Normalizing flows for atomic solids

  title={Normalizing flows for atomic solids},
  author={Peter Wirnsberger and George Papamakarios and Borja Ibarz and S{\'e}bastien Racani{\`e}re and Andy Ballard and Alexander Pritzel and Charles Blundell},
  journal={Machine Learning: Science and Technology},
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… 

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  • 153, 144112
  • 2020


  • Rev. E 65, 046122
  • 2002


  • 129, 124105
  • 2008

Understanding Molecular Simulation


  • 22, 245
  • 1976


  • Rev. Lett. 79, 3002
  • 1997


  • 127, 154113
  • 2007

Equivariant Flows: exact likelihood generative learning for symmetric densities

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  • C. Calì
  • Computer Science
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  • 2020
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Normalizing Flows on Tori and Spheres

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