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

arXiv: https://arxiv.org/abs/2502.05411 · HTML: https://mlsanity.com/paper/2502.05411/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 114 references that are in the map.
- First posted on arXiv 8 Feb 2025.

## Builds on (most cited first, 6 of 6)

- [AI Feynman: a Physics-Inspired Method for Symbolic Regression](https://mlsanity.com/paper/1905.11481/) (2019)
- [By-passing the Kohn-Sham equations with machine learning](https://mlsanity.com/paper/1609.02815/) (2016)
- [Learning the exchange-correlation functional from nature with fully differentiable density functional theory](https://mlsanity.com/paper/2102.04229/) (2021)
- [Toward Physically Plausible Data-Driven Models: A Novel Neural Network Approach to Symbolic Regression](https://mlsanity.com/paper/2302.00773/) (2023)
- [Orbital-free Bond Breaking via Machine Learning](https://mlsanity.com/paper/1306.1812/) (2013)
- [Degeneration of kernel regression with Matern kernels into low-order polynomial regression in high dimension](https://mlsanity.com/paper/2311.10790/) (2023)

## Built on by (most cited first, 2 of 2)

- [Gaussian Process Regression -- Neural Network Hybrid with Optimized Redundant Coordinates](https://mlsanity.com/paper/2509.08457/) (2025)
- [All-electron machine-learned orbital-free density functional for periodic CHNO systems](https://mlsanity.com/paper/2610.17975/) (2026)

## Abstract (arXiv)

Machine learning (ML) of kinetic energy functionals (KEF) for orbital-free density functional theory (OF-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 machine learning of crystal cell-averaged kinetic energy densities (τ) 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 (GPR-NN) method was used to understand the type of functional dependence of τ on the features which contained cell-averaged terms of the 4th order gradient expansion and the product of the electron density and Kohn-Sham effective potential. Based on this analysis, an analytic model is constructed that can reproduce Kohn-Sham 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.