Recursive evaluation and iterative contraction of N-body equivariant features

Abstract

Mapping an atomistic configuration to a symmetrized N-point correlation of a field associated with the atomic positions (e.g., an atomic density) has emerged as an elegant and effective solution to represent structures as the input of machine-learning algorithms. While it has become clear that low-order density correlations do not provide a complete representation of an atomic environment, the exponential increase in the number of possible N-body invariants makes it difficult to design a concise and effective representation. We discuss how to exploit recursion relations between equivariant features of different order (generalizations of N-body invariants that provide a complete representation of the symmetries of improper rotations) to compute high-order terms efficiently. In combination with the automatic selection of the most expressive combination of features at each order, this approach provides a conceptual and practical framework to generate systematically improvable, symmetry adapted representations for atomistic machine learning.

Gaussian Process Regression for Materials and Molecules

“… 78 One can go further in body order explicitly while continuing to keep the regression linear. 64 , 65 , 67 , 68 , 73 , 79 …”

“…For each value of the training set size, we show the mean absolute error (MAE) evaluated on the test set which consists of the remaining structures from the full dataset. Models based on FCHL (2018), 233 SOAP (2018), 69 aSLATM, 234 Coulomb Matrix (CM), 235 and Bag-of-bonds (BOB) 236 representations use Gaussian process/kernel ridge regression, whereas NICE 73 and MTP 67 use linear ridge regression, and SchNet 237 and PhysNet 238 are graph neural networks. GM-sNN uses a representation similar in spirit to MTP but based on a Gaussian radial basis set and a feed-forward neural network for regression.…”

Multi-scale approach for the prediction of atomic scale properties

“… 16 We indicate these as the two-point density correlation to stress that similar results are to be expected from any equivalent local featurization 17 such as atom-centered symmetry functions, 78 SNAP, 79 MTP, 80 ACE, 72 NICE. 69 We report errors in terms of the root mean square error (RMSE), or the percentage RMSE (RMSE%), which is expressed as a percentage of the standard deviation of the target properties.…”

“…A particularly clean, efficient, recursive formulation can be derived exploiting the fact that the equivariant features behave as angular momenta, and can then be combined using Clebsch–Gordan coefficients to build higher-order correlations. 69 In analytical derivations we use a partially-discretized basis, in which the radial contribution is kept as a continuous index, corresponding to with 〈 x̂ | lm 〉 ≡ Y m l ( x̂ ). Written in this basis, 〈 arlm | ρ i 〉 expresses the decomposition of the density in independent angular momentum channels, evaluated at a distance r from the central atom.…”

“…To see this, let us start by representing the energy prediction in terms of the partially-discretized basis of eqn (8) . Upon symmetrization of the tensor product between ρ and V , and going to the coupled angular momentum basis, 69 one obtains a set of invariants that can be expressed using the basis 〈 X | ≡ 〈 a 1 r 1 ; a 2 r 2 ; l | where 〈 a 1 r 1 lm | ρ < i 〉 and 〈 a 2 r 2 lm | V i 〉 indicate the spherical harmonics projections of the local density and the local potential fields respectively, and we omit the indication of the structure A, for brevity. Eqn (12) shows explicitly how contains information on the correlation between the value of the atom density | ρ i 〉 and the potential | V i 〉, each evaluated at a given distance from the central atom.…”