Symmetry-Adapted Machine Learning for Tensorial Properties of Atomistic Systems

Andrea Grisafi*, David M. Wilkins, Gábor Csányi, Michele Ceriotti

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

194 Citations (Scopus)
469 Downloads (Pure)


Statistical learning methods show great promise in providing an accurate prediction of materials and molecular properties, while minimizing the need for computationally demanding electronic structure calculations. The accuracy and transferability of these models are increased significantly by encoding into the learning procedure the fundamental symmetries of rotational and permutational invariance of scalar properties. However, the prediction of tensorial properties requires that the model respects the appropriate geometric transformations, rather than invariance, when the reference frame is rotated. We introduce a formalism that extends existing schemes and makes it possible to perform machine learning of tensorial properties of arbitrary rank, and for general molecular geometries. To demonstrate it, we derive a tensor kernel adapted to rotational symmetry, which is the natural generalization of the smooth overlap of atomic positions kernel commonly used for the prediction of scalar properties at the atomic scale. The performance and generality of the approach is demonstrated by learning the instantaneous response to an external electric field of water oligomers of increasing complexity, from the isolated molecule to the condensed phase.

Original languageEnglish
Article number036002
JournalPhysical Review Letters
Issue number3
Publication statusPublished - 19 Jan 2018


  • Machine Learning
  • Symmetry
  • Gaussian Process Regression
  • Coupled Cluster Theory
  • Dielectric Response

ASJC Scopus subject areas

  • General Physics and Astronomy


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