# D4FT: A Deep Learning Approach to Kohn-Sham Density Functional Theory

**Authors:** Tianbo Li, Min Lin, Zheyuan Hu, Kunhao Zheng, Giovanni Vignale, Kenji Kawaguchi, A. H. Castro Neto, Kostya S. Novoselov, Shuicheng Yan  
**arXiv:** [2303.00399](https://arxiv.org/abs/2303.00399)  
**Date:** 2023-03-01  
**Summary page:** https://www.summarizepaper.com/en/paper/2303.00399v1/

## Key points

- Traditional method for solving KS-DFT equations is through the SCF approach
- Proposed deep learning approach to KS-DFT
- Reparameterization of orthogonal constraint as a feed-forward computation
- Same expressivity as SCF method but reduces computational complexity from O(N^4) to O(N^3)
- Incorporates numerical integration by amortizing it into optimization steps
- Stochastic gradient descent with sampled minibatch of quadrature grids improves efficiency and stability
- Extensive experiments demonstrate advantages in efficiency and stability
- Enables exploration of more complex neural-based wave functions
- Paper accepted by ICLR 2023 and recognized as one of the top 25% papers
- Authors: Tianbo Li, Min Lin, Zheyuan Hu, Kunhao Zheng, Giovanni Vignale, Kenji Kawaguchi, A.H. Castro Neto, Kostya S. Novoselov, Shuicheng Yan

## Summary

In the field of Kohn-Sham Density Functional Theory (KS-DFT), the traditional method for solving the equations is through the Self-Consistent Field (SCF) approach. This involves iteratively solving a system of non-interactive single-electron wave functions under an effective potential. However, in this work, a deep learning approach to KS-DFT is proposed. The first key contribution of this work is a reparameterization of the orthogonal constraint as a feed-forward computation, which allows for direct minimization of the total energy. It has been proven that this approach has the same expressivity as the SCF method but significantly reduces computational complexity from O(N^4) to O(N^3), where N represents the number of atomic orbitals. Additionally, this deep learning approach incorporates numerical integration by amortizing it into optimization steps. Stochastic gradient descent (SGD) is performed with a sampled minibatch of quadrature grids at each step, improving both efficiency and stability. Extensive experiments have been conducted to demonstrate the advantages of this approach in terms of both efficiency and stability. Furthermore, it has been shown that this deep learning approach enables exploration of more complex neural-based wave functions. The paper has been accepted by The Eleventh International Conference on Learning Representations (ICLR 2023) and has received notable recognition as one of the top 25% papers. The authors include Tianbo Li, Min Lin, Zheyuan Hu, Kunhao Zheng, Giovanni Vignale, Kenji Kawaguchi, A.H. Castro Neto, Kostya S. Novoselov, and Shuicheng Yan. Overall, this work presents a novel deep learning approach to KS-DFT that offers improved computational efficiency and stability while allowing for exploration of more complex wave functions based on neural networks.