  Deep-learning density functional theory (DFT) shows great promise to
significantly accelerate material discovery and potentially revolutionize
materials research. However, current research in this field primarily relies on
data-driven supervised learning, making the developments of neural networks and
DFT isolated from each other. In this work, we present a theoretical framework
of neural-network DFT, which unifies the optimization of neural networks with
the variational computation of DFT, enabling physics-informed unsupervised
learning. Moreover, we develop a differential DFT code incorporated with
deep-learning DFT Hamiltonian, and introduce algorithms of automatic
differentiation and backpropagation into DFT, demonstrating the capability of
neural-network DFT. The physics-informed neural-network architecture not only
surpasses conventional approaches in accuracy and efficiency, but also offers a
new paradigm for developing deep-learning DFT methods.
