Building nonparametric nn-body force fields using Gaussian process regression

Constructing a classical potential suited to simulate a given atomic system is a remarkably difficult task. This chapter presents a framework under which this problem can be tackled, based on the Bayesian construction of nonparametric force fields of a given order using Gaussian process (GP) priors. The formalism of GP regression is first reviewed, particularly in relation to its application in learning local atomic energies and forces. For accurate regression it is fundamental to incorporate prior knowledge into the GP kernel function. To this end, this chapter details how properties of smoothness, invariance and interaction order of a force field can be encoded into corresponding kernel properties. A range of kernels is then proposed, possessing all the required properties and an adjustable parameter $n$ governing the interaction order modelled. The order $n$ best suited to describe a given system can be found automatically within the Bayesian framework by maximisation of the marginal likelihood. The procedure is first tested on a toy model of known interaction and later applied to two real materials described at the DFT level of accuracy. The models automatically selected for the two materials were found to be in agreement with physical intuition. More in general, it was found that lower order (simpler) models should be chosen when the data are not sufficient to resolve more complex interactions. Low $n$ GPs can be further sped up by orders of magnitude by constructing the corresponding tabulated force field, here named "MFF".Comment: 31 pages, 11 figures, book chapte

Band gap prediction for large organic crystal structures with machine learning

Machine-learning models are capable of capturing the structure-property relationship from a dataset of computationally demanding ab initio calculations. Over the past two years, the Organic Materials Database (OMDB) has hosted a growing number of calculated electronic properties of previously synthesized organic crystal structures. The complexity of the organic crystals contained within the OMDB, which have on average 82 atoms per unit cell, makes this database a challenging platform for machine learning applications. In this paper, the focus is on predicting the band gap which represents one of the basic properties of a crystalline materials. With this aim, a consistent dataset of 12 500 crystal structures and their corresponding DFT band gap are released, freely available for download at https://omdb.mathub.io/dataset. An ensemble of two state-of-the-art models reach a mean absolute error (MAE) of 0.388 eV, which corresponds to a percentage error of 13% for an average band gap of 3.05 eV. Finally, the trained models are employed to predict the band gap for 260 092 materials contained within the Crystallography Open Database (COD) and made available online so that the predictions can be obtained for any arbitrary crystal structure uploaded by a user.Comment: 10 pages, 6 figure

Representations of molecules and materials for interpolation of quantum-mechanical simulations via machine learning

Computational study of molecules and materials from first principles is a cornerstone of physics, chemistry and materials science, but limited by the cost of accurate and precise simulations. In settings involving many simulations, machine learning can reduce these costs, sometimes by orders of magnitude, by interpolating between reference simulations. This requires representations that describe any molecule or material and support interpolation. We review, discuss and benchmark state-of-the-art representations and relations between them, including smooth overlap of atomic positions, many-body tensor representation, and symmetry functions. For this, we use a unified mathematical framework based on many-body functions, group averaging and tensor products, and compare energy predictions for organic molecules, binary alloys and Al-Ga-In sesquioxides in numerical experiments controlled for data distribution, regression method and hyper-parameter optimization

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

Machine learning with systematic density-functional theory calculations: Application to melting temperatures of single and binary component solids

A combination of systematic density functional theory (DFT) calculations and machine learning techniques has a wide range of potential applications. This study presents an application of the combination of systematic DFT calculations and regression techniques to the prediction of the melting temperature for single and binary compounds. Here we adopt the ordinary least-squares regression (OLSR), partial least-squares regression (PLSR), support vector regression (SVR) and Gaussian process regression (GPR). Among the four kinds of regression techniques, the SVR provides the best prediction. In addition, the inclusion of physical properties computed by the DFT calculation to a set of predictor variables makes the prediction better. Finally, a simulation to find the highest melting temperature toward the efficient materials design using kriging is demonstrated. The kriging design finds the compound with the highest melting temperature much faster than random designs. This result may stimulate the application of kriging to efficient materials design for a broad range of applications

Machine learning reveals orbital interaction in crystalline materials

We propose a novel representation of crystalline materials named orbital-field matrix (OFM) based on the distribution of valence shell electrons. We demonstrate that this new representation can be highly useful in mining material data. Our experiment shows that the formation energies of crystalline materials, the atomization energies of molecular materials, and the local magnetic moments of the constituent atoms in transition metal--rare-earth metal bimetal alloys can be predicted with high accuracy using the OFM. Knowledge regarding the role of coordination numbers of transition-metal and rare-earth metal elements in determining the local magnetic moment of transition metal sites can be acquired directly from decision tree regression analyses using the OFM.Comment: 10 page