Few-step flow-map generators, such as consistency models and MeanFlow, accelerate sampling by directly learning long-range transport maps between noise and data. However, these models are typically deterministic, which makes them difficult to optimize with reinforcement learning (RL) post-training methods that require stochastic trajectories and well-defined likelihood ratios. Existing SDE-based stochasticization techniques are designed for velocity-based samplers with infinitesimal or finely discretized transitions, and therefore do not directly apply to long-range flow maps. In this work, we propose Flow-Map GRPO, an online RL post-training framework for deterministic few-step flow-map generators. The key component is Anchored Stochastic Flow Map Composition (ASFMC), a path-preserving stochasticization mechanism that introduces randomness through anchor-based conditional resampling while preserving the original marginal probability path of the deterministic flow map. We derive GRPO objectives for both single-time and two-time flow-map parameterizations. Experiments on few-step FLUX-based text-to-image generators, including MeanFlow and sCM, show that Flow-Map GRPO improves pretrained deterministic flow-map models across reward-based, perceptual, and task-level evaluation metrics. Our results demonstrate that deterministic few-step flow-map generators can be effectively aligned with RL post-training without modifying their original model parameterization or retraining them as native stochastic models.
Flow and diffusion models generate high-quality samples in many modalities; however, many network evaluations are required during inference due to numerical integration of an underlying differential equation. Flow maps alleviate this problem by learning the solution map of the differential equation directly, enabling few-step sampling. Yet, current methods are restricted to approximating the solution map of ODEs. These methods can be used to learn the transition kernel of an SDE, thereby obtaining a solution map that recovers the marginal distributions of the process (weak convergence) rather than the solution path (strong convergence). We propose Strong Stochastic Flow Maps (SSFMs) as a novel framework for learning the strong solution map of additive-noise SDEs, directly generalizing deterministic flow maps to the stochastic setting. Further, a polynomial approximation to Brownian motion is introduced and shown to converge pathwise. These results enable a simulation-free training objective for the solution map of diffusion models. We demonstrate that SSFMs outperform previous stochastic flow map methods on image generation and enable few-step sampling of molecular systems.
Sam McCallum, Zander W. Blasingame, Timothy Herschell +3
In generative modeling, we often wish to produce samples that maximize a user-specified reward such as aesthetic quality or alignment with human preferences, a problem known as \textit{guidance}. Despite their widespread use, existing guidance methods either require expensive multi-particle, many-step schemes or rely on poorly understood approximations. We reformulate guidance as a \textit{deterministic optimal control problem}, yielding a hierarchy of algorithms that subsumes existing approaches at the coarsest level. We show that the \textit{flow map}, an object of significant recent interest for its role in fast inference, arises naturally in the optimal solution. Based on this observation, we propose \textbf{Flow Map Reward Guidance (FMRG)}: a training-free, \textit{single-trajectory} framework that uses the flow map to both integrate and guide the flow. At text-to-image scale, FMRG matches or surpasses baselines across inverse problems and reward-guided generation with \textbf{as few as 3 NFEs}, giving at least an order-of-magnitude speedup in comparison to prior state of the art. Code is available at https://github.com/jrrhuang/fmrg.
Reinforcement learning is increasingly used to align image generators with reward signals, and Flow-GRPO recently extended this paradigm to flow-matching models by treating the denoising sampler as a stochastic policy that can be optimized from reward feedback. Training in this setting is unstable in a way specific to multi-step denoising: the policy update changes systematically across denoising steps, with importance ratios drifting below one, becoming increasingly dispersed, clipping at different rates, and leaving fewer usable samples late in training. Prior work treats these effects as separate failure modes and addresses each with a hand-tuned stabilizer. We show instead that they arise from a single per-step quantity, which we call path variance. This quantity is determined exactly by the sampler's Gaussian transition kernel and can be estimated cheaply during training. This reframes instability as a resource that can be measured and budgeted rather than a collection of symptoms to repair. Our method, λ-Controlled GRPO, calibrates importance-ratio behavior from this predicted law rather than from noisy empirical statistics, and allocates gradient effort across denoising steps according to their predicted cost. The two scales governing the update are fixed by standard policy choices rather than introduced as free tuning parameters. On a text-to-image model under two reward settings, rendering difficult target text scored by optical character recognition and matching human preferences scored by a preference model, λ-Controlled GRPO improves both text accuracy and preference reward over the strongest empirical stabilizer. It also keeps late-step path variance within its intended budget, precisely where the baseline systematically overshoots. The result is a Flow-GRPO update calibrated by its own transition law rather than stabilized after instability appears.