# Kohn-Sham equations as regularizer: building prior knowledge into machine-learned physics

arXiv: https://arxiv.org/abs/2009.08551 · HTML: https://mlsanity.com/paper/2009.08551/

## Key facts

- Cited by 36 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Sep 2020.

## Built on by (most cited first, 30 of 36)

- [Machine learning accelerated computational fluid dynamics](https://mlsanity.com/paper/2102.01010/) (2021)
- [Physics-Integrated Variational Autoencoders for Robust and Interpretable Generative Modeling](https://mlsanity.com/paper/2102.13156/) (2021)
- [Informing Geometric Deep Learning with Electronic Interactions to Accelerate Quantum Chemistry](https://mlsanity.com/paper/2105.14655/) (2021)
- [SchNetPack 2.0: A neural network toolbox for atomistic machine learning](https://mlsanity.com/paper/2212.05517/) (2022)
- [Partial Differential Equations Meet Deep Neural Networks: A Survey](https://mlsanity.com/paper/2211.05567/) (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)
- [Known Operator Learning and Hybrid Machine Learning in Medical Imaging -- A Review of the Past, the Present, and the Future](https://mlsanity.com/paper/2108.04543/) (2021)
- [Overcoming the Barrier of Orbital-Free Density Functional Theory for Molecular Systems Using Deep Learning](https://mlsanity.com/paper/2309.16578/) (2023)
- [Differentiable Physics: A Position Piece](https://mlsanity.com/paper/2109.07573/) (2021)
- [Physics perception in sloshing scenes with guaranteed thermodynamic consistency](https://mlsanity.com/paper/2106.13301/) (2021)
- [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)
- [Constants of motion network](https://mlsanity.com/paper/2208.10387/) (2022)
- [A composable machine-learning approach for steady-state simulations on high-resolution grids](https://mlsanity.com/paper/2210.05837/) (2022)
- [Image Super-resolution Inspired Electron Density Prediction](https://mlsanity.com/paper/2402.12335/) (2024)
- [Accurate and scalable exchange-correlation with deep learning](https://mlsanity.com/paper/2506.14665/) (2025)
- [DQC: a Python program package for Differentiable Quantum Chemistry](https://mlsanity.com/paper/2110.11678/) (2021)
- [D4FT: A Deep Learning Approach to Kohn-Sham Density Functional Theory](https://mlsanity.com/paper/2303.00399/) (2023)
- [Learning Equivariant Non-Local Electron Density Functionals](https://mlsanity.com/paper/2410.07972/) (2024)
- [Reducing the Cost of Quantum Chemical Data By Backpropagating Through Density Functional Theory](https://mlsanity.com/paper/2402.04030/) (2024)
- [A composable autoencoder-based iterative algorithm for accelerating numerical simulations](https://mlsanity.com/paper/2110.03780/) (2021)
- [How Well Does Kohn-Sham Regularizer Work for Weakly Correlated Systems?](https://mlsanity.com/paper/2110.14846/) (2021)
- [A Transferable Recommender Approach for Selecting the Best Density Functional Approximations in Chemical Discovery](https://mlsanity.com/paper/2207.10747/) (2022)
- [Variational principle to regularize machine-learned density functionals: the non-interacting kinetic-energy functional](https://mlsanity.com/paper/2306.17587/) (2023)
- [Derivative Informed Learning of Exchange-Correlation Functionals](https://mlsanity.com/paper/2606.04279/) (2026)
- [Grad DFT: a software library for machine learning enhanced density functional theory](https://mlsanity.com/paper/2309.15127/) (2023)
- [Physics-integrated generative modeling using attentive planar normalizing flow based variational autoencoder](https://mlsanity.com/paper/2404.12267/) (2024)
- [NeuralSCF: Neural network self-consistent fields for density functional theory](https://mlsanity.com/paper/2406.15873/) (2024)
- [Data-driven Modeling in Metrology -- A Short Introduction, Current Developments and Future Perspectives](https://mlsanity.com/paper/2406.16659/) (2024)
- [Self-consistent Gradient-like Eigen Decomposition in Solving Schr"odinger Equations](https://mlsanity.com/paper/2202.01388/) (2022)

## Abstract (arXiv)

Including prior knowledge is important for effective machine learning models in physics, and is usually achieved by explicitly adding loss terms or constraints on model architectures. Prior knowledge embedded in the physics computation itself rarely draws attention. We show that solving the Kohn-Sham equations when training neural networks for the exchange-correlation functional provides an implicit regularization that greatly improves generalization. Two separations suffice for learning the entire one-dimensional H2 dissociation curve within chemical accuracy, including the strongly correlated region. Our models also generalize to unseen types of molecules and overcome self-interaction error.