Unlocking the Duality between Flow and Field Matching

Daniil Shlenskii, Alexander Varlamov, Nazar Buzun, Alexander Korotin

Keywords: Generative Models Theory, Flow Matching, Field Matching

Abstract:

Conditional Flow Matching (CFM) unifies conventional generative paradigms such as diffusion and flow-based models. Interaction Field Matching (IFM) is a recently proposed 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, which we call *forward-only* IFM. Specifically, we construct mappings between CFM and forward-only IFM and show that they induce the same generative dynamics. 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.
Our findings suggest developing generative models using both interpretations rather than treating them separately. Moreover, they highlight a novel direction for generative modeling based on backward-oriented field lines, which lies outside the conventional CFM formalism and may lead to new generative properties.

Submission Number: 121