LyapuFlow: Controlling Generative Flows with Lyapunov Feedback for Inverse Problems

2026-10-03

AI Summary

This study addresses the poor test-time constraint enforcement of pretrained generative flow models in inverse problems by proposing LyapuFlow. Grounded in Lyapunov feedback control theory, the framework predicts terminal samples and solves for minimum-norm controls to strictly satisfy physical or observational constraints while preserving pretrained dynamics. Its core innovation lies in introducing a Lyapunov descent condition coupled with a trust-region mechanism, enabling precise test-time guidance without retraining. Experimental results demonstrate that LyapuFlow significantly outperforms prevailing state-of-the-art methods in both data and latent spaces across scientific machine learning and image inverse problem benchmarks.

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.

Problem

• Generative Flows
• Inverse Problems
• Test-time Constraints
• Lyapunov Feedback Control

Innovation

• Lyapunov feedback control
• Generative flow models
• Inverse problems
• Minimum-norm control
• Test-time guidance