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

arXiv: https://arxiv.org/abs/2411.07627 · HTML: https://mlsanity.com/paper/2411.07627/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- First posted on arXiv 12 Nov 2024.

## Placed in

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

## Built on by (most cited first, 3 of 3)

- [UniEdit-Flow: Unleashing Inversion and Editing in the Era of Flow Models](https://mlsanity.com/paper/2504.13109/) (2025)
- [Steering Guidance for Personalized Text-to-Image Diffusion Models](https://mlsanity.com/paper/2508.00319/) (2025)
- [Memory-Efficient Fine-Tuning Diffusion Transformers via Dynamic Patch Sampling and Block Skipping](https://mlsanity.com/paper/2603.20755/) (2026)

## Abstract (arXiv)

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 NFE=10 on CIFAR-10 and LSUN-Church, respectively.