# NeuralSCF: Neural network self-consistent fields for density functional theory

arXiv: https://arxiv.org/abs/2406.15873 · HTML: https://mlsanity.com/paper/2406.15873/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 62 references that are in the map.
- First posted on arXiv 22 Jun 2024.

## Builds on (most cited first, 25 of 25)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Fast Graph Representation Learning with PyTorch Geometric](https://mlsanity.com/paper/1903.02428/) (2019)
- [Sigmoid-Weighted Linear Units for Neural Network Function Approximation in Reinforcement Learning](https://mlsanity.com/paper/1702.03118/) (2017)
- [E(3)-Equivariant Graph Neural Networks for Data-Efficient and Accurate Interatomic Potentials](https://mlsanity.com/paper/2101.03164/) (2021)
- [Deep Equilibrium Models](https://mlsanity.com/paper/1909.01377/) (2019)
- [SchNet: A continuous-filter convolutional neural network for modeling quantum interactions](https://mlsanity.com/paper/1706.08566/) (2017)
- [MACE: Higher Order Equivariant Message Passing Neural Networks for Fast and Accurate Force Fields](https://mlsanity.com/paper/2206.07697/) (2022)
- [Deep Potential Molecular Dynamics: a scalable model with the accuracy of quantum mechanics](https://mlsanity.com/paper/1707.09571/) (2017)
- [EquiformerV2: Improved Equivariant Transformer for Scaling to Higher-Degree Representations](https://mlsanity.com/paper/2306.12059/) (2023)
- [Artificial Intelligence for Science in Quantum, Atomistic, and Continuum Systems](https://mlsanity.com/paper/2307.08423/) (2023)
- [Reducing SO(3) Convolutions to SO(2) for Efficient Equivariant GNNs](https://mlsanity.com/paper/2302.03655/) (2023)
- [By-passing the Kohn-Sham equations with machine learning](https://mlsanity.com/paper/1609.02815/) (2016)
- [SE(3)-equivariant prediction of molecular wavefunctions and electronic densities](https://mlsanity.com/paper/2106.02347/) (2021)
- [Efficient and Equivariant Graph Networks for Predicting Quantum Hamiltonian](https://mlsanity.com/paper/2306.04922/) (2023)
- [Kohn-Sham equations as regularizer: building prior knowledge into machine-learned physics](https://mlsanity.com/paper/2009.08551/) (2020)
- [Informing Geometric Deep Learning with Electronic Interactions to Accelerate Quantum Chemistry](https://mlsanity.com/paper/2105.14655/) (2021)
- [Equivariant graph neural networks for fast electron density estimation of molecules, liquids, and solids](https://mlsanity.com/paper/2112.00652/) (2021)
- [Learning the exchange-correlation functional from nature with fully differentiable density functional theory](https://mlsanity.com/paper/2102.04229/) (2021)
- [Equivariant representations for molecular Hamiltonians and N-center atomic-scale properties](https://mlsanity.com/paper/2109.12083/) (2021)
- [DeePKS: a comprehensive data-driven approach towards chemically accurate density functional theory](https://mlsanity.com/paper/2008.00167/) (2020)
- [Overcoming the Barrier of Orbital-Free Density Functional Theory for Molecular Systems Using Deep Learning](https://mlsanity.com/paper/2309.16578/) (2023)
- [Deep Density: circumventing the Kohn-Sham equations via symmetry preserving neural networks](https://mlsanity.com/paper/1912.00775/) (2019)
- [KineticNet: Deep learning a transferable kinetic energy functional for orbital-free density functional theory](https://mlsanity.com/paper/2305.13316/) (2023)
- [Electronic-structure properties from atom-centered predictions of the electron density](https://mlsanity.com/paper/2206.14087/) (2022)

## Built on by (most cited first, 1 of 1)

- [Efficient Prediction of SO(3)-Equivariant Hamiltonian Matrices via SO(2) Local Frames](https://mlsanity.com/paper/2506.09398/) (2025)

## Abstract (arXiv)

Kohn-Sham density functional theory (KS-DFT) has found widespread application in accurate electronic structure calculations. However, it can be computationally demanding especially for large-scale simulations, motivating recent efforts toward its machine-learning (ML) acceleration. We propose a neural network self-consistent fields (NeuralSCF) framework that establishes the Kohn-Sham density map as a deep learning objective, which encodes the mechanics of the Kohn-Sham equations. Modeling this map with an SE(3)-equivariant graph transformer, NeuralSCF emulates the Kohn-Sham self-consistent iterations to obtain electron densities, from which other properties can be derived. NeuralSCF achieves state-of-the-art accuracy in electron density prediction and derived properties, featuring exceptional zero-shot generalization to a remarkable range of out-of-distribution systems. NeuralSCF reveals that learning from KS-DFT's intrinsic mechanics significantly enhances the model's accuracy and transferability, offering a promising stepping stone for accelerating electronic structure calculations through mechanics learning.