  The most successful and popular machine learning models of atomic-scale
properties derive their transferability from a locality ansatz. The properties
of a large molecule or a bulk material are written as a sum over contributions
that depend on the configurations within finite atom-centered environments. The
obvious downside of this approach is that it cannot capture non-local,
non-additive effects such as those arising due to long-range electrostatics or
quantum interference. We propose a solution to this problem by introducing
non-local representations of the system that are remapped as feature vectors
that are defined locally and are equivariant in O(3). We consider in particular
one form that has the same asymptotic behavior as the electrostatic potential.
We demonstrate that this framework can capture non-local, long-range physics by
building a model for the electrostatic energy of randomly distributed
point-charges, for the unrelaxed binding curves of charged organic molecular
dimers, and for the electronic dielectric response of liquid water. By
combining a representation of the system that is sensitive to long-range
correlations with the transferability of an atom-centered additive model, this
method outperforms current state-of-the-art machine-learning schemes, and
provides a conceptual framework to incorporate non-local physics into atomistic
machine learning.
