The electronic structure of molecules and materials determines chemical reactivity. If we could only compute it accurately and efficiently, we could accelerate molecular research and help solve some of society's biggest problems. One prominent approach to electronic structure is Density Functional Theory (DFT), at the heart of which are the Kohn-Sham (KS) equations. These equations are a nonlinear eigenvalue problem of the form H[rho] Psi = E Psi, where H is a real symmetric matrix called the Hamiltonian, Psi is an eigenvector called the wave function, E is an eigenvalue called the energy, and rho is a real-valued field called the charge density, which is unknown a priori.

In this thesis, we investigate the use of machine-learning models for reducing the amount of computation to solve the KS equations. Our strategy is to develop models to predict the charge density using equivariant graph-neural-networks. We show on materials and molecules that our method may obtain highly-accurate results leading to computational savings, sometimes obtaining chemical accuracy, commonly defined to be 1 kcal/mol, using a single step of KS-DFT. Our results demonstrate that density learning is a reliable means of speeding up DFT computations, and represents a step forward in speeding up electronic structure calculations.