Unlocking the Duality between Flow and Field Matching

TL;DR

This paper explores the relationship between Flow Matching and Interaction Field Matching frameworks, revealing a duality that enhances understanding and offers new techniques for generative modeling.

Contribution

It establishes a bijection between conditional flow matching and a subclass of interaction field matching, and demonstrates that general IFM is more expressive than CFM.

Findings

CFM and forward-only IFM are equivalent under a bijection.

General IFM includes models beyond CFM, such as electrostatic field matching.

The duality provides new probabilistic interpretations and techniques for generative models.

Abstract

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.