发表机构
School of Mathematical Sciences, Shanghai Jiao Tong University（上海交通大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究提出PMOT势匹配最优传输框架，结合CNF与广义Benamou-Brenier形式，实现$p$-最优传输的精确动力学建模，在合成基准、高维表格数据及颜色变换任务中表现出良好性能。
AI 中文摘要
我们提出了势匹配最优传输（PMOT），这是一种针对具有$c_p(x,y)=\|x-y\|^p$的一般$p$-代价最优传输的势流框架。PMOT以所选指数$p$的广义Benamou-Brenier形式，用标量势参数化连续归一化流（CNF）速度场。它沿由模型自身端点确定的直桥，通过自诱导匹配损失训练势梯度，同时允许灵活的终端分布匹配。我们的主要结果确立了零损失精确性：在给定的正则性、精确终端匹配和唯一性假设下，任何零损失解都满足广义Benamou-Brenier最优系统，并恢复相应的$p$-最优传输映射和动力学。在合成基准上，PMOT学习到的$p$-特定映射与对应的$p$-匹配OT参考一致；它作为基于似然的密度模型在高维表格数据上仍具竞争力；基于最大均值差异（MMD）的颜色变换实验则展示了灵活的基于样本的终端匹配能力。
英文摘要
We introduce Potential Matching Optimal Transport (PMOT), a potential-flow framework for general $p$-cost optimal transport with $c_p(x,y)=\|x-y\|^p$. PMOT parameterizes the CNF velocity field with a scalar potential in the generalized Benamou--Brenier form for the chosen exponent $p$. It trains the potential gradient with a self-induced matching loss along straight bridges determined by the model's own endpoints, while allowing flexible terminal distribution matching. Our main result establishes zero-loss exactness: under the stated regularity, exact terminal matching, and uniqueness assumptions, any zero-loss solution satisfies the generalized Benamou--Brenier optimality system and recovers the corresponding $p$-optimal transport map and dynamics. On synthetic benchmarks, PMOT learns $p$-specific maps that agree with the corresponding $p$-matched OT references. It also remains competitive as a likelihood-based density model on high-dimensional tabular data, and an MMD-based color transformation experiment demonstrates flexible sample-based terminal matching.
