MintFlow: Minimal Trajectory Intervention for Constrained Flow Matching

2026-09-30

AI Summary

This study addresses the issue that enforcing constraints during sampling often causes severe deviations from the pretrained distribution. To this end, it proposes a training-free framework that formulates constraint enforcement as minimal intervention on flow trajectories. By deriving closed-form solutions via adjoint equations, the method eliminates iterative optimization overhead and adaptively selects optimal intervention timings to balance constraint satisfaction with distribution fidelity in perturbed intermediate states. This flow matching-based minimal trajectory intervention technique requires no additional training. It significantly improves constraint satisfaction rates across visual generation and physical system tasks while preserving generative distribution quality superior to existing state-of-the-art methods.

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}. 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. 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. 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.

Problem

• constrained sampling
• flow matching
• distribution preservation
• generative modeling

Innovation

• Constrained Flow Matching
• Training-free Sampling
• Minimal Trajectory Intervention
• Adjoint Formulation
• Adaptive Intervention Time