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Lyapunov Guidance: A Unified Framework for Stabilizing Generative Flows

This paper introduces LyaGuide, a unified framework that stabilizes generative flow matching by formulating guidance as a Lyapunov control problem, thereby providing explicit stability guarantees and unifying various existing guidance strategies through a novel pseudo-projection operator.

Original authors: Jingdong Zhang, Xinze Li, Yize Jiang, Luan Yang, Minkai Xu, Junhong Liu

Published 2026-07-17

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CC BY 4.0

Original authors: Jingdong Zhang, Xinze Li, Yize Jiang, Luan Yang, Minkai Xu, Junhong Liu

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to teach a robot to draw a perfect picture of a cat. You don't just want any cat; you want a fluffy, orange tabby sitting on a windowsill. In the world of artificial intelligence, this is called "generative modeling." Scientists have built powerful tools, like Flow Matching, that act like a magical river. This river starts with a chaotic, swirling mist (random noise) and flows smoothly until it settles into a clear, beautiful image of a cat. The river's path is pre-calculated by a smart computer program.

But what happens when you want to change the destination? Maybe you want a black cat, or a cat wearing a hat, or you need the image to fit a specific puzzle piece (like fixing a blurry photo). Usually, to change the river's destination, you have to drain the whole river, rebuild the banks, and retrain the computer from scratch. This is slow, expensive, and frustrating. Alternatively, you can try to gently push the river in the right direction while it's flowing. This is called "guidance." However, current ways of pushing the river are often like guessing in the dark; you might push too hard and crash the boat, or not hard enough and miss the target. There's no guarantee the river will stay on course.

This is where a new idea called Lyapunov Control comes in. Think of it as a mathematical "safety net" or a "gravity map." In physics, a Lyapunov function is like a landscape with hills and valleys. If you roll a ball down a hill, it naturally rolls toward the bottom (the valley). If you can prove that your river is always rolling "downhill" toward your desired target, you know it will never get lost or spin out of control. It guarantees stability.

The paper "Lyapunov Guidance: A Unified Framework for Stabilizing Generative Flows" introduces a new tool called LyaGuide. The researchers propose that instead of guessing how to steer the generative river, we should treat the steering as a control problem with a built-in safety guarantee. They discovered that many different ways of guiding AI (like using labels, rewards, or energy scores) are actually all doing the same thing: they are all trying to make the AI's path roll down a specific "Lyapunov hill" toward the answer.

The team's main finding is that they can mathematically prove that guiding these AI flows is exactly the same as controlling a system to be stable. They didn't just say this; they proved it with a theorem. To make this work in real life, they invented a "pseudo-projection" operator. Imagine you are driving a car and you want to turn left, but your steering wheel is sticky and might overcorrect. The pseudo-projection is like a smart mechanic who instantly tweaks your steering angle just enough to ensure you stay on the road, without needing to rebuild the whole car. This tweak is a simple, closed-form math formula that can be added to existing AI systems without needing to retrain them.

The paper shows that this method works in two main ways. First, if you are an expert who knows the rules (like a physicist who knows the laws of gravity), you can write down the "hill" yourself. Second, if you don't know the rules but have a few examples of good outcomes (like a few pictures of the cat you want), the system can learn the shape of the hill from those few examples.

The researchers tested LyaGuide on everything from simple 2D shapes to complex tasks like fixing blurry photos, planning robot movements, and generating molecular structures. In every case, adding this "safety net" made the AI faster, more accurate, and much more reliable. The AI reached the target destination with fewer steps and fewer mistakes. The paper suggests that this approach is a universal fix that can be plugged into almost any existing flow-matching system to make it safer and smarter, without the heavy cost of retraining. It turns the chaotic art of AI guidance into a stable, predictable science.

Technical Summary: Lyapunov Guidance (LyaGuide)

Problem Statement

Flow matching has established itself as a computationally efficient and mathematically principled framework for learning complex data distributions, serving as a deterministic alternative to stochastic diffusion models. However, adapting pretrained flow models to new downstream tasks (e.g., incorporating class labels, structural constraints, or reward functions) remains challenging. Direct retraining or fine-tuning is often computationally expensive. While post-training guidance offers a flexible alternative by modifying inference dynamics without retraining, existing methods are largely heuristic or approximation-based. Crucially, these methods lack explicit theoretical guarantees regarding the stability and reliability of the guided generation process, particularly when integrating heterogeneous forms of prior information.

Methodology

The authors propose LyaGuide, a unified framework that reformulates guided flow matching as a Lyapunov control problem. The core methodology rests on three pillars:

1. Theoretical Equivalence

The paper establishes a theoretical equivalence between guided flow matching and Lyapunov control.

Concept: The energy or potential function defining a conditional distribution is interpreted as a Lyapunov function ( $V$ ), and the added guidance term is viewed as a control input ( $c_t$ ) that stabilizes the generative dynamics toward task-preferred regions.
Equivalence (Theorem 2): Finding a guidance term to perform conditional sampling is equivalent to finding a controller that satisfies a local Lyapunov condition. This unifies diverse strategies—such as classifier guidance, reward guidance, and energy-based guidance—under a single control-theoretic interpretation where the guidance minimizes $V(x)$ during generation.

2. Pseudo-Projection Operator

To address the limitation that learned or heuristic guidance terms may not rigorously satisfy stability conditions across the entire state space, the authors introduce a pseudo-projection operator with a closed-form expression (Theorem 3).

Function: Given a candidate guidance term $c_t$ , the operator projects it onto the admissible set of Lyapunov-stable controls.
Mechanism: It modifies the component of the guidance along the gradient of the Lyapunov function ( $\nabla V$ ) to ensure the Lyapunov decrease condition ( $\nabla V \cdot (u_t + c_t) < -\delta V$ ) is met.
Advantage: Unlike exact projection methods that require solving online quadratic programs (computationally expensive in high dimensions), this pseudo-projection is analytically explicit, efficient, and easy to integrate into existing pipelines.

3. Practical Settings

The framework supports two distinct scenarios for obtaining the Lyapunov function $V$ :

Model-Driven (Explicit Priors): Domain knowledge is explicitly formulated as an analytical potential $V(x)$ (e.g., $V(x) = -\log p(y|x)$ for classifier guidance or $V(x) = -R(x)$ for reward guidance).
Data-Driven (Implicit Priors): When explicit formulations are unavailable, $V$ is learned from limited task-specific data (few-shot preference pairs) via supervised regression. The authors propose a "local-minima-aware" training strategy using soft importance weighting to ensure the learned potential correctly identifies high-density regions.

Key Contributions

Unified Theoretical Framework: The paper provides a control-theoretic interpretation of guided flow matching, proving its equivalence to Lyapunov control. This unifies representative guidance settings (classifier, reward, energy-based, and inverse problems) under a single stability analysis.
Stabilization Mechanism: The introduction of the pseudo-projection operator provides a principled, closed-form method to enforce Lyapunov stability on any candidate guidance term, offering explicit guarantees for conditional sampling.
LyaGuide Implementation: The development of a practical framework supporting both model-driven and data-driven settings. It is designed to be compatible with existing guidance methods, introducing minimal computational overhead.

Experimental Results

The authors evaluate LyaGuide across synthetic benchmarks, image inverse problems, reinforcement learning (RL) planning, and energy-based modeling (EBM).

Synthetic Benchmarks: On 2D tasks (e.g., 8-Gaussian mixture, S-curve), LyaGuide consistently improves sample quality and trajectory stability. It eliminates spurious samples and noise observed in baseline methods (e.g., contrastive energy guidance, learned guidance) and accelerates convergence, achieving closer proximity to the target distribution at earlier inference times.
Image Inverse Problems: On CelebA-HQ tasks (box inpainting, super-resolution, deblurring), LyaGuide-enhanced baselines show consistent improvements in FID, LPIPS, PSNR, and SSIM compared to original guidance methods, demonstrating robustness without additional hyperparameter tuning.
RL Planning: In D4RL locomotion tasks, LyaGuide improves normalized returns across various settings (Medium-Expert, Medium, Medium-Replay) for both CFM and OT-CFM backbones, indicating more reliable trajectory optimization.
Energy-Based Modeling: Applied to Energy Matching (EM) tasks, LyaGuide reduces the number of integration steps required for convergence and accelerates the decay of Wasserstein-2 distance, improving sampling efficiency.

Significance and Claims

The paper claims that LyaGuide bridges the gap between theoretical stability analysis and practical generative modeling. By interpreting guidance as a control input and enforcing Lyapunov conditions, the framework offers explicit stability guarantees that are currently missing in heuristic guidance methods.

The authors emphasize that their approach:

Unifies disparate guidance strategies under a common mathematical principle.
Stabilizes generative flows without requiring model retraining.
Accelerates inference by inducing contraction toward high-density regions.
Generalizes to scenarios where priors are either analytically defined or must be learned from sparse data.

The work suggests that this control-theoretic perspective opens new opportunities for deploying pre-trained generative models in domains requiring rigorous reliability, such as molecular design and safety-critical planning, while maintaining computational efficiency.