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Leveraging Previous Steps: A Training-free Fast Solver for Flow Diffusion

Kaiyu Song, Hanjiang Lai

TL;DR

This work tackles the slow sampling of flow diffusion models (FDMs) by introducing a training-free flow-solver that leverages information from previous steps to approximate high-order derivatives via Taylor expansion and polynomial interpolation. The solver uses a cache of past evaluations to compute $x_{t_n}$ with a single function evaluation per step, achieving an effective local error of $O(h_n^{p})$ and, with a predictor-corrector trick, $O(h_n^{p+1})$. The method substantially reduces NFEs while maintaining or improving sample quality across a wide range of datasets, including CIFAR-10, CelebA-HQ, LSUN variants, ImageNet, and text-to-image generation, and works in both pixel-space and latent-space FDMs. These results demonstrate the practical impact of a training-free, step-reusing approach for accelerating flow-based diffusion generation in real-world applications.

Abstract

Flow diffusion models (FDMs) have recently shown potential in generation tasks due to the high generation quality. However, the current ordinary differential equation (ODE) solver for FDMs, e.g., the Euler solver, still suffers from slow generation since ODE solvers need many number function evaluations (NFE) to keep high-quality generation. In this paper, we propose a novel training-free flow-solver to reduce NFE while maintaining high-quality generation. The key insight for the flow-solver is to leverage the previous steps to reduce the NFE, where a cache is created to reuse these results from the previous steps. Specifically, the Taylor expansion is first used to approximate the ODE. To calculate the high-order derivatives of Taylor expansion, the flow-solver proposes to use the previous steps and a polynomial interpolation to approximate it, where the number of orders we could approximate equals the number of previous steps we cached. We also prove that the flow-solver has a more minor approximation error and faster generation speed. Experimental results on the CIFAR-10, CelebA-HQ, LSUN-Bedroom, LSUN-Church, ImageNet, and real text-to-image generation prove the efficiency of the flow-solver. Specifically, the flow-solver improves the FID-30K from 13.79 to 6.75, from 46.64 to 19.49 with $\text{NFE}=10$ on CIFAR-10 and LSUN-Church, respectively.

Leveraging Previous Steps: A Training-free Fast Solver for Flow Diffusion

TL;DR

This work tackles the slow sampling of flow diffusion models (FDMs) by introducing a training-free flow-solver that leverages information from previous steps to approximate high-order derivatives via Taylor expansion and polynomial interpolation. The solver uses a cache of past evaluations to compute with a single function evaluation per step, achieving an effective local error of and, with a predictor-corrector trick, . The method substantially reduces NFEs while maintaining or improving sample quality across a wide range of datasets, including CIFAR-10, CelebA-HQ, LSUN variants, ImageNet, and text-to-image generation, and works in both pixel-space and latent-space FDMs. These results demonstrate the practical impact of a training-free, step-reusing approach for accelerating flow-based diffusion generation in real-world applications.

Abstract

Flow diffusion models (FDMs) have recently shown potential in generation tasks due to the high generation quality. However, the current ordinary differential equation (ODE) solver for FDMs, e.g., the Euler solver, still suffers from slow generation since ODE solvers need many number function evaluations (NFE) to keep high-quality generation. In this paper, we propose a novel training-free flow-solver to reduce NFE while maintaining high-quality generation. The key insight for the flow-solver is to leverage the previous steps to reduce the NFE, where a cache is created to reuse these results from the previous steps. Specifically, the Taylor expansion is first used to approximate the ODE. To calculate the high-order derivatives of Taylor expansion, the flow-solver proposes to use the previous steps and a polynomial interpolation to approximate it, where the number of orders we could approximate equals the number of previous steps we cached. We also prove that the flow-solver has a more minor approximation error and faster generation speed. Experimental results on the CIFAR-10, CelebA-HQ, LSUN-Bedroom, LSUN-Church, ImageNet, and real text-to-image generation prove the efficiency of the flow-solver. Specifically, the flow-solver improves the FID-30K from 13.79 to 6.75, from 46.64 to 19.49 with on CIFAR-10 and LSUN-Church, respectively.

Paper Structure

This paper contains 12 sections, 8 theorems, 33 equations, 21 figures, 7 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Preliminary
  4. Methodology
  5. Increasing the solver order
  6. Experimental Results
  7. Implementation Details
  8. Qualitative Results
  9. Ablation Study
  10. Conclusion
  11. Detailed Proof
  12. More Quantitative Examples

Key Result

Lemma 1

The continuous integral in Eq. eq:sub-sequance could be solved by the high-order derivative of $v_{\theta}(\ast,\ast)$ as follows: where $p$ is the order for the solver. $v_{\theta}(x_{t_{n-1},t_{n-1}})$ is the $i$-th order partial derivatives of $t$ for $v_{\theta}(x_{t_{n-1},t_{n-1}})$ at $t_{n-1}$, where $v_{\theta}^{0}(\ast,\ast) = v_{\theta}(\ast,\ast)$. $C_i$ is the factor and could be calc

Figures (21)

  • Figure 1: Conditional generation results on real text-to-image generations. We compared the FID $\downarrow$ among different sampling solvers for FDMs with different NEFs. We also report the influence of the different strengths of classifier-free guidance (CFG).
  • Figure 2: Qualitative examples for the unconditional generation. We mainly report the results on LSUN-Bedroom and LSUN-Church under with $\text{NEF}=10$.
  • Figure 3: Qualitative examples for the conditional generation. We mainly report the text-to-image generation under different prompts with $\text{NEF}=6$ and $\text{CFG}=2$.
  • Figure 4: Qualitative examples for the unconditional generation on CIFAR-10 under $\text{NEF}=10$.
  • Figure 5: Qualitative examples for the unconditional generation on LSUN-Church under $\text{NEF}=8$.
  • ...and 16 more figures

Theorems & Definitions (12)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • ...and 2 more