# LyapuFlow: Controlling Generative Flows with Lyapunov Feedback for Inverse Problems

arXiv: https://arxiv.org/abs/2610.04326 · HTML: https://mlsanity.com/paper/2610.04326/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 34 of its 45 references that are in the map.
- First posted on arXiv 3 Oct 2026.

## Builds on (most cited first, 30 of 34)

- [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021)
- [Classifier-Free Diffusion Guidance](https://mlsanity.com/paper/2207.12598/) (2022)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Scaling Rectified Flow Transformers for High-Resolution Image Synthesis](https://mlsanity.com/paper/2403.03206/) (2024)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Diffusion Posterior Sampling for General Noisy Inverse Problems](https://mlsanity.com/paper/2209.14687/) (2022)
- [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023)
- [PixArt-α: Fast Training of Diffusion Transformer for Photorealistic Text-to-Image Synthesis](https://mlsanity.com/paper/2310.00426/) (2023)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [SeeSR: Towards Semantics-Aware Real-World Image Super-Resolution](https://mlsanity.com/paper/2311.16518/) (2023)
- [Poseidon: Efficient Foundation Models for PDEs](https://mlsanity.com/paper/2405.19101/) (2024)
- [A Survey on Diffusion Models for Inverse Problems](https://mlsanity.com/paper/2410.00083/) (2024)
- [Improving Diffusion Inverse Problem Solving with Decoupled Noise Annealing](https://mlsanity.com/paper/2407.01521/) (2024)
- [Solving Linear Inverse Problems Provably via Posterior Sampling with Latent Diffusion Models](https://mlsanity.com/paper/2307.00619/) (2023)
- [D-Flow: Differentiating through Flows for Controlled Generation](https://mlsanity.com/paper/2402.14017/) (2024)
- [Constrained Synthesis with Projected Diffusion Models](https://mlsanity.com/paper/2402.03559/) (2024)
- [FlowDPS: Flow-Driven Posterior Sampling for Inverse Problems](https://mlsanity.com/paper/2503.08136/) (2025)
- [Functional Flow Matching](https://mlsanity.com/paper/2305.17209/) (2023)
- [Learning Physical Models that Can Respect Conservation Laws](https://mlsanity.com/paper/2302.11002/) (2023)
- [xDiT: an Inference Engine for Diffusion Transformers (DiTs) with Massive Parallelism](https://mlsanity.com/paper/2411.01738/) (2024)
- [Physics-Constrained Flow Matching: Sampling Generative Models with Hard Constraints](https://mlsanity.com/paper/2506.04171/) (2025)
- [Gradient-Free Generation for Hard-Constrained Systems](https://mlsanity.com/paper/2412.01786/) (2024)
- [Training Free Guided Flow Matching with Optimal Control](https://mlsanity.com/paper/2410.18070/) (2024)
- [Solving Inverse Problems with FLAIR](https://mlsanity.com/paper/2506.02680/) (2025)
- [Rectified-CFG++ for Flow Based Models](https://mlsanity.com/paper/2510.07631/) (2025)
- [Toward Diffusible High-Dimensional Latent Spaces: A Frequency Perspective](https://mlsanity.com/paper/2511.22249/) (2025)
- [End-to-End Probabilistic Framework for Learning with Hard Constraints](https://mlsanity.com/paper/2506.07003/) (2025)
- [FlowLPS: Langevin-Proximal Sampling for Flow-based Inverse Problem Solvers](https://mlsanity.com/paper/2512.07150/) (2025)
- [UniConFlow: A Unified Constrained Flow-Matching Framework for Certified Motion Planning](https://mlsanity.com/paper/2506.02955/) (2025)
- [SAD-Flower: Flow Matching for Safe, Admissible, and Dynamically Consistent Planning](https://mlsanity.com/paper/2511.05355/) (2025)

## Abstract (arXiv)

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.