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

arXiv: https://arxiv.org/abs/2504.10612 · HTML: https://mlsanity.com/paper/2504.10612/

## Key facts

- Cited by 27 papers in the map (data through 14 Oct 2026).
- Placed in Flow matching, time series and science, one of 14 directions in Diffusion models.
- Builds on 46 of its 74 references that are in the map.
- First posted on arXiv 14 Apr 2025.

## Placed in

- Diffusion models / [Flow matching, time series and science](https://mlsanity.com/diffusion/flow-matching-time-series-and-science/)

## Builds on (most cited first, 30 of 46)

- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Deep Learning Face Attributes in the Wild](https://mlsanity.com/paper/1411.7766/) (2014)
- [Diffusion Posterior Sampling for General Noisy Inverse Problems](https://mlsanity.com/paper/2209.14687/) (2022)
- [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (2022)
- [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023)
- [A Downsampled Variant of ImageNet as an Alternative to the CIFAR datasets](https://mlsanity.com/paper/1707.08819/) (2017)
- [How to Train Your Energy-Based Models](https://mlsanity.com/paper/2101.03288/) (2021)
- [Inference-Time Scaling for Diffusion Models beyond Scaling Denoising Steps](https://mlsanity.com/paper/2501.09732/) (2025)
- [A Variational Perspective on Solving Inverse Problems with Diffusion Models](https://mlsanity.com/paper/2305.04391/) (2023)
- [A Survey on Diffusion Models for Inverse Problems](https://mlsanity.com/paper/2410.00083/) (2024)
- [Biological Sequence Design with GFlowNets](https://mlsanity.com/paper/2203.04115/) (2022)
- [Improving Diffusion Inverse Problem Solving with Decoupled Noise Annealing](https://mlsanity.com/paper/2407.01521/) (2024)
- [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)
- [D-Flow: Differentiating through Flows for Controlled Generation](https://mlsanity.com/paper/2402.14017/) (2024)
- [Conditioning by adaptive sampling for robust design](https://mlsanity.com/paper/1901.10060/) (2019)
- [Diffusion-based Molecule Generation with Informative Prior Bridges](https://mlsanity.com/paper/2209.00865/) (2022)
- [Action Matching: Learning Stochastic Dynamics from Samples](https://mlsanity.com/paper/2210.06662/) (2022)
- [Soft Truncation: A Universal Training Technique of Score-based Diffusion Model for High Precision Score Estimation](https://mlsanity.com/paper/2106.05527/) (2021)
- [Oops I Took A Gradient: Scalable Sampling for Discrete Distributions](https://mlsanity.com/paper/2102.04509/) (2021)
- [AdaLead: A simple and robust adaptive greedy search algorithm for sequence design](https://mlsanity.com/paper/2010.02141/) (2020)
- [On conditional diffusion models for PDE simulations](https://mlsanity.com/paper/2410.16415/) (2024)
- [Flow Priors for Linear Inverse Problems via Iterative Corrupted Trajectory Matching](https://mlsanity.com/paper/2405.18816/) (2024)
- [Is Noise Conditioning Necessary for Denoising Generative Models?](https://mlsanity.com/paper/2502.13129/) (2025)
- [A Geometric View of Data Complexity: Efficient Local Intrinsic Dimension Estimation with Diffusion Models](https://mlsanity.com/paper/2406.03537/) (2024)

## Built on by (most cited first, 27 of 27)

- [GuideFlow: Constraint-Guided Flow Matching for Planning in End-to-End Autonomous Driving](https://mlsanity.com/paper/2511.18729/) (2025)
- [Equilibrium Matching: Generative Modeling with Implicit Energy-Based Models](https://mlsanity.com/paper/2510.02300/) (2025)
- [Pinet: Optimizing hard-constrained neural networks with orthogonal projection layers](https://mlsanity.com/paper/2508.10480/) (2025)
- [Counterfactual Identifiability via Dynamic Optimal Transport](https://mlsanity.com/paper/2510.08294/) (2025)
- [Good Things Come in Pairs: Paired Autoencoders for Inverse Problems](https://mlsanity.com/paper/2505.06549/) (2025)
- [Think Consistently, Reason Efficiently: Energy-Based Calibration for Implicit Chain-of-Thought](https://mlsanity.com/paper/2511.07124/) (2025)
- [GT-SVJ: Generative-Transformer-Based Self-Supervised Video Judge For Efficient Video Reward Modeling](https://mlsanity.com/paper/2602.05202/) (2026)
- [Lagrangian Flow Matching: A Least-Action Framework for Principled Path Design](https://mlsanity.com/paper/2605.15419/) (2026)
- [Beyond Imitation: Constraint-Aware Trajectory Generation with Flow Matching For End-to-End Autonomous Driving](https://mlsanity.com/paper/2510.26292/) (2025)
- [Sketch2Colab: Sketch-Conditioned Multi-Human Animation via Controllable Flow Distillation](https://mlsanity.com/paper/2603.02190/) (2026)
- [Graph Energy Matching: Transport-Aligned Energy-Based Modeling for Graph Generation](https://mlsanity.com/paper/2603.23398/) (2026)
- [Generative Pseudo-Force Fields for Molecular Generation](https://mlsanity.com/paper/2605.19050/) (2026)
- [Energy-Guided Generative Modeling for Low-Energy Molecular Structure Discovery](https://mlsanity.com/paper/2512.22597/) (2025)
- [Repurposing Protein Language Models for Latent Flow-Based Fitness Optimization](https://mlsanity.com/paper/2602.02425/) (2026)
- [Distance Marching for Generative Modeling](https://mlsanity.com/paper/2602.02928/) (2026)
- [Recovering Hidden Reward in Diffusion-Based Policies](https://mlsanity.com/paper/2605.00623/) (2026)
- [Energy Generative Modeling: A Lyapunov-based Energy Matching Perspective](https://mlsanity.com/paper/2605.05530/) (2026)
- [Learning Normalized Energy Models for Linear Inverse Problems](https://mlsanity.com/paper/2605.15487/) (2026)
- [Generating Physically Consistent Molecules with Energy-Based Models](https://mlsanity.com/paper/2605.18381/) (2026)
- [Conflict-Aware Additive Guidance for Flow Models under Compositional Rewards](https://mlsanity.com/paper/2605.20758/) (2026)
- [MATCH: Flow Matching for Multi-View Anomaly Detection](https://mlsanity.com/paper/2606.24375/) (2026)
- [Lyapunov Guidance: A Unified Framework for Stabilizing Generative Flows](https://mlsanity.com/paper/2607.14272/) (2026)
- [Composing Flow-Matching Energies with Known Physics: Generation, OOD Detection, and Inversion on PDE Fields](https://mlsanity.com/paper/2608.18004/) (2026)
- [Observation-Conditioned Latent Energy Priors for Sparse Implicit Neural Shape Completion](https://mlsanity.com/paper/2609.03694/) (2026)
- [Flow-Matched Motion Priors: Online Optimal-Transport Rewards for Imitation Learning](https://mlsanity.com/paper/2609.15631/) (2026)
- [Manifold-Stable Flow Matching](https://mlsanity.com/paper/2609.35454/) (2026)
- [PotentialEP: Backprop-Free Training of CIFAR-Scale Generative Models via Equilibrium Propagation](https://mlsanity.com/paper/2610.14409/) (2026)

## Abstract (arXiv)

Current state-of-the-art generative models map noise to data distributions by matching flows or scores. A key limitation of these models is their inability to readily integrate available partial observations and additional priors. In contrast, energy-based models (EBMs) address this by incorporating corresponding scalar energy terms. Here, we propose Energy Matching, a framework that endows flow-based approaches with the flexibility of EBMs. Far from the data manifold, samples move from noise to data along irrotational, optimal transport paths. 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 these dynamics with a single time-independent scalar field, which serves as both a powerful generator and a flexible prior for effective regularization of inverse problems. The present 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 flexibility of the method to introduce an interaction energy that supports the exploration of diverse modes, which we demonstrate in a controlled protein generation setting. This approach learns a scalar potential energy, without time conditioning, auxiliary generators, or additional networks, marking a significant departure from recent EBM methods. We believe this simplified yet rigorous formulation significantly advances EBMs capabilities and paves the way for their wider adoption in generative modeling in diverse domains.