# PhD Proposal: Speeding up Density Functional Theory with Machine Learning

IRB IRB-4105

The Kohn-Sham (KS) equations are a nonlinear eigenvalue problem of the form $H[\rho]\Psi = E\Psi$, where $H$ is a real symmetric matrix called the \textit{Hamiltonian}, $\Psi$ is an eigenvector called the \textit{wave function}, $E$ is an eigenvalue called the \textit{energy}, and $\rho(\mathbf{r}) = \sum_i |\Psi_i(\mathbf{r})|^2$ is an real-valued field called the \textit{charge density}, which is unknown a priori. The KS equations are nonlinear in the sense that the matrix $H$ depends on the charge density $\rho$, which in turn depends on the eigenvectors $\Psi$ of $H$. Typically it is solved by fixed-point iteration, where an initial guess for $\rho$ is made and then a sequence of \textit{linear} eigenvalue problems are solved until convergence. The cost of the iteration is dominated by the eigenvalue problem. Consequently methods of reducing the number of requisite iterations are of great interest.

In this thesis, we investigate the use of machine-learning models for solving the KS equations. Our strategy is to develop machine-learning models which approximate numerical quantities in existing algorithms for electronic structure. In particular we use equivariant graph-neural-networks to predict the Kohn-Sham charge density. We show such methods obtain an average savings of $13\%$ in the number of iterations needed to reach convergence. Our methods are general, but we focus on learning data from catalysis, an application with potentially large environmental impact.