Flow matching models excel at generative modeling, and many downstream tasks require samples to satisfy prescribed constraints such as observed measurements or physical laws. Existing constrained samplers, however, often face a trade‑off: enforcing constraints can substantially displace samples from the pretrained data distribution. To address this, we introduce MintFlow, a training‑free constrained sampling framework that treats constraint enforcement as a minimal intervention on the pretrained flow trajectory. MintFlow seeks the smallest perturbation of an intermediate flow state so that its subsequent evolution under the pretrained flow field satisfies the target constraint. By perturbing only the flow state while keeping the flow field unchanged, MintFlow enforces constraints while minimizing deviation from the pretrained distribution. An adjoint formulation provides a closed‑form expression for this perturbation, eliminating costly iterative optimization. Moreover, MintFlow adaptively selects the intervention time to balance the required perturbation magnitude against its amplification by the remaining flow. Experiments on generative vision tasks and physical system modeling show that MintFlow achieves competitive constraint satisfaction while preserving the pretrained generative distribution better than state‑of‑the‑art constrained methods.
Review: MintFlow achieves a compelling balance between constraint enforcement and distribution fidelity, demonstrating practical effectiveness across diverse generative scenarios.