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Introduction to invariant-based machine learning for periodic crystals

Machine learning can be justified only if input descriptors are crystal invariants independent of accidental choices. To use a household analogy, the average color of human clothes can be the easiest descriptor extracted from images but cannot be seriously used to predict height or any reliable data about people. Similarly, no properties of crystals can be reliably predicted from ambiguous parameters of a unit cell and a motif. Since crystal structures are determined in a rigid form, they should be considered equivalent modulo rigid motion or isometry , which preserves all interpoint distances. Then crystals can be justifiably distinguished only by isometry invariants that are independent of a unit cell and are preserved under any translations and rotations. Though Niggli ’ s reduced cell is unique, it is discontinuous under atomic perturbations, which are always present in real crystals. This continuity of invariants is important to quantify similarities between near identical crystals obtained by Crystal Structure Prediction [1] as approximations to energy minima. The pictures on the right of Figure 1 show that almost any small perturbation of points breaks most past invariants: symmetry groups and descriptors depending on primitive or reduced cells, because the volume of the primitive cell doubles. The density was used to represent a crystal structure in energy landscapes but is constant under perturbations and not enough to distinguish dense (really, almost all) crystals. But density functions [2] depending on a variable radius give more information about relative positions of atoms. All machine learning approaches implicitly assume that a target property continuously depends on a given input, for example similar crystals should have close values of their lattice energy. We experimentally tested that the lattice energy is discontinuous