Error propagation in exchange-correlation functionals from machine learned quantities

Many accurate and broadly applicable density functional approximations (DFAs) have traditionally been developed from approximate exchange-correlation holes. In the regime of machine learned DFAs (ML-DFAs), several attempts have been made to fit functionals to reference energies. However, models can be made more physically interpretable by training on quantities that are transformed into density functionals or by implementing exact constraints. Here, we construct DFAs based on electron gas data from quantum Monte Carlo (QMC) simulations through the lens of machine learning techniques. Fundamental objects in electronic structure theory, such as the exchange-correlation hole, are approximated by feed-forward neural networks and genetic algorithms. They are then transformed and evaluated as exchange-correlation energies. These learned objects and resulting quantities are used to understand the origin of significant errors in the development of ML-DFAs. We also consider the implications of these findings for how functionals can be trained at the accuracy of wavefunction methods, but at the scaling of density functional theory (DFT).