Energy Matching: Unifying Flow Matching and Energy-Based Models for Generative Modeling

Michal Balcerak, Tamaz Amiranashvili, Antonio Terpin, Suprosanna Shit, Sebastian Kaltenbach, Petros Koumoutsakos, Bjoern Menze

Published 2025-04-14, updated 2025-05-14Version 2

The most widely used generative models map noise and data distributions by matching flows or scores. However, they struggle to incorporate partial observations and additional priors--something energy-based models (EBMs) handle elegantly by simply adding corresponding scalar energy terms. We address this issue by proposing Energy Matching, a framework that endows flow-based approaches with the flexibility of EBMs. Far from the data manifold, samples move along curl-free, optimal transport paths from noise to data. As they approach the data manifold, an entropic energy term guides the system into a Boltzmann equilibrium distribution, explicitly capturing the underlying likelihood structure of the data. We parameterize this dynamic with a single time-independent scalar field, which serves as both a powerful generator and a flexible prior for effective regularization of inverse problems. Our method substantially outperforms existing EBMs on CIFAR-10 and ImageNet generation in terms of fidelity, while retaining simulation-free training of transport-based approaches away from the data manifold. Furthermore, we leverage the method's flexibility to introduce an interaction energy that supports diverse mode exploration, which we demonstrate in a controlled protein-generation setting. Our approach focuses on learning a scalar potential energy--without time-conditioning, auxiliary generators, or additional networks--which marks a significant departure from recent EBM methods. We believe that this simplified framework significantly advances EBMs capabilities and paves the way for their wider adoption in generative modeling across diverse domains.