mlsanityData to 7 Oct 2026

Normalizing flows for atomic solids

arXiv 2111.08696 ↗First posted 16 Nov 2021physics.comp-ph cond-mat.stat-mech stat.MLcited by 39

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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.

Key facts. Cited by 39 papers in the map (data through 7 Oct 2026). Builds on 5 of its 19 references that are in the map. First posted on arXiv 16 Nov 2021.

Builds on

5 of 19 references.

References · 5

  1. 2017Attention Is All You Needcited by 59,932
  2. 2014Adam: A Method for Stochastic Optimizationcited by 36,960
  3. 2015Variational Inference with Normalizing Flowscited by 2,161
  4. 2020Equivariant Flows: Exact Likelihood Generative Learning for Symmetric Densitiescited by 248
  5. 2020Normalizing Flows on Tori and Spherescited by 117

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2021: 1; 2022: 5; 2023: 5; 2024: 11; 2025: 12; 2026: 5.

39 papers.

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  1. 2022Flow Annealed Importance Sampling Bootstrapcited by 124
  2. 2024FlowMM: Generating Materials with Riemannian Flow Matchingcited by 105
  3. 2024FlowLLM: Flow Matching for Material Generation with Large Language Models as Base Distributionscited by 52
  4. 2024Space Group Informed Transformer for Crystalline Materials Generationcited by 40
  5. 2022Physics Guided Deep Learning for Generative Design of Crystal Materials with Symmetry Constraintscited by 30
  6. 2023Rigid Body Flows for Sampling Molecular Crystal Structurescited by 27
  7. 2025Position: The Future of Bayesian Prediction Is Prior-Fittedcited by 19
  8. 2024Efficient mapping of phase diagrams with conditional Boltzmann Generatorscited by 18
  9. 2024Iterated Energy-based Flow Matching for Sampling from Boltzmann Densitiescited by 12
  10. 2025Scalable Boltzmann Generators for equilibrium sampling of large-scale materialscited by 11
  11. 2021Machine Learning of Thermodynamic Observables in the Presence of Mode Collapsecited by 10
  12. 2024Scalable Normalizing Flows Enable Boltzmann Generators for Macromoleculescited by 10
27 more
  1. 2025Symmetry-Aware Bayesian Flow Networks for Crystal Generationcited by 9
  2. 2025Space Group Conditional Flow Matchingcited by 7
  3. 2022Deep Variational Free Energy Approach to Dense Hydrogencited by 6
  4. 2024AdvNF: Reducing Mode Collapse in Conditional Normalising Flows using Adversarial Learningcited by 6
  5. 2022Conditioning Normalizing Flows for Rare Event Samplingcited by 5
  6. 2025Boltzmann generators for amorphous particle systemscited by 5
  7. 2022Atomic structure generation from reconstructing structural fingerprintscited by 4
  8. 2023Laser: Latent Set Representations for 3D Generative Modelingcited by 3
  9. 2023Understanding Deep Generative Models with Generalized Empirical Likelihoodscited by 3
  10. 2025Generative Models for Crystalline Materialscited by 3
  11. 2024Active learning of Boltzmann samplers and potential energies with quantum mechanical accuracycited by 2
  12. 2024Deep generative modelling of canonical ensemble with differentiable thermal propertiescited by 2
  13. 2025Energy-Based Coarse-Graining in Molecular Dynamics: A Flow-Based Framework without Datacited by 2
  14. 2025SESaMo: Symmetry-Enforcing Stochastic Modulation for Normalizing Flowscited by 2
  15. 2023Enhanced Sampling with Machine Learning: A Reviewcited by 1
  16. 2023Learned Mappings for Targeted Free Energy Perturbation between Peptide Conformationscited by 1
  17. 2024Ratio Divergence Learning Using Target Energy in Restricted Boltzmann Machines: Beyond Kullback--Leibler Divergence Learningcited by 1
  18. 2025Estimating Solvation Free Energies with Boltzmann Generatorscited by 1
  19. 2025Assessing generative modeling approaches for free energy estimates in condensed mattercited by 1
  20. 2026Boltzmann Generators for Condensed Matter via Riemannian Flow Matchingcited by 1
  21. 2026Differentiable free energy surface: a variational approach to directly observing rare events using generative deep-learning modelscited by 1
  22. 2024Flow Perturbation to Accelerate Unbiased Sampling of Boltzmann distributioncited by 0
  23. 2025Accurate and thermodynamically consistent hydrogen equation of state for planetary modeling with flow matchingcited by 0
  24. 2025Fundamental Limits of Crystalline Equivariant Graph Neural Networks: A Circuit Complexity Perspectivecited by 0
  25. 2026Generative Nested Sampling of Atomistic Thermodynamic Landscapescited by 0
  26. 2026Correlation-Free Transition Path Sampling through Shooting Point Generation Guided by Committor Learningcited by 0
  27. 2026Grand Canonical Generatorscited by 0