Data-driven kinetic energy density fitting for orbital-free DFT: Linear vs Gaussian process regression

Abstract

We study the dependence of kinetic energy densities (KED) on density-dependent variables that have been suggested in previous works on kinetic energy functionals (KEF) for orbitalfree DFT (OF-DFT). We focus on the role of data distribution and on data and regressor selection. We compare unweighted and weighted linear and Gaussian process regressions of KED for light metals and a semiconductor. We find that good quality linear regression resulting in good energy-volume dependence is possible over density-dependent variables suggested in previous literature. This is achieved with weighted fitting based on KED histogram. With Gaussian process regressions, excellent KED fit quality well exceeding that of linear regressions is obtained as well as a good energy-volume dependence which was somewhat better than that of best linear regressions. We find that while the use of the effective potential as a descriptor improves linear KED fitting, it does not improve the quality of the energy-volume dependence with linear regressions but substantially improves it with Gaussian process regression. Gaussian process regression is also able to perform well without data weighting.

Easy representation of multivariate functions with low-dimensional terms via Gaussian process regression kernel design: applications to machine learning of potential energy surfaces and kinetic energy densities from sparse data

“…Similar to what was found with RS-HDMR-GPR in [13], variables x 1 , x 2 , x 3 , x 7 are seen as most important, and their length parameters are relatively small. The importance of x 7 (which is the product of the electron density and Kohn-Sham effective potential) in capturing most of the variance of the KED is also consistent with the result of [22]. The dwindling of the variance of f 4 (x 4 ) and f 5 (x 5 ) corresponds to their optimized length parameters becoming large, indicating their low relevance, in a way similar to ARD.…”

“…The PES data were normalized to unit variance before fitting; we therefore use isotropic kernels. The KED data were scaled to [0,1] for the same reason; as their distributions are extremely uneven (see [22]), we found that scaling is preferred over normalization in this case. The length parameter l is 5.47 (d = 4-6)-8.17 .…”

“…We fitted the potential energy surfaces (PES) of H 2 CO and UF 6 molecules and the kinetic energy densities (KEDs) of aluminium, magnesium, and silicon crystals at equilibrium and strained geometries. For the description of the datasets and information about the applications of these functions, see [20] for H 2 CO, [21] for UF 6 , and [22] for the KED data. We chose these applications, as in them, high fitting accuracy is required (with errors much smaller than 1% of the data range for the model to be useful at all and with desirable accuracy of better than 0.01% [23][24][25]).…”

Modeling of plasmonic properties of nanostructures for next generation solar cells and beyond

“…Moreover, in recent years, significant advances in KEF development, including advances in semi-local (i.e. linear-scaling) KEFs [182,183] as well as machine-learned KEFs [194][195][196][197][198][199][200], are being made. The newest functionals appearing in recent years can treat systems with more non-uniform densities and give hope that in relatively near future many materials used in plasmonics (metals as well as non-metals) can be modeled with OFDFT with at least semi-quantitative accuracy.…”

Kinetic Energy Density Functionals Based on a Generalized Screened Coulomb Potential: Linear Response and Future Perspectives

“…Nevertheless, accurate approximations of T s [n] are hard to obtain because this quantity usually gives a dominant contribution to the ground-state energy [3] and because of the highly non-local nature of the KE functional [5,51,[73][74][75][76][77][78]. For this reason, more recently, machine-learning methods have also been used to develop KE functionals [79][80][81][82][83][84][85][86][87].…”