# Energy-Weighted Flow Matching: Unlocking Continuous Normalizing Flows for Efficient and Scalable Boltzmann Sampling

arXiv: https://arxiv.org/abs/2509.03726 · HTML: https://mlsanity.com/paper/2509.03726/

## Key facts

- Cited by 7 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 37 of its 56 references that are in the map.
- First posted on arXiv 3 Sep 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 37)

- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [Diffusion Models: A Comprehensive Survey of Methods and Applications](https://mlsanity.com/paper/2209.00796/) (2022)
- [E(n) Equivariant Graph Neural Networks](https://mlsanity.com/paper/2102.09844/) (2021)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [Flow Matching Guide and Code](https://mlsanity.com/paper/2412.06264/) (2024)
- [Equivariant Flows: Exact Likelihood Generative Learning for Symmetric Densities](https://mlsanity.com/paper/2006.02425/) (2020)
- [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)
- [Improved sampling via learned diffusions](https://mlsanity.com/paper/2307.01198/) (2023)
- [Adjoint Sampling: Highly Scalable Diffusion Samplers via Adjoint Matching](https://mlsanity.com/paper/2504.11713/) (2025)
- [Energy-Weighted Flow Matching for Offline Reinforcement Learning](https://mlsanity.com/paper/2503.04975/) (2025)
- [NETS: A Non-Equilibrium Transport Sampler](https://mlsanity.com/paper/2410.02711/) (2024)
- [Sequential Controlled Langevin Diffusions](https://mlsanity.com/paper/2412.07081/) (2024)
- [Transferable Boltzmann Generators](https://mlsanity.com/paper/2406.14426/) (2024)
- [SE(3) Equivariant Augmented Coupling Flows](https://mlsanity.com/paper/2308.10364/) (2023)
- [Underdamped Diffusion Bridges with Applications to Sampling](https://mlsanity.com/paper/2503.01006/) (2025)
- [Progressive Inference-Time Annealing of Diffusion Models for Sampling from Boltzmann Densities](https://mlsanity.com/paper/2506.16471/) (2025)
- [Liouville Flow Importance Sampler](https://mlsanity.com/paper/2405.06672/) (2024)
- [Trust Region Constrained Measure Transport in Path Space for Stochastic Optimal Control and Inference](https://mlsanity.com/paper/2508.12511/) (2025)
- [A Framework for Adaptive MCMC Targeting Multimodal Distributions](https://mlsanity.com/paper/1812.02609/) (2018)
- [Training Neural Samplers with Reverse Diffusive KL Divergence](https://mlsanity.com/paper/2410.12456/) (2024)
- [BNEM: A Boltzmann Sampler Based on Bootstrapped Noised Energy Matching](https://mlsanity.com/paper/2409.09787/) (2024)

## Built on by (most cited first, 7 of 7)

- [Scalable Boltzmann Generators for equilibrium sampling of large-scale materials](https://mlsanity.com/paper/2509.25486/) (2025)
- [Diffusion LLMs are Natural Adversaries for any LLM](https://mlsanity.com/paper/2511.00203/) (2025)
- [Enhanced Sampling for Efficient Learning of Coarse-Grained Machine Learning Potentials](https://mlsanity.com/paper/2510.11148/) (2025)
- [Learning Structure, Energy, and Dynamics: A Survey of Artificial Intelligence for Protein Dynamics](https://mlsanity.com/paper/2604.25244/) (2026)
- [Neural Non-Equilibrium Hamiltonian Monte Carlo for Corrected Boltzmann Sampling](https://mlsanity.com/paper/2607.15682/) (2026)
- [Neural Transport Nested Sampling](https://mlsanity.com/paper/2609.29413/) (2026)
- [Reinforcement Learning via Fixed-point Flow Matching](https://mlsanity.com/paper/2610.12848/) (2026)

## Abstract (arXiv)

Sampling from unnormalized target distributions, e.g. Boltzmann distributions μtarget(x) propto exp(-E(x)/T), is fundamental to many scientific applications yet computationally challenging due to complex, high-dimensional energy landscapes. Existing approaches applying modern generative models to Boltzmann distributions either require large datasets of samples drawn from the target distribution or, when using only energy evaluations for training, cannot efficiently leverage the expressivity of advanced architectures like continuous normalizing flows that have shown promise for molecular sampling. To address these shortcomings, we introduce Energy-Weighted Flow Matching (EWFM), a novel training objective enabling continuous normalizing flows to model Boltzmann distributions using only energy function evaluations. Our objective reformulates conditional flow matching via importance sampling, allowing training with samples from arbitrary proposal distributions. Based on this objective, we develop two algorithms: iterative EWFM (iEWFM), which progressively refines proposals through iterative training, and annealed EWFM (aEWFM), which additionally incorporates temperature annealing for challenging energy landscapes. On benchmark systems, including challenging 55-particle Lennard-Jones clusters, our algorithms demonstrate sample quality competitive with established energy-only methods while requiring up to three orders of magnitude fewer energy evaluations.