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We introduce a scheme based on machine learning and deep neural networks tomodel the environmental dependence of the electronic polarizability ininsulating materials. Application to liquid water shows that training thenetwork with a relatively small number of molecular configurations issufficient to predict the polarizability of arbitrary liquid configurations inclose agreement with ab initio density functional theory calculations. Incombination with a neural network representation of the interatomic potentialenergy surface,the scheme allows us to calculate the Raman spectra along2-nanosecond classical trajectories at different temperatures for H2O and D2O.The vast gains in efficiency provided by the machine learning approach enablelonger trajectories and larger system sizes relative to ab initio methods,reducing the statistical error and improving the resolution of thelow-frequency Raman spectra. Decomposing the spectra into intramolecular andintermolecular contributions elucidates the mechanisms behind the temperaturedependence of the low-frequency and stretch modes.

Raman Spectrum and Polarizability of Liquid Water from Deep Neural Networks