How to guide your flow: Steering flow maps for rapid test-time alignment

In generative modeling, we often wish to produce samples that satisfy a user-specified reward such as measurement consistency, aesthetic quality, or alignment with human intent, a problem known as inference-time guidance. While flow-based models enable high-quality generation, existing guidance methods either require expensive multi-particle, many-step schemes to sample from a reward-tilted distribution or rely on heuristic approximations. To design efficient algorithms, we instead reformulate guidance as a deterministic optimal control problem, rather than the stochastic control problem from which the reward tilt emerges. We find that the flow map arises naturally in the optimal solution, and that many existing flow-based guidance methods are best understood as coarse approximations that replace it with a single Euler step. Rather than relying on these approximations, we propose Flow Map Trajectory Guidance (FMTG): a principled framework that uses the flow map to both integrate and guide, enabling training-free alignment in just a few network evaluations. We demonstrate FMTG at text-to-image scale using a FLUX.1-distilled flow map, showing that it achieves comparable performance to baselines across inverse problems and image editing tasks with up to 10 times fewer function evaluations.