# Iterated Energy-based Flow Matching for Sampling from Boltzmann Densities

arXiv: https://arxiv.org/abs/2408.16249 · HTML: https://mlsanity.com/paper/2408.16249/

## Key facts

- Cited by 13 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 13 of its 18 references that are in the map.
- First posted on arXiv 29 Aug 2024.

## 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, 13 of 13)

- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [E(n) Equivariant Graph Neural Networks](https://mlsanity.com/paper/2102.09844/) (2021)
- [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023)
- [Flow Network based Generative Models for Non-Iterative Diverse Candidate Generation](https://mlsanity.com/paper/2106.04399/) (2021)
- [Path Integral Sampler: a stochastic control approach for sampling](https://mlsanity.com/paper/2111.15141/) (2021)
- [Denoising Diffusion Samplers](https://mlsanity.com/paper/2302.13834/) (2023)
- [An optimal control perspective on diffusion-based generative modeling](https://mlsanity.com/paper/2211.01364/) (2022)
- [Equivariant flow matching](https://mlsanity.com/paper/2306.15030/) (2023)
- [Flow Annealed Importance Sampling Bootstrap](https://mlsanity.com/paper/2208.01893/) (2022)
- [Iterated Denoising Energy Matching for Sampling from Boltzmann Densities](https://mlsanity.com/paper/2402.06121/) (2024)
- [A theory of continuous generative flow networks](https://mlsanity.com/paper/2301.12594/) (2023)
- [Normalizing flows for atomic solids](https://mlsanity.com/paper/2111.08696/) (2021)

## Built on by (most cited first, 13 of 13)

- [Feynman-Kac Correctors in Diffusion: Annealing, Guidance, and Product of Experts](https://mlsanity.com/paper/2503.02819/) (2025)
- [BNEM: A Boltzmann Sampler Based on Bootstrapped Noised Energy Matching](https://mlsanity.com/paper/2409.09787/) (2024)
- [Temperature-Annealed Boltzmann Generators](https://mlsanity.com/paper/2501.19077/) (2025)
- [Energy-Weighted Flow Matching: Unlocking Continuous Normalizing Flows for Efficient and Scalable Boltzmann Sampling](https://mlsanity.com/paper/2509.03726/) (2025)
- [Neural Flow Samplers with Shortcut Models](https://mlsanity.com/paper/2502.07337/) (2025)
- [Energy-based generator matching: A neural sampler for general state space](https://mlsanity.com/paper/2505.19646/) (2025)
- [FlowSynth: Instrument Generation Through Distributional Flow Matching and Test-Time Search](https://mlsanity.com/paper/2510.21667/) (2025)
- [Ambient Dataloops: Generative Models for Dataset Refinement](https://mlsanity.com/paper/2601.15417/) (2026)
- [Potential Score Matching: Debiasing Molecular Structure Sampling with Potential Energy Guidance](https://mlsanity.com/paper/2503.14569/) (2025)
- [FES-FM: Free Energy Surface Sampling via Reduced Flow Matching](https://mlsanity.com/paper/2605.00337/) (2026)
- [Neural Non-Equilibrium Hamiltonian Monte Carlo for Corrected Boltzmann Sampling](https://mlsanity.com/paper/2607.15682/) (2026)
- [Mode Coverage in Normalizing Flow Boltzmann Generators via Log-Ratio Variation](https://mlsanity.com/paper/2609.09473/) (2026)
- [Reinforcement Learning via Fixed-point Flow Matching](https://mlsanity.com/paper/2610.12848/) (2026)

## Abstract (arXiv)

In this work, we consider the problem of training a generator from evaluations of energy functions or unnormalized densities. This is a fundamental problem in probabilistic inference, which is crucial for scientific applications such as learning the 3D coordinate distribution of a molecule. To solve this problem, we propose iterated energy-based flow matching (iEFM), the first off-policy approach to train continuous normalizing flow (CNF) models from unnormalized densities. We introduce the simulation-free energy-based flow matching objective, which trains the model to predict the Monte Carlo estimation of the marginal vector field constructed from known energy functions. Our framework is general and can be extended to variance-exploding (VE) and optimal transport (OT) conditional probability paths. We evaluate iEFM on a two-dimensional Gaussian mixture model (GMM) and an eight-dimensional four-particle double-well potential (DW-4) energy function. Our results demonstrate that iEFM outperforms existing methods, showcasing its potential for efficient and scalable probabilistic modeling in complex high-dimensional systems.