One-Step Flow Matching: A Generative Framework

• One-step flow matching is a generative modeling framework that directly maps distributions in a single non-iterative step using principles from optimal transport.
• It leverages techniques such as convex potential parameterization and average-velocity fields to ensure precise, straight-line data transformations.
• Empirical benchmarks demonstrate significant speedups and competitive FID scores across images, audio, and other modalities, proving its practical efficacy.

One-step flow matching is a generative modeling framework designed to learn mappings between distributions (e.g., from noise or a structured prior to complex data) such that generation can be performed in a single, non-iterative step. Unlike classical flow-based or diffusion-based methods that require sequential integration of a learned transformation (often requiring dozens to hundreds of steps), one-step flow matching develops specialized training objectives and parameterizations to compress the entire transport into a single function evaluation, often leveraging theoretical insights from optimal transport, average-velocity flow fields, or straight-line flow trajectories. This paradigm has rapidly advanced in both theory and application, with significant empirical results across modalities including images, audio, text, graph data, and reinforcement learning.

1. Mathematical Foundations and Theoretical Guarantees

One-step flow matching builds on the theory of continuous optimal transport (OT) and flow matching (FM), particularly for quadratic cost. The key goal is to learn a deterministic or controlled stochastic mapping $T^*$ that realizes the displacement between an initial distribution $p_0$ (e.g., noise) and a target $p_1$ (e.g., images) in a single “straight” path.

• For quadratic cost, Brenier’s theorem ensures that the optimal transport map is the gradient of a convex potential: $T^* = \nabla \Psi^*$.
• In optimal flow matching (OFM), the trajectory is constrained to be linear for all $t \in [0,1]$: $z_t = (1-t)z_0 + t \nabla \Psi(z_0)$, where $\Psi$ is a convex potential (Kornilov et al., 19 Mar 2024).
• The general framework proceeds by learning a vector field (velocity function) $u_{t, \Psi}$ or, in mean flow methods, an average-velocity field $u(z, r, t)$, such that the entire displacement can be computed in one step (Geng et al., 19 May 2025).

A one-step objective is constructed so that, under minimization, the learned transport exactly matches the OT solution or an optimal “straight” displacement (in the mean flow or average velocity sense). Theoretical results show equivalence between optimizing loss functions over convex potentials and minimizing the dual optimal transport loss (Kornilov et al., 19 Mar 2024), or ensure that the Wasserstein distance between the model’s single-step output and the true data distribution is upper-bounded by the variance of the target (Chen et al., 31 Jul 2025).

2. Parameterization and Loss Design

Several approaches structure the parameterization and training loss to guarantee straight-line flows or analytic invertibility:

• OFM restricts the flow field to gradients of convex potentials, which are commonly parameterized by Input Convex Neural Networks (ICNNs). The OFM loss has an explicit convex conjugate duality, leading to efficient and unbiased learning of the OT map (Kornilov et al., 19 Mar 2024).
• MeanFlow models depart from learning instantaneous velocity fields and instead regress an average-velocity field over the full [0,1] interval, using identities:

$$u(z_t, r, t) = v(z_t, t) - (t-r)\frac{d}{dt} u(z_t, r, t)$$

The loss penalizes deviations from a self-consistency condition, allowing sampling by $z_0 = z_1 - u_\theta(z_1, 0,1)$ in one step (Geng et al., 19 May 2025).

• Block Flow and OT-enhanced Mean Flows use label-based or optimal transport-based couplings to match source and target via minimal-cost, straight interpolations (Wang et al., 20 Jan 2025, Akbari et al., 26 Sep 2025).
• In discrete domains, such as retrosynthesis or text/token modeling, convex combinations and discrete transport paths are employed to model the transition in one step (Yadav et al., 4 Jun 2025, Liu et al., 22 Dec 2024).

The core theme is that the learned map (either direct or via average velocity/convex potential) allows generation by a single application of the learned network, as opposed to ODE/SDE integration.

3. Algorithmic Implementations

The realization of one-step flow matching involves specific sampling, inversion, or trajectory recovery algorithms:

Approach	Core Mechanism	Sampling Formula or Step
OFM (Kornilov et al., 19 Mar 2024)	Linearization via convex potentials	$x_1 = \nabla \Psi^*(x_0)$
MeanFlow (Geng et al., 19 May 2025)	Average velocity identity	$z_0 = z_1 - u_\theta(z_1,0,1)$
Block Flow (Wang et al., 20 Jan 2025)	Label-prior aligned linear interpolation	Use sample-pair with blockwise prior
OT-MeanFlow (Akbari et al., 26 Sep 2025)	OT pairing + mean flow	$x_1 \approx x_0 + u_{1,0}(x_0)$
Discrete flows (Yadav et al., 4 Jun 2025)	Convex comb. or bridge in token space	Markovian bridge with synthon prior

In all cases, the sampling cost is reduced from many neural forward passes to a single evaluation, as the entire transport is integrated analytically or parameterized directly.

4. Empirical Performance and Benchmarks

One-step flow matching models set new performance standards in generative modeling efficiency:

• FGM records an FID of 3.08 on CIFAR10, outperforming original 50-step models while requiring only one generation step (Huang et al., 25 Oct 2024).
• MeanFlow achieves an FID of 3.43 on ImageNet 256×256 with 1-NFE, eclipsing other one-step models and closing the gap to multi-step methods (Geng et al., 19 May 2025).
• Block Flow and OT-based mean flows demonstrate reductions in curvature, numerically quantifiable by derived bounds, and show lower FID/Inception/Wasserstein distances compared to both vanilla mean flow and multi-step baselines (Wang et al., 20 Jan 2025, Akbari et al., 26 Sep 2025).
• Downstream, in speech, text-to-speech, retrosynthesis, and trajectory forecasting, one-step flow matching and its variants achieve state-of-the-art exact match, FID, and round-trip feasibility metrics, causing dramatic speedups (100× or more) without degradation in output fidelity (Yadav et al., 4 Jun 2025, Huynh-Nguyen et al., 19 May 2025, Fu et al., 13 Mar 2025).

An overview of reported metrics:

Domain	Model	Steps Used	FID / Main Metric	Notable
Images	FGM	1	3.08 (CIFAR10)	Below 50-step teacher
Images	MeanFlow-XL/2	1	3.43 (ImageNet 256)	Above prior one-step SOTA
Speech (TTS)	OZSpeech	1	WER 0.05, High SIM	2.7–6.5× faster, SOTA content/naturalness/prosody metrics
Retrosynthesis	RSF	1	Top-1 = 60%, Round-trip	20% above prior SOTA, +19% in top-5 round-trip
RL	FPMD-R/FPMD-M	1	Returns ≈ SOTA	100–1000× function eval reduction versus diffusion RL policies

5. Generalizations and Extensions

One-step flow matching has been generalized and extended in several directions:

• Multi-modal and structured data: Discrete flows for molecule graphs (Yadav et al., 4 Jun 2025), auto-regressive model distillation for token sequences (Liu et al., 22 Dec 2024), and trajectory prediction (Fu et al., 13 Mar 2025).
• Policy learning in RL: Direct policy distribution transport using flow mirror descent or mean-flow policy parameterization, with performance theoretically justified by discretization error bounds in terms of 2-Wasserstein distance and conditional variance (Chen et al., 31 Jul 2025).
• Speaker and attribute conditioning in zero-shot TTS: Optimal transport conditional flow matching with learned priors to enable accurate, attribute-controllable, single-step speech synthesis (Huynh-Nguyen et al., 19 May 2025).
• Incorporation of average-velocity and velocity composition identities to avoid expensive JVPs during training, as in speech enhancement (Yang et al., 19 Sep 2025).
• Optimal transport couplings in high-dimensional generative problems to ensure both fidelity and sample diversity in one-step models (Akbari et al., 26 Sep 2025).

Open problems include extending these theoretical constructions to general entropic transport and stochastic bridges, as described in unified frameworks encompassing both flow matching and Schrödinger bridge matching approaches (Kim, 27 Mar 2025).

6. Limitations, Trade-Offs, and Future Directions

While one-step flow matching achieves remarkable speedups and often competitive or superior sample quality, several limitations and avenues for further research remain:

• The straight-line or average-velocity constraint may, in complex data domains, capture only a portion of the full generative complexity, potentially struggling with highly non-linear data manifolds or multimodal transitions.
• Estimating convex potentials or mean velocities reliably in high dimension requires careful regularization, selection of priors, or design of objective functions (e.g., regularization terms to balance trajectory straightness vs. sample diversity (Wang et al., 20 Jan 2025)).
• Some empirical studies indicate that mean flow-based one-step generation may deviate from the full multi-step process in certain datasets, motivating the integration of OT couplings or distributionally faithful trajectory matching (Akbari et al., 26 Sep 2025, You et al., 26 Mar 2025).
• The sharpness and domain coverage of outputs can depend on teacher quality (in distillation approaches), the design of attribute factorization (e.g., speech), or the quality of block/prior alignment.
• Theoretical guarantees require ideal optimization and may be subject to model misspecification or richness of the neural function class.

Future directions include the unification of deterministic and stochastic bridge algorithms, further analysis of error bounds in single-step discretization, application to new problem classes (e.g., multimodal fusion), and the development of fully teacher-free, self-contained single-step models.

7. Summary Table of Core One-Step Flow Matching Methods

Method	Key Mechanism	Typical Domains	Notable Result
OFM (Kornilov et al., 19 Mar 2024)	Convex potential, OT straight	All	Exact OT in 1 step
MeanFlow (Geng et al., 19 May 2025)	Average velocity field	Images, Audio	3.43 ImageNet FID
FGM (Huang et al., 25 Oct 2024)	Teacher distillation via flow	Images, T2I	FID 3.08 @ 1 step
Block/OT Mean Flow	Label/OT-aligned trajectories	Images, 3D	Lower FID/W₂, high diversity
SFMSE/COSE	Step-invariant/vel. comp. avg.	Speech Enhancement	5× faster, strong PESQ
DD (Liu et al., 22 Dec 2024)	Distilled ODE to AR model	Image/text AR	217.8× faster, FID 11.35
RSF (Yadav et al., 4 Jun 2025)	Synthon/discrete flow + SMC	Chemistry	Top-1=60%, +19% round-trip

One-step flow matching thus represents a collection of theoretically justified, computationally efficient techniques for generative modeling, unifying and extending ideas from optimal transport, ODE bridge problems, and consistency frameworks, with strong applicability across data modalities and tasks.