  We introduce a local machine-learning method for predicting the electron
densities of periodic systems. The framework is based on a numerical,
atom-centred auxiliary basis, which enables an accurate expansion of the
all-electron density in a form suitable for learning isolated and periodic
systems alike. We show that using this formulation the electron densities of
metals, semiconductors and molecular crystals can all be accurately predicted
using symmetry-adapted Gaussian process regression models, properly adjusted
for the non-orthogonal nature of the basis. These predicted densities enable
the efficient calculation of electronic properties which present errors on the
order of tens of meV/atom when compared to ab initio density-functional
calculations. We demonstrate the key power of this approach by using a model
trained on ice unit cells containing only 4 water molecules to predict the
electron densities of cells containing up to 512 molecules, and see no increase
in the magnitude of the errors of derived electronic properties when increasing
the system size. Indeed, we find that these extrapolated derived energies are
more accurate than those predicted using a direct machine-learning model.
Finally, on heterogeneous datasets SALTED can predict electron densities with
errors below 4%.
