Using Machine Learning to Construct and Categorize Density Functionals

Density functional theory (DFT), combined with standard exchange-correlation approximations, is a usefully accurate and efficient tool to generate computational predictions in chemistry and material sciences. In the past decade, machine learning has been used extensively to build density functional approximations that concur with human-defined standards. This thesis details the effort to construct and characterize exchange-correlation approximations in DFT with physics-informed machine learning. The Kohn-Sham regularizer (KSR) is a differentiable approach for making machine-learned density functionals. It allows approximating the exchange-correlation functional while self-consistently solving the Kohn-Sham equations. It was initially formulated to generate accurate predictions for strongly-correlated molecules. Here I discuss a spin-adapted extension of the KSR that machine-learns the exchange-correlation energy densities as a functional of the spin densities and substantially improves generalizability for weakly correlated molecules. With a neural network approximation that accounts for nonlocal interactions, a training set of just five atoms and ions in 1D can predict the ground-state properties of several molecules with near chemical accuracy. The differentiability of spin-adapted KSR ensures a fast convergence during training and yields accurate predictions of the exchange-correlation potentials and other properties, often complying with known exact behaviors. While this serves as a proof of concept for what machine learning can achieve, such methods, in principle, can add to the complexity of the existing diverse approaches for designing exchange-correlation approximations, further deluding the existence of a unified scheme for systematic improvement of density functionals. On the other hand, machine learning, especially unsupervised learning algorithms, can help categorize different exchange-correlation approximations without introducing human bias or considering any absolute errors. To answer the question of how several exchange-correlation functionals are similar or different from each other, we propose a novel approach to group these functionals based on statistical learning tools. This approach does not use any exact information, accounts for density-driven differences in approximations based on the theory of density-corrected DFT, and avoids any form of bias between empirical, partially-empirical, and non-empirical approximations. It sorts exchange-correlation functionals based on similarities predicted using a novel, parameter-free unsupervised clustering algorithm. For 33 popular exchange-correlation functionals and the MGAE109 dataset, this scheme generates categories of functionals that somewhat mimic the popular Jacob’s ladder categorization while depicting that Minnesota functionals of recent vintage might have strayed far from the path of typical functional development