Osmotic Guidance for Generative Flows

This paper studies guidance for flow-based generative models from the perspective of intervention efficiency, an important aspect that is often less explicitly considered in existing guidance research. To address the question of how to achieve the same instantaneous guidance effect with minimal modification to the pretrained dynamics, we formulate guidance as a local constrained optimization problem that minimizes intervention energy subject to a prescribed objective-change rate, yielding a unique canonical minimum-energy solution. Using the marginal equivalence between deterministic and stochastic dynamics, we further show that this solution admits a relative osmotic representation. This leads to osmotic guidance (OG), a training-free framework that provides a unified view of guidance and supports diverse guidance objectives through objective-induced osmotic fields with a shared energy-calibration rule. We instantiate OG for entropy, reward, target-density, and conditional guidance, and introduce a two-anchor preview for efficient strength selection. Experiments on synthetic flow models and image datasets demonstrate effective guidance, improved intervention efficiency, and empirical support for the proposed canonical minimum-energy principle.