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Diffusion models (DMs) have made significant progress in the fields of image,audio, and video generation. One downside of DMs is their slow iterativeprocess. Recent algorithms for fast sampling are designed from the perspectiveof differential equations. However, in higher-order algorithms based on Taylorexpansion, estimating the derivative of the score function becomes intractabledue to the complexity of large-scale, well-trained neural networks. Driven bythis motivation, in this work, we introduce the recursive difference (RD)method to calculate the derivative of the score function in the realm of DMs.Based on the RD method and the truncated Taylor expansion of score-integrand,we propose SciRE-Solver with the convergence order guarantee for acceleratingsampling of DMs. To further investigate the effectiveness of the RD method, wealso propose a variant named SciREI-Solver based on the RD method andexponential integrator. Our proposed sampling algorithms with RD method attainstate-of-the-art (SOTA) FIDs in comparison to existing training-free samplingalgorithms, across both discrete-time and continuous-time pre-trained DMs,under various number of score function evaluations (NFE). Remarkably,SciRE-Solver using a small NFEs demonstrates promising potential to surpass theFID achieved by some pre-trained models in their original papers using no fewerthan $1000$ NFEs. For example, we reach SOTA value of $2.40$ FID with $100$ NFEfor continuous-time DM and of $3.15$ FID with $84$ NFE for discrete-time DM onCIFAR-10, as well as of $2.17$ (2.02) FID with $18$ (50) NFE for discrete-timeDM on CelebA 64$\times$64.

SciRE-Solver: Accelerating Diffusion Models Sampling by Score-integrand Solver with Recursive Difference