Unlocking the Duality between Flow and Field Matching

TL;DR

The paper addresses whether Conditional Flow Matching (CFM) and Interaction Field Matching (IFM) describe the same underlying generative dynamics or are fundamentally different. It proves a constructive duality: CFM is equivalent to forward-only IFM in terms of the induced transport, with $\frac{d\mathbf{x}}{dt} = \mathbf{v}_t(\mathbf{x}) = \frac{\mathbf{E}(\mathbf{x},t)_\mathbf{x}}{\mathbf{E}(\mathbf{x},t)_t}$ and the global field $\mathbf{E}(\mathbf{x},t) = (\mathbf{E}_\mathbf{x}, \mathbf{E}_t)$ satisfying $\mathbf{E}_t = p_t$ and $\mathbf{E}_\mathbf{x} = \mathbf{v}_t p_t$. The work also shows that general IFM is strictly more expressive than CFM, as it accommodates backward-oriented field lines (e.g., EFM), expanding the class of admissible dynamics. It highlights conceptual and practical implications, including a probabilistic interpretation of forward-only IFM and IFM-inspired techniques for enhancing CFM, such as multi-sample velocity estimation and volume-coverage strategies, with Poisson Flow Generative Models (PFGM) illustrating a unifying flow-field perspective. Overall, the results unify flow- and field-based generative paradigms and suggest a cohesive pipeline for leveraging insights across both frameworks.

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.