Machine learning-guided construction of an analytic kinetic energy functional for orbital free density functional theory

Abstract Machine learning (ML) of kinetic energy functionals (KEF) for orbital-free density functional theory (DFT) holds the promise of addressing an important bottleneck in large-scale ab initio materials modeling where sufficiently accurate analytic KEFs are lacking. However, ML models are not as easily handled as analytic expressions; they need to be provided in the form of algorithms and associated data. Here, we bridge the two approaches and construct an analytic expression for a KEF guided by interpretative ML of crystal cell-averaged kinetic energy densities ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mrow> <mml:mover> <mml:mi>τ</mml:mi> <mml:mo stretchy="false">¯</mml:mo> </mml:mover> </mml:mrow> </mml:mrow> </mml:mrow> </mml:math> ) of several hundred materials. A previously published dataset including multiple phases of 433 unary, binary, and ternary compounds containing Li, Al, Mg, Si, As, Ga, Sb, Na, Sn, P, and In was used for training, including data at the equilibrium geometry as well as strained structures. A hybrid Gaussian process regression—neural network method was used to understand the type of functional dependence of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mover> <mml:mi>τ</mml:mi> <mml:mo accent="true">―</mml:mo> </mml:mover> </mml:mrow> </mml:math> on the features which contained cell-averaged terms of the 4th order gradient expansion and the product of the electron density and Kohn–Sham (KS) effective potential. Based on this analysis, an analytic model is constructed that can reproduce KS DFT energy–volume curves with sufficient accuracy (pronounced minima that are sufficiently close to the minima of the Kohn–Sham DFT-based curves and with sufficiently close curvatures) to enable structure optimizations and elastic response calculations.