Understanding machine-learned density functionals

Machine learning (ML) is an increasingly popular statistical tool for analyzing either measured or calculated data sets. Here, we explore its application to a well‐defined physics problem, investigating issues of how the underlying physics is handled by ML, and how self‐consistent solutions can be f...

Full description

Saved in:
Bibliographic Details
Published in:International journal of quantum chemistry 2016-06, Vol.116 (11), p.819-833
Main Authors: Li, Li, Snyder, John C., Pelaschier, Isabelle M., Huang, Jessica, Niranjan, Uma-Naresh, Duncan, Paul, Rupp, Matthias, Müller, Klaus-Robert, Burke, Kieron
Format: Article
Language:eng ; fre ; ger
Subjects:
Chemistry
density functional theory
kinetic energy functional
machine learning
orbital free
Physical chemistry
Quantum physics
self-consistent calculation
Citations: Items that this one cites
Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:Machine learning (ML) is an increasingly popular statistical tool for analyzing either measured or calculated data sets. Here, we explore its application to a well‐defined physics problem, investigating issues of how the underlying physics is handled by ML, and how self‐consistent solutions can be found by limiting the domain in which ML is applied. The particular problem is how to find accurate approximate density functionals for the kinetic energy (KE) of noninteracting electrons. Kernel ridge regression is used to approximate the KE of non‐interacting fermions in a one dimensional box as a functional of their density. The properties of different kernels and methods of cross‐validation are explored, reproducing the physics faithfully in some cases, but not others. We also address how self‐consistency can be achieved with information on only a limited electronic density domain. Accurate constrained optimal densities are found via a modified Euler‐Lagrange constrained minimization of the machine‐learned total energy, despite the poor quality of its functional derivative. A projected gradient descent algorithm is derived using local principal component analysis. Additionally, a sparse grid representation of the density can be used without degrading the performance of the methods. The implications for machine‐learned density functional approximations are discussed. © 2015 Wiley Periodicals, Inc. Machine learning has been used to find accurate approximate density functionals. In this approach, kernel ridge regression is used to approximate the kinetic energy of noninteracting fermions in a one dimensional box as a functional of their density. Key procedures in this method including the choice of kernel, different cross‐validation, representation of sparse grid, manifold reconstruction and projected gradient descent algorithm are explored and documented.
ISSN:0020-7608
1097-461X
DOI:10.1002/qua.25040