# Scalable Boltzmann Generators for equilibrium sampling of large-scale materials

arXiv: https://arxiv.org/abs/2509.25486 · HTML: https://mlsanity.com/paper/2509.25486/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 65 references that are in the map.
- First posted on arXiv 29 Sep 2025.

## Builds on (most cited first, 27 of 27)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [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)
- [NICE: Non-linear Independent Components Estimation](https://mlsanity.com/paper/1410.8516/) (2014)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [Neural Spline Flows](https://mlsanity.com/paper/1906.04032/) (2019)
- [Learning Local Equivariant Representations for Large-Scale Atomistic Dynamics](https://mlsanity.com/paper/2204.05249/) (2022)
- [Equivariant Flows: Exact Likelihood Generative Learning for Symmetric Densities](https://mlsanity.com/paper/2006.02425/) (2020)
- [E(n) Equivariant Normalizing Flows](https://mlsanity.com/paper/2105.09016/) (2021)
- [Equivariant flow matching](https://mlsanity.com/paper/2306.15030/) (2023)
- [Flow Annealed Importance Sampling Bootstrap](https://mlsanity.com/paper/2208.01893/) (2022)
- [Adjoint Sampling: Highly Scalable Diffusion Samplers via Adjoint Matching](https://mlsanity.com/paper/2504.11713/) (2025)
- [Targeted free energy estimation via learned mappings](https://mlsanity.com/paper/2002.04913/) (2020)
- [Augmented Normalizing Flows: Bridging the Gap Between Generative Flows and Latent Variable Models](https://mlsanity.com/paper/2002.07101/) (2020)
- [Transferable Boltzmann Generators](https://mlsanity.com/paper/2406.14426/) (2024)
- [Normalizing flows for atomic solids](https://mlsanity.com/paper/2111.08696/) (2021)
- [Scalable Equilibrium Sampling with Sequential Boltzmann Generators](https://mlsanity.com/paper/2502.18462/) (2025)
- [SE(3) Equivariant Augmented Coupling Flows](https://mlsanity.com/paper/2308.10364/) (2023)
- [Rigid Body Flows for Sampling Molecular Crystal Structures](https://mlsanity.com/paper/2301.11355/) (2023)
- [A practical guide to machine learning interatomic potentials -- Status and future](https://mlsanity.com/paper/2503.09814/) (2025)
- [Consistent Sampling and Simulation: Molecular Dynamics with Energy-Based Diffusion Models](https://mlsanity.com/paper/2506.17139/) (2025)
- [Efficient mapping of phase diagrams with conditional Boltzmann Generators](https://mlsanity.com/paper/2406.12378/) (2024)
- [Energy-Weighted Flow Matching: Unlocking Continuous Normalizing Flows for Efficient and Scalable Boltzmann Sampling](https://mlsanity.com/paper/2509.03726/) (2025)
- [Estimating Gibbs free energies via isobaric-isothermal flows](https://mlsanity.com/paper/2305.13233/) (2023)

## Built on by (most cited first, 11 of 11)

- [Enhanced Sampling for Efficient Learning of Coarse-Grained Machine Learning Potentials](https://mlsanity.com/paper/2510.11148/) (2025)
- [Assessing generative modeling approaches for free energy estimates in condensed matter](https://mlsanity.com/paper/2512.23930/) (2025)
- [Few-Step Boltzmann Generators via Scalable Likelihood Flow Maps](https://mlsanity.com/paper/2606.29110/) (2026)
- [Boltzmann-Expected Molecular Design with Decoupled Annealing Flows](https://mlsanity.com/paper/2607.19519/) (2026)
- [Estimating Solvation Free Energies with Boltzmann Generators](https://mlsanity.com/paper/2512.18147/) (2025)
- [Boltzmann Generators for Condensed Matter via Riemannian Flow Matching](https://mlsanity.com/paper/2602.18482/) (2026)
- [Coarse-Grained Boltzmann Generators](https://mlsanity.com/paper/2602.10637/) (2026)
- [Jeffreys Flow: Robust Boltzmann Generators for Rare Event Sampling via Parallel Tempering Distillation](https://mlsanity.com/paper/2604.05303/) (2026)
- [StAD: Stein Amortized Divergence for Fast Likelihoods with Diffusion and Flow](https://mlsanity.com/paper/2605.16486/) (2026)
- [Generative Nested Sampling of Atomistic Thermodynamic Landscapes](https://mlsanity.com/paper/2609.03193/) (2026)
- [Correlation-Free Transition Path Sampling through Shooting Point Generation Guided by Committor Learning](https://mlsanity.com/paper/2609.20461/) (2026)

## Abstract (arXiv)

The use of generative models to sample equilibrium distributions of many-body systems, as first demonstrated by Boltzmann Generators, has attracted substantial interest due to their ability to produce unbiased and uncorrelated samples in `one shot'. Despite their promise and impressive results across the natural sciences, scaling these models to large systems remains a major challenge. In this work, we introduce a Boltzmann Generator architecture that addresses this scalability bottleneck with a focus on applications in materials science. We leverage augmented coupling flows in combination with graph neural networks to base the generation process on local environmental information, while allowing for energy-based training and fast inference. Compared to previous architectures, our model trains significantly faster, requires far less computational resources, and achieves superior sampling efficiencies. Crucially, the architecture is transferable to larger system sizes, which allows for the efficient sampling of materials with simulation cells of unprecedented size. We demonstrate the potential of our approach by applying it to several materials systems, including Lennard-Jones crystals, ice phases of mW water, and the phase diagram of silicon, for system sizes well above one thousand atoms. The trained Boltzmann Generators produce highly accurate equilibrium ensembles for various crystal structures, as well as Helmholtz and Gibbs free energies across a range of system sizes, able to reach scales where finite-size effects become negligible.