Conditional Flow Matching (CFM) unifies conventional generative paradigms such as diffusion models and flow matching. Interaction Field Matching (IFM) is a newer framework that generalizes Electrostatic Field Matching (EFM) rooted in Poisson Flow Generative Models (PFGM). While both frameworks define generative dynamics, they start from different objects: CFM specifies a conditional probability path in data space, whereas IFM specifies a physics-inspired interaction field in an augmented data space. This raises a basic question: are CFM and IFM genuinely different, or are they two descriptions of the same underlying dynamics? We show that they coincide for a natural subclass of IFM that we call forward-only IFM. Specifically, we construct a bijection between CFM and forward-only IFM. We further show that general IFM is strictly more expressive: it includes EFM and other interaction fields that cannot be realized within the standard CFM formulation. Finally, we highlight how this duality can benefit both frameworks: it provides a probabilistic interpretation of forward-only IFM and yields novel, IFM-driven techniques for CFM.

翻译：条件流匹配（CFM）统一了传统的生成范式，如扩散模型和流匹配。交互场匹配（IFM）是一个较新的框架，它推广了根植于泊松流生成模型（PFGM）的静电场匹配（EFM）。虽然这两个框架都定义了生成动力学，但它们始于不同的对象：CFM在数据空间中指定了一个条件概率路径，而IFM在一个增广数据空间中指定了一个受物理学启发的交互场。这引出了一个基本问题：CFM和IFM是真正不同的，还是它们是对同一底层动力学的两种描述？我们证明，对于一个自然的IFM子类——我们称之为仅向前IFM——它们是重合的。具体而言，我们在CFM与仅向前IFM之间构建了一个双射。我们进一步证明，一般IFM的表达能力严格更强：它包含了EFM以及其他无法在标准CFM公式中实现的交互场。最后，我们强调了这种对偶性如何使两个框架受益：它为仅向前IFM提供了概率论解释，并为CFM带来了新颖的、由IFM驱动的技术。