# Equivariant graph neural networks for fast electron density estimation of molecules, liquids, and solids

arXiv: https://arxiv.org/abs/2112.00652 · HTML: https://mlsanity.com/paper/2112.00652/

## Key facts

- Cited by 25 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 74 references that are in the map.
- First posted on arXiv 1 Dec 2021.

## Builds on (most cited first, 15 of 15)

- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds](https://mlsanity.com/paper/1802.08219/) (2018)
- [E(3)-Equivariant Graph Neural Networks for Data-Efficient and Accurate Interatomic Potentials](https://mlsanity.com/paper/2101.03164/) (2021)
- [Directional Message Passing for Molecular Graphs](https://mlsanity.com/paper/2003.03123/) (2020)
- [Equivariant message passing for the prediction of tensorial properties and molecular spectra](https://mlsanity.com/paper/2102.03150/) (2021)
- [Cormorant: Covariant Molecular Neural Networks](https://mlsanity.com/paper/1906.04015/) (2019)
- [Machine Learning Force Fields](https://mlsanity.com/paper/2010.07067/) (2020)
- [Clebsch-Gordan Nets: a Fully Fourier Space Spherical Convolutional Neural Network](https://mlsanity.com/paper/1806.09231/) (2018)
- [By-passing the Kohn-Sham equations with machine learning](https://mlsanity.com/paper/1609.02815/) (2016)
- [Informing Geometric Deep Learning with Electronic Interactions to Accelerate Quantum Chemistry](https://mlsanity.com/paper/2105.14655/) (2021)
- [DeepDFT: Neural Message Passing Network for Accurate Charge Density Prediction](https://mlsanity.com/paper/2011.03346/) (2020)
- [Quantum deep field: data-driven wave function, electron density generation, and atomization energy prediction and extrapolation with machine learning](https://mlsanity.com/paper/2011.07923/) (2020)
- [Deep Density: circumventing the Kohn-Sham equations via symmetry preserving neural networks](https://mlsanity.com/paper/1912.00775/) (2019)

## Built on by (most cited first, 25 of 25)

- [Higher-Order Equivariant Neural Networks for Charge Density Prediction in Materials](https://mlsanity.com/paper/2312.05388/) (2023)
- [Subequivariant Graph Reinforcement Learning in 3D Environments](https://mlsanity.com/paper/2305.18951/) (2023)
- [Equivariant Neural Operator Learning with Graphon Convolution](https://mlsanity.com/paper/2311.10908/) (2023)
- [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)
- [Image Super-resolution Inspired Electron Density Prediction](https://mlsanity.com/paper/2402.12335/) (2024)
- [Towards A Transferable Acceleration Method for Density Functional Theory](https://mlsanity.com/paper/2509.25724/) (2025)
- [EDBench: Large-Scale Electron Density Data for Molecular Modeling](https://mlsanity.com/paper/2505.09262/) (2025)
- [Global Plane Waves From Local Gaussians: Periodic Charge Densities in a Blink](https://mlsanity.com/paper/2601.19966/) (2026)
- [Surrogate Functionals for Machine-Learned Orbital-Free Density Functional Theory](https://mlsanity.com/paper/2604.20458/) (2026)
- [Learning the Kohn-Sham map with neural operators for quasi-linear scaling density functional theory](https://mlsanity.com/paper/2608.23895/) (2026)
- [NeuralSCF: Neural network self-consistent fields for density functional theory](https://mlsanity.com/paper/2406.15873/) (2024)
- [FIRE-GNN: Force-informed, Relaxed Equivariance Graph Neural Network for Rapid and Accurate Prediction of Surface Properties](https://mlsanity.com/paper/2508.16012/) (2025)
- [M=oLe-Lambda: Learning the Coupled-Cluster Response State for Energies, Gradients, and Properties](https://mlsanity.com/paper/2605.29622/) (2026)
- [Implicit Machine Learning Force Fields Accelerate Molecular Dynamics Simulations](https://mlsanity.com/paper/2607.29158/) (2026)
- [ED-DiT: Physics-Guided Diffusion Pretraining for Transferable Molecular Representations from Electron Density](https://mlsanity.com/paper/2608.03260/) (2026)
- [Neural Polarization: Toward Electron Density for Molecules by Extending Equivariant Networks](https://mlsanity.com/paper/2406.00441/) (2024)
- [E3STO: Orbital Inspired SE(3)-Equivariant Molecular Representation for Electron Density Prediction](https://mlsanity.com/paper/2410.06119/) (2024)
- [Surprisingly High Redundancy in Electronic Structure Data Across Materials Explained by Low Intrinsic Dimensionality](https://mlsanity.com/paper/2507.09001/) (2025)
- [Coupled Cluster con M=oLe: Molecular Orbital Learning for Neural Wavefunctions](https://mlsanity.com/paper/2602.20232/) (2026)
- [ChargeFlow: Flow-Matching Refinement of Charge-Conditioned Electron Densities](https://mlsanity.com/paper/2603.23943/) (2026)
- [A Fixed-Point Neural Operator for Size- and Functional-Transferable Hamiltonian Prediction](https://mlsanity.com/paper/2606.14498/) (2026)
- [Neural-Network Solutions to Real-Space Charge Density and Generalization](https://mlsanity.com/paper/2609.14906/) (2026)
- [Complete Neural Electronic Initialization Accelerates Materials DFT](https://mlsanity.com/paper/2609.21759/) (2026)
- [Equivariant Flow Matching for Electron Density Prediction](https://mlsanity.com/paper/2610.02651/) (2026)

## Abstract (arXiv)

Electron density ρ(vecr) is the fundamental variable in the calculation of ground state energy with density functional theory (DFT). Beyond total energy, features and changes in ρ(vecr) distributions are often used to capture critical physicochemical phenomena in functional materials. We present a machine learning framework for the prediction of ρ(vecr). The model is based on equivariant graph neural networks and the electron density is predicted at special query point vertices that are part of the message passing graph, but only receive messages. The model is tested across multiple data sets of molecules (QM9), liquid ethylene carbonate electrolyte (EC) and LixNiyMnzCo(1-y-z)O2 lithium ion battery cathodes (NMC). For QM9 molecules, the accuracy of the proposed model exceeds typical variability in ρ(vecr) obtained from DFT done with different exchange-correlation functionals. The accuracy on all three datasets is beyond state of the art and the computation time is orders of magnitude faster than DFT.