用拉格朗日对偶流实现生成过程的非线性约束，无需复杂优化。
Constrained Flow Matching via Lagrangian Dual Flows
- 在去噪过程中同步流动对偶状态变量，自动满足约束
- 避免投影、伪逆和子优化问题，计算开销更低
- 适用于机器人、物理模拟等需复杂约束的场景
流匹配是强大的生成建模工具，但机器人、规划和物理模拟等新兴应用要求生成结果在推理时满足特定约束。这类约束通常复杂且高度非线性，现有针对线性约束（如图像修复）的方法难以适用，而基于投影或优化的替代方案代价高昂。本文提出拉格朗日对偶流，一种基于拉格朗日对偶动力学的新型约束生成方法。通过在生成样本的同时流动对偶协态变量，可保证非线性约束的严格满足，无需在去噪过程中进行昂贵的优化子问题、伪逆或投影操作。所得算法简单高效，并在流匹配与数值优化中的原-对偶方法间建立了新的理论联系。
原文摘要 · Abstract (English)
Flow matching is a powerful tool for generative modeling, but emerging applications in robotics, planning, and physics require inference-time constraints on generated outputs. Such constraints are often complex and highly nonlinear. As a result, methods designed for linear constraints like image inpainting are rarely sufficient, and projection or optimization-based alternatives can be prohibitively expensive. In this paper, we introduce Lagrangian Dual Flows, a new family of constrained generation techniques based on Lagrangian dual dynamics. By simply flowing a dual co-state alongside generated samples, we can guarantee nonlinear constraint satisfaction without expensive optimization subproblems, pseudoinverses, or projection steps during the denoising process. The resulting constrained generation algorithms are simple, effective, and open new theoretical connections between flow matching and primal-dual methods in numerical optimization.
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