# By-passing the Kohn-Sham equations with machine learning

arXiv: https://arxiv.org/abs/1609.02815 · HTML: https://mlsanity.com/paper/1609.02815/

## Key facts

- Cited by 67 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Sep 2016.

## Built on by (most cited first, 30 of 67)

- [SE(3)-equivariant prediction of molecular wavefunctions and electronic densities](https://mlsanity.com/paper/2106.02347/) (2021)
- [SchNetPack 2.0: A neural network toolbox for atomistic machine learning](https://mlsanity.com/paper/2212.05517/) (2022)
- [Equivariant graph neural networks for fast electron density estimation of molecules, liquids, and solids](https://mlsanity.com/paper/2112.00652/) (2021)
- [DiffMD: A Geometric Diffusion Model for Molecular Dynamics Simulations](https://mlsanity.com/paper/2204.08672/) (2022)
- [Learning the exchange-correlation functional from nature with fully differentiable density functional theory](https://mlsanity.com/paper/2102.04229/) (2021)
- [Evolving symbolic density functionals](https://mlsanity.com/paper/2203.02540/) (2022)
- [Machine learning nonequilibrium electron forces for adiabatic spin dynamics](https://mlsanity.com/paper/2112.12124/) (2021)
- [Descriptors for Machine Learning Model of Generalized Force Field in Condensed Matter Systems](https://mlsanity.com/paper/2201.00798/) (2022)
- [Physics-inspired Equivariant Descriptors of Non-bonded Interactions](https://mlsanity.com/paper/2308.13208/) (2023)
- [Overcoming the Barrier of Orbital-Free Density Functional Theory for Molecular Systems Using Deep Learning](https://mlsanity.com/paper/2309.16578/) (2023)
- [Higher-Order Equivariant Neural Networks for Charge Density Prediction in Materials](https://mlsanity.com/paper/2312.05388/) (2023)
- [Equivariant Neural Operator Learning with Graphon Convolution](https://mlsanity.com/paper/2311.10908/) (2023)
- [Audacity of huge: overcoming challenges of data scarcity and data quality for machine learning in computational materials discovery](https://mlsanity.com/paper/2111.01905/) (2021)
- [ELECTRA: A Cartesian Network for 3D Charge Density Prediction with Floating Orbitals](https://mlsanity.com/paper/2503.08305/) (2025)
- [A Recipe for Charge Density Prediction](https://mlsanity.com/paper/2405.19276/) (2024)
- [Bayesian neural networks for weak solution of PDEs with uncertainty quantification](https://mlsanity.com/paper/2101.04879/) (2021)
- [Geometric Transformer for End-to-End Molecule Properties Prediction](https://mlsanity.com/paper/2110.13721/) (2021)
- [Image Super-resolution Inspired Electron Density Prediction](https://mlsanity.com/paper/2402.12335/) (2024)
- [Scalable Training of Trustworthy and Energy-Efficient Predictive Graph Foundation Models for Atomistic Materials Modeling: A Case Study with HydraGNN](https://mlsanity.com/paper/2406.12909/) (2024)
- [Wigner kernels: body-ordered equivariant machine learning without a basis](https://mlsanity.com/paper/2303.04124/) (2023)
- [3DReact: Geometric deep learning for chemical reactions](https://mlsanity.com/paper/2312.08307/) (2023)
- [Prediction of the electron density of states for crystalline compounds with Atomistic Line Graph Neural Networks (ALIGNN)](https://mlsanity.com/paper/2201.08348/) (2022)
- [Embed and Emulate: Learning to estimate parameters of dynamical systems with uncertainty quantification](https://mlsanity.com/paper/2211.01554/) (2022)
- [Towards Combinatorial Generalization for Catalysts: A Kohn-Sham Charge-Density Approach](https://mlsanity.com/paper/2310.18702/) (2023)
- [Electronic-structure properties from atom-centered predictions of the electron density](https://mlsanity.com/paper/2206.14087/) (2022)
- [Deep learning for surrogate modelling of 2D mantle convection](https://mlsanity.com/paper/2108.10105/) (2021)
- [How Well Does Kohn-Sham Regularizer Work for Weakly Correlated Systems?](https://mlsanity.com/paper/2110.14846/) (2021)
- [Accurate Molecular-Orbital-Based Machine Learning Energies via Unsupervised Clustering of Chemical Space](https://mlsanity.com/paper/2204.09831/) (2022)
- [Putting Density Functional Theory to the Test in Machine-Learning-Accelerated Materials Discovery](https://mlsanity.com/paper/2205.02967/) (2022)
- [Degeneration of kernel regression with Matern kernels into low-order polynomial regression in high dimension](https://mlsanity.com/paper/2311.10790/) (2023)

## Abstract (arXiv)

Last year, at least 30,000 scientific papers used the Kohn-Sham scheme of density functional theory to solve electronic structure problems in a wide variety of scientific fields, ranging from materials science to biochemistry to astrophysics. Machine learning holds the promise of learning the kinetic energy functional via examples, by-passing the need to solve the Kohn-Sham equations. This should yield substantial savings in computer time, allowing either larger systems or longer time-scales to be tackled, but attempts to machine-learn this functional have been limited by the need to find its derivative. The present work overcomes this difficulty by directly learning the density-potential and energy-density maps for test systems and various molecules. Both improved accuracy and lower computational cost with this method are demonstrated by reproducing DFT energies for a range of molecular geometries generated during molecular dynamics simulations. Moreover, the methodology could be applied directly to quantum chemical calculations, allowing construction of density functionals of quantum-chemical accuracy.