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

arXiv: https://arxiv.org/abs/2308.07896 · HTML: https://mlsanity.com/paper/2308.07896/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 18 of its 26 references that are in the map.
- First posted on arXiv 15 Aug 2023.

## Placed in

- Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/)

## Builds on (most cited first, 18 of 18)

- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Hierarchical Text-Conditional Image Generation with CLIP Latents](https://mlsanity.com/paper/2204.06125/) (2022)
- [Generative Adversarial Networks](https://mlsanity.com/paper/2203.00667/) (2022)
- [DPM-Solver: A Fast ODE Solver for Diffusion Probabilistic Model Sampling in Around 10 Steps](https://mlsanity.com/paper/2206.00927/) (2022)
- [SDEdit: Guided Image Synthesis and Editing with Stochastic Differential Equations](https://mlsanity.com/paper/2108.01073/) (2021)
- [Consistency Models](https://mlsanity.com/paper/2303.01469/) (2023)
- [Imagen Video: High Definition Video Generation with Diffusion Models](https://mlsanity.com/paper/2210.02303/) (2022)
- [DPM-Solver++: Fast Solver for Guided Sampling of Diffusion Probabilistic Models](https://mlsanity.com/paper/2211.01095/) (2022)
- [Pseudo Numerical Methods for Diffusion Models on Manifolds](https://mlsanity.com/paper/2202.09778/) (2022)
- [Score-based Generative Modeling in Latent Space](https://mlsanity.com/paper/2106.05931/) (2021)
- [WaveGrad: Estimating Gradients for Waveform Generation](https://mlsanity.com/paper/2009.00713/) (2020)
- [Fast Sampling of Diffusion Models with Exponential Integrator](https://mlsanity.com/paper/2204.13902/) (2022)
- [Tackling the Generative Learning Trilemma with Denoising Diffusion GANs](https://mlsanity.com/paper/2112.07804/) (2021)
- [UniPC: A Unified Predictor-Corrector Framework for Fast Sampling of Diffusion Models](https://mlsanity.com/paper/2302.04867/) (2023)
- [Analytic-DPM: an Analytic Estimate of the Optimal Reverse Variance in Diffusion Probabilistic Models](https://mlsanity.com/paper/2201.06503/) (2022)
- [ERA-Solver: Error-Robust Adams Solver for Fast Sampling of Diffusion Probabilistic Models](https://mlsanity.com/paper/2301.12935/) (2023)

## Built on by (most cited first, 10 of 10)

- [Accelerating Diffusion Sampling with Optimized Time Steps](https://mlsanity.com/paper/2402.17376/) (2024)
- [Dequantified Diffusion-Schr"odinger Bridge for Density Ratio Estimation](https://mlsanity.com/paper/2505.05034/) (2025)
- [EVODiff: Entropy-aware Variance Optimized Diffusion Inference](https://mlsanity.com/paper/2509.26096/) (2025)
- [Variational Learning of Gaussian Process Latent Variable Models through Stochastic Gradient Annealed Importance Sampling](https://mlsanity.com/paper/2408.06710/) (2024)
- [Diffusion Bridge Variational Inference for Deep Gaussian Processes](https://mlsanity.com/paper/2509.19078/) (2025)
- [Large Language Model as Token Compressor and Decompressor](https://mlsanity.com/paper/2603.25340/) (2026)
- [A Minimum Variance Path Principle for Accurate and Stable Score-Based Density Ratio Estimation](https://mlsanity.com/paper/2602.00834/) (2026)
- [Mitigating the Contractivity Trap in Diffusion ODEs via Stein Stabilization](https://mlsanity.com/paper/2606.07835/) (2026)
- [Towards Disentangled Preference Optimization Dynamics: Suppress the Loser, Preserve the Winner](https://mlsanity.com/paper/2604.18239/) (2026)
- [DiFA: Inference-Time Forward-Process Alignment for Diffusion Models](https://mlsanity.com/paper/2607.17972/) (2026)

## Abstract (arXiv)

Diffusion models (DMs) have made significant progress in the fields of image, audio, and video generation. One downside of DMs is their slow iterative process. Recent algorithms for fast sampling are designed from the perspective of differential equations. However, in higher-order algorithms based on Taylor expansion, estimating the derivative of the score function becomes intractable due to the complexity of large-scale, well-trained neural networks. Driven by this 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 accelerating sampling of DMs. To further investigate the effectiveness of the RD method, we also propose a variant named SciREI-Solver based on the RD method and exponential integrator. Our proposed sampling algorithms with RD method attain state-of-the-art (SOTA) FIDs in comparison to existing training-free sampling algorithms, 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 the FID achieved by some pre-trained models in their original papers using no fewer than 1000 NFEs. For example, we reach SOTA value of 2.40 FID with 100 NFE for continuous-time DM and of 3.15 FID with 84 NFE for discrete-time DM on CIFAR-10, as well as of 2.17 (2.02) FID with 18 (50) NFE for discrete-time DM on CelebA 64×64.