  Electronic nearsightedness is one of the fundamental principles governing the
behavior of condensed matter and supporting its description in terms of local
entities such as chemical bonds. Locality also underlies the tremendous success
of machine-learning schemes that predict quantum mechanical observables -- such
as the cohesive energy, the electron density, or a variety of response
properties -- as a sum of atom-centred contributions, based on a short-range
representation of atomic environments. One of the main shortcomings of these
approaches is their inability to capture physical effects, ranging from
electrostatic interactions to quantum delocalization, which have a long-range
nature. Here we show how to build a multi-scale scheme that combines in the
same framework local and non-local information, overcoming such limitations. We
show that the simplest version of such features can be put in formal
correspondence with a multipole expansion of permanent electrostatics. The
data-driven nature of the model construction, however, makes this simple form
suitable to tackle also different types of delocalized and collective effects.
We present several examples that range from molecular physics, to surface
science and biophysics, demonstrating the ability of this multi-scale approach
to model interactions driven by electrostatics, polarization and dispersion, as
well as the cooperative behavior of dielectric response functions.
