Understanding machine-learned density functionals

Li Li, John C. Snyder, Isabelle M. Pelaschier, Jessica Huang, Uma Naresh Niranjan, Paul Duncan, Matthias Rupp, Klaus Muller, Kieron Burke

Research output: Contribution to journalArticle

39 Citations (Scopus)

Abstract

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.

Original languageEnglish
JournalInternational Journal of Quantum Chemistry
DOIs
Publication statusAccepted/In press - 2015

Fingerprint

functionals
Learning systems
Physics
Kinetic energy
machine learning
Fermions
Principal component analysis
physics
kinetic energy
Derivatives
Electrons
descent
principal components analysis
boxes
ridges
regression analysis
fermions
grids
gradients
optimization

Keywords

  • Density functional theory
  • Kinetic energy functional
  • Machine learning
  • Orbital free
  • Self-consistent calculation

ASJC Scopus subject areas

  • Condensed Matter Physics
  • Atomic and Molecular Physics, and Optics
  • Physical and Theoretical Chemistry

Cite this

Li, L., Snyder, J. C., Pelaschier, I. M., Huang, J., Niranjan, U. N., Duncan, P., ... Burke, K. (Accepted/In press). Understanding machine-learned density functionals. International Journal of Quantum Chemistry. https://doi.org/10.1002/qua.25040

Understanding machine-learned density functionals. / Li, Li; Snyder, John C.; Pelaschier, Isabelle M.; Huang, Jessica; Niranjan, Uma Naresh; Duncan, Paul; Rupp, Matthias; Muller, Klaus; Burke, Kieron.

In: International Journal of Quantum Chemistry, 2015.

Research output: Contribution to journalArticle

Li, L, Snyder, JC, Pelaschier, IM, Huang, J, Niranjan, UN, Duncan, P, Rupp, M, Muller, K & Burke, K 2015, 'Understanding machine-learned density functionals', International Journal of Quantum Chemistry. https://doi.org/10.1002/qua.25040
Li, Li ; Snyder, John C. ; Pelaschier, Isabelle M. ; Huang, Jessica ; Niranjan, Uma Naresh ; Duncan, Paul ; Rupp, Matthias ; Muller, Klaus ; Burke, Kieron. / Understanding machine-learned density functionals. In: International Journal of Quantum Chemistry. 2015.
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