically, we figure out how to solve
differential equations on manifolds and show that DDIMs are simple cases of
pseudo numerical methods. We change several classical numerical methods to
corresponding pseudo numerical methods and find that the pseudo linear
multi-step method is the best in most situations. According to our experiments,
by directly using pre-trained models on Cifar10, CelebA and LSUN, PNDMs can
generate higher quality synthetic images with only 50 steps compared with
1000-step DDIMs (20x speedup), significantly outperform DDIMs with 250 steps
(by around 0.4 in FID) and have good generalization on different variance
schedules. Our implementation is available at
https://github.com/luping-liu/PNDM.
