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MintFlow: Minimal Trajectory Intervention for Constrained Flow Matching

Yesom Park, Kelvin Kan, Qifan Chen, Thomas Flynn, Hayden Schaeffer, Xihaier Luo

arXiv:2610.02260v2cs.AI

TL;DR

Constrained samplers can satisfy prescribed conditions while displacing samples from the pretrained flow distribution. MintFlow minimally perturbs an intermediate state and follows the unchanged flow, using adjoint sensitivity and adaptive timing. Across vision and physical-system tasks, it reports constraint satisfaction alongside substantially better preservation of the pretrained distribution than existing constrained methods.

  • Problem

    Constrained generation must satisfy application-specific conditions without substantially shifting samples from the pretrained flow distribution.

    Source: §1 · p. 1, §2 · p. 3

  • Method

    MintFlow uses adjoint sensitivity to compute a minimum-norm intermediate trajectory perturbation, keeps the pretrained flow unchanged, and adaptively selects when to intervene.

    Source: §1 · p. 2

  • Results

    Across image and physics applications, MintFlow effectively satisfies constraints while preserving the learned distribution across a broad range of tasks.

    Source: §1 · p. 2

  • Takeaways & Limitations

    The framework offers a training-free way to balance constraint satisfaction and fidelity to the pretrained distribution across diverse applications.

    Source: §1 · p. 2

    MintFlow relies on a first-order linearization; highly nonlinear constraints may call for iterative refinement or higher-order approximations.

    Source: §5 · p. 10

Abstract

from arXiv · show

Flow matching models excel at generative modeling, and many downstream applications require their samples to satisfy prescribed constraints, such as observed measurements and physical laws. However, existing constrained samplers often face a trade-off: \textit{enforcing constraints can substantially displace samples from the pretrained data distribution}. To address this trade-off, we introduce \textbf{MintFlow}, a training-free constrained sampling framework that formulates constraint enforcement as a minimal intervention on the pretrained flow trajectory. MintFlow seeks the minimal perturbation of an intermediate flow state such that its subsequent evolution under the pretrained flow field satisfies the target constraint. By minimally perturbing the flow state while keeping the pretrained flow field unchanged, MintFlow enforces the constraint while minimizing unnecessary deviation from the pretrained distribution. An adjoint formulation yields a closed-form expression for this perturbation, eliminating expensive iterative optimization. Furthermore, MintFlow adaptively selects the intervention time to balance the required perturbation magnitude with its amplification by the remaining flow. Across a range of tasks in generative vision and physical system modeling, MintFlow achieves competitive constraint satisfaction while preserving the pretrained generative distribution substantially better than state-of-the-art constrained methods.

1 INTRODUCTION

Constrained generation must satisfy application-specific requirements without losing the distribution learned by a pretrained flow. MintFlow addresses this trade-off by minimally intervening on a trajectory while leaving the flow field unchanged.

1 INTRODUCTION

  • Existing constrained samplers can satisfy constraints yet substantially shift samples away from the pretrained distribution.The introduction groups inference-time approaches as guidance, trajectory optimization, and post-hoc projection, alongside training-time methods that require retraining for new constraints.Source: §1 · p. 1
  • MintFlow finds the smallest intermediate-state perturbation that satisfies the terminal constraint under the unchanged pretrained flow, using adjoint sensitivity and adaptive intervention-time selection.The adjoint formulation avoids explicit construction of the high-dimensional flow Jacobian and costly optimization through the full trajectory.Source: §1 · p. 2
  • Across image and physics applications, MintFlow satisfies constraints while preserving the learned distribution across a broad range of tasks.The paper also characterizes constraint error and bounds the Wasserstein distance between corrected and pretrained distributions by intervention magnitude and remaining flow time.Source: §1 · p. 2

2 BACKGROUND

The background frames constrained generation as producing samples from a pretrained flow that satisfy a prescribed condition while remaining close to its terminal distribution. MintFlow casts this as a minimal-intervention problem.

2 BACKGROUND

  • The pretrained flow transports a base distribution through continuous dynamics to a terminal distribution treated as an approximation of the target data distribution.The flow map denotes movement of a state from one time to another along the pretrained dynamics.Source: §2 · p. 2
  • The generation objective is to obtain terminal samples that satisfy a differentiable constraint using the pretrained flow and constraint operator, without additional training.The pretrained velocity field is not explicitly trained to enforce that constraint.Source: §2 · pp. 2–3
  • Constraint satisfaction alone may move samples away from the pretrained distribution, so the goal is the smallest necessary perturbation to intermediate flow states.The formulation treats proximity to the pretrained terminal distribution as a co-objective with satisfying the constraint.Source: §2 · p. 3

3 METHOD

MintFlow minimally perturbs an intermediate state and then uses the original flow to reach the terminal constraint. An adjoint computes the correction, while intervention-time selection balances correction size and remaining propagation.

3.1 TRAJECTORY INTERVENTION FOR CONSTRAINED GENERATION

  • MintFlow perturbs an intermediate trajectory state only as much as needed, then propagates it to the terminal constraint under the unchanged pretrained flow.This keeps the flow dynamics fixed while targeting the constraint manifold at the terminal time.Source: §3.1 · p. 3

3.2 ADJOINT-BASED OPTIMAL CORRECTION

  • The minimum-norm correction is obtained from a linearized terminal constraint, with adjoint sensitivity providing the needed vector-Jacobian product without forming the full flow Jacobian.The correction is also interpretable as a single Newton step for the nonlinear constraint equation.Source: §3.2 · pp. 3–4
  • Under mild regularity conditions, the linearized correction has a second-order terminal constraint residual, and distributional deviation is bounded by correction magnitude and remaining integration time.The distributional bound also assumes the flow field is Lipschitz in the state.Source: §3.2 · p. 4

3.3 ADAPTIVE INTERVENTION TIME SELECTION

  • Intervention time balances correction magnitude against remaining propagation time; the practical criterion also penalizes later integration cost, and iterative refinement can continue until a prescribed residual tolerance is met.Refinement restricts successive intervention times to increase, shortening each subsequent backward adjoint integration.Source: §3.3 · pp. 4–5

4 EXPERIMENTS

MintFlow is evaluated on image inverse problems, image editing, and physical-system modeling, with comparisons measuring constraint satisfaction, distribution preservation, and task-specific performance. Across these settings, results show a balance between constraint satisfaction and fidelity to the learned distribution, with physical-system results also exposing a trade-off between residual size and distributional and dynamical metrics.

4.1 IMAGE INVERSE PROBLEMS

  • Across all evaluated image inverse tasks and datasets, MintFlow achieves the best reported constraint error, FID, PSNR, SSIM, and LPIPS.The reported balance preserves reconstruction quality alongside constraint satisfaction; competing methods can reduce constraint error while producing higher FID and inferior reconstruction quality.Source: §4.1 · pp. 5–6, Table 1 · p. 6
  • Figure 3 compares ground truth, degraded inputs, and method outputs; MintFlow’s representative reconstructions are described as natural-looking and faithful to measurement constraints.The comparisons cover inpainting and super resolution on AFHQ-Cat and deblurring on FFHQ.Source: §4.1 · p. 6, Figure 3 · p. 6

4.2 IMAGE EDITING

  • For PIE-Bench image editing, MintFlow attains the best constraint satisfaction and prompt adherence, while Freedom has higher image similarity at lower prompt adherence.MintFlow is also reported as the most computationally efficient method, with the lowest inference time and least memory use.Source: §4.2 · pp. 7–8, Table 2 · p. 7
  • Figure 4 compares edits for input images and text prompts; MintFlow is described as following prompts while preserving visual coherence and image quality.Strikethrough and red text mark original and edited prompt text, respectively.Source: Figure 4 · p. 7, §4.2 · p. 8

4.3 PHYSICAL SYSTEM MODELING

  • MintFlow has the lowest FID, MMD, W2, and simulation-error point estimates on both physical systems, while ECI and PCFM achieve the smallest constraint residuals.The results place MintFlow on a distinct Pareto point, not in uniform dominance; ECI and PCFM remain preferable when minimizing explicit residual alone.Source: §4.3 · p. 8, Table 3 · p. 8
  • On Heat, finer sampling improves distributional fidelity and constraint satisfaction, while λ trades constraint satisfaction against distribution preservation.The method degrades noticeably only below 100 sampling steps and remains stable over a moderate range of λ; larger λ favors later interventions.Source: §4.3 · p. 9, Figure 6 · p. 9

5 CONCLUSION

MintFlow is a training-free framework for constrained generation that uses adjoint sensitivity to compute minimum-norm intermediate perturbations while preserving pretrained dynamics. Its correction is the first-order solution of the linearized KKT system for the minimum-perturbation endpoint-constrained problem.

5 CONCLUSION

  • Experiments across domains, constraints, and backbone models show that MintFlow effectively satisfies constraints while preserving the pretrained distribution.Source: §5 · p. 9
  • The first-order correction can be complemented by iterative refinements or higher-order approximations for highly nonlinear constraints.Source: §5 · p. 10

A.1 CONTROL-THEORETIC INTERPRETATION OF THE MINIMUM-NORM CORRECTION

  • The adjoint correction is the solution of the linearized KKT system for the minimum-perturbation endpoint-constrained problem.It approximates the nonlinear KKT system around the generated trajectory.Source: §A.1 · p. 13

A.2 VALIDITY OF THE LINEARIZED CORRECTION

This appendix examines the accuracy and stability of MintFlow’s linearized correction, iterative refinement, sampler implementation, and related constraint-enforcement approaches. It also evaluates MintFlow on image-generation tasks with color-intensity and nonlinear autoencoder-fidelity constraints, measuring constraint satisfaction and preservation of the pretrained flow-matching distribution.

A.2 VALIDITY OF THE LINEARIZED CORRECTION

  • The minimum-norm linearized correction satisfies the nonlinear constraint up to a second-order residual under the stated smoothness and full-row-rank conditions.The full-row-rank condition is a local regularity assumption that may fail for nonlinear constraints; the analysis is therefore limited to locally regular configurations.Source: §A.2 · pp. 13, 15
  • The corrected terminal distribution’s deviation from the pretrained-flow distribution is controlled by the intermediate correction magnitude and remaining integration time.The stability result assumes a globally Lipschitz flow field, which the appendix interprets as needing to hold on the region explored by trajectories.Source: §A.2 · pp. 15–16

A.3 CONVERGENCE OF ITERATIVE REFINEMENT

  • Iterative refinement converges under the condition K|F(u(0))| < 1, with the theorem assuming smooth dynamics and constraints on the relevant reachable set.The theorem states an iteration bound and an asymptotic estimate.Source: §A.3 · pp. 17, 19

A.4 SPECIALIZATION TO RECTIFIED FLOW

  • For Euler-sampled rectified flows, the correction uses local velocity Jacobians, computed by a single vector–Jacobian product for one step or discrete adjoints across multiple steps.When the velocity field is locally state-insensitive, the correction reduces to a terminal linearized projection; otherwise it accounts for how the dynamics transform the perturbation.Source: §A.4 · pp. 19–20

B ALGORITHM

  • The single-correction algorithm samples a trajectory, solves the adjoint backward once, evaluates candidate intervention times, and reintegrates from the selected correction.The iterative extension repeats candidate-time selection and correction over multiple refinement iterations.Source: B ALGORITHM · pp. 21–22

C RELATED WORK

  • Prior constraint-enforcement methods include velocity modification, projection onto constraint manifolds, and source or intermediate-state optimization; velocity-modification methods alter dynamics, while MintFlow leaves the pretrained field unchanged.Velocity-modification approaches require constraint-specific velocity corrections during sampling.Source: C RELATED WORK · p. 22
  • Projection methods correct intermediate or terminal states, while optimization methods adjust noise, states, or controls; MintFlow instead uses minimal perturbations and a single adjoint system.Projection methods include ECI (Cheng et al., 2025) and PCFM (Utkarsh et al., 2026); optimization methods include D-Flow (Ben-Hamu et al., 2024) and FlowChef (Patel et al., 2025). The passage contrasts their explicit corrections or computationally demanding iterative optimization with MintFlow's approach.Source: C RELATED WORK · pp. 22–23

D APPLICATION 1: IMAGE GENERATION

  • The image experiments extend MintFlow evaluation to standard flow-matching models, color-intensity constraints, and nonlinear fidelity constraints defined through a pretrained autoencoder.The appendix describes these as additional image-generation experiments and says, together with the main experiments, they demonstrate generalization across flow-matching models and image-domain constraints.Source: D APPLICATION 1: IMAGE GENERATION · p. 23

D.1 DATASETS AND CONSTRAINTS

  • Color-intensity targets control digit stroke thickness on MNIST and RGB color composition on CIFAR-10, and are sampled from the 20th–80th percentiles of each dataset's intensity distribution.The central percentile range avoids overly rare intensity levels that only a small fraction of the data naturally satisfy.Source: §D.1 · p. 24
  • A pretrained autoencoder's reconstruction constraint tests MintFlow on nonlinear neural-network constraints and aims to preserve data-manifold characteristics; test reconstruction losses are 0.0008 on MNIST and 0.00139 on CIFAR-10.The experiment assumes the pretrained autoencoder is meaningful for the underlying data distribution; it does not claim reconstruction error is a universally optimal fidelity measure.Source: §D.1 · p. 24

D.2 FLOW MATCHING MODEL AND BASELINES

  • Image comparisons include DiffusionPDE, D-Flow, ECI, and PCFM, and MintFlow tests 10 uniformly spaced candidate intervention times across the sampling trajectory.The baselines guide velocity, optimize latent noise, or apply final projections; they are included to examine applicability across image-domain constraints, including global color intensity and nonlinear autoencoder fidelity.Source: §D.2 · pp. 24–25

D.3 EVALUATION PROTOCOL

  • Evaluation separately measures constraint satisfaction and conditional-distribution preservation, comparing 2,000 generated samples with 2,000 pretrained-flow samples satisfying the constraint using FID, KID, MMD, and W2.The reference is the pretrained flow's constraint-conditioned distribution, not the ground-truth data distribution; W2 is computed between Gaussian approximations in feature space to avoid prohibitive empirical computation.Source: §D.3 · p. 25

D.4.1 MNIST

MNIST experiments show MintFlow balancing constraint satisfaction and distributional fidelity, including under nonlinear autoencoder constraints. The appendix also examines adaptive intervention times, color-tone constraints on CIFAR-10, and image inverse-problem settings.

D.4.1 MNIST

  • MintFlow offers a better balance between constraint satisfaction and distributional fidelity than projection-based methods on MNIST intensity constraints.Its constraint error is slightly higher than PCFM and ECI, but its FID, KID, MMD, and W2 scores are substantially better; post-hoc projection can alter image appearance.Source: §D.4.1 · p. 26
  • For MNIST autoencoder fidelity constraints, MintFlow reduces reconstruction error and achieves the best distribution-matching performance while remaining faster than D-Flow and PCFM.The experiment demonstrates learned nonlinear constraint integration, but its effectiveness depends on the auxiliary autoencoder's quality and generalization.Source: §D.4.1 · p. 28
  • The selected intervention times adapt to constraint structure: linear intensity constraints favor earlier times, while nonlinear autoencoder fidelity constraints yield a more dispersed distribution.The authors relate this difference to the constraints' geometry and sensitivity, rather than a fixed intervention schedule.Source: §D.4.1 · p. 31, Figure 10 · p. 31

D.4.2 CIFAR10

  • On CIFAR-10 color-tone constraints, MintFlow provides a favorable balance between constraint satisfaction and distribution preservation, shifting color appearance while retaining semantic content and visual quality.The experiments use 10,000 samples; explicit projection methods tend to introduce visible artifacts, especially with higher resolution and complex image structure.Source: §D.4.2 · pp. 31–32, Table 8 · p. 32, Figure 11 · p. 33
  • For CIFAR-10 autoencoder fidelity constraints, MintFlow improves distribution-matching metrics over unconstrained flow matching, while existing constrained methods show limited performance.The experiment evaluates learned-constraint integration while preserving the pretrained flow distribution, rather than optimizing unconditional CIFAR-10 generation.Source: §D.4.2 · p. 32, Table 9 · p. 32, Figure 12 · p. 34

E.1 DATASETS AND CONSTRAINTS

  • The image inverse-problem benchmarks evaluate 64×64 AFHQ-Cat and FFHQ images using inpainting, 4× super-resolution, and Gaussian deblurring constraints.Inpainting observes pixels outside a centered 20×20 mask; super-resolution requires reproducing a 16×16 observation after downsampling, while deblurring uses an 11-pixel kernel with σ = 1.0.Source: §E.1 · p. 35

E.2 FLOW-MATCHING MODEL AND BASELINES

  • The inverse-problem comparisons use the same pretrained Rectified Flow++ models and 10-step sampling budget across methods.The RFM implementation exploits the linearity of its flow path to obtain the optimal correction without explicitly solving the adjoint equation backward in time.Source: §E.2 · pp. 35–36

E.3 EVALUATION PROTOCOL

The appendix specifies evaluation measures and additional image inverse-problem and editing setups. It also reports further qualitative results and explains that D-Flow was excluded from InstaFlow editing because its memory use exceeded the experimental setup's available memory.

E.3 EVALUATION PROTOCOL

  • Inverse-problem evaluation measures constraint residual, pairwise reconstruction fidelity with PSNR, SSIM, and LPIPS, distributional fidelity with FID, and runtime and peak GPU memory.Lower constraint residual and FID indicate better constraint satisfaction and distributional fidelity; higher PSNR and SSIM and lower LPIPS indicate better pairwise fidelity.Source: §E.3 · p. 36

E.4 ADDITIONAL QUALITATIVE RESULTS

  • Additional uncurated results compare ground-truth images, degraded inputs, and method outputs for inpainting, super-resolution, and deblurring on AFHQ-Cat and FFHQ.The AFHQ-Cat results are in Figures 13–15 and the FFHQ results in Figures 16–18.Source: §E.4 · p. 36, Figure 13 · p. 37, Figure 14 · p. 38

F.1 DATASETS AND CONSTRAINTS

  • The image-editing benchmark uses PIE-Bench images and masks, requiring unmasked regions to remain unchanged while the masked region follows text guidance.This combines preservation and semantic-edit requirements; the benchmark evaluates constraint satisfaction, prompt adherence, image similarity, and efficiency.Source: §F.1 · p. 43, F APPLICATION 3: IMAGE EDITING · p. 43

F.2 FLOW-MATCHING MODEL AND BASELINES

  • Editing experiments use InstaFlow, a Stable Diffusion v1.5 variant, and exclude D-Flow because its memory overhead exceeds the available memory in the experimental setup.D-Flow requires differentiating through the full forward flow ODE integration.Source: §F.2 · p. 43

F.3 EVALUATION PROTOCOL

The editing evaluation measures preservation outside the mask, text-prompt alignment, similarity to the reference image, and computational cost.

F.3 EVALUATION PROTOCOL

  • Editing evaluation measures unmasked-region preservation with MSE, prompt adherence with CLIP-T, reference similarity with CLIP-I, and efficiency with inference time and peak GPU memory.Higher CLIP-T indicates better text-image alignment, while higher CLIP-I indicates greater visual similarity to the reference.Source: §F.3 · pp. 43–44

F.4 ADDITIONAL QUALITATIVE RESULTS

Additional editing examples show MintFlow modifying images according to prompts while preserving input context. The physical-system appendix defines heat and reaction-diffusion benchmarks, joint constraints, and matched sampling configurations for method comparisons.

F.4 ADDITIONAL QUALITATIVE RESULTS

  • Additional image-editing results show MintFlow following prompts while preserving the input image, whereas baselines tend to lose background, introduce artifacts, or produce blurry outputs.Figures 19 and 20 compare editing methods for an input image and its text prompt.Source: §F.4 · p. 44, Figure 19 · p. 45, Figure 20 · p. 46

G APPLICATION 4: PHYSICAL SYSTEM MODELING

  • The physical-system appendix evaluates conservation, initial- and boundary-condition satisfaction, agreement with numerical-solution distributions, local PDE consistency, and sensitivity to resolution and candidate-time selection.It distinguishes physical PDE time τ from generative-flow time t.Source: G APPLICATION 4: PHYSICAL SYSTEM… · p. 47, §G.1 · p. 47

G.1 PHYSICAL SYSTEMS

  • Heat training trajectories use the analytic family u(x,τ)=sin(x+ϕ)exp(−ντ), with ν∼U[1,5] and ϕ∼U[0,π], distinct from the PDE-solved benchmark reference pool.Heat fields use a 100×100 grid. RD1D follows ∂τu=ν∂xxu+ρu(1−u), with ν=5×10−3 and ρ=10−2; fields use a 128×100 grid with constant Neumann fluxes gL∈[0,0.05] and gR∈[−0.05,0].Source: §G.1 · p. 47

G.2 PHYSICAL CONSTRAINT

  • MintFlow applies one trajectory intervention to the joint physical constraint operator rather than correcting its components independently.The Heat operator combines conservation and initial-condition residuals; the RD1D operator includes global-balance, initial-condition, and boundary-condition residuals.Source: §G.2 · pp. 47–48

G.3 FLOW-MATCHING MODEL AND BASELINES

  • Physical-system comparisons use the same checkpoint, source noise, and sample-indexed constraint target for each method within an equation.Fixed-step samplers use 100 Euler steps for Heat and 200 for RD1D; the reported MintFlow configuration uses 10 candidate times and correction scale γ = 1.Source: §G.3 · p. 49

G.4 EVALUATION PROTOCOL

The evaluation compares generated physical trajectories with a numerical solver distribution conditioned on the same FFM-derived inputs and parameters. MintFlow leads on distribution and physical-consistency metrics, while ECI and PCFM minimize constraint residuals.

G.4 EVALUATION PROTOCOL

  • The numerical reference solves 1,000 physical problems conditioned on the shared bank of FFM-derived initial conditions, boundary conditions, and fitted physical parameters.It is neither the unconditional training distribution nor an unconstrained FFM sample pool.Source: §G.4 · p. 49
  • Constraint residual measures the maximum normalized constraint violation, while PDE residual and same-initial-condition simulation error assess physical consistency rather than target-IC matching.The main table reports mean constraint residual and simulation error; Table 10 also reports standard deviations and PDE residuals.Source: §G.4 · pp. 49–50
  • Set-level distances use a frozen, quadrature-weighted, mean-centered POD representation, retaining 5 Heat and 14 RD1D components at a 99.9% variance threshold.The retained dimensions explain 99.9463% of Heat variance and 99.9429% of RD1D variance.Source: §G.4 · pp. 49–50
  • FID, KID, MMD, and empirical W2 measure agreement with the conditioned numerical distribution, not divergence from the unconditional pretrained FFM distribution.KID is an unbiased estimator and can be slightly negative; empirical W2 uses Hungarian assignment for equal-sized sets.Source: §G.4 · p. 50

G.5 FULL QUANTITATIVE RESULTS

  • MintFlow has the smallest point estimate on all four distribution and both physical-consistency metrics for both systems, while ECI and PCFM have the smallest constraint residuals.The distributional comparisons use a conditioned numerical reference, not the unconditional FFM distribution.Source: §G.5 · p. 50

G.6 ADDITIONAL QUALITATIVE RESULTS

The additional physical-system examples compare generated fields with numerical rollouts initialized from each field’s own initial condition. Selected panels are diagnostic rather than evidence of set-level dominance.

G.6 ADDITIONAL QUALITATIVE RESULTS

  • Table 10 covers 1,000 generated samples for Heat and RD1D under initial-condition, boundary-condition, and conservation constraints.Its caption frames the benchmark as a trade-off between constraint satisfaction and distribution preservation.Source: Table 10 · p. 51
  • In selected examples, ECI and PCFM show broad errors, whereas MintFlow is closer to the numerical rollout from its own generated initial condition.Because panels show selected samples and omit some variants, they do not establish set-level dominance.Source: §G.6 · pp. 50–51
  • The reported physical metrics distinguish explicit constraint satisfaction from dynamical self-consistency: a field can have a small constraint residual yet depart from its own implied PDE solution.Simulation error measures relative trajectory error after integrating a numerical solver from the generated initial condition.Source: §G.6 · p. 51
  • Figure 21’s first row shows generated spacetime fields and its second row shows pointwise absolute error to a numerical Heat trajectory initialized from each sample’s initial condition.Figures 22 and 23 provide additional Heat and RD1D qualitative results.Source: Figure 21 · p. 51, Figure 22 · p. 52, Figure 23 · p. 52

H.1 ABLATION ON INTEGRATION RESOLUTION

The ablations examine sensitivity to integration resolution and to the regularization used for intervention-time selection. Both affect the balance between constraint satisfaction and distribution preservation.

H.1 ABLATION ON INTEGRATION RESOLUTION

  • Increasing Euler steps from 50 to 100, 200, and 400 monotonically improves FID, constraint residual, and same-IC simulation error, with more noticeable degradation below 100 steps.The ablation varies integration resolution while holding the other sampler settings fixed.Source: §H.1 · p. 53
  • The finite-difference PDE residual improves through 200 steps but rises slightly at 400; the headline Heat comparison uses 100 steps to match fixed-step methods.Higher resolutions quantify numerical sensitivity rather than define the reported operating point.Source: §H.1 · p. 53

H.2 ABLATION ON CANDIDATE TIME SELECTION AND OPTIMAL-TIME REGULARIZATION.

  • Varying λ trades off constraint satisfaction against distribution preservation: larger λ favors later interventions, which leave less time for the pretrained flow to absorb the correction.Performance is relatively stable across a moderate λ range; at small λ, the candidate grid governs selected times more than the regularizer.Source: §H.2 · p. 53

H.3 ABLATION ON INTERVENTION STEPS

The image-editing ablation varies the number of trajectory interventions to balance prompt adherence with preservation of the input. Moderate intervention counts are sufficient once results stabilize.

H.3 ABLATION ON INTERVENTION STEPS

  • Too few interventions can produce prompt-aligned images that deviate substantially from the input, while increasing N progressively improves input consistency while retaining prompt adherence.A single or small number of interventions is often insufficient to satisfy both the prompt and the input constraint.Source: §H.3 · p. 53
  • After a certain number of interventions, editing results converge and remain largely unchanged, suggesting a moderate N is sufficient.The figure describes improved background preservation and changes concentrated in masked regions as N increases.Source: §H.3 · p. 54, Figure 24 · p. 54
  • Even with many interventions, the method continues to generate plausible images within the learned distribution.The authors interpret this as evidence that the minimal-intervention strategy avoids excessive perturbations.Source: §H.3 · p. 54

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