  Machine learning has demonstrated great power in materials design, discovery,
and property prediction. However, despite the success of machine learning in
predicting discrete properties, challenges remain for continuous property
prediction. The challenge is aggravated in crystalline solids due to
crystallographic symmetry considerations and data scarcity. Here we demonstrate
the direct prediction of phonon density of states using only atomic species and
positions as input. We apply Euclidean neural networks, which by construction
are equivariant to 3D rotations, translations, and inversion and thereby
capture full crystal symmetry, and achieve high-quality prediction using a
small training set of $\sim 10^{3}$ examples with over 64 atom types. Our
predictive model reproduces key features of experimental data and even
generalizes to materials with unseen elements,and is naturally suited to
efficiently predict alloy systems without additional computational cost. We
demonstrate the potential of our network by predicting a broad number of high
phononic specific heat capacity materials. Our work indicates an efficient
approach to explore materials' phonon structure, and can further enable rapid
screening for high-performance thermal storage materials and phonon-mediated
superconductors.
