LyapuFlow:Controlling Generative Flows with
Lyapunov Feedback for Inverse Problems

Minseon Gwak1,4, Hans Hao-Hsun Hsu2, Danielle C. Maddix3, N. Benjamin Erichson1,4

1 Lawrence Berkeley National Lab · 2 Georgia Institute of Technology · 3 Siemens · 4 International Computer Science Institute

Paper arXiv BibTeX

Abstract

Pretrained flow models provide powerful generative priors for scientific and image inverse problems, but their samples must also satisfy constraints supplied at inference time. LyapuFlow guides generation with Lyapunov feedback: at each sampling step, it predicts the terminal sample, measures its constraint violation, and computes the smallest correction needed to meet a prescribed decrease condition. The controller stays inactive when the original flow already makes sufficient progress, and a trust region keeps corrections from overwhelming the pretrained dynamics. Experiments in data and latent spaces show improved sample fidelity with competitive constraint satisfaction, without retraining the generative model.

Method Overview

LyapuFlow adds inference-time feedback to a pretrained flow model. It evaluates constraints on a predicted final sample, then chooses a minimum-norm correction using a Lyapunov decrease condition. A trust region limits the correction relative to the learned flow. No retraining is required.

01

Predict

Estimate the final sample from the current state and pretrained velocity.

02

Evaluate

Measure the predicted sample’s constraint violation.

03

Control

Apply bounded feedback only when the nominal dynamics need correction.

Results · Data space

Scientific inverse problems

LyapuFlow guides generation under physical constraints across heat, Burgers, and reaction–diffusion equations.

1D Heat

Under initial-condition and conservation-law constraints, the heat-equation examples compare generated fields and their absolute errors across guidance methods.

Qualitative comparison of generated one-dimensional heat solutions under initial-condition and conservation-law constraints.

1D Burgers

The Burgers examples show average generated solutions and pointwise absolute error under initial-condition, local-dynamics, and conservation constraints.

Comparison of generated Burgers solutions and absolute errors across LyapuFlow and baseline methods.

Guidance sensitivity

The comparison shows how seven PDE samplers respond to their main guidance hyperparameters.

Hyperparameter sensitivity across seven PDE samplers, including LyapuFlow and DPS.

Results · Latent space

Image inverse problems

Latent-space experiments study super-resolution and deblurring while keeping reconstructions consistent with the measurements.

Reconstruction fidelity

Super-resolution and motion-deblurring examples compare the measurement, competing reconstructions, LyapuFlow, and ground truth. Enlarged regions make fine-detail differences easier to see.

Image reconstruction comparisons showing observations, baseline reconstructions, LyapuFlow, and ground truth with detail crops.

Sampling budget

Gaussian deblurring reconstructions compare FlowDPS and LyapuFlow at 28 and 40 function evaluations, showing how the restored details change with the sampling budget.

Gaussian deblurring comparison of FlowDPS and LyapuFlow with 28 and 40 function evaluations, alongside measurements and ground truth.

BibTeX

@article{lyapuflow,
  title={LyapuFlow: Controlling Generative Flows with Lyapunov Feedback for Inverse Problems},
  author={Gwak, Minseon and Hsu, Hans Hao-Hsun and Maddix, Danielle C. and Erichson, N. Benjamin},
  journal={arXiv preprint arXiv:2610.04326},
  year={2026}
}