Equilibrium Matching: Generative Modeling with Implicit Energy-Based
  Models
Authors: name: Runqian Wang; name: Yilun Du
Published: 2025-10-02 17:59:06

Abstract: We introduce Equilibrium Matching (EqM), a generative modeling framework
built from an equilibrium dynamics perspective. EqM discards the
non-equilibrium, time-conditional dynamics in traditional diffusion and
flow-based generative models and instead learns the equilibrium gradient of an
implicit energy landscape. Through this approach, we can adopt an
optimization-based sampling process at inference time, where samples are
obtained by gradient descent on the learned landscape with adjustable step
sizes, adaptive optimizers, and adaptive compute. EqM surpasses the generation
performance of diffusion/flow models empirically, achieving an FID of 1.90 on
ImageNet 256times256. EqM is also theoretically justified to learn and
sample from the data manifold. Beyond generation, EqM is a flexible framework
that naturally handles tasks including partially noised image denoising, OOD
detection, and image composition. By replacing time-conditional velocities with
a unified equilibrium landscape, EqM offers a tighter bridge between flow and
energy-based models and a simple route to optimization-driven inference.