Machine learning optimization of the collocation point set for solving the Kohn-Sham equation

TL;DR

This paper demonstrates that machine learning techniques can significantly reduce the number of collocation points needed to solve the Kohn-Sham equations in density functional theory, maintaining high accuracy while improving computational efficiency.

Contribution

The authors introduce a machine learning approach combining Gaussian process regression and genetic algorithms to optimize collocation point sets for the Kohn-Sham equation, reducing their size by over an order of magnitude.

Findings

Reduced collocation points from ~51,000 to 2,000 while maintaining accuracy.

Applied method successfully to CO and H2O molecules.

Enhanced efficiency in solving Kohn-Sham equations.

Abstract

The rectangular collocation approach makes it possible to solve the Schr\"odinger equation with basis functions that do not have amplitude in all regions in which wavefunctions have significant amplitude. Collocation points can be restricted to a small region of space. As no integrals are computed, there are no problems due to discontinuities in the potential, and there is no need to use integrable basis functions. In this paper, we show, for the Kohn-Sham equation, that machine learning can be used to drastically reduce the size of the collocation point set. This is demonstrated by solving the Kohn-Sham equations for CO and H2O. We solve the Kohn-Sham equation on a given effective potential which is a critical part of all DFT calculations, and monitor orbital energies and orbital shapes. We use a combination of Gaussian process regression and a genetic algorithm to reduce the collocation point set size by more than an order of magnitude (from about 51,000 points to 2,000 points) while retaining mHartree accuracy.