# Higher-Order Equivariant Neural Networks for Charge Density Prediction in Materials

arXiv: https://arxiv.org/abs/2312.05388 · HTML: https://mlsanity.com/paper/2312.05388/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 88 references that are in the map.
- First posted on arXiv 8 Dec 2023.

## Builds on (most cited first, 22 of 22)

- [Gaussian Error Linear Units (GELUs)](https://mlsanity.com/paper/1606.08415/) (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)
- [SchNet: A continuous-filter convolutional neural network for modeling quantum interactions](https://mlsanity.com/paper/1706.08566/) (2017)
- [Equivariant message passing for the prediction of tensorial properties and molecular spectra](https://mlsanity.com/paper/2102.03150/) (2021)
- [3D Steerable CNNs: Learning Rotationally Equivariant Features in Volumetric Data](https://mlsanity.com/paper/1807.02547/) (2018)
- [Learning Local Equivariant Representations for Large-Scale Atomistic Dynamics](https://mlsanity.com/paper/2204.05249/) (2022)
- [The Open Catalyst 2020 (OC20) Dataset and Community Challenges](https://mlsanity.com/paper/2010.09990/) (2020)
- [e3nn: Euclidean Neural Networks](https://mlsanity.com/paper/2207.09453/) (2022)
- [Reducing SO(3) Convolutions to SO(2) for Efficient Equivariant GNNs](https://mlsanity.com/paper/2302.03655/) (2023)
- [The Open Catalyst 2022 (OC22) Dataset and Challenges for Oxide Electrocatalysts](https://mlsanity.com/paper/2206.08917/) (2022)
- [By-passing the Kohn-Sham equations with machine learning](https://mlsanity.com/paper/1609.02815/) (2016)
- [SE(3)-equivariant prediction of molecular wavefunctions and electronic densities](https://mlsanity.com/paper/2106.02347/) (2021)
- [Informing Geometric Deep Learning with Electronic Interactions to Accelerate Quantum Chemistry](https://mlsanity.com/paper/2105.14655/) (2021)
- [Equivariant graph neural networks for fast electron density estimation of molecules, liquids, and solids](https://mlsanity.com/paper/2112.00652/) (2021)
- [Equivariant representations for molecular Hamiltonians and N-center atomic-scale properties](https://mlsanity.com/paper/2109.12083/) (2021)
- [Incompleteness of graph neural networks for points clouds in three dimensions](https://mlsanity.com/paper/2201.07136/) (2022)
- [The Open DAC 2023 Dataset and Challenges for Sorbent Discovery in Direct Air Capture](https://mlsanity.com/paper/2311.00341/) (2023)
- [DeepDFT: Neural Message Passing Network for Accurate Charge Density Prediction](https://mlsanity.com/paper/2011.03346/) (2020)
- [Machine-learning-accelerated simulations to enable automatic surface reconstruction](https://mlsanity.com/paper/2305.07251/) (2023)
- [Beyond MD17: the reactive xxMD dataset](https://mlsanity.com/paper/2308.11155/) (2023)
- [Towards Combinatorial Generalization for Catalysts: A Kohn-Sham Charge-Density Approach](https://mlsanity.com/paper/2310.18702/) (2023)

## Built on by (most cited first, 13 of 13)

- [An Efficient Sparse Kernel Generator for O(3)-Equivariant Deep Networks](https://mlsanity.com/paper/2501.13986/) (2025)
- [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)
- [Transferable SCF-Acceleration through Solver-Aligned Initialization Learning](https://mlsanity.com/paper/2604.21657/) (2026)
- [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)
- [HPC-AI Coupling Methodology for Scientific Applications](https://mlsanity.com/paper/2507.01025/) (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)
- [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)

The calculation of electron density distribution using density functional theory (DFT) in materials and molecules is central to the study of their quantum and macro-scale properties, yet accurate and efficient calculation remains a long-standing challenge. We introduce ChargE3Net, an E(3)-equivariant graph neural network for predicting electron density in atomic systems. ChargE3Net enables the learning of higher-order equivariant feature to achieve high predictive accuracy and model expressivity. We show that ChargE3Net exceeds the performance of prior work on diverse sets of molecules and materials. When trained on the massive dataset of over 100K materials in the Materials Project database, our model is able to capture the complexity and variability in the data, leading to a significant 26.7% reduction in self-consistent iterations when used to initialize DFT calculations on unseen materials. Furthermore, we show that non-self-consistent DFT calculations using our predicted charge densities yield near-DFT performance on electronic and thermodynamic property prediction at a fraction of the computational cost. Further analysis attributes the greater predictive accuracy to improved modeling of systems with high angular variations. These results illuminate a pathway towards a machine learning-accelerated ab initio calculations for materials discovery.