Abstract
Background: The increasingly common applications of machine-learning schemes to atomic-scale simulations have triggered efforts to better understand the mathematical properties of the mapping between the Cartesian coordinates of the atoms and the variety of representations that can be used to convert them into a finite set of symmetric descriptors or features. Methods: Here, we analyze the sensitivity of the mapping to atomic displacements, using a singular value decomposition of the Jacobian of the transformation to quantify the sensitivity for different configurations, choice of representations and implementation details. Results: We show that the combination of symmetry and smoothness leads to mappings that have singular points at which the Jacobian has one or more null singular values (besides those corresponding to infinitesimal translations and rotations). This is in fact desirable, because it enforces physical symmetry constraints on the values predicted by regression models constructed using such representations.However, besides these symmetry-induced singularities, there are also spurious singular points, that we find to be linked to the incompleteness of the mapping, i.e. the fact that, for certain classes of representations, structurally distinct configurations are not guaranteed to be mapped onto different feature vectors. Additional singularities can be introduced by a too aggressive truncation of the infinite basis set that is used to discretize the representations. Conclusions: These results exemplify the subtle issues that arise when constructing symmetric representations of atomic structures, and provide conceptual and numerical tools to identify and investigate them in both benchmark and realistic applications.
Highlights
There has been a tidal wave of interest in the last decade in applying machine learning tools to atomistic modelling problems
In the present paper we will continue our theoretical investigation of representations of local atomic environments, Ai, given in terms of a feature vector ξ = ξ(Ai) = {ξq(Ai)}q=1...nfeat that is invariant under rotations, reflections and permutations of like atoms
〈x1; . . . xν|ρi⊗ν〉 = d R 〈x1| R |ρi〉 · · · 〈xν| R |ρi〉. These definitions are quite abstract, but encompass a majority of the representations that have been used in the application of machine learning to atomistic problems – including atom-centered symmetry functions[6, 12, 13], smooth overlap of atomic positions (SOAP) powerspectrum[14], bispectrum [7], FCHL descriptors[15], all of which are limited to low correlation orders, as well as representations for which ν can be increased systematically, such as the moment tensor potential (MTP)[16], the atomic cluster expansion (ACE)[4] and the N -body iterative contraction of equivariants (NICE)[17]
Summary
There has been a tidal wave of interest in the last decade in applying machine learning tools to atomistic modelling problems. In the present paper we will continue our theoretical investigation of representations of local atomic environments, Ai, given in terms of a feature vector ξ = ξ(Ai) = {ξq(Ai)}q=1...nfeat that is invariant under rotations, reflections and permutations of like atoms Such a representation immediately leads to the question whether the atomic environment Ai can be reconstructed from the features ξ, up to symmetries. These observations apply to the vast majority of descriptors used in the field, including in particular [6, 7]; see [2] for an extensive discussion This challenge points to the fundamental question under which conditions the feature vector ξ is a coordinate system, or in other words, whether its image is a smooth manifold. We will demonstrate that spurious singularities (loss of sensitivity in non-symmetric structures) arise by two mechanisms: (1) a lack of numerical resolution in the discretisation of the atomic density, which is remedied; and (2) as intersections of degenerate pair manifolds, which are more fundamental and non-trivial to remove or rule out
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
