# Equilibrium Matching: Generative Modeling with Implicit Energy-Based Models

arXiv: https://arxiv.org/abs/2510.02300 · HTML: https://mlsanity.com/paper/2510.02300/

## Key facts

- Cited by 21 papers in the map (data through 14 Oct 2026).
- Builds on 30 of its 39 references that are in the map.
- First posted on arXiv 2 Oct 2025.

## Builds on (most cited first, 30 of 30)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Scalable Diffusion Models with Transformers](https://mlsanity.com/paper/2212.09748/) (2022)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Improved Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2102.09672/) (2021)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [DPM-Solver: A Fast ODE Solver for Diffusion Probabilistic Model Sampling in Around 10 Steps](https://mlsanity.com/paper/2206.00927/) (2022)
- [Glow: Generative Flow with Invertible 1x1 Convolutions](https://mlsanity.com/paper/1807.03039/) (2018)
- [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (2022)
- [SiT: Exploring Flow and Diffusion-based Generative Models with Scalable Interpolant Transformers](https://mlsanity.com/paper/2401.08740/) (2024)
- [StyleGAN-XL: Scaling StyleGAN to Large Diverse Datasets](https://mlsanity.com/paper/2202.00273/) (2022)
- [PixelCNN++: Improving the PixelCNN with Discretized Logistic Mixture Likelihood and Other Modifications](https://mlsanity.com/paper/1701.05517/) (2017)
- [Your Classifier is Secretly an Energy Based Model and You Should Treat it Like One](https://mlsanity.com/paper/1912.03263/) (2019)
- [Understanding Diffusion Objectives as the ELBO with Simple Data Augmentation](https://mlsanity.com/paper/2303.00848/) (2023)
- [Flow Matching Guide and Code](https://mlsanity.com/paper/2412.06264/) (2024)
- [Reduce, Reuse, Recycle: Compositional Generation with Energy-Based Diffusion Models and MCMC](https://mlsanity.com/paper/2302.11552/) (2023)
- [A Theory of Generative ConvNet](https://mlsanity.com/paper/1602.03264/) (2016)
- [Improved Contrastive Divergence Training of Energy Based Models](https://mlsanity.com/paper/2012.01316/) (2020)
- [Learning Energy-Based Models by Diffusion Recovery Likelihood](https://mlsanity.com/paper/2012.08125/) (2020)
- [On the Anatomy of MCMC-Based Maximum Likelihood Learning of Energy-Based Models](https://mlsanity.com/paper/1903.12370/) (2019)
- [Diffuse and Disperse: Image Generation with Representation Regularization](https://mlsanity.com/paper/2506.09027/) (2025)
- [Is Noise Conditioning Necessary for Denoising Generative Models?](https://mlsanity.com/paper/2502.13129/) (2025)
- [Energy Matching: Unifying Flow Matching and Energy-Based Models for Generative Modeling](https://mlsanity.com/paper/2504.10612/) (2025)
- [Energy-Based Models for Anomaly Detection: A Manifold Diffusion Recovery Approach](https://mlsanity.com/paper/2310.18677/) (2023)
- [Generative Modeling by Inclusive Neural Random Fields with Applications in Image Generation and Anomaly Detection](https://mlsanity.com/paper/1806.00271/) (2018)

## Built on by (most cited first, 21 of 21)

- [Diffusion Models in Simulation-Based Inference: A Tutorial Review](https://mlsanity.com/paper/2512.20685/) (2025)
- [Blind denoising diffusion models and the blessings of dimensionality](https://mlsanity.com/paper/2602.09639/) (2026)
- [The Geometry of Noise: Why Diffusion Models Don't Need Noise Conditioning](https://mlsanity.com/paper/2602.18428/) (2026)
- [Think Consistently, Reason Efficiently: Energy-Based Calibration for Implicit Chain-of-Thought](https://mlsanity.com/paper/2511.07124/) (2025)
- [Speedrunning ImageNet Diffusion](https://mlsanity.com/paper/2512.12386/) (2025)
- [Generative Control as Optimization: Time Unconditional Flow Matching for Adaptive and Robust Robotic Control](https://mlsanity.com/paper/2603.17834/) (2026)
- [π0-EqM: Equilibrium Matching for Closed-Loop Vision-Language-Action Control](https://mlsanity.com/paper/2605.23128/) (2026)
- [DVD: Discrete Voxel Diffusion for 3D Generation and Editing](https://mlsanity.com/paper/2605.07971/) (2026)
- [What Time Is It? How Data Geometry Makes Time Conditioning Optional for Flow Matching](https://mlsanity.com/paper/2605.08344/) (2026)
- [Beckmann Transport Models: From Autonomous Flows to One-Step Maps](https://mlsanity.com/paper/2608.01692/) (2026)
- [Generative Pseudo-Force Fields for Molecular Generation](https://mlsanity.com/paper/2605.19050/) (2026)
- [Equilibrium Forcing: Adaptive Video Generation Without Noise Conditioning](https://mlsanity.com/paper/2608.14706/) (2026)
- [Conditional Denoising Model as a Physical Surrogate Model](https://mlsanity.com/paper/2601.21021/) (2026)
- [Distance Marching for Generative Modeling](https://mlsanity.com/paper/2602.02928/) (2026)
- [TerraBind: Fast and Accurate Binding Affinity Prediction through Coarse Structural Representations](https://mlsanity.com/paper/2602.07735/) (2026)
- [Recovering Hidden Reward in Diffusion-Based Policies](https://mlsanity.com/paper/2605.00623/) (2026)
- [Generating Physically Consistent Molecules with Energy-Based Models](https://mlsanity.com/paper/2605.18381/) (2026)
- [Advancing Optimal Subset Oracle via Learning Relaxation of Neural Set Functions](https://mlsanity.com/paper/2607.11555/) (2026)
- [Does Uniform Discrete Diffusion Need Time?](https://mlsanity.com/paper/2609.30977/) (2026)
- [Jumping up and down: Denoiser diffusion models for discrete ordinal data](https://mlsanity.com/paper/2610.02754/) (2026)
- [PotentialEP: Backprop-Free Training of CIFAR-Scale Generative Models via Equilibrium Propagation](https://mlsanity.com/paper/2610.14409/) (2026)

## Abstract (arXiv)

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 256×256. 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.