mlsanityData to 7 Oct 2026

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

arXiv 2112.00652 ↗First posted 1 Dec 2021physics.comp-ph stat.MLcited by 24

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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.

Key facts. Cited by 24 papers in the map (data through 7 Oct 2026). Builds on 15 of its 74 references that are in the map. First posted on arXiv 1 Dec 2021.

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15 of 74 references.

References · 15

  1. 2015U-Net: Convolutional Networks for Biomedical Image Segmentationcited by 19,604
  2. 2017Neural Message Passing for Quantum Chemistrycited by 3,460
  3. 2016Group Equivariant Convolutional Networkscited by 1,154
  4. 2018Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point cloudscited by 735
  5. 2021E(3)-Equivariant Graph Neural Networks for Data-Efficient and Accurate Interatomic Potentialscited by 701
  6. 2020Directional Message Passing for Molecular Graphscited by 556
  7. 2021Equivariant message passing for the prediction of tensorial properties and molecular spectracited by 438
  8. 2019Cormorant: Covariant Molecular Neural Networkscited by 262
  9. 2020Machine Learning Force Fieldscited by 204
  10. 2018Clebsch-Gordan Nets: a Fully Fourier Space Spherical Convolutional Neural Networkcited by 168
  11. 2016By-passing the Kohn-Sham equations with machine learningcited by 65
  12. 2021Informing Geometric Deep Learning with Electronic Interactions to Accelerate Quantum Chemistrycited by 27
3 more
  1. 2020DeepDFT: Neural Message Passing Network for Accurate Charge Density Predictioncited by 12
  2. 2020Quantum deep field: data-driven wave function, electron density generation, and atomization energy prediction and extrapolation with machine learningcited by 11
  3. 2019Deep Density: circumventing the Kohn-Sham equations via symmetry preserving neural networkscited by 10

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2023: 3; 2024: 5; 2025: 5; 2026: 11.

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  1. 2023Higher-Order Equivariant Neural Networks for Charge Density Prediction in Materialscited by 12
  2. 2023Subequivariant Graph Reinforcement Learning in 3D Environmentscited by 10
  3. 2023Equivariant Neural Operator Learning with Graphon Convolutioncited by 10
  4. 2025ELECTRA: A Cartesian Network for 3D Charge Density Prediction with Floating Orbitalscited by 9
  5. 2024Image Super-resolution Inspired Electron Density Predictioncited by 8
  6. 2024A Recipe for Charge Density Predictioncited by 8
  7. 2025EDBench: Large-Scale Electron Density Data for Molecular Modelingcited by 2
  8. 2025Towards A Transferable Acceleration Method for Density Functional Theorycited by 2
  9. 2024NeuralSCF: Neural network self-consistent fields for density functional theorycited by 1
  10. 2025FIRE-GNN: Force-informed, Relaxed Equivariance Graph Neural Network for Rapid and Accurate Prediction of Surface Propertiescited by 1
  11. 2026Global Plane Waves From Local Gaussians: Periodic Charge Densities in a Blinkcited by 1
  12. 2026M=oLe-Lambda: Learning the Coupled-Cluster Response State for Energies, Gradients, and Propertiescited by 1
12 more
  1. 2026Implicit Machine Learning Force Fields Accelerate Molecular Dynamics Simulationscited by 1
  2. 2026ED-DiT: Physics-Guided Diffusion Pretraining for Transferable Molecular Representations from Electron Densitycited by 1
  3. 2026Learning the Kohn-Sham map with neural operators for quasi-linear scaling density functional theorycited by 1
  4. 2024Neural Polarization: Toward Electron Density for Molecules by Extending Equivariant Networkscited by 0
  5. 2024E3STO: Orbital Inspired SE(3)-Equivariant Molecular Representation for Electron Density Predictioncited by 0
  6. 2025Surprisingly High Redundancy in Electronic Structure Data Across Materials Explained by Low Intrinsic Dimensionalitycited by 0
  7. 2026Coupled Cluster con M=oLe: Molecular Orbital Learning for Neural Wavefunctionscited by 0
  8. 2026ChargeFlow: Flow-Matching Refinement of Charge-Conditioned Electron Densitiescited by 0
  9. 2026Surrogate Functionals for Machine-Learned Orbital-Free Density Functional Theorycited by 0
  10. 2026A Fixed-Point Neural Operator for Size- and Functional-Transferable Hamiltonian Predictioncited by 0
  11. 2026Neural-Network Solutions to Real-Space Charge Density and Generalizationcited by 0
  12. 2026Complete Neural Electronic Initialization Accelerates Materials DFTcited by 0