LyapuFlow: Controlling Generative Flows with Lyapunov Feedback for Inverse Problems

Minseon Gwak; Hans Hao-Hsun Hsu; Danielle C. Maddix; N. Benjamin Erichson

cs.LGOct 3, 2026

LyapuFlow: Controlling Generative Flows with Lyapunov Feedback for Inverse Problems

Authors: Minseon Gwak, Hans Hao-Hsun Hsu, Danielle C. Maddix, N. Benjamin Erichson

Organizations: Lawrence Berkeley National Lab · International Computer Science Institute · Georgia Institute of Technology · Siemens

Abstract

Pretrained flow models are now widely used as generative priors in science and vision, where inference-time guidance enables test-time constraints without retraining. Existing methods use projection, posterior sampling, or iterative optimization of the generative trajectory. We propose LyapuFlow, an alternative based on Lyapunov feedback control. At each sampling step, LyapuFlow predicts the terminal sample towards which the current flow is evolving, and evaluates the constraint violation on this prediction. Then, we compute the minimum-norm control that satisfies a prescribed Lyapunov decrease condition. The resulting control remains inactive when the uncontrolled dynamics already reduce the constraint violation at the prescribed rate. Otherwise, it provides a corrective update within a feedback trust region that prevents the control from dominating the pretrained dynamics. We demonstrate LyapuFlow in both data and latent spaces, outperforming alternatives spanning different mechanisms for test-time constraint enforcement in scientific machine learning and image inverse problems.

Figures & tables

Figure 1: Test-time constraints for scientific and image inverse problems. (a) Physical constraints guide PDE solution generation. (b) Measurements guide image super-resolution and deblurring.

Figure 2: Illustration of LyapuFlow. At each integration step, LyapuFlow evaluates the pretrained velocity, predicts the terminal sample, computes the minimum-norm feedback from the Lyapunov decrease condition, bounds the feedback using the trust region, and advances the controlled flow. (a) When the pretrained dynamics already satisfy the Lyapunov decrease condition, no feedback is applied. (b) Otherwise, LyapuFlow applies the bounded control.

Figure 3

Dataset	Metric $\downarrow$	LyapuFlow	DPS	SAD-Flower	FlowDPS	ProFlow	PCFM	OC-Flow	FFM [-2pt]w/ Proj.
1D Heat	$\mathrm{MMSE}/10^{-2}$	0.012	0.221	3.463	2.127	0.485	0.115	0.134	5.206
	$\mathrm{SMSE}/10^{-2}$	0.004	0.002	1.835	1.234	0.335	0.700	0.339	2.809
	FPD	0.006	0.093	1.064	0.826	0.377	0.639	0.353	1.481
	$\mathrm{CE(IC)}/10^{-2}$	0	0	0	0	0	0	0	0
	$\mathrm{CE(CL)}/10^{-2}$	0	0	0.003	0.001	0.002	0	0.001	0.001
1D Burgers	$\mathrm{CMMSE}/10^{-2}$	0.163	0.254	6.874	7.578	9.326	4.303	4.172	8.266

Table 1: Quantitative evaluation for physics-constrained generation. Bold: best; underline: second-best. All methods apply the same number of final projections (Burgers: 3; other tasks: 1).

Figure 3: Average 1D Burgers solutions within a fixed-initial-condition group (top) and pointwise absolute error to the ground truth (bottom), subject to the IC, ND constraints over the first five time steps, and CL constraints. See Section B.7 for results on other PDE datasets.

	AFHQ 1k (768 $\times$ 768)
	Super-resolution x4 (Bicubic)				Deblurring (Gauss)				Deblurring (Motion)
Method	PSNR $\uparrow$	SSIM $\uparrow$	FID $\downarrow$	LPIPS $\downarrow$	PSNR $\uparrow$	SSIM $\uparrow$	FID $\downarrow$	LPIPS $\downarrow$	PSNR $\uparrow$	SSIM $\uparrow$	FID $\downarrow$	LPIPS $\downarrow$
FlowChef	29.43	0.798	27.30	0.095	25.22	0.717	40.81	0.274	26.33	0.739	40.83	0.230
FlowDPS	28.24	0.746	16.05	0.122	25.41	0.680	26.87	0.244	25.95	0.693	22.78	0.212
LyapuFlow (Ours)	30.05	0.829	9.68	0.045	26.88	0.723	20.21	0.172	27.03	0.740	20.47	0.166
	FFHQ 1k (768 $\times$ 768)

Table 2: Quantitative evaluation for image inverse problems. All experiments use Stable Diffusion 3.5-Medium with 28 NFEs. Bold: best; Underline: second best.

Figure 4: Qualitative comparison of reconstruction fidelity. LyapuFlow avoids hallucinated structures (top) while better preserving fine details (bottom).

Method	$\frac{\mathrm{CMMSE}}{10^{-2}}$	$\frac{\mathrm{CSMSE}}{10^{-2}}$	FPD	$\frac{\mathrm{CE(IC)}}{10^{-2}}$	$\frac{\mathrm{CE(ND)}}{10^{-2}}$	$\frac{\mathrm{CE(CL)}}{10^{-2}}$	$\frac{\mathrm{CMMSE}}{10^{-2}}$	$\frac{\mathrm{CSMSE}}{10^{-2}}$	FPD	$\frac{\mathrm{CE(IC)}}{10^{-2}}$	$\frac{\mathrm{CE(CL)}}{10^{-2}}$
	1D Burgers						2D NS
LyapuFlow (Ours)	0.165	0.213	0.200	0	1.532	0.978	5.457	2.480	0.568	0	0
w/o feedback trust region	0.285	0.237	0.474	0	1.303	0.896	5.497	2.473	0.576	0	0
w/o time derivative	0.574	0.401	1.890	0	4.757	3.067	6.038	2.712	0.761	0	0

Table 3: Ablations on feedback trust region and time derivative.

Figure 5: Task adaptability of DPS and LyapuFlow to base step size and decay rate, respectively.

Figure 6: Qualitative comparison of reconstruction fidelity at different NFE budgets.

Appendix figures & tables26 assets

Supplementary material from the paper’s appendix.

Appendix

Task	Spatial dimension	Temporal dimension	$d$	Train samples
1D Heat	$100$	$100$	$10{,}000$	5,000
1D Burgers	$101$	$101$	$10{,}201$	6,400
1D RD	$128$	$100$	$12{,}800$	6,400
2D NS	$64\times 64$	$50$	$204{,}800$	10,000

Appendix

Table 4: Summary of PDE datasets.

Task	Type	$m$	Residual
1D Heat	Linear IC	100	$x(:,0)-x^{\rm ref}(:,0)$
	Linear CL	99	$\sum_{s}x(s,\tau)\Delta s-\sum_{s}x(s,0)\Delta s$ , $\tau=1,\ldots,99$
	Total	199
1D Burgers	Linear IC	101	$x(:,0)-x^{\rm ref}(:,0)$
	Nonlinear Dynamics	495	$\mathcal{R}_{j}^{n}=x_{j}^{n+1}-x_{j}^{n}+\frac{\Delta\tau}{\Delta s}[G(x_{j}^{n},x_{j+1}^{n})-G(x_{j-1}^{n},x_{j}^{n})]$ , $n=0,\ldots,4$
	Nonlinear CL	100	$M^{n}-M^{0}+\Delta\tau\sum_{q=0}^{n-1}(\Phi^{q}+\Phi^{q+1})/2$ , $n=1,\ldots,100$

Appendix

Table 5: Summary of the test-time PDE constraints.

Dataset	Selected conditions	Reference fetch per condition	Total
1D Heat	1 IC ( $\phi=\pi/4$ )	512 analytic $\nu$ -trajectories, $\nu\sim\mathcal{U}[1,5]$	512
1D Reaction–Diffusion	16 IC	32 BC trajectories per IC	512
1D Burgers	16 IC	32 BC trajectories per IC	512
2D Navier–Stokes	10 IC (vorticity)	10 forcing trajectories per IC (vorticity)	100

Appendix

Table 6: Construction of the test-reference data used for PDE simulation. For each selected condition, reference trajectories are fetched from the indicated complementary condition set.

Dataset	LyapuFlow	DPS	SAD-Flower	FlowDPS	ProFlow	OC-Flow
1D Heat	$\alpha=50$ ; $c_{\mathrm{tr}}=10$	$\eta_{\mathrm{DPS}}=0.2$	$\alpha_{\mathrm{SF}}=14$	$K_{iter}=3$ ; $\eta_{\mathrm{FD}}=0.02$	$K_{iter}=3$ ; $\eta_{\mathrm{PF}}=0.1$	$K_{iter}=3$ ; $\beta=0.995$ ; $\eta_{\mathrm{OC}}=0.5$
1D Burgers	$\alpha=10$ ; $c_{\mathrm{tr}}=1$	$\eta_{\mathrm{DPS}}=0.4$	$\alpha_{\mathrm{SF}}=5$	$K_{iter}=3$ ; $\eta_{\mathrm{FD}}=0.06$	$K_{iter}=3$ ; $\eta_{\mathrm{PF}}=0.1$	$K_{iter}=3$ ; $\beta=0.995$ ; $\eta_{\mathrm{OC}}=2.5$
1D RD	$\alpha=30$ ; $c_{\mathrm{tr}}=1$	$\eta_{\mathrm{DPS}}=0.2$	$\alpha_{\mathrm{SF}}=5$	$K_{iter}=3$ ; $\eta_{\mathrm{FD}}=0.06$	$K_{iter}=3$ ; $\eta_{\mathrm{PF}}=0.5$	$K_{iter}=3$ ; $\beta=0.995$ ; $\eta_{\mathrm{OC}}=0.5$
2D NS	$\alpha=50$ ; $c_{\mathrm{tr}}=1$	$\eta_{\mathrm{DPS}}=1$	$\alpha_{\mathrm{SF}}=14$	$K_{iter}=3$ ; $\eta_{\mathrm{FD}}=0.08$	$K_{iter}=3$ ; $\eta_{\mathrm{PF}}=0.5$	$K_{iter}=3$ ; $\beta=0.995$ ; $\eta_{\mathrm{OC}}=0.5$

Appendix

Table 7: Hyperparameters used in physics-constrained generation. Hyperparameter sweeps use

N=64

samples per dataset and

N_{\mathrm{idx}}\in\{16,10\}

selected indices for multi-condition datasets.

\alpha

and

\alpha_{\mathrm{SF}}

are the LyapuFlow and SAD-Flower gains, respectively;

\eta_{\mathrm{DPS}}

is the DPS base step size for

\eta_{\mathrm{DPS}}/\|{\bm{h}}({\bm{x}})\|

;

K_{iter}

is the number of iterations;

\eta_{\mathrm{FD}}

\eta_{\mathrm{PF}}

, and

\eta_{\mathrm{OC}}

are the FlowDPS, ProFlow, and OC-Flow step sizes, respectively.

Dataset	Metric $\downarrow$	LyapuFlow (Ours)	DPS	SAD-Flower	FlowDPS	ProFlow	PCFM	OC-Flow
1D Heat	Second / sample	1.106	0.802	0.401	0.71	0.774	0.618	2.992
1D Heat	Peak memory (GiB)	0.167	0.157	0.115	0.115	0.833	0.493	0.545
1D Burgers	Second / sample	1.87	1.29	0.579	2.242	2.295	1.712	3
1D Burgers	Peak memory (GiB)	0.168	0.158	0.141	0.141	0.862	0.62	0.561
1D RD	Second / sample	1.351	0.959	0.475	1.297	1.375	0.987	2.867
1D RD	Peak memory (GiB)	0.186	0.174	0.119	0.119	1.31	0.748	0.805

Appendix

Table 8: Sampling efficiency measured on the same NVIDIA A100 GPU with batch size 1 and 100 sampling steps. Wall time is the mean elapsed time per generated sample over 5 timed runs after 1 warm-up run; peak allocated memory is the maximum CUDA allocation across those runs. Both measurements include one-step final projection.

Method	$\frac{\mathrm{MMSE}}{10^{-2}}$	$\frac{\mathrm{SMSE}}{10^{-2}}$	FPD	$\frac{\mathrm{CE(IC)}}{10^{-2}}$	$\frac{\mathrm{CE(CL)}}{10^{-2}}$	$\frac{\mathrm{CMMSE}}{10^{-2}}$	$\frac{\mathrm{CSMSE}}{10^{-2}}$	FPD	$\frac{\mathrm{CE(IC)}}{10^{-2}}$	$\frac{\mathrm{CE(ND)}}{10^{-2}}$	$\frac{\mathrm{CE(CL)}}{10^{-2}}$
	1D Heat					1D Burgers
LyapuFlow (Ours)	0.012	0.004	0.006	0	0	0.163	0.212	0.204	0	0.010	0.005
w/o trust region	0.012	0.004	0.006	0	0	0.283	0.235	0.458	0	0.008	0.004
w/o time derivative	0.067	0.018	0.043	0	0	0.584	0.398	2.240	0	2.208	4.800

Appendix

Table 9: Ablations on feedback trust region and time derivative across all PDE tasks.

Figure 7: Hyperparameter sensitivity across the evaluated PDE samplers. Each method is evaluated over five values of its principal guidance parameter:

\alpha

for LyapuFlow,

\eta_{\mathrm{DPS}}

for DPS,

\alpha_{\mathrm{SF}}

for SAD-Flower,

\eta_{\mathrm{FD}}

for FlowDPS,

\eta_{\mathrm{PF}}

for ProFlow, and

\eta_{\mathrm{OC}}

for OC-Flow. PCFM has no tuned guidance parameter and is shown as a horizontal reference. The 1D tasks use a logarithmic vertical scale, and non-finite evaluations are marked with

\times

Figure 8: Qualitative comparison on 1D Heat under initial-condition and conservation-law constraints.

Panel of Figure 9 — Figure 9: Qualitative comparisons on 1D Burgers under initial-condition, local-dynamics, and mass-balance constraints.

Panel of Figure 10 — Figure 10: Qualitative comparisons on 1D Reaction–Diffusion under initial-condition and balance-law constraints.

Figure 11: Qualitative comparisons on 2D Navier–Stokes under initial-vorticity and total-vorticity constraints.

Figure 12: Qualitative comparisons on 2D Navier–Stokes under initial-vorticity and total-vorticity constraints.

Dataset	Task	$\alpha$	$k_{\mathrm{sg}}$	$c_{\mathrm{tr}}$	$\alpha$	$k_{\mathrm{sg}}$	$c_{\mathrm{tr}}$
		28 NFEs			40 NFEs
AFHQ	Super-resolution (Bicubic)	5	28	1.0	5	40	1.0
AFHQ	Deblurring (Gaussian)	5	3	1.0	5	40	1.0
AFHQ	Deblurring (Motion)	5	7	1.0	5	40	1.0
FFHQ	Super-resolution (Bicubic)	5	28	1.0	5	40	1.0
FFHQ	Deblurring (Gaussian)	5	3	1.0	5	7	1.0

Appendix

Table 10: Task-specific LyapuFlow settings for 28 and 40 NFEs.

k_{\mathrm{sg}}

denotes the control step at which stop-gradient on

v_{\theta}

is turned off (zero-indexed);

k_{\mathrm{sg}}

equal to the number of NFEs keeps it on throughout sampling.

Sampler	$N_{iter}$	NFE	SR $\times 4$		Deblur (Gaussian)		Deblur (Motion)		Second / image
			PSNR	LPIPS	PSNR	LPIPS	PSNR	LPIPS
LyapuFlow (Ours)	1	40	30.63	0.041	28.76	0.117	28.87	0.120	13.8
FlowChef	1	40	30.90	0.060	26.62	0.219	28.52	0.175	11.9
FlowDPS	3	40	30.09	0.101	26.57	0.208	27.83	0.171	26.6
FlowLPS	15	40	24.86	0.060	28.87	0.075	30.08	0.065	112.6
FLAIR	15	40	22.84	0.044	26.29	0.094	26.93	0.088	112.8

Appendix

Table 11: Trade-off between sampling speed and quality, evaluated on the first 10 FFHQ images.

	AFHQ 1k (768 $\times$ 768)
Method	Super-resolution x4 (Bicubic)				Deblurring (Gauss)				Deblurring (Motion)
Method	PSNR $\uparrow$	SSIM $\uparrow$	FID $\downarrow$	LPIPS $\downarrow$	PSNR $\uparrow$	SSIM $\uparrow$	FID $\downarrow$	LPIPS $\downarrow$	PSNR $\uparrow$	SSIM $\uparrow$	FID $\downarrow$	LPIPS $\downarrow$
FlowChef	29.67	0.807	22.34	0.077	25.42	0.719	38.55	0.256	26.56	0.743	36.37	0.2107
FlowDPS	28.51	0.764	18.45	0.124	25.67	0.697	29.65	0.244	26.20	0.710	25.21	0.2106
LyapuFlow (Ours)	30.35	0.833	8.93	0.040	27.42	0.732	17.86	0.137	27.57	0.750	17.06	0.1429
	FFHQ 1k (768 $\times$ 768)

Appendix

Table 12: Quantitative evaluation for image inverse problems. All experiments use Stable Diffusion 3.5-Medium with 40 NFEs. Bold: best; Underline: second best.

Figure 13: Ablation on gradient switch step

k_{\mathrm{sg}}

on 10 FFHQ images.

Figure 14: Spectral-density comparisons across datasets and inverse-problem tasks for $28$ NFEs.

Figure 15: AFHQ bicubic super-resolution using Stable Diffusion 3.5-Medium for 28 NFEs.

Figure 16: AFHQ Gaussian deblurring using Stable Diffusion 3.5-Medium for 28 NFEs.

Figure 17: AFHQ motion deblurring using Stable Diffusion 3.5-Medium for 28 NFEs.

Figure 18: FFHQ bicubic super-resolution using Stable Diffusion 3.5-Medium for 28 NFEs.

Figure 19: FFHQ Gaussian deblurring using Stable Diffusion 3.5-Medium for 28 NFEs.

Figure 20: FFHQ motion deblurring using Stable Diffusion 3.5-Medium for 28 NFEs.

Figure 21: DIV2K bicubic super-resolution using Stable Diffusion 3.5-Medium for 28 NFEs.

Figure 22: DIV2K Gaussian deblurring using Stable Diffusion 3.5-Medium for 28 NFEs.

Figure 23: DIV2K motion deblurring using Stable Diffusion 3.5-Medium for 28 NFEs.

Constrained Generative Modeling

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

See all 182 papers

Lyapunov Guidance: A Unified Framework for Stabilizing Generative Flows
Jul 15, 2026Jingdong Zhang, Xinze Li, Yize Jiang +3Flow Matching Lyapunov-Based Control
Department of Mathematics and I-X, Imperial College London, London, United Kingdom · MicroCyto, Beijing, China · Research Institute of Intelligent Complex Systems, Fudan University, Shanghai, China +1
MintFlow: Minimal Trajectory Intervention for Constrained Flow Matching
Sep 30, 2026Yesom Park, Kelvin Kan, Qifan Chen +2Constrained Generative Modeling Constrained Sampling
Department of Mathematics, University of California, Los Angeles · Brookhaven National Laboratory
ConicFlow: constrained Inference-time optimal control for flow matching models
Oct 9, 2026Shervin Khalafi, Kasra Fallah, Sung Cho +3Constrained Generative Modeling Constrained Sampling
University of Pennsylvania · Columbia University · University of Tennessee, Knoxville

Jul 15, 2026J/K move · Enter open · S save

Lyapunov Guidance: A Unified Framework for Stabilizing Generative Flows

Jingdong Zhang, Xinze Li, Yize Jiang, Luan Yang+2

Department of Mathematics and I-X, Imperial College London, London, United Kingdom · MicroCyto, Beijing, China · Research Institute of Intelligent Complex Systems, Fudan University, Shanghai, China+1

Flow Matching Lyapunov-Based Control Inference-Time Guidance

Open paper AskPDF arXiv

Flow matching has emerged as an effective framework for learning complex data distributions, but adapting pretrained flow models to new tasks often requires computationally expensive retraining. Post-training guidance provides a more efficient alternative, but existing methods are largely heuristic and offer no explicit stability guarantees. We address this limitation by proposing LyaGuide, a unified Lyapunov-guided framework that formulates flow guidance as a Lyapunov control problem. Our main theoretical result establishes an equivalence between guided flow matching and Lyapunov control, thereby unifying common guidance strategies, such as classifier guidance, reward guidance, and energy-based guidance, within a single control-theoretic framework. To enforce the Lyapunov condition, we introduce a pseudo-projection operator with a closed-form expression that endows learned or heuristic guidance terms with explicit stability guarantees. LyaGuide supports two practical settings: a model-driven setting, where the target guidance distribution is specified through a known Lyapunov function, and a data-driven setting, where the guidance is adapted from task-specific downstream data. LyaGuide is compatible with existing guidance methods, introduces minimal additional computational overhead, and is straightforward to integrate in practice. Extensive experiments on synthetic benchmarks, image inverse problems, reinforcement learning planning, and energy-based modeling demonstrate consistent improvements in sample quality, guidance fidelity, and robustness, while maintaining computational efficiency.

Abstract

Constrained Generative Modeling

Explore similar work

Lyapunov Guidance: A Unified Framework for Stabilizing Generative Flows

MintFlow: Minimal Trajectory Intervention for Constrained Flow Matching

ConicFlow: constrained Inference-time optimal control for flow matching models