Applying Exact Conditions to Machine Learned Density Functionals

Historical methods of functional development in density functional theory have been largely guided by analytic conditions that constrain the exact functional one is trying to approximate. Recently, machine learning functionals, which are formed by extrapolating the results from a small number of exactly solved systems to unsolved systems that are similar in nature, have turned away from utilizing constraints on the exact functional. In this work, we show that imposing these exact conditions onto machine learning functionals can improve the ability of the machine learning functionals to extrapolate to unsolved systems.

In particular, we machine learn the non-interacting kinetic energy of a set of densities in two ways. First, we train on the densities without any applied scaling. Second, we train on the densities scaled so that the root mean square of the density is unity, which relates to the unscaled non-interacting kinetic energy through exact conditions. We compare the performance of these machine learning functionals, showing that the latter tends to produce lower errors given an equal number of training data.