MintFlow: Minimal Trajectory Intervention for Constrained Flow Matching

MintFlow：用于约束流匹配的最小轨迹干预

Abstract: Flow matching models excel at generative modeling, and many downstream applications require their samples to satisfy prescribed constraints, such as observed measurements and physical laws. However, existing constrained samplers often face a trade-off: \textit{enforcing constraints can substantially displace samples from the pretrained data distribution}.

摘要： 流匹配（Flow matching）模型在生成建模方面表现出色，而许多下游应用要求其生成的样本满足预设的约束条件，例如观测测量值和物理定律。然而，现有的约束采样器往往面临一种权衡：强制执行约束可能会导致样本严重偏离预训练的数据分布。

To address this trade-off, we introduce \textbf{MintFlow}, a training-free constrained sampling framework that formulates constraint enforcement as a minimal intervention on the pretrained flow trajectory. MintFlow seeks the minimal perturbation of an intermediate flow state such that its subsequent evolution under the pretrained flow field satisfies the target constraint.

为了解决这一权衡问题，我们引入了 MintFlow，这是一个无需训练的约束采样框架，它将约束执行表述为对预训练流轨迹的最小干预。MintFlow 旨在寻找中间流状态的最小扰动，使得该状态在预训练流场下的后续演化能够满足目标约束。

By minimally perturbing the flow state while keeping the pretrained flow field unchanged, MintFlow enforces the constraint while minimizing unnecessary deviation from the pretrained distribution. An adjoint formulation yields a closed-form expression for this perturbation, eliminating expensive iterative optimization.

通过在保持预训练流场不变的同时对流状态进行最小化扰动，MintFlow 在执行约束的同时，最大限度地减少了对预训练分布的不必要偏离。伴随公式（Adjoint formulation）为这种扰动提供了闭式解，从而消除了昂贵的迭代优化过程。

Furthermore, MintFlow adaptively selects the intervention time to balance the required perturbation magnitude with its amplification by the remaining flow. Across a range of tasks in generative vision and physical system modeling, MintFlow achieves competitive constraint satisfaction while preserving the pretrained generative distribution substantially better than state-of-the-art constrained methods.

此外，MintFlow 能够自适应地选择干预时间，以平衡所需的扰动幅度与剩余流对扰动的放大效应。在生成式视觉和物理系统建模的一系列任务中，MintFlow 在实现具有竞争力的约束满足度的同时，比现有的最先进约束方法更好地保留了预训练的生成分布。