# Understanding Machine-learned Density Functionals

arXiv: https://arxiv.org/abs/1404.1333 · HTML: https://mlsanity.com/paper/1404.1333/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Apr 2014.

## Built on by (most cited first, 8 of 8)

- [Review of Pedestrian Trajectory Prediction Methods: Comparing Deep Learning and Knowledge-based Approaches](https://mlsanity.com/paper/2111.06740/) (2021)
- [Evolving symbolic density functionals](https://mlsanity.com/paper/2203.02540/) (2022)
- [How Well Does Kohn-Sham Regularizer Work for Weakly Correlated Systems?](https://mlsanity.com/paper/2110.14846/) (2021)
- [Degeneration of kernel regression with Matern kernels into low-order polynomial regression in high dimension](https://mlsanity.com/paper/2311.10790/) (2023)
- [Machine learning for accuracy in density functional approximations](https://mlsanity.com/paper/2311.00196/) (2023)
- [Grad DFT: a software library for machine learning enhanced density functional theory](https://mlsanity.com/paper/2309.15127/) (2023)
- [Enhancing molecular dynamics with equivariant machine-learned densities](https://mlsanity.com/paper/2604.24563/) (2026)
- [Predicting fermionic densities using a Projected Quantum Kernel method](https://mlsanity.com/paper/2504.14002/) (2025)

## Abstract (arXiv)

Kernel ridge regression is used to approximate the kinetic energy of non-interacting fermions in a one-dimensional box as a functional of their density. The properties of different kernels and methods of cross-validation are explored, and highly accurate energies are achieved. Accurate em constrained optimal densities are found via a modified Euler-Lagrange constrained minimization of the total energy. A projected gradient descent algorithm is derived using local principal component analysis. Additionally, a sparse grid representation of the density can be used without degrading the performance of the methods. The implications for machine-learned density functional approximations are discussed.