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Mapping an atomistic configuration to a symmetrized N-point correlation of a field associated with the atomic positions (e.g., an atomic density) has emerged as an elegant and effective solution to represent structures as the input of machine-learning algorithms. While it has become clear that low-order density correlations do not provide a complete representation of an atomic environment, the exponential increase in the number of possible N-body invariants makes it difficult to design a concise and effective representation. We discuss how to exploit recursion relations between equivariant features of different order (generalizations of N-body invariants that provide a complete representation of the symmetries of improper rotations) to compute high-order terms efficiently. In combination with the automatic selection of the most expressive combination of features at each order, this approach provides a conceptual and practical framework to generate systematically improvable, symmetry adapted representations for atomistic machine learning.

The input of almost every machine learning algorithm targeting the properties of matter at the atomic scale involves a transformation of the list of Cartesian atomic coordinates into a more symmetric representation. Many of the most popular representations can be seen as an expansion of the symmetrized correlations of the atom density and differ mainly by the choice of basis. Considerable effort has been dedicated to the optimization of the basis set, typically driven by heuristic considerations on the behavior of the regression target. Here, we take a different, unsupervised viewpoint, aiming to determine the basis that encodes in the most compact way possible the structural information that is relevant for the dataset at hand. For each training dataset and number of basis functions, one can build a unique basis that is optimal in this sense and can be computed at no additional cost with respect to the primitive basis by approximating it with splines. We demonstrate that this construction yields representations that are accurate and computationally efficient, particularly when working with representations that correspond to high-body order correlations. We present examples that involve both molecular and condensed-phase machine-learning models.

Data-driven schemes that associate molecular and crystal structures with their microscopic properties share the need for a concise, effective description of the arrangement of their atomic constituents. Many types of models rely on descriptions of atom-centered environments, that are associated with an atomic property or with an atomic contribution to an extensive macroscopic quantity. Frameworks in this class can be understood in terms of atom-centered density correlations (ACDC), that are used as a basis for a body-ordered, symmetry-adapted expansion of the targets.Several other schemes, that gather information on the relationship between neighboring atoms using ``message-passing' ideas, cannot be directly mapped to correlations centered around a single atom. We generalize the ACDC framework to include multi-centered information, generating representations that provide a complete linear basis to regress symmetric functions of atomic coordinates, and provides a coherent foundation to systematize our understanding of both atom-centered and message-passing, invariant and equivariant machine-learning schemes.

Graph neural networks (GNN) are very popular methods in machine learning and have been applied very successfully to the prediction of the properties of molecules and materials. First-order GNNs are well known to be incomplete, i.e., there exist graphs that are distinct but appear identical when seen through the lens of the GNN. More complicated schemes have thus been designed to increase their resolving power. Applications to molecules (and more generally, point clouds), however, add a geometric dimension to the problem. The most straightforward and prevalent approach to construct graph representation for molecules regards atoms as vertices in a graph and draws a bond between each pair of atoms within a chosen cutoff. Bonds can be decorated with the distance between atoms, and the resulting ``distance graph NNs'' (dGNN) have empirically demonstrated excellent resolving power and are widely used in chemical ML, {with all known indistinguishable configurations being resolved in the fully-connected limit, which is equivalent to infinite or sufficiently large cutoff.} Here we {present a counterexample that proves that dGNNs are not complete even for the restricted case of fully-connected graphs induced by 3D atom clouds.} We construct pairs of distinct point clouds whose associated graphs are, for any cutoff radius, equivalent based on a first-order Weisfeiler-Lehman test. This class of degenerate structures includes chemically-plausible configurations, {both for isolated structures and for infinite structures that are periodic in 1, 2, and 3 dimensions.} The existence of indistinguishable configurations sets an ultimate limit to the expressive power of some of the well-established GNN architectures for atomistic machine learning. Models that explicitly use angular or directional information in the description of atomic environments can resolve this class of degeneracies.