Variational principle to regularize machine-learned density functionals: the non-interacting kinetic-energy functional

TL;DR

This paper introduces a novel regularization method for training neural network-based density functionals, focusing on the non-interacting kinetic energy in density functional theory, demonstrating high accuracy and generalizability on one-dimensional systems and atoms.

Contribution

The paper presents a new regularization technique for deep learning of density functionals, particularly improving the modeling of the kinetic-energy functional in DFT.

Findings

Effective training of kinetic-energy functionals on 1D systems.

High accuracy in modeling hydrogen chains and atoms.

Demonstrated generalization to exchange-correlation functionals.

Abstract

Practical density functional theory (DFT) owes its success to the groundbreaking work of Kohn and Sham that introduced the exact calculation of the non-interacting kinetic energy of the electrons using an auxiliary mean-field system. However, the full power of DFT will not be unleashed until the exact relationship between the electron density and the non-interacting kinetic energy is found. Various attempts have been made to approximate this functional, similar to the exchange--correlation functional, with much less success due to the larger contribution of kinetic energy and its more non-local nature. In this work we propose a new and efficient regularization method to train density functionals based on deep neural networks, with particular interest in the kinetic-energy functional. The method is tested on (effectively) one-dimensional systems, including the hydrogen chain, non-interacting electrons, and atoms of the first two periods, with excellent results. For the atomic systems, the generalizability of the regularization method is demonstrated by training also an exchange--correlation functional, and the contrasting nature of the two functionals is discussed from a machine-learning perspective.