# Learning the exchange-correlation functional from nature with fully differentiable density functional theory

arXiv: https://arxiv.org/abs/2102.04229 · HTML: https://mlsanity.com/paper/2102.04229/

## Key facts

- Cited by 23 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 79 references that are in the map.
- First posted on arXiv 8 Feb 2021.

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

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [On the Variance of the Adaptive Learning Rate and Beyond](https://mlsanity.com/paper/1908.03265/) (2019)
- [Deep Equilibrium Models](https://mlsanity.com/paper/1909.01377/) (2019)
- [By-passing the Kohn-Sham equations with machine learning](https://mlsanity.com/paper/1609.02815/) (2016)
- [Kohn-Sham equations as regularizer: building prior knowledge into machine-learned physics](https://mlsanity.com/paper/2009.08551/) (2020)
- [DeePKS: a comprehensive data-driven approach towards chemically accurate density functional theory](https://mlsanity.com/paper/2008.00167/) (2020)
- [Orbital-free Bond Breaking via Machine Learning](https://mlsanity.com/paper/1306.1812/) (2013)
- [xi-torch: differentiable scientific computing library](https://mlsanity.com/paper/2010.01921/) (2020)

## Built on by (most cited first, 23 of 23)

- [Evolving symbolic density functionals](https://mlsanity.com/paper/2203.02540/) (2022)
- [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)
- [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 Pair Potentials using Differentiable Simulations](https://mlsanity.com/paper/2209.07679/) (2022)
- [Reducing the Cost of Quantum Chemical Data By Backpropagating Through Density Functional Theory](https://mlsanity.com/paper/2402.04030/) (2024)
- [How Well Does Kohn-Sham Regularizer Work for Weakly Correlated Systems?](https://mlsanity.com/paper/2110.14846/) (2021)
- [Variational principle to regularize machine-learned density functionals: the non-interacting kinetic-energy functional](https://mlsanity.com/paper/2306.17587/) (2023)
- [Exploiting Ligand Additivity for Transferable Machine Learning of Multireference Character Across Known Transition Metal Complex Ligands](https://mlsanity.com/paper/2205.02879/) (2022)
- [Machine learning-guided construction of an analytic kinetic energy functional for orbital free density functional theory](https://mlsanity.com/paper/2502.05411/) (2025)
- [Learning Transport Processes with Machine Intelligence](https://mlsanity.com/paper/2109.13096/) (2021)
- [Grad DFT: a software library for machine learning enhanced density functional theory](https://mlsanity.com/paper/2309.15127/) (2023)
- [Open Source Infrastructure for Differentiable Density Functional Theory](https://mlsanity.com/paper/2309.15985/) (2023)
- [Differentiable Modeling and Optimization of Battery Electrolyte Mixtures Using Geometric Deep Learning](https://mlsanity.com/paper/2310.03047/) (2023)
- [NeuralSCF: Neural network self-consistent fields for density functional theory](https://mlsanity.com/paper/2406.15873/) (2024)
- [Machine Learning - Driven Materials Discovery: Unlocking Next-Generation Functional Materials - A review](https://mlsanity.com/paper/2503.18975/) (2025)
- [Self-consistent Gradient-like Eigen Decomposition in Solving Schr"odinger Equations](https://mlsanity.com/paper/2202.01388/) (2022)
- [AD-NEGF: An End-to-End Differentiable Quantum Transport Simulator for Sensitivity Analysis and Inverse Problems](https://mlsanity.com/paper/2202.05098/) (2022)
- [A fully differentiable framework for training proxy Exchange Correlation Functionals for periodic systems](https://mlsanity.com/paper/2602.15923/) (2026)
- [Benchmarking Optimizers to Solve Inverse Problems with Differentiable Physics Simulators](https://mlsanity.com/paper/2609.13819/) (2026)

## Abstract (arXiv)

Improving the predictive capability of molecular properties in ab initio simulations is essential for advanced material discovery. Despite recent progress making use of machine learning, utilizing deep neural networks to improve quantum chemistry modelling remains severely limited by the scarcity and heterogeneity of appropriate experimental data. Here we show how training a neural network to replace the exchange-correlation functional within a fully-differentiable three-dimensional Kohn-Sham density functional theory (DFT) framework can greatly improve simulation accuracy. Using only eight experimental data points on diatomic molecules, our trained exchange-correlation networks enable improved prediction accuracy of atomization energies across a collection of 104 molecules containing new bonds and atoms that are not present in the training dataset.