  We introduce a scheme based on machine learning and deep neural networks to
model the environmental dependence of the electronic polarizability in
insulating materials. Application to liquid water shows that training the
network with a relatively small number of molecular configurations is
sufficient to predict the polarizability of arbitrary liquid configurations in
close agreement with ab initio density functional theory calculations. In
combination with a neural network representation of the interatomic potential
energy surface,the scheme allows us to calculate the Raman spectra along
2-nanosecond classical trajectories at different temperatures for H2O and D2O.
The vast gains in efficiency provided by the machine learning approach enable
longer trajectories and larger system sizes relative to ab initio methods,
reducing the statistical error and improving the resolution of the
low-frequency Raman spectra. Decomposing the spectra into intramolecular and
intermolecular contributions elucidates the mechanisms behind the temperature
dependence of the low-frequency and stretch modes.
