# Learning the Kohn-Sham map with neural operators for quasi-linear scaling density functional theory

arXiv: https://arxiv.org/abs/2608.23895 · HTML: https://mlsanity.com/paper/2608.23895/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 53 references that are in the map.
- First posted on arXiv 24 Aug 2026.

## Builds on (most cited first, 9 of 9)

- [Chain-of-Thought Prompting Elicits Reasoning in Large Language Models](https://mlsanity.com/paper/2201.11903/) (2022)
- [Neural Operators for Accelerating Scientific Simulations and Design](https://mlsanity.com/paper/2309.15325/) (2023)
- [Machine Learning Force Fields](https://mlsanity.com/paper/2010.07067/) (2020)
- [By-passing the Kohn-Sham equations with machine learning](https://mlsanity.com/paper/1609.02815/) (2016)
- [Equivariant graph neural networks for fast electron density estimation of molecules, liquids, and solids](https://mlsanity.com/paper/2112.00652/) (2021)
- [Overcoming the Barrier of Orbital-Free Density Functional Theory for Molecular Systems Using Deep Learning](https://mlsanity.com/paper/2309.16578/) (2023)
- [Image Super-resolution Inspired Electron Density Prediction](https://mlsanity.com/paper/2402.12335/) (2024)
- [Learning from the electronic structure of molecules across the periodic table](https://mlsanity.com/paper/2510.00224/) (2025)
- [OrbitAll: A Unified Quantum Mechanical Representation Deep Learning Framework for All Molecular Systems](https://mlsanity.com/paper/2507.03853/) (2025)

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

- [Euclidean Fourier Neural Operators](https://mlsanity.com/paper/2608.28425/) (2026)
- [Beyond DFT electronic band structures via dual-space orbital learning](https://mlsanity.com/paper/2610.19919/) (2026)

## Abstract (arXiv)

Kohn--Sham density functional theory (DFT) underpins electronic-structure simulations, but repeated orbital diagonalizations lead to cubic scaling, restricting quantum calculations to modest scales only. Eliminating these auxiliary orbitals while retaining Kohn--Sham accuracy is the central goal of orbital-free DFT, but both analytical and machine-learning methods have so far fallen short. Prior learning approaches either try to learn the variational kinetic-energy functionals, which are ill-conditioned, or directly predict the ground state, which extrapolate poorly to larger systems. Instead, we identify the Kohn--Sham map as the right learning target for orbital-free DFT. It maps a Kohn--Sham potential directly to the corresponding density and noninteracting kinetic energy, quantities otherwise obtained through an orbital diagonalization. Focusing on the density component in this work, a domain-invariant SE(3)-equivariant Fourier neural operator learns to predict it from the potential as input on real-space grids, enabling stable quasi-linear scaling SCFs. Trained jointly on 8,504 molecules and solids, a single model generalizes to out-of-distribution organic molecules, insulators, and metals. For the first time, the same method converges SCFs across these systems without explicitly constructing Kohn--Sham orbitals, while reproducing densities, electronic spectra, and structural observables at Kohn--Sham DFT accuracy. Linear-scaling SCFs additionally allow converging magnesium dislocation densities containing up to 82,500 valence electrons on a single GPU.