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

arXiv: https://arxiv.org/abs/2303.00399 · HTML: https://mlsanity.com/paper/2303.00399/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 52 references that are in the map.
- First posted on arXiv 1 Mar 2023.

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

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [Optimization Methods for Large-Scale Machine Learning](https://mlsanity.com/paper/1606.04838/) (2016)
- [Cheap Orthogonal Constraints in Neural Networks: A Simple Parametrization of the Orthogonal and Unitary Group](https://mlsanity.com/paper/1901.08428/) (2019)
- [Optimal whitening and decorrelation](https://mlsanity.com/paper/1512.00809/) (2015)
- [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)
- [DeePKS: a comprehensive data-driven approach towards chemically accurate density functional theory](https://mlsanity.com/paper/2008.00167/) (2020)
- [Accelerating Finite-temperature Kohn-Sham Density Functional Theory with Deep Neural Networks](https://mlsanity.com/paper/2010.04905/) (2020)

## Built on by (most cited first, 6 of 6)

- [Reducing the Cost of Quantum Chemical Data By Backpropagating Through Density Functional Theory](https://mlsanity.com/paper/2402.04030/) (2024)
- [Grad DFT: a software library for machine learning enhanced density functional theory](https://mlsanity.com/paper/2309.15127/) (2023)
- [Self-Refining Training for Amortized Density Functional Theory](https://mlsanity.com/paper/2506.01225/) (2025)
- [Generating QM1B with PySCFIPU](https://mlsanity.com/paper/2311.01135/) (2023)
- [Hamiltonian Neural PDE Solvers through Functional Approximation](https://mlsanity.com/paper/2505.13275/) (2025)
- [Scaling Density Functional Theory with Gaussian Splatting](https://mlsanity.com/paper/2609.31483/) (2026)

## Abstract (arXiv)

Kohn-Sham Density Functional Theory (KS-DFT) has been traditionally solved by the Self-Consistent Field (SCF) method. Behind the SCF loop is the physics intuition of solving a system of non-interactive single-electron wave functions under an effective potential. In this work, we propose a deep learning approach to KS-DFT. First, in contrast to the conventional SCF loop, we propose to directly minimize the total energy by reparameterizing the orthogonal constraint as a feed-forward computation. We prove that such an approach has the same expressivity as the SCF method, yet reduces the computational complexity from O(N^4) to O(N^3). Second, the numerical integration which involves a summation over the quadrature grids can be amortized to the optimization steps. At each step, stochastic gradient descent (SGD) is performed with a sampled minibatch of the grids. Extensive experiments are carried out to demonstrate the advantage of our approach in terms of efficiency and stability. In addition, we show that our approach enables us to explore more complex neural-based wave functions.