MintFlow Minimizes Trajectory Changes in Constrained Flow Matching

Summary

Flow matching models are effective generative models, but applying constraints such as observed measurements or physical laws can move their samples away from the pretrained distribution. The paper introduces MintFlow, a training-free constrained sampling framework that treats constraint enforcement as a minimal intervention on the pretrained flow trajectory. It searches for the smallest perturbation to an intermediate flow state whose later evolution under the unchanged pretrained flow field satisfies the target constraint. An adjoint formulation provides a closed-form perturbation, removing the need for expensive iterative optimization. MintFlow also chooses the intervention time adaptively, balancing the size of the required perturbation against how strongly the remaining flow amplifies it. Experiments across generative vision and physical-system modeling show competitive constraint satisfaction while preserving the pretrained generative distribution substantially better than existing constrained samplers.