Delta Rectified Flow Sampling (DRFS)

• Delta Rectified Flow Sampling (DRFS) is a rectified flow-based method that leverages inversion-free, path-aware editing by explicitly modeling the difference between source and target velocity fields.
• It employs a time-dependent shift term, such as c(t) = t(1-t), to steer noisy latents closer to the target trajectory, reducing over-smoothing and preserving fine details.
• Empirical results show DRFS achieves higher editing quality and fidelity—evidenced by superior CLIP similarity and lower LPIPS—while maintaining efficient runtime compared to similar methods.

Delta Rectified Flow Sampling (DRFS) is an inversion-free, path-aware editing framework within rectified flow models for text-to-image editing. In its specific formulation, DRFS is a distillation-based method that explicitly models the discrepancy between the source and target velocity fields in order to mitigate over-smoothing artifacts rampant in prior distillation sampling approaches, while a time-dependent shift term pushes noisy latents closer to the target trajectory and enhances alignment with the target distribution (Beaudouin et al., 1 Sep 2025). Within the broader rectified-flow literature, the same phrase is also used more loosely for extremely straight, low-NFE, or one-step rectified-flow sampling after rectification (Zhu et al., 2024, Zhang et al., 28 Nov 2025). This suggests a distinction between DRFS as a named editing method and DRFS as a broader sampling regime.

1. Rectified-flow foundations

Rectified flow models constrain transport trajectories to be linear from the base distribution to the data distribution. The canonical interpolation is

$X_t = (1-t)X_0 + t X_1,$

and the model is trained with the squared loss

$$\mathcal{L}(\theta) = \mathbb{E}_{t, X_0, X_1} \big\| v_\theta(X_t, t) - (X_1 - X_0) \big\|^2.$$

This structural restriction greatly accelerates sampling, often enabling high-quality generation with a single Euler step, and under standard assumptions rectified flows achieve sample complexity $\tilde{O}(\varepsilon^{-2})$, improving on the best known $\tilde{O}(\varepsilon^{-4})$ bounds for diffusion and general flow matching models (Sahoo et al., 28 Jan 2026).

The same straight-path structure is the geometric background for DRFS. In rectified-flow-based generation and editing, a learned velocity field $v_\theta(x_t, t, \varphi)$ transports noisy latents toward the data distribution under a text condition $\varphi$ (Beaudouin et al., 1 Sep 2025). A plausible implication is that DRFS inherits the efficiency incentives of rectified flow—namely, editing by manipulating a comparatively straight trajectory rather than by relying on a long stochastic denoising chain.

2. Core formulation in text-to-image editing

DRFS is defined in the editing setting where a source image $x_0^{src}$ with prompt $\varphi^{src}$ is edited under a target prompt $\varphi^{tgt}$. Standard flow trajectories use linear schedules of the form

$x_t = (1-t)x_0 + t x_1,$

and DRFS specializes this to source and target noisy latents

$$\mathcal{L}(\theta) = \mathbb{E}_{t, X_0, X_1} \big\| v_\theta(X_t, t) - (X_1 - X_0) \big\|^2.$$0

Its basic energy is

$$\mathcal{L}(\theta) = \mathbb{E}_{t, X_0, X_1} \big\| v_\theta(X_t, t) - (X_1 - X_0) \big\|^2.$$1

The defining move is that DRFS optimizes the difference between source and target velocity residuals rather than optimizing only the target branch (Beaudouin et al., 1 Sep 2025).

The stated intuition is that subtracting residuals—velocity minus data derivative—for source and target penalizes only the differences associated with the semantic edit, leaving image regions shared by both prompts untouched (Beaudouin et al., 1 Sep 2025). In the terminology of the paper, this explicit differencing is central to reducing out-of-scope collateral changes. The method was motivated in part by the observation that RFDS, while having strong editing capability, excessively smooths images and fails to preserve fine details (Beaudouin et al., 1 Sep 2025).

3. Time-dependent shift and the unifying role of $$\mathcal{L}(\theta) = \mathbb{E}_{t, X_0, X_1} \big\| v_\theta(X_t, t) - (X_1 - X_0) \big\|^2.$$2

DRFS augments the target trajectory with a time-dependent shift term,

$$\mathcal{L}(\theta) = \mathbb{E}_{t, X_0, X_1} \big\| v_\theta(X_t, t) - (X_1 - X_0) \big\|^2.$$3

and the final energy becomes

$$\mathcal{L}(\theta) = \mathbb{E}_{t, X_0, X_1} \big\| v_\theta(X_t, t) - (X_1 - X_0) \big\|^2.$$4

The shift term is introduced because the partially edited image $$\mathcal{L}(\theta) = \mathbb{E}_{t, X_0, X_1} \big\| v_\theta(X_t, t) - (X_1 - X_0) \big\|^2.$$5 may not yet follow the intended edit path; the correction term pushes the trajectory closer to the ideal semantic path, reduces model-data mismatch, and leads to more precise editing (Beaudouin et al., 1 Sep 2025).

The choice of $$\mathcal{L}(\theta) = \mathbb{E}_{t, X_0, X_1} \big\| v_\theta(X_t, t) - (X_1 - X_0) \big\|^2.$$6 determines how DRFS relates to earlier optimization-based and ODE-based editing procedures. The paper states that $$\mathcal{L}(\theta) = \mathbb{E}_{t, X_0, X_1} \big\| v_\theta(X_t, t) - (X_1 - X_0) \big\|^2.$$7 is typically chosen to increase with $$\mathcal{L}(\theta) = \mathbb{E}_{t, X_0, X_1} \big\| v_\theta(X_t, t) - (X_1 - X_0) \big\|^2.$$8 and decay to zero at the beginning and end, for example $$\mathcal{L}(\theta) = \mathbb{E}_{t, X_0, X_1} \big\| v_\theta(X_t, t) - (X_1 - X_0) \big\|^2.$$9, because this stabilizes optimization and avoids error amplification at noisy early steps (Beaudouin et al., 1 Sep 2025).

Choice of $\tilde{O}(\varepsilon^{-2})$0	Relation stated in the paper	Reported interpretation
$\tilde{O}(\varepsilon^{-2})$1	DRFS reduces to DDS	Pure delta, no shift
$\tilde{O}(\varepsilon^{-2})$2	DRFS = FlowEdit	Linear shift, ODE trajectory
$\tilde{O}(\varepsilon^{-2})$3	Example schedule used in analysis	Increases with $\tilde{O}(\varepsilon^{-2})$4, zero at endpoints

The paper formally positions DRFS as a unifying framework. If $\tilde{O}(\varepsilon^{-2})$5, the energy reduces to a form analogous to Delta Denoising Score (DDS); if $\tilde{O}(\varepsilon^{-2})$6, the method recovers the inversion-free method FlowEdit as a strict special case (Beaudouin et al., 1 Sep 2025). The accompanying theoretical analysis states that if the shift is too large, errors at high-noise steps can amplify along the editing path, and that an effective schedule should approach zero near both $\tilde{O}(\varepsilon^{-2})$7 and $\tilde{O}(\varepsilon^{-2})$8 (Beaudouin et al., 1 Sep 2025).

4. Empirical behavior and benchmarked properties

On the widely used PIE Benchmark, DRFS is reported to achieve superior editing quality, fidelity, and controllability while requiring no architectural modifications (Beaudouin et al., 1 Sep 2025). The summary provided for the paper states that DRFS is inversion-free, operates as post-hoc optimization over the pre-trained flow prior, and has similar runtime as FlowEdit while being significantly faster than distillation-based iRFDS (Beaudouin et al., 1 Sep 2025).

Several concrete outcomes are highlighted for SD3-based evaluation on PIE (Beaudouin et al., 1 Sep 2025):

Reported quantity	Value	Comparison stated
Edited-region CLIP similarity	23.83	Highest among all
LPIPS	93.81	vs. 186.39 for iRFDS, 105.46 for FlowEdit
Runtime	Similar to FlowEdit	Significantly faster than iRFDS

The paper also attributes DRFS’s fidelity gains to its path-aware differencing. Figure-level analysis in the summary states that DRFS gradients vanish in un-edited regions, unlike RFDS which edits indiscriminately (Beaudouin et al., 1 Sep 2025). Qualitatively, DRFS is described as handling object replacement, attribute changes, and style transfer while avoiding the background destruction observed in prior methods; the “brown horse” to “zebra” example is given as a representative case (Beaudouin et al., 1 Sep 2025). The same source notes that the time-dependent shift provides a tunable tradeoff between strong edits and fidelity.

5. Relation to adjacent rectified-flow sampling methods

DRFS belongs to a broader family of rectified-flow methods that alter either the sampling dynamics or the training target. One adjacent direction replaces deterministic sampling by a family of stochastic samplers derived from the same marginals. For a deterministic flow $\tilde{O}(\varepsilon^{-2})$9, the construction

$\tilde{O}(\varepsilon^{-4})$0

yields valid stochastic samplers controlled by $\tilde{O}(\varepsilon^{-4})$1; these improve diversity and robustness without retraining the flow model, provided the flow field and score function are available (Singh et al., 2024). This is not DRFS, but it addresses a nearby problem: sensitivity of deterministic rectified flows to estimation and discretization error.

A separate line of work studies few-step or one-step rectified-flow distillation. TraFlow defines a trajectory function

$\tilde{O}(\varepsilon^{-4})$2

and jointly optimizes output reconstruction, velocity alignment, and self-consistency. In the DRFS context described by the summary, one-step generation is obtained as $\tilde{O}(\varepsilon^{-4})$3, while multi-step propagation iterates the same consistency function over time (Wu et al., 24 Feb 2025). SC-Flow takes a different route by jointly training a single network to predict both local velocity and the data endpoint with a consistency penalty

$\tilde{O}(\varepsilon^{-4})$4

with the aim of stabilizing optimization and improving the straightness of generation paths (Han et al., 13 Jul 2026).

These methods occupy adjacent but distinct design spaces. DRFS, in the specific sense of (Beaudouin et al., 1 Sep 2025), is an editing objective defined by source–target residual differencing and a shift schedule; TraFlow and SC-Flow are trajectory-distillation or parameterization methods for efficient generation rather than path-aware editing.

6. Broader usage, one-step sampling, and limitations

Outside the text-to-image editing paper that explicitly names DRFS, the phrase also appears in a broader rectified-flow efficiency discourse. In the model-collapse study on Reflow, real-data augmented variants are reported to preserve high-quality sample generation even with fewer sampling steps, and the summary explicitly connects this to “Delta Rectified Flow Sampling” understood as using large steps with low NFE (Zhu et al., 2024). In the Rectified MeanFlow work, DRFS is described as referring to non-iterative, single-step flow sampling after rectification, and Re-Meanflow is presented as a practical route to that goal by training a MeanFlow model on rectified, truncated couplings (Zhang et al., 28 Nov 2025).

The same broader literature also emphasizes constraints. Reflow trained only on self-generated samples is reported to suffer model collapse, with performance degrading over repeated recursive training; real-data integration is presented as the mechanism that prevents collapse and preserves rank and spectral support (Zhu et al., 2024). This suggests that when DRFS is used in the broader one-step-sampling sense, its practical viability depends not only on straight trajectories but also on how those trajectories are produced and maintained during recursive rectification.

Taken together, the literature supports two closely related readings. The narrow reading identifies DRFS with the specific inversion-free, path-aware editing framework of (Beaudouin et al., 1 Sep 2025). The broader reading treats DRFS as a label for extremely efficient rectified-flow sampling with very few function evaluations (Zhu et al., 2024, Zhang et al., 28 Nov 2025). The overlap between the two is the central role of rectified trajectories: in the editing setting they support path-aware source–target differencing, and in the generative setting they support low-NFE or one-step transport.