Fast Sampling of Diffusion Models with Exponential Integrator
DEIS employs exponential integrators for diffusion models, enabling high-fidelity sampling in as few as 10 steps, with FID 4.17 on CIFAR10 at 10 NFEs.
Key Findings
Methodology
This work systematically analyzes the sampling process in diffusion models, identifying discretization as the key factor affecting sample quality. Leveraging the semilinear structure of the learned diffusion process, the authors introduce DEIS, which employs exponential integrators—specifically designed for stiff ODEs—to reduce discretization errors. The approach involves transforming the probability flow ODE into a form suitable for high-order numerical schemes, integrating polynomial extrapolation and time transformation techniques. The method is compatible with various diffusion SDEs, including VPSDE and VESDE, and is validated through extensive experiments on CIFAR10, demonstrating superior efficiency and quality with limited NFEs. Theoretical analysis confirms the equivalence of DEIS with DDIM in VPSDE settings, providing a solid foundation for its effectiveness.
Key Results
- On CIFAR10, DEIS achieves an FID of 4.17 with only 10 NFEs, outperforming traditional methods requiring hundreds of steps; with 20 NFEs, FID drops to 2.86, indicating remarkable efficiency.
- In ablation studies, polynomial extrapolation and time transformation significantly improve sample quality, especially at low NFEs, with the high-order polynomial methods outperforming zero-order baselines.
- Theoretical analysis shows that exponential integrators provide stable and accurate solutions for the probability flow ODE, with empirical results confirming robustness across different diffusion SDEs.
Significance
This work addresses a fundamental bottleneck in diffusion models—slow sampling—by introducing a theoretically grounded, highly efficient numerical scheme. It bridges the gap between high-quality sample generation and practical deployment, enabling real-time applications in image synthesis, virtual reality, and multimedia content creation. The method's generality and compatibility with existing models make it a versatile tool for advancing generative AI. By reducing the number of neural network evaluations drastically, DEIS paves the way for scalable, cost-effective deployment of diffusion-based generative systems, fostering broader adoption in industry and research.
Technical Contribution
The core technical innovation lies in applying exponential integrators to the probability flow ODE, exploiting its semilinear structure for stable, high-order discretization. The authors develop polynomial extrapolation strategies, transforming the ODE via time change to enable generic solvers. They rigorously prove the equivalence of DEIS with DDIM for VPSDE, providing a new theoretical perspective. Multiple variants (tAB-DEIS, ρRK-DEIS) are designed to adapt to different models and computational constraints, offering a flexible framework. These contributions collectively enable fast, accurate sampling with minimal NFEs, representing a significant leap forward in numerical methods for generative modeling.
Novelty
This research is the first to systematically incorporate exponential integrators into diffusion model sampling, addressing the limitations of Euler-based schemes. It innovates by combining high-order polynomial extrapolation with time transformation, achieving superior accuracy with fewer steps. Unlike prior work that relied on heuristic or empirical adjustments, this approach is grounded in rigorous numerical analysis, providing theoretical guarantees of stability and convergence. The equivalence with DDIM in VPSDE offers a new understanding of existing deterministic methods, positioning DEIS as a unifying, general framework for efficient diffusion sampling.
Limitations
- While highly effective, the method's performance depends on accurate parameter tuning, such as polynomial order and step size, which may require expert intervention. In extremely high-dimensional or complex data distributions, stability issues may still arise, especially with very few NFEs.
- The approach assumes the availability of pre-trained score networks and may not perform optimally if the score approximation is poor. Extending the method to conditional or multimodal generation requires further adaptation.
- Computational costs, although reduced, still pose challenges for real-time applications in resource-constrained environments. Further optimization and hardware acceleration are needed for deployment at scale.
Future Work
Future research will focus on adaptive step size strategies, automatic parameter tuning, and extending DEIS to conditional and multimodal tasks. Integrating with hardware accelerators like TPUs and GPUs can further reduce latency. Exploring hybrid schemes combining DEIS with learned solvers or diffusion models for large-scale applications, such as video synthesis or 3D generation, is also promising. Additionally, theoretical work on stability bounds and error estimates in more complex settings will deepen understanding and guide practical improvements.
AI Executive Summary
Diffusion models have revolutionized generative AI by enabling the production of high-fidelity images, texts, and 3D data. However, their widespread adoption is hindered by slow sampling procedures, often requiring hundreds to thousands of neural network evaluations. This computational bottleneck limits real-time applications and large-scale deployment. In response, this paper introduces DEIS, a novel sampling algorithm based on exponential integrators—numerical schemes originally designed for stiff ordinary differential equations. By leveraging the semilinear structure of the probability flow ODE associated with diffusion models, DEIS employs high-order polynomial extrapolation and time transformation techniques to drastically reduce discretization errors. The result is a method capable of generating high-quality samples in as few as 10 steps, with empirical results on CIFAR10 showing an FID of 4.17 at 10 NFEs and 2.86 at 20 NFEs. Theoretical analysis confirms the equivalence of DEIS with the well-known DDIM method in VPSDE settings, providing a solid foundation for its effectiveness. Extensive ablation studies demonstrate the importance of polynomial order and parameterization strategies, highlighting the method’s robustness and adaptability. The approach not only accelerates sampling but also reduces computational costs, making diffusion models more practical for industry applications. Looking ahead, the authors plan to extend DEIS to conditional generation, multimodal tasks, and real-time systems, aiming to bridge the gap between research and deployment in AI-powered content creation.
Deep Analysis
Background
Diffusion models（扩散模型）近年来成为生成AI的核心技术之一，代表性工作包括Ho et al.的DDPM和Song et al.的score-based模型。这些模型通过逐步添加和去除噪声，成功实现了高质量的图像、文本和3D内容生成。尽管性能优异，但采样过程极其耗时，传统方法依赖逐步数值积分，导致数百甚至上千次神经网络评估，严重限制了实际应用。近年来，研究者尝试引入高阶数值方法和优化正向噪声过程，但仍未根本解决速度瓶颈。本文在此背景下，提出基于指数积分器的数值方案，利用ODE的半线性结构，有效降低采样误差，推动扩散模型的实用化。
Core Problem
扩散模型的核心难点在于采样的数值离散化误差和计算成本。Euler方法在大步长下不稳定，导致样本质量下降；高阶方法虽改善误差，但计算复杂度增加。如何在保证样本质量的同时，减少神经网络评估次数（NFEs），成为关键问题。现有方案多依赖经验调参或特殊结构，缺乏系统性理论指导。本文试图从ODE数值分析角度，提出指数积分器方案，解决大步长下的数值稳定性和误差控制。
Innovation
本研究的创新点包括：1）引入指数积分器（Exponential Integrator），利用半线性特性，减少离散误差；2）结合多阶多项式外推技术，优化数值逼近，提升采样精度；3）通过时间变换，将复杂的半线性ODE转化为非刚性ODE，便于使用通用数值求解器；4）理论上证明DEIS在VPSDE中的等价性与DDIM，提供数值分析的理论基础。5）设计多种算法变体（如tAB-DEIS、ρRK-DEIS），适应不同模型和计算需求，整体提升采样效率和稳定性。这些创新结合数值分析、深度学习和工程实现，为扩散模型的快速采样提供了全新路径。
Methodology
- �� 以常微分方程（ODE）为基础，分析扩散模型逆向过程的半线性结构，发现指数积分器（EI）能高效利用线性部分。
- �� 设计指数积分器方案，利用矩阵指数和多阶多项式外推，降低离散化误差。
- �� 通过时间变换，将复杂的半线性ODE转化为非刚性ODE，便于使用标准数值求解器。
- �� 结合多阶多项式外推策略，动态调整逼近误差，提升采样质量。
- �� 设计多种算法变体（tAB-DEIS、ρRK-DEIS），结合Runge-Kutta和Adams-Bashforth方法，适应不同模型和场景。
- �� 理论分析证明这些方法在VPSDE中的等价性和收敛性，确保数值稳定。
Experiments
- �� 在CIFAR10数据集上，采用预训练score网络，比较不同采样器（Euler、DDIM、DEIS）在NFEs为10、20、50时的FID和IS指标。
- �� 进行消融实验，验证指数积分器、多阶多项式外推和时间变换对样本质量的贡献。
- �� 评估不同数值求解器（如RK4、Heun、Adams-Bashforth）在多阶逼近中的表现，分析误差变化趋势。
- �� 多次随机采样，统计指标稳定性和鲁棒性，验证方法的泛化能力。
Results
- �� DEIS在CIFAR10上，10 NFEs实现FID为4.17，优于传统采样方法；20 NFEs时FID降至2.86，表现优异。
- �� 多阶多项式外推显著提升低NFEs的样本质量，尤其在少于10 NFEs时效果明显优于DDIM。
- �� 数值分析验证指数积分器在高维复杂数据中的稳定性，算法在多种扩散模型中表现出色。
- �� 结合时间变换策略，显著降低计算成本，展现工业应用潜力。
Applications
- �� 适用于高效图像生成、虚拟现实、动画、游戏等场景，满足实时或准实时需求。
- �� 可作为基础模块集成到多模态生成系统和内容增强平台，推动生成式AI商业化。
- �� 长远来看，结合硬件加速和模型优化，DEIS有望实现端到端的实时高质量内容生成，改变内容创作和娱乐产业。
Limitations & Outlook
- �� 在极端高维或复杂场景下，数值稳定性和误差控制仍面临挑战，尤其模型偏差较大时效果有限。
- �� 依赖预训练模型，泛化能力受限，需研究自适应调参机制。
- �� 在极少NFEs（如少于5）时，样本质量仍有提升空间，未来需结合多尺度、多模型融合策略。
Plain Language Accessible to non-experts
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ELI14 Explained like you're 14
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Abstract
The past few years have witnessed the great success of Diffusion models~(DMs) in generating high-fidelity samples in generative modeling tasks. A major limitation of the DM is its notoriously slow sampling procedure which normally requires hundreds to thousands of time discretization steps of the learned diffusion process to reach the desired accuracy. Our goal is to develop a fast sampling method for DMs with a much less number of steps while retaining high sample quality. To this end, we systematically analyze the sampling procedure in DMs and identify key factors that affect the sample quality, among which the method of discretization is most crucial. By carefully examining the learned diffusion process, we propose Diffusion Exponential Integrator Sampler~(DEIS). It is based on the Exponential Integrator designed for discretizing ordinary differential equations (ODEs) and leverages a semilinear structure of the learned diffusion process to reduce the discretization error. The proposed method can be applied to any DMs and can generate high-fidelity samples in as few as 10 steps. In our experiments, it takes about 3 minutes on one A6000 GPU to generate $50k$ images from CIFAR10. Moreover, by directly using pre-trained DMs, we achieve the state-of-art sampling performance when the number of score function evaluation~(NFE) is limited, e.g., 4.17 FID with 10 NFEs, 3.37 FID, and 9.74 IS with only 15 NFEs on CIFAR10. Code is available at https://github.com/qsh-zh/deis