Unlocking the Duality between Flow and Field Matching

2026-02-02

AI Summary

This work investigates whether conditional flow matching (CFM) and interacting field matching (IFM) are fundamentally equivalent. By establishing a bijection between forward-unique IFM and CFM, the study reveals their equivalence within a specific subclass for the first time, while demonstrating that general IFM possesses strictly greater expressivity—encompassing, for instance, energy-based flow matching (EFM). Integrating probabilistic path modeling, physics-inspired field theory, and generative dynamics analysis, the paper provides a probabilistic interpretation of IFM within the Poisson flow framework and introduces a novel CFM training approach grounded in IFM principles. These contributions extend the theoretical foundations of generative modeling and clarify the relationship between distinct flow-matching paradigms.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Relational Probabilistic Models
• Computer Vision: Diffusion Models for Vision

Application Category

• User Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Social Networks and Social Media: Generative AI / large language models and their impact on social systems

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.

Problem

• Conditional Flow Matching
• Interaction Field Matching
• generative dynamics
• duality
• field matching

Innovation

• Conditional Flow Matching
• Interaction Field Matching
• generative modeling
• duality
• Poisson Flow Generative Models