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

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.

References in corpus (16)

• PyTorch: An Imperative Style, High-Performance Deep Learning Library
• Decoupled Weight Decay Regularization
• Recent developments in the PySCF program package
• Skillful Precipitation Nowcasting using Deep Generative Models of Radar
• Finding Density Functionals with Machine Learning
• Deep neural network solution of the electronic Schrödinger equation
• Ab-Initio Solution of the Many-Electron Schrödinger Equation with Deep Neural Networks
• Fermionic neural-network states for ab-initio electronic structure
• Kohn-Sham equations as regularizer: building prior knowledge into machine-learned physics
• Laplacian-level density functionals for the kinetic energy density and exchange-correlation energy
• Orbital-free Bond Breaking via Machine Learning
• Learning the exchange-correlation functional from nature with fully differentiable density functional theory
• Nuclear energy density functionals from machine learning
• Conjugate-Gradient Optimization Method for Orbital-free Density Functional Calculations
• One Dimensional Mimicking of Electronic Structure: The Case for Exponentials
• Order-$N$ orbital-free density-functional calculations with machine learning of functional derivatives for semiconductors and metals

Cited by in corpus (6)

• Bridging electronic and classical density-functional theory using universal machine-learned functional approximations
• Machine learning based nonlocal kinetic energy density functional for simple metals and alloys
• Multi-channel machine learning based nonlocal kinetic energy density functional for semiconductors
• Leveraging Normalizing Flows for Orbital-Free Density Functional Theory
• Solving deep-learning density functional theory via variational autoencoders
• Neural network distillation of orbital dependent density functional theory