# Normalizing flows for atomic solids

arXiv: https://arxiv.org/abs/2111.08696 · HTML: https://mlsanity.com/paper/2111.08696/

## Key facts

- Cited by 39 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 19 references that are in the map.
- First posted on arXiv 16 Nov 2021.

## Builds on (most cited first, 5 of 5)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Equivariant Flows: Exact Likelihood Generative Learning for Symmetric Densities](https://mlsanity.com/paper/2006.02425/) (2020)
- [Normalizing Flows on Tori and Spheres](https://mlsanity.com/paper/2002.02428/) (2020)

## Built on by (most cited first, 30 of 39)

- [Flow Annealed Importance Sampling Bootstrap](https://mlsanity.com/paper/2208.01893/) (2022)
- [FlowMM: Generating Materials with Riemannian Flow Matching](https://mlsanity.com/paper/2406.04713/) (2024)
- [FlowLLM: Flow Matching for Material Generation with Large Language Models as Base Distributions](https://mlsanity.com/paper/2410.23405/) (2024)
- [Space Group Informed Transformer for Crystalline Materials Generation](https://mlsanity.com/paper/2403.15734/) (2024)
- [Physics Guided Deep Learning for Generative Design of Crystal Materials with Symmetry Constraints](https://mlsanity.com/paper/2203.14352/) (2022)
- [Rigid Body Flows for Sampling Molecular Crystal Structures](https://mlsanity.com/paper/2301.11355/) (2023)
- [Position: The Future of Bayesian Prediction Is Prior-Fitted](https://mlsanity.com/paper/2505.23947/) (2025)
- [Efficient mapping of phase diagrams with conditional Boltzmann Generators](https://mlsanity.com/paper/2406.12378/) (2024)
- [Iterated Energy-based Flow Matching for Sampling from Boltzmann Densities](https://mlsanity.com/paper/2408.16249/) (2024)
- [Scalable Boltzmann Generators for equilibrium sampling of large-scale materials](https://mlsanity.com/paper/2509.25486/) (2025)
- [Machine Learning of Thermodynamic Observables in the Presence of Mode Collapse](https://mlsanity.com/paper/2111.11303/) (2021)
- [Scalable Normalizing Flows Enable Boltzmann Generators for Macromolecules](https://mlsanity.com/paper/2401.04246/) (2024)
- [Symmetry-Aware Bayesian Flow Networks for Crystal Generation](https://mlsanity.com/paper/2502.03146/) (2025)
- [Deep Variational Free Energy Approach to Dense Hydrogen](https://mlsanity.com/paper/2209.06095/) (2022)
- [Space Group Conditional Flow Matching](https://mlsanity.com/paper/2509.23822/) (2025)
- [AdvNF: Reducing Mode Collapse in Conditional Normalising Flows using Adversarial Learning](https://mlsanity.com/paper/2401.15948/) (2024)
- [Boltzmann generators for amorphous particle systems](https://mlsanity.com/paper/2512.16607/) (2025)
- [Conditioning Normalizing Flows for Rare Event Sampling](https://mlsanity.com/paper/2207.14530/) (2022)
- [Atomic structure generation from reconstructing structural fingerprints](https://mlsanity.com/paper/2207.13227/) (2022)
- [Generative Models for Crystalline Materials](https://mlsanity.com/paper/2511.22652/) (2025)
- [Laser: Latent Set Representations for 3D Generative Modeling](https://mlsanity.com/paper/2301.05747/) (2023)
- [Understanding Deep Generative Models with Generalized Empirical Likelihoods](https://mlsanity.com/paper/2306.09780/) (2023)
- [Assessing generative modeling approaches for free energy estimates in condensed matter](https://mlsanity.com/paper/2512.23930/) (2025)
- [Active learning of Boltzmann samplers and potential energies with quantum mechanical accuracy](https://mlsanity.com/paper/2401.16487/) (2024)
- [Deep generative modelling of canonical ensemble with differentiable thermal properties](https://mlsanity.com/paper/2404.18404/) (2024)
- [Energy-Based Coarse-Graining in Molecular Dynamics: A Flow-Based Framework without Data](https://mlsanity.com/paper/2504.20940/) (2025)
- [SESaMo: Symmetry-Enforcing Stochastic Modulation for Normalizing Flows](https://mlsanity.com/paper/2505.19619/) (2025)
- [Enhanced Sampling with Machine Learning: A Review](https://mlsanity.com/paper/2306.09111/) (2023)
- [Learned Mappings for Targeted Free Energy Perturbation between Peptide Conformations](https://mlsanity.com/paper/2306.14010/) (2023)
- [Ratio Divergence Learning Using Target Energy in Restricted Boltzmann Machines: Beyond Kullback--Leibler Divergence Learning](https://mlsanity.com/paper/2409.07679/) (2024)

## Abstract (arXiv)

We present a machine-learning approach, based on normalizing flows, for modelling atomic solids. Our model transforms an analytically tractable base distribution into the target solid without requiring ground-truth samples for training. We report Helmholtz free energy estimates for cubic and hexagonal ice modelled as monatomic water as well as for a truncated and shifted Lennard-Jones system, and find them to be in excellent agreement with literature values and with estimates from established baseline methods. We further investigate structural properties and show that the model samples are nearly indistinguishable from the ones obtained with molecular dynamics. Our results thus demonstrate that normalizing flows can provide high-quality samples and free energy estimates without the need for multi-staging.