# Sharpen Your Flow: Sharpness-Aware Sampling for Flow Matching

arXiv: https://arxiv.org/abs/2605.11547 · HTML: https://mlsanity.com/paper/2605.11547/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Placed in Flow matching, time series and science, one of 14 directions in Diffusion models.
- Builds on 34 of its 41 references that are in the map.
- First posted on arXiv 12 May 2026.

## Placed in

- Diffusion models / [Flow matching, time series and science](https://mlsanity.com/diffusion/flow-matching-time-series-and-science/)

## Builds on (most cited first, 30 of 34)

- [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Scaling Rectified Flow Transformers for High-Resolution Image Synthesis](https://mlsanity.com/paper/2403.03206/) (2024)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [DPM-Solver: A Fast ODE Solver for Diffusion Probabilistic Model Sampling in Around 10 Steps](https://mlsanity.com/paper/2206.00927/) (2022)
- [Consistency Models](https://mlsanity.com/paper/2303.01469/) (2023)
- [Improved Precision and Recall Metric for Assessing Generative Models](https://mlsanity.com/paper/1904.06991/) (2019)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [Latent Consistency Models: Synthesizing High-Resolution Images with Few-Step Inference](https://mlsanity.com/paper/2310.04378/) (2023)
- [UniPC: A Unified Predictor-Corrector Framework for Fast Sampling of Diffusion Models](https://mlsanity.com/paper/2302.04867/) (2023)
- [Reliable Fidelity and Diversity Metrics for Generative Models](https://mlsanity.com/paper/2002.09797/) (2020)
- [Pyramidal Flow Matching for Efficient Video Generative Modeling](https://mlsanity.com/paper/2410.05954/) (2024)
- [Flow Matching Guide and Code](https://mlsanity.com/paper/2412.06264/) (2024)
- [Nearly d-Linear Convergence Bounds for Diffusion Models via Stochastic Localization](https://mlsanity.com/paper/2308.03686/) (2023)
- [Learning Fast Samplers for Diffusion Models by Differentiating Through Sample Quality](https://mlsanity.com/paper/2202.05830/) (2022)
- [Learning to Efficiently Sample from Diffusion Probabilistic Models](https://mlsanity.com/paper/2106.03802/) (2021)
- [Align Your Steps: Optimizing Sampling Schedules in Diffusion Models](https://mlsanity.com/paper/2404.14507/) (2024)
- [Minimizing Trajectory Curvature of ODE-based Generative Models](https://mlsanity.com/paper/2301.12003/) (2023)
- [Accelerating Diffusion Sampling with Optimized Time Steps](https://mlsanity.com/paper/2402.17376/) (2024)
- [On the Trajectory Regularity of ODE-based Diffusion Sampling](https://mlsanity.com/paper/2405.11326/) (2024)
- [FlowLLM: Flow Matching for Material Generation with Large Language Models as Base Distributions](https://mlsanity.com/paper/2410.23405/) (2024)
- [FlowTS: Time Series Generation via Rectified Flow](https://mlsanity.com/paper/2411.07506/) (2024)
- [On Numerical Integration in Neural Ordinary Differential Equations](https://mlsanity.com/paper/2206.07335/) (2022)
- [Lipschitz Singularities in Diffusion Models](https://mlsanity.com/paper/2306.11251/) (2023)
- [Score-Optimal Diffusion Schedules](https://mlsanity.com/paper/2412.07877/) (2024)
- [Entropic Time Schedulers for Generative Diffusion Models](https://mlsanity.com/paper/2504.13612/) (2025)
- [Scaling Atomistic Protein Binder Design with Generative Pretraining and Test-Time Compute](https://mlsanity.com/paper/2603.27950/) (2026)
- [Optimal Scheduling of Dynamic Transport](https://mlsanity.com/paper/2504.14425/) (2025)

## Built on by (most cited first, 2 of 2)

- [Geometry-Aware Time Reparameterization for Flow-Map Distillation](https://mlsanity.com/paper/2610.02427/) (2026)
- [LyapuFlow: Controlling Generative Flows with Lyapunov Feedback for Inverse Problems](https://mlsanity.com/paper/2610.04326/) (2026)

## Abstract (arXiv)

Flow matching models generate samples by numerically integrating a learned velocity field, with each integration step requiring a neural network evaluation. Fast generation therefore requires using a small fixed evaluation budget effectively: the key question is not only how to integrate the flow, but where the sampler should spend its steps. We propose SharpEuler, a training-free sampler that profiles a pretrained model offline by estimating where the learned velocity field changes most rapidly along calibration trajectories. This finite-difference estimate defines a solver-aware sharpness profile, which is smoothed and converted by a quantile transform into a timestep grid for any desired inference budget. At test time, sampling remains ordinary Euler integration with the same number of model evaluations as a uniform schedule. We justify SharpEuler using three principles: a numerical principle identifying trajectory acceleration as the leading source of Euler discretization error, a variational principle deriving sharpness-based power-law timestep densities, and a statistical guarantee showing that the finite-sample calibrated sampler is stable at the terminal distribution level. Our experiments show that SharpEuler improves sample quality at fixed budgets, reducing inter-mode leakage and increasing mode coverage.