# Variational principle to regularize machine-learned density functionals: the non-interacting kinetic-energy functional

arXiv: https://arxiv.org/abs/2306.17587 · HTML: https://mlsanity.com/paper/2306.17587/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 50 references that are in the map.
- First posted on arXiv 30 Jun 2023.

## Builds on (most cited first, 7 of 7)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Skillful Precipitation Nowcasting using Deep Generative Models of Radar](https://mlsanity.com/paper/2104.00954/) (2021)
- [Ab-Initio Solution of the Many-Electron Schr"odinger Equation with Deep Neural Networks](https://mlsanity.com/paper/1909.02487/) (2019)
- [Deep neural network solution of the electronic Schr"odinger equation](https://mlsanity.com/paper/1909.08423/) (2019)
- [Kohn-Sham equations as regularizer: building prior knowledge into machine-learned physics](https://mlsanity.com/paper/2009.08551/) (2020)
- [Learning the exchange-correlation functional from nature with fully differentiable density functional theory](https://mlsanity.com/paper/2102.04229/) (2021)
- [Orbital-free Bond Breaking via Machine Learning](https://mlsanity.com/paper/1306.1812/) (2013)

## Built on by (most cited first, 3 of 3)

- [Overcoming the Barrier of Orbital-Free Density Functional Theory for Molecular Systems Using Deep Learning](https://mlsanity.com/paper/2309.16578/) (2023)
- [Self-Consistency Training for Density-Functional-Theory Hamiltonian Prediction](https://mlsanity.com/paper/2403.09560/) (2024)
- [All-electron machine-learned orbital-free density functional for periodic CHNO systems](https://mlsanity.com/paper/2610.17975/) (2026)

## Abstract (arXiv)

Practical density functional theory (DFT) owes its success to the groundbreaking work of Kohn and Sham that introduced the exact calculation of the non-interacting kinetic energy of the electrons using an auxiliary mean-field system. However, the full power of DFT will not be unleashed until the exact relationship between the electron density and the non-interacting kinetic energy is found. Various attempts have been made to approximate this functional, similar to the exchange--correlation functional, with much less success due to the larger contribution of kinetic energy and its more non-local nature. In this work we propose a new and efficient regularization method to train density functionals based on deep neural networks, with particular interest in the kinetic-energy functional. The method is tested on (effectively) one-dimensional systems, including the hydrogen chain, non-interacting electrons, and atoms of the first two periods, with excellent results. For the atomic systems, the generalizability of the regularization method is demonstrated by training also an exchange--correlation functional, and the contrasting nature of the two functionals is discussed from a machine-learning perspective.