Symmetry-Adapted Machine Learning for Tensorial Properties of Atomistic Systems

Abstract

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.

From DFT to machine learning: recent approaches to materials science–a review

“…We now present a relatively chronological list used in recent materials research, which is considerable but not exhaustive. These include: bondorientational order parameters (BOP) [243]; Behler-Parrinello atom-centered symmetry functions (ACSF) [233,244], and its modified [245] and weighted (wACSF) [246] versions; Gaussian Approximation Potentials (GAP) [212,232] using smooth overlap of atomic positions (SOAP) [213] also extended for tensorial properties [247]; Coulomb matrix [248] and Bag of Bonds (BOB) [249], and the subsequent interatomic many body expansions (MBE) [250,251] like the so-called BAML (bonds, angles machine learning) [252] and fixed-size inverse distances [253]; metric fingerprints [238]; bispectrum [213]; atomic local frame (ALF) [254]; partial radial and angular distribution functions (PRDF, ADF) [255] and generalized radial distribution functions (GRDF) [224]; Fourier series of radial distribution functions [256]; force vectors representations [257]; spectral neighbor analysis potential (SNAP) [258]; permutation invariant polynomials [245]; particle densities [259]; angular Fourier series (AFS) [213]; topological polyhedra [260], Voronoi [261] and Voronoi-Dirichlet [262] tessellations; spherical harmonics [263]; histogram of distances, angles, or dihedral angles [264]; classical forcefield-inspired descriptors (CFID) [209]; graph-based such as Graph Approximated Energy (GRAPE) [265]; constant complexity descriptors based on Chebyshev polynomials [266]; symmetrized gradient-domain machine learning ...…”

Non-covalent interactions across organic and biological subsets of chemical space: Physics-based potentials parametrized from machine learning

“…though only applied to dipoles so far. 52,53 In Appendix B, we extend Glielmo et al's covariant-kernel description to quadrupoles using atom-centered Gaussian functions. Tests on small training sets indicated results on par with Fig.…”

Discovering a Transferable Charge Assignment Model Using Machine Learning

“…[12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27] They have been 3 used to discover materials [28][29][30][31][32][33][34][35][36][37] and study dynamical processes such as charge and exciton transfer. [38][39][40][41] Most related to this work are ML models of existing charge models, [9,[42][43][44] which are orders of magnitude faster than ab initio calculation.…”