Source-linked AI summary
Finding Density Functionals with Machine Learning
John C. Snyder, Matthias Rupp, Katja Hansen, Klaus-Robert Müller, Kieron Burke
TL;DR
The paper develops a machine-learning approximation to the kinetic-energy functional for non-interacting spinless fermions in a one-dimensional box. It generates densities by numerically solving the Schrödinger equation, uses PCA to represent local density variation, and provides reproducibility details for the resulting MLA.
Problem
The paper addresses constructing a kinetic-energy density functional that bypasses repeatedly solving the Schrödinger equation for the one-dimensional fermion model.
Method
The approach generates densities and kinetic energies with Numerov’s method, trains an MLA, and uses local PCA projections for self-consistent density searches.
Results
The reported PCA example retains 99.98% of local density variance with m = 30 and ℓ = 5.
Takeaways & Limitations
A small PCA subspace can represent the local density variation used in the paper’s self-consistent search.
Abstract
from arXiv · showhide
Machine learning is used to approximate density functionals. For the model problem of the kinetic energy of non-interacting fermions in 1d, mean absolute errors below 1 kcal/mol on test densities similar to the training set are reached with fewer than 100 training densities. A predictor identifies if a test density is within the interpolation region. Via principal component analysis, a projected functional derivative finds highly accurate self-consistent densities. Challenges for application of our method to real electronic structure problems are discussed.
Supplementary Information for “Finding Density Functionals with Machine Learning”
The supplementary material documents the information needed to reproduce the machine-learning approximation to the kinetic-energy functional and provides additional MLA details.
- The supplement is intended to enable exact reproduction of the machine-learning approximation to the kinetic-energy functional.It supplements the main-paper equations and Numerov-method details.
- The listed potential parameters can be used with Numerov’s method to reconstruct the training densities.
SELF-CONSISTENT DENSITIES
PCA projects local density variation onto a small number of directions, retaining nearly all variance in the example reported and optimizing projection parameters for self-consistent searches.
- SELF-CONSISTENT DENSITIES: PCA retains the dominant local variation by keeping eigenvectors with the largest eigenvalues and discarding directions below a cutoff.
- SELF-CONSISTENT DENSITIES: 99.98% of the variance is retained for N = 1 with m = 30 neighboring training densities and ℓ = 5 retained components.
- SELF-CONSISTENT DENSITIES: The PCA projection uses G = 500 eigenvalues, although only a few are needed to represent local density variation.
NUMEROV’S METHOD
Numerov’s method discretizes the one-dimensional Schrödinger problem, locates the lowest eigenvalues, and uses normalized eigenfunctions to obtain densities and kinetic energies.
- NUMEROV’S METHOD: Numerov’s method solves the Schrödinger equation for non-interacting spinless fermions in a one-dimensional box with hard-wall boundary conditions.
- NUMEROV’S METHOD: The wavefunction and potential are discretized on a uniform grid with spacing Δx = 1/(G − 1).
- NUMEROV’S METHOD: For self-consistent-density tests, the PCA search uses m = 30 and ℓ = 5; for ℓ ≥ 7, gradient descent fails to converge in some cases.
- NUMEROV’S METHOD: The lowest N eigenvalues are found by scanning energy intervals and refining each root with binary search until the interval is shorter than 10^-14.
- NUMEROV’S METHOD: After eigenfunction normalization, the resulting orbitals determine the density and kinetic energy used by the MLA.