Generative models often map noise to data by matching flows or scores, but these approaches become cumbersome for incorporating partial observations or additional priors. Inspired by recent advances in Wasserstein gradient flows, we propose Energy Matching, a framework that unifies flow-based approaches with the flexibility of energy-based models (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 generation (FID 3.97 compared to 8.61), while retaining the simulation-free training of transport-based approaches away from the data manifold. Additionally, we exploit the flexibility of our method and introduce an interaction energy for diverse mode exploration. Our approach focuses on learning a static scalar potential energy -- without time conditioning, auxiliary generators, or additional networks -- marking a significant departure from recent EBM methods. We believe this simplified framework significantly advances EBM capabilities and paves the way for their broader adoption in generative modeling across diverse domains.

本研究解决了现有生成模型在处理部分观测或额外先验时的局限性，提出了一个名为“能量匹配”的新框架，结合了基于流的方法与能量模型的灵活性。研究表明，该方法在CIFAR-10生成任务上取得了显著的性能提升，同时简化了训练过程，这为基于能量的模型在生成建模领域的广泛应用奠定了基础。

能量匹配：统一流匹配和基于能量的生成建模