  Representation learning for the electronic structure problem is a major
challenge of machine learning in computational condensed matter and materials
physics. Within quantum mechanical first principles approaches, Kohn-Sham
density functional theory (DFT) is the preeminent tool for understanding
electronic structure, and the high-dimensional wavefunctions calculated in this
approach serve as the building block for downstream calculations of correlated
many-body excitations and related physical observables. Here, we use
variational autoencoders (VAE) for the unsupervised learning of
high-dimensional DFT wavefunctions and show that these wavefunctions lie in a
low-dimensional manifold within the latent space. Our model autonomously
determines the optimal representation of the electronic structure, avoiding
limitations due to manual feature engineering and selection in prior work. To
demonstrate the utility of the latent space representation of the DFT
wavefunction, we use it for the supervised training of neural networks (NN) for
downstream prediction of the quasiparticle bandstructures within the GW
formalism, which includes many-electron correlations beyond DFT. The GW
prediction achieves a low error of 0.11 eV for a combined test set of metals
and semiconductors drawn from the Computational 2D Materials Database (C2DB),
suggesting that latent space representation captures key physical information
from the original data. Finally, we explore the interpretability of the VAE
representation and show that the successful representation learning and
downstream prediction by our model is derived from the smoothness of the VAE
latent space, which also enables the generation of wavefunctions on arbitrary
points in latent space. Our work provides a novel and general machine-learning
framework for investigating electronic structure and many-body physics.
