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

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.

@article{balcerak2025_2504.10612,
  title={ Energy Matching: Unifying Flow Matching and Energy-Based Models for Generative Modeling },
  author={ Michal Balcerak and Tamaz Amiranashvili and Antonio Terpin and Suprosanna Shit and Sebastian Kaltenbach and Petros Koumoutsakos and Bjoern Menze },
  journal={arXiv preprint arXiv:2504.10612},
  year={ 2025 }
}