Atom-Density Representations for Machine Learning

The applications of machine learning techniques to chemistry and materials science become more numerous by the day. The main challenge is to devise representations of atomic systems that are at the same time complete and concise, so as to reduce the number of reference calculations that are needed to predict the properties of different types of materials reliably. This has led to a proliferation of alternative ways to convert an atomic structure into an input for a machine-learning model. We introduce an abstract definition of chemical environments that is based on a smoothed atomic density, using a bra-ket notation to emphasize basis set independence and to highlight the connections with some popular choices of representations for describing atomic systems. The correlations between the spatial distribution of atoms and their chemical identities are computed as inner products between these feature kets, which can be given an explicit representation in terms of the expansion of the atom density on orthogonal basis functions, that is equivalent to the smooth overlap of atomic positions (SOAP) power spectrum, but also in real space, corresponding to $n$-body correlations of the atom density. This formalism lays the foundations for a more systematic tuning of the behavior of the representations, by introducing operators that represent the correlations between structure, composition, and the target properties. It provides a unifying picture of recent developments in the field and indicates a way forward towards more effective and computationally affordable machine-learning schemes for molecules and materials

Feature Optimization for Atomistic Machine Learning Yields A Data-Driven Construction of the Periodic Table of the Elements

Machine-learning of atomic-scale properties amounts to extracting correlations between structure, composition and the quantity that one wants to predict. Representing the input structure in a way that best reflects such correlations makes it possible to improve the accuracy of the model for a given amount of reference data. When using a description of the structures that is transparent and well-principled, optimizing the representation might reveal insights into the chemistry of the data set. Here we show how one can generalize the SOAP kernel to introduce a distance-dependent weight that accounts for the multi-scale nature of the interactions, and a description of correlations between chemical species. We show that this improves substantially the performance of ML models of molecular and materials stability, while making it easier to work with complex, multi-component systems and to extend SOAP to coarse-grained intermolecular potentials. The element correlations that give the best performing model show striking similarities with the conventional periodic table of the elements, providing an inspiring example of how machine learning can rediscover, and generalize, intuitive concepts that constitute the foundations of chemistry.Comment: 9 pages, 4 figure

Atomic-scale representation and statistical learning of tensorial properties

This chapter discusses the importance of incorporating three-dimensional symmetries in the context of statistical learning models geared towards the interpolation of the tensorial properties of atomic-scale structures. We focus on Gaussian process regression, and in particular on the construction of structural representations, and the associated kernel functions, that are endowed with the geometric covariance properties compatible with those of the learning targets. We summarize the general formulation of such a symmetry-adapted Gaussian process regression model, and how it can be implemented based on a scheme that generalizes the popular smooth overlap of atomic positions representation. We give examples of the performance of this framework when learning the polarizability and the ground-state electron density of a molecule

Efficient implementation of atom-density representations

Physically motivated and mathematically robust atom-centered representations of molecular structures are key to the success of modern atomistic machine learning. They lie at the foundation of a wide range of methods to predict the properties of both materials and molecules and to explore and visualize their chemical structures and compositions. Recently, it has become clear that many of the most effective representations share a fundamental formal connection. They can all be expressed as a discretization of n-body correlation functions of the local atom density, suggesting the opportunity of standardizing and, more importantly, optimizing their evaluation. We present an implementation, named librascal, whose modular design lends itself both to developing refinements to the density-based formalism and to rapid prototyping for new developments of rotationally equivariant atomistic representations. As an example, we discuss smooth overlap of atomic position (SOAP) features, perhaps the most widely used member of this family of representations, to show how the expansion of the local density can be optimized for any choice of radial basis sets. We discuss the representation in the context of a kernel ridge regression model, commonly used with SOAP features, and analyze how the computational effort scales for each of the individual steps of the calculation. By applying data reduction techniques in feature space, we show how to reduce the total computational cost by a factor of up to 4 without affecting the model’s symmetry properties and without significantly impacting its accuracy

Boltzmann-conserving classical dynamics in quantum time-correlation functions: "Matsubara dynamics".

We show that a single change in the derivation of the linearized semiclassical-initial value representation (LSC-IVR or "classical Wigner approximation") results in a classical dynamics which conserves the quantum Boltzmann distribution. We rederive the (standard) LSC-IVR approach by writing the (exact) quantum time-correlation function in terms of the normal modes of a free ring-polymer (i.e., a discrete imaginary-time Feynman path), taking the limit that the number of polymer beads N → ∞, such that the lowest normal-mode frequencies take their "Matsubara" values. The change we propose is to truncate the quantum Liouvillian, not explicitly in powers of ħ(2) at ħ(0) (which gives back the standard LSC-IVR approximation), but in the normal-mode derivatives corresponding to the lowest Matsubara frequencies. The resulting "Matsubara" dynamics is inherently classical (since all terms O(ħ(2)) disappear from the Matsubara Liouvillian in the limit N → ∞) and conserves the quantum Boltzmann distribution because the Matsubara Hamiltonian is symmetric with respect to imaginary-time translation. Numerical tests show that the Matsubara approximation to the quantum time-correlation function converges with respect to the number of modes and gives better agreement than LSC-IVR with the exact quantum result. Matsubara dynamics is too computationally expensive to be applied to complex systems, but its further approximation may lead to practical methods.T.J.H.H., M.J.W., and S.C.A. acknowledge funding from the U.K. Engineering and Physical Sciences Research Council. A.M. acknowledges the European Lifelong Learning Programme (LLP) for an Erasmus student placement scholarship. T.J.H.H. also acknowledges a Research Fellowship from Jesus College, Cambridge and helpful discussions with Dr. Adam Harper.This is the author accepted manuscript. The final version is available from AIP via http://dx.doi.org/10.1063/1.491631

Path-integral dynamics of water using curvilinear centroids

We develop a path-integral dynamics method for water that resembles centroid molecular dynamics (CMD), except that the centroids are averages of curvilinear, rather than cartesian, bead coordinates. The curvilinear coordinates are used explicitly only when computing the potential of mean force, the components of which are re-expressed in terms of cartesian 'quasi-centroids' (so-called because they are close to the cartesian centroids). Cartesian equations of motion are obtained by making small approximations to the quantum Boltzmann distribution. Simulations of the infrared spectra of various water models over 150-600 K show these approximations to be justified: for a two-dimensional OH-bond model, the quasi-centroid molecular dynamics (QCMD) spectra lie close to the exact quantum spectra, and almost on top of the Matsubara dynamics spectra; for gas-phase water, the QCMD spectra are close to the exact quantum spectra; for liquid water and ice (using the q-TIP4P/F surface), the QCMD spectra are close to the CMD spectra at 600 K, and line up with the results of thermostatted ring-polymer molecular dynamics and approximate quantum calculations at 300 and 150 K. The QCMD spectra show no sign of the CMD 'curvature problem' (of erroneous red shifts and broadening). In the liquid and ice simulations, the potential of mean force was evaluated on the fly by generalising an adiabatic CMD algorithm to curvilinear coordinates; the full limit of adiabatic separation needed to be taken, which made the QCMD calculations 8 times more expensive than partially adiabatic CMD at 300 K, and 32 times at 150 K (and the intensities may still not be converged at this temperature). The QCMD method is probably generalisable to many other systems, provided collective bead-coordinates can be identified that yield compact mean-field ring-polymer distributions.Cambridge University Vice Chancellor's award

Snow property controls on modelled Ku-band altimeter estimates of first-year sea ice thickness: Case studies from the Canadian and Norwegian Arctic

Uncertainty in snow properties impacts the accuracy of Arctic sea ice thickness estimates from radar altimetry. On firstyear sea ice (FYI), spatiotemporal variations in snow properties can cause the Ku-band main radar scattering horizon to appear above the snow/sea ice interface. This can increase the estimated sea ice freeboard by several centimeters, leading to FYI thickness overestimations. This study examines the expected changes in Kuband main scattering horizon and its impact on FYI thickness estimates, with variations in snow temperature, salinity and density derived from 10 naturally occurring Arctic FYI Cases encompassing saline/non-saline, warm/cold, simple/complexly layered snow (4 cm to 45 cm) overlying FYI (48 cm to 170 cm). Using a semi-empirical modeling approach, snow properties from these Cases are used to derive layer-wise brine volume and dielectric constant estimates, to simulate the Ku-band main scattering horizon and delays in radar propagation speed. Differences between modeled and observed FYI thickness are calculated to assess sources of error. Under both cold and warm conditions, saline snow covers are shown to shift the main scattering horizon above from the snow/sea ice interface, causing thickness retrieval errors. Overestimates in FYI thicknesses of up to 65% are found for warm, saline snow overlaying thin sea ice. Our simulations exhibited a distinct shift in the main scattering horizon when the snow layer densities became greater than 440 kg/m3 , especially under warmer snow conditions. Our simulations suggest a mean Ku-band propagation delay for snow of 39%, which is higher than 25%, suggested in previous studies

Machine-learning of atomic-scale properties based on physical principles

We briefly summarize the kernel regression approach, as used recently in materials modelling, to fitting functions, particularly potential energy surfaces, and highlight how the linear algebra framework can be used to both predict and train from linear functionals of the potential energy, such as the total energy and atomic forces. We then give a detailed account of the Smooth Overlap of Atomic Positions (SOAP) representation and kernel, showing how it arises from an abstract representation of smooth atomic densities, and how it is related to several popular density-based representations of atomic structure. We also discuss recent generalisations that allow fine control of correlations between different atomic species, prediction and fitting of tensorial properties, and also how to construct structural kernels---applicable to comparing entire molecules or periodic systems---that go beyond an additive combination of local environments