MintFlow: Minimal Trajectory Intervention for Constrained Flow Matching

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

cs.AISep 30, 2026

MintFlow: Minimal Trajectory Intervention for Constrained Flow Matching

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

Organizations: Department of Mathematics, University of California, Los Angeles · Brookhaven National Laboratory

Abstract

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: enforcing constraints can substantially displace samples from the pretrained data distribution. To address this trade-off, we introduce 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.

Figures & tables

Figure 1: Balancing constraint satisfaction and distributional fidelity. Unconstrained sampling (A) generates samples from the pretrained distribution but often violates the constraints, while prior constrained sampling methods (B) enforce the constraints at the cost of large distributional shifts. The proposed method (C) achieves constraint satisfaction while better preserving the pretrained distribution. Image inpainting examples illustrate this trade-off.

Panel of Figure 2 — Figure 2: Comparison of constrained generation strategies. Steer Velocity Field modifies the flow by learning an additional velocity field. Terminal Projection projects the generated sample onto the constraint manifold. Trajectory Optimization iteratively optimizes states along the flow. In contrast, our Trajectory Intervention minimally perturbs the trajectory at an intermediate time then follows the original flow.

	AFHQ-Cat					FFHQ					Efficiency
Method	$H\downarrow$	FID $\downarrow$	PSNR $\uparrow$	SSIM $\uparrow$	LPIPS $\downarrow$	$H\downarrow$	FID $\downarrow$	PSNR $\uparrow$	SSIM $\uparrow$	LPIPS $\downarrow$	Time $\downarrow$	Memory $\downarrow$
Task: Inpainting
D-Flow	1.775E-2	23.662	20.8597	0.7180	0.1578	1.592E-2	29.692	22.7378	0.7910	0.0915	153.47	3597.02
DPS	2.583E-2	14.532	20.9537	0.6232	0.1806	2.897E-2	33.498	21.3944	0.6913	0.1146	6.61	646.74
Freedom	6.013E-2	19.999	17.8908	0.4377	0.2502	6.789E-2	39.998	18.0828	0.5334	0.1605	4.60	637.36
PNPFlow	1.926E-2	24.063	22.2547	0.6877	0.1897	2.126E-2	63.579	22.3143	0.7388	0.1281	5.56	338.28

Table 1: Quantitative comparison of methods for image inverse problems on AFHQ-Cat and FFHQ. We report constraint error

H

, FID, PSNR (dB), SSIM, LPIPS, inference time (s), and peak GPU memory (MB). Light green ( green ) indicates the top performer, while light red ( red ) represents the second-best performer across all methods for each metric (applies to all tables herein).

Panel of Figure 3 — Figure 3: Representative qualitative results for image inverse problems. We compare ground-truth images (GT), degraded inputs, and results from different methods. Inpainting and super resolution are evaluated on AFHQ-Cat, while deblurring is evaluated on FFHQ.

Panel of Figure 4 — Figure 4: Qualitative comparison of image editing results. Given an input image and its corresponding text prompt, we compare the editing results produced by different methods. Strikethrough and red text indicate the original and edited text, respectively.

Method	H $\downarrow$	CLIP-T $\uparrow$	CLIP-I $\uparrow$	Time $\downarrow$	Memory $\downarrow$
DPS	0.0115	0.2916	0.7911	9.17	8881.75
Freedom	0.0071	0.2756	0.8466	17.65	8122.47
PNPFlow	0.0503	0.2918	0.6897	42.45	6507.95
FlowChef	0.0136	0.2891	0.7939	9.29	6507.26
MintFlow	0.0069	0.2926	0.8151	8.79	6507.26

Table 2: Comparison on image editing task.

	System 1: Heat Equation					System 2: Reaction-Diffusion Equation
Method	FID $\downarrow$	MMD $\downarrow$	$\mathbf{W_{2}}\downarrow$	$\mathbf{H}\downarrow$	Sim. Error $\downarrow$	FID $\downarrow$	MMD $\downarrow$	$\mathbf{W_{2}}\downarrow$	$\mathbf{H}\downarrow$	Sim. Error $\downarrow$
Vanilla FFM	$1.255\mathrm{E}{-3}$	$2.496\mathrm{E}{-2}$	$4.872\mathrm{E}{-2}$	$5.16\mathrm{E}{-3}$	$5.57\mathrm{E}{-2}$	$1.74\mathrm{E}{-4}$	$2.130\mathrm{E}{-2}$	$2.078\mathrm{E}{-2}$	$1.907\mathrm{E}{0}$	$3.55\mathrm{E}{-2}$
DiffusionPDE	$4.712\mathrm{E}{-3}$	$5.361\mathrm{E}{-2}$	$7.431\mathrm{E}{-2}$	$1.86\mathrm{E}{-2}$	$8.49\mathrm{E}{-2}$	$9.34\mathrm{E}{-4}$	$6.063\mathrm{E}{-2}$	$3.712\mathrm{E}{-2}$	$9.56\mathrm{E}{-1}$	$3.29\mathrm{E}{-2}$
D-Flow	$1.290\mathrm{E}{-3}$	$2.554\mathrm{E}{-2}$	$4.098\mathrm{E}{-2}$	$4.71\mathrm{E}{-3}$	$4.34\mathrm{E}{-2}$	$3.95\mathrm{E}{-4}$	$3.861\mathrm{E}{-2}$	$4.085\mathrm{E}{-2}$	$4.44\mathrm{E}{-1}$	$3.02\mathrm{E}{-2}$
ECI	$1.865\mathrm{E}{-1}$	$2.067\mathrm{E}{-1}$	$4.502\mathrm{E}{-1}$	$2.27\mathrm{E}{-7}$	$4.47\mathrm{E}{-1}$	$1.829\mathrm{E}{-2}$	$1.299\mathrm{E}{-1}$	$1.647\mathrm{E}{-1}$	$8.02\mathrm{E}{-6}$	$2.28\mathrm{E}{-1}$
PCFM	$1.851\mathrm{E}{-1}$	$2.062\mathrm{E}{-1}$	$4.485\mathrm{E}{-1}$	$1.30\mathrm{E}{-7}$	$4.45\mathrm{E}{-1}$	$1.831\mathrm{E}{-2}$	$1.301\mathrm{E}{-1}$	$1.648\mathrm{E}{-1}$	$8.06\mathrm{E}{-6}$	$2.28\mathrm{E}{-1}$

Table 3: Physical system modeling benchmark results. Performance evaluation across

N=1000

generated samples comparing System 1 (Heat Equation) and System 2 (Reaction-Diffusion Equation). Lower values (

\downarrow

) indicate superior performance.

Figure 5: Qualitative physical consistency under physical constraints. The top row shows samples generated by ECI, PCFM, and MintFlow, while the bottom row illustrates the pointwise absolute discrepancy between the generated field and this sample-conditioned numerical rollout.

Panel of Figure 6 — Figure 6: Sensitivity of MintFlow to sampling steps and regularization parameter $\lambda$ .

Appendix figures & tables27 assets

Supplementary material from the paper’s appendix.

Appendix

\dot{u}=f_{\theta}(u,t),\qquad u(0)=u_{0},

Appendix

Algorithm 1 Adjoint-Based Optimal Correction

s_{i-1}<s_{1}<\cdots<s_{K}<T

Appendix

Algorithm 2 Iterative Adjoint-Based Optimal Correction

Ablation studies: App. H
Task	Details	Data	FM Method	Constraint
Image generation	App. D	MNIST, CIFAR-10	Flow Matching	Intensity, fidelity
Image inverse problems	App. E (Sec. 4.1 )	AFHQ-Cat, FFHQ	Rectified Flow	Observation consistency
Image editing	App. F (Sec. 4.2 )	PIE-Bench	Latent text-conditioned FM	Text-guided generation with region preservation
Physics-informed generation	App. G (Sec. 4.3 )	Heat, RD1D PDE	Functional Flow Matching	Boundary condition, conservation laws

Appendix

Table 4: Overview of experiments and ablation studies. We evaluate our method across diverse tasks, datasets, and flow matching (FM) models, and further conduct ablation studies. Detailed experimental settings and results are provided in the corresponding sections.

Dataset	Channels	ResBlocks	Attention	Dropout	Channel Mult.
MNIST	64	2	$\{2\}$	0.1	$(1,2,2)$
CIFAR-10	128	4	$\{2\}$	0.3	$(2,2,2)$

Appendix

Table 5: Flow Matching model configurations for image datasets.

	Distribution Metrics				Constraint	Efficiency
Method	FID $\downarrow$	KID $\downarrow$	MMD $\downarrow$	$W_{2}$ $\downarrow$	$H$ $\downarrow$	Time (s) $\downarrow$	Memory (GB) $\downarrow$
Unconstrained	0.879	0.02215	0.03158	0.938	2.79E-1	5.07	0.10
DiffusionPDE	0.899	0.06860	0.04427	1.064	5.36E-2	4.29	5.10
D-Flow	0.813	0.05342	0.04726	0.902	5.49E-2	703.40	125.27
ECI	1.851	0.09982	0.07863	1.361	3.55E-15	8.89	0.35
PCFM	1.736	0.12379	0.12064	1.317	1.43E-12	18.37	10.12

Appendix

Table 6: Quantitative results on the average-intensity constrained MNIST dataset. Light green ( green ) indicates the top performer, while light red ( red ) represents the second-best performer across all methods for each metric.

Panel of Figure 7 — Figure 7: Controlling MNIST digit stroke thickness with MintFlow using average intensity constraints. Starting from unconstrained samples, our method modifies the generated digits according to different target intensity constraints, producing thicker or thinner strokes while preserving distribution.

	Distribution Metrics				Constraint	Efficiency
Method	FID $\downarrow$	KID $\downarrow$	MMD $\downarrow$	$W_{2}$ $\downarrow$	$H$ $\downarrow$	Time (s) $\downarrow$	Memory (GB) $\downarrow$
Unconstrained	0.261	0.0089	0.0093	0.0093	4.54E-3	4.45	0.12
DiffusionPDE	0.242	0.0074	0.0078	0.0069	3.59E-3	6.98	5.15
D-Flow	0.257	0.0087	0.0092	0.5076	3.91E-3	1674.73	63.76
ECI	0.932	0.0079	0.0164	0.9659	3.43E-2	11.91	0.40
PCFM	10.225	0.3210	0.3336	3.1977	1.02E-1	79.56	10.17

Appendix

Table 7: Quantitative results on the AE-based fidelity constrained MNIST dataset.

Figure 8: Uncurated results for intensity constrained generation on MNIST. We compare unconstrained samples with constrained samples generated by different methods. All samples in each row are generated from the same initial noise across methods.

Figure 9: Uncurated results for AE-based fidelity-constrained generation on MNIST. We compare unconstrained samples with constrained samples generated by different methods. All samples in each row are generated from the same initial noise across methods.

Panel of Figure 10 — Figure 10: Distribution of the selected intervention time $s^{\ast}$ on MNIST. Each histogram shows the intervention times selected by ( 8 ) across generated samples for the corresponding constraint. (a) The linear intensity constraint tends to favor earlier intervention times, whereas (b) the nonlinear fidelity constraint defined by a trained autoencoder results in a more dispersed distribution of selected intervention times.

	Distribution Metrics				Constraint	Efficiency
Method	FID $\downarrow$	KID $\downarrow$	MMD $\downarrow$	$W_{2}$ $\downarrow$	$H$ $\downarrow$	Time (s) $\downarrow$	Memory (GB) $\downarrow$
Unconstrained	11.1568	0.0421	0.0518	5.354	1.02E-1	14.17	0.44
DiffusionPDE	8.4464	0.0014	0.0012	2.9062	4.66E-2	12.08	2.25
D-Flow	9.8484	0.0019	0.0015	3.138	5.789E-4	787.30	101.01
ECI	322.500	0.3596	0.29951	17.9582	2.53E-16	19.81	0.34
PCFM	125.262	0.0950	0.0727	11.1920	1.72E-12	27.10	4.26

Appendix

Table 8: Quantitative results on the color-style constrained CIFAR10 dataset.

	Distribution Metrics				Constraint	Efficiency
Method	FID $\downarrow$	KID $\downarrow$	MMD $\downarrow$	$W_{2}$ $\downarrow$	$H$ $\downarrow$	Time (s) $\downarrow$	Memory (GB) $\downarrow$
Unconstrained	11.797	0.0053	0.0043	3.4347	2.18E-5	13.91	0.72
DiffusionPDE	10.779	0.0058	0.0042	3.3389	1.06E-5	14.82	2.59
D-Flow	9.984	0.0057	0.0042	3.3438	9.09E-6	1618.61	103.74
ECI	310.353	0.3391	0.3083	17.6227	3.17E-4	30.80	0.68
PCFM	116.953	0.0729	0.0578	10.8794	6.78E-4	49.53	4.60

Appendix

Table 9: Quantitative results on the AE-based fidelity constrained CIFAR10 dataset.

Panel of Figure 11 — Figure 11: Qualitative comparison on CIFAR-10 under different color constraints. We compare unconstrained samples with constrained samples generated by different methods. All samples in each row are generated from the same initial noise across methods.

Figure 12: Uncurated results for AE-based fidelity-constrained generation on CIFAR10. We compare unconstrained samples with constrained samples generated by different methods. All samples in each row are generated from the same initial noise across methods.

Figure 13: Uncurated inpainting results on the AFHQ-Cat dataset. The figure compares ground-truth images (GT), degraded inputs, and corresponding inpainting results from different methods.

Figure 14: Uncurated super resolution (SR) results on the AFHQ-Cat dataset. The figure compares ground-truth images (GT), degraded inputs, and corresponding SR results from different methods.

Figure 15: Uncurated deblurring results on the AFHQ-Cat dataset. The figure compares ground-truth images (GT), degraded inputs, and corresponding deblurred results from different methods.

Figure 16: Uncurated inpainting results on the FFHQ dataset. The figure compares ground-truth images (GT), degraded inputs, and corresponding inpainting results from different methods.

Figure 17: Uncurated super resolution (SR) results on the FFHQ dataset. The figure compares ground-truth images (GT), degraded inputs, and corresponding SR results from different methods.

Figure 18: Uncurated deblurring results on the FFHQ dataset. The figure compares ground-truth images (GT), degraded inputs, and corresponding deblurred results from different methods.

Panel of Figure 19 — Figure 19: Additional qualitative comparison of image editing results. Given an input image and its corresponding text prompt, we compare the editing results produced by different methods. Strikethrough and red text indicate the original and edited text, respectively.

Panel of Figure 20 — Figure 20: Additional qualitative comparison of image editing results. Given an input image and its corresponding text prompt, we compare the editing results produced by different methods. Strikethrough and red text indicate the original and edited text, respectively.

System 1: Heat Equation
	Distribution Fidelity †				Constraint ††	Physical Consistency †††
Method	FID $\downarrow$	KID $\downarrow$	MMD $\downarrow$	$\mathbf{W_{2}}\downarrow$	$\mathbf{H}\downarrow$	PDE Res. $\downarrow$	Sim. Error $\downarrow$
Vanilla FFM	$1.255\mathrm{E}{-3}$	$-3.17\mathrm{E}{-4}$	$2.496\mathrm{E}{-2}$	$4.872\mathrm{E}{-2}$	$5.16\mathrm{E}{-3}\!\pm\!2.19\mathrm{E}{-3}$	$1.103\mathrm{E}{0}\!\pm\!5.08\mathrm{E}{-1}$	$5.57\mathrm{E}{-2}\!\pm\!6.15\mathrm{E}{-2}$
DiffusionPDE	$4.712\mathrm{E}{-3}$	$1.019\mathrm{E}{-3}$	$5.361\mathrm{E}{-2}$	$7.431\mathrm{E}{-2}$	$1.86\mathrm{E}{-2}\!\pm\!2.20\mathrm{E}{-2}$	$1.044\mathrm{E}{0}\!\pm\!6.98\mathrm{E}{-1}$	$8.49\mathrm{E}{-2}\!\pm\!2.95\mathrm{E}{-2}$
D-Flow	$1.290\mathrm{E}{-3}$	$-2.10\mathrm{E}{-4}$	$2.554\mathrm{E}{-2}$	$4.098\mathrm{E}{-2}$	$4.71\mathrm{E}{-3}\!\pm\!2.08\mathrm{E}{-3}$	$7.82\mathrm{E}{-1}\!\pm\!3.76\mathrm{E}{-1}$	$4.34\mathrm{E}{-2}\!\pm\!2.46\mathrm{E}{-2}$
ECI	$1.865\mathrm{E}{-1}$	$4.004\mathrm{E}{-2}$	$2.067\mathrm{E}{-1}$	$4.502\mathrm{E}{-1}$	$2.27\mathrm{E}{-7}\!\pm\!6.45\mathrm{E}{-8}$	$1.554\mathrm{E}{0}\!\pm\!8.78\mathrm{E}{-1}$	$4.47\mathrm{E}{-1}\!\pm\!2.86\mathrm{E}{-1}$

Appendix

Table 10: Physical System Modeling Benchmark under initial conditions, boundary conditions, and conservation constraints. Performance evaluation across

N=1000

generated samples on the Heat Equation (System 1) and Reaction-Diffusion Equation (System 2). The benchmark highlights the trade-off between Constraint Satisfaction and Distribution Preservation . While strict-constraint baselines (ECI, PCFM) achieve near-zero constraint residuals, they severely degrade distributional quality. In contrast, MintFlow achieves an optimal Pareto frontier, enforcing physical constraints while preserving the pre-trained data distribution. For all metrics, lower values (

\downarrow

) indicate superior performance.

Figure 21: Additional Heat qualitative results. For each example, the first row shows the generated spacetime field for each method. The second row shows the pointwise absolute error to a numerical Heat trajectory initialized from that generated sample’s own initial condition.

Figure 22: Additional Heat qualitative results.

Panel of Figure 23 — Figure 23: Additional RD1D qualitative results.

Panel of Figure 24 — Figure 24: Comparison of image editing results with different numbers of interventions $N$ . When $N$ is too small, the generated images align with the target prompt but fail to satisfy the constraint, resulting in substantial deviations from the input image. As $N$ increases, the method better preserves the background and overall appearance of the input image while naturally modifying only the masked regions to satisfy the target prompt. Beyond a certain $N$ , the results remain largely unchanged, suggesting that a moderate number of interventions is sufficient, while even with more interventions, our method remains within the learned distribution, demonstrating its stability.

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Constrained generative modeling develops methods that produce samples satisfying structural, physical, semantic, or user-defined requirements.

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Constraint-Aware Flow Matching: Decision Aligned End-to-End Training for Constrained Sampling

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Deep generative models provide state-of-the-art performance across a wide array of applications, with recent studies showing increasing applicability for science and engineering. Despite a growing corpus of literature focused on the integration of physics-based constraints into the generation process, existing approaches fail to enforce strict constraint satisfaction while maintaining sample quality. In particular, training-free constrained sampling methods, while providing per-sample feasibility guarantees, introduce a fundamental mismatch between the training objective and the constrained sampling procedure, often leading to performance degradation. Identifying this training-sampling misalignment as a central limitation of current constrained generative modeling approaches, this paper proposes Constraint-Aware Flow Matching, a novel end-to-end framework that explicitly incorporates constraint projections into the training objective. By aligning the model's learned dynamics with the constrained sampling process, the proposed method mitigates distributional shift induced by projection-based corrections, enabling high-quality constrained generation. The proposed approach is evaluated on three challenging real-world benchmarks, illustrating the generality and efficacy of the method.

Abstract

Constrained Generative Modeling

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