One of the most exciting tools that have entered the material science toolbox in recent years is machine learning. This collection of statistical methods has already proved to be capable of considerably speeding up both fundamental and applied research. At present, we are witnessing an explosion of works that develop and apply machine learning to solid-state systems. We provide a comprehensive overview and analysis of the most recent research in this topic. As a starting point, we introduce machine learning principles, algorithms, descriptors, and databases in materials science. We continue with the description of different machine learning approaches for the discovery of stable materials and the prediction of their crystal structure. Then we discuss research in numerous quantitative structure-property relationships and various approaches for the replacement of first-principle methods by machine learning. We review how active learning and surrogate-based optimization can be applied to improve the rational design process and related examples of applications. Two major questions are always the interpretability of and the physical understanding gained from machine learning models. We consider therefore the different facets of interpretability and their importance in materials science. Finally, we propose solutions and future research paths for various challenges in computational materials science.

A very popular ab-initio scheme to calculate electronic properties in solids is the use of hybrid functionals in density functional theory (DFT) that mixes a portion of Fock exchange with DFT functionals. In spite of their success, a major problem still remains, related to the use of one single mixing parameter for all materials. Guided by physical arguments that connect the mixing parameter to the dielectric properties of the solid, and ultimately to its band gap, we propose a method to calculate this parameter from the electronic density alone. This method is able to cut significantly the error of traditional hybrid functionals for large and small gap materials, while retaining a good description of structural properties. Moreover, its implementation is simple and leads to a negligible increase of the computational time.Density functional theory (DFT) is one of the major achievements of theoretical physics in the last decades. It is now routinely used to interpret experiments or to predict properties of novel materials. The success of DFT relies on the Kohn-Sham (KS) scheme and the existence of good approximations for the unknown exchange and correlation (xc) functional. In the standard KS formulation the xc potential is local and static. Since the original suggestion of the local-density approximation (LDA) [1], a swarm of functionals has been proposed in the literature [2]. In the ab-initio study of solids, the Perdew, Burke and Ernzerhof [5] (PBE) parametrization of the xc functional has been for many years the default choice for many applications. A good functional must yield ground states properties (like structural parameters), while it is expected that the KS gap and true quasiparticle gap differ by the derivative discontinuity [3]. Indeed, for semiconductors and insulators PBE yields good structural properties and KS band-gap energies that are at best half of their experimental value. To obtain both the ground state and quasiparticle energies correctly within one and the same formalism, one can resort to, e.g., a many-body GW calculation [6,7]. However, GW is by all measures an expensive technique, with a very unfavorable scaling with the number of atoms in the unit cell. It is therefore unpractical for the study of band structures of large systems and clearly prohibitive regarding total energy calculations even for simple realistic systems.Much of the computational effort in GW comes from the dynamically screened Coulomb interaction W . It has therefore been crucial to explore to which extent dynamical effects are mandatory, or whether non-locality is the dominating characteristic. The move from local KS potential to non-local functionals has first been pushed forward in Quantum Chemistry, where today the so-called hybrid functionals are very popular. These functionals mix a fraction α of Fock exchange with a combination of LDA and generalized gradient (GGA) functionals. The application of hybrid functionals to the solid state had a much slower start [8,9]. The situation changed recently, helped by th...