Diffusion and flow-based generative models dominate visual synthesis, with guidance aligning samples to user input and improving perceptual quality. However, Classifier-Free Guidance (CFG) and extrapolation-based methods are heuristic linear combinations of velocities/scores that ignore the generative manifold geometry, breaking probability conservation and driving samples off the learned manifold under strong guidance. We analyse guidance through the continuity equation and show its effect decomposes into a divergence term and a score-parallel term defined invariantly across parameterisations. We prove the divergence term blows up structurally as sampling approaches the data manifold, motivating a time-dependent schedule alongside score-parallel attenuation. The resulting plug-and-play rule, Adaptive Manifold Guidance (AdaMaG), bounds both terms at no additional inference cost. Finally, we show that most empirical heuristics for reducing saturation or improving generation quality correspond directly to the two terms in our decomposition. Across image generation benchmarks, AdaMaG improves realism, reduces hallucinations, and induces controlled desaturation in high-guidance regimes.
Flow-based generative methods offer a simple and effective framework for high-fidelity generation, yet pretrained flow models are rarely used in their vanilla conditional form: in image generation, samples without guidance often appear diffuse and lack fine-grained detail. Existing guidance techniques such as classifier-free guidance (CFG) improve fidelity but reduce sample diversity. We introduce Momentum Guidance (MG), a guidance method that improves sample quality by extrapolating the current velocity away from an exponential moving average of past velocities along the ODE trajectory, while preserving the standard one-evaluation-per-step cost. MG provides gains beyond CFG, improving the precision-recall Pareto frontier. Experiments demonstrate the effectiveness of MG across benchmarks. On ImageNet-256, MG improves FID by 36.54% without CFG and 25.42% with CFG on average across sampling settings, attaining an FID of 1.553 at 16 sampling steps. Evaluations on large flow-based models, including Stable Diffusion 3 and FLUX.1-dev, further confirm improvements across standard metrics.
Inference-time guided sampling steers state-of-the-art diffusion and flow models without fine-tuning by interpreting the generation process as a controllable trajectory. This provides a simple and flexible way to inject external constraints (e.g., cost functions or pre-trained verifiers) for controlled generation. However, existing methods often fail when composing multiple constraints simultaneously, which leads to deviations from the true data manifold. In this work, we identify root causes of this off-manifold drift and find that the approximation error scales severely with gradient misalignment. Building on these findings, we propose Conflict-Aware Additive Guidance (gcar), a lightweight and learnable method, which actively rectifies off-manifold drift by dynamically detecting and resolving gradient conflicts. We validate gcar across diverse domains, ranging from synthetic datasets and image editing to generative decision-making for planning and control. Our results demonstrate that gcar effectively rectifies off-manifold drift, surpassing baselines in generation fidelity while using light compute. Code is available at https://github.com/yuxuehui/CAR-guidance.
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.