# Image Super-resolution Inspired Electron Density Prediction

arXiv: https://arxiv.org/abs/2402.12335 · HTML: https://mlsanity.com/paper/2402.12335/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 87 references that are in the map.
- First posted on arXiv 19 Feb 2024.

## Builds on (most cited first, 27 of 27)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Photo-Realistic Single Image Super-Resolution Using a Generative Adversarial Network](https://mlsanity.com/paper/1609.04802/) (2016)
- [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)
- [MACE: Higher Order Equivariant Message Passing Neural Networks for Fast and Accurate Force Fields](https://mlsanity.com/paper/2206.07697/) (2022)
- [Equivariant message passing for the prediction of tensorial properties and molecular spectra](https://mlsanity.com/paper/2102.03150/) (2021)
- [Deep Learning for Image Super-resolution: A Survey](https://mlsanity.com/paper/1902.06068/) (2019)
- [Learning Local Equivariant Representations for Large-Scale Atomistic Dynamics](https://mlsanity.com/paper/2204.05249/) (2022)
- [Fast and Uncertainty-Aware Directional Message Passing for Non-Equilibrium Molecules](https://mlsanity.com/paper/2011.14115/) (2020)
- [Deep Potential Molecular Dynamics: a scalable model with the accuracy of quantum mechanics](https://mlsanity.com/paper/1707.09571/) (2017)
- [TorchMD-NET: Equivariant Transformers for Neural Network based Molecular Potentials](https://mlsanity.com/paper/2202.02541/) (2022)
- [Less is more: sampling chemical space with active learning](https://mlsanity.com/paper/1801.09319/) (2018)
- [Unifying machine learning and quantum chemistry -- a deep neural network for molecular wavefunctions](https://mlsanity.com/paper/1906.10033/) (2019)
- [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)
- [Real-World Single Image Super-Resolution: A Brief Review](https://mlsanity.com/paper/2103.02368/) (2021)
- [Kohn-Sham equations as regularizer: building prior knowledge into machine-learned physics](https://mlsanity.com/paper/2009.08551/) (2020)
- [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)
- [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)
- [Higher-Order Equivariant Neural Networks for Charge Density Prediction in Materials](https://mlsanity.com/paper/2312.05388/) (2023)
- [DeepDFT: Neural Message Passing Network for Accurate Charge Density Prediction](https://mlsanity.com/paper/2011.03346/) (2020)
- [Deep Density: circumventing the Kohn-Sham equations via symmetry preserving neural networks](https://mlsanity.com/paper/1912.00775/) (2019)
- [Electronic-structure properties from atom-centered predictions of the electron density](https://mlsanity.com/paper/2206.14087/) (2022)
- [Learning the electronic density of states in condensed matter](https://mlsanity.com/paper/2006.11803/) (2020)

## Built on by (most cited first, 8 of 8)

- [ELECTRA: A Cartesian Network for 3D Charge Density Prediction with Floating Orbitals](https://mlsanity.com/paper/2503.08305/) (2025)
- [Towards A Transferable Acceleration Method for Density Functional Theory](https://mlsanity.com/paper/2509.25724/) (2025)
- [Global Plane Waves From Local Gaussians: Periodic Charge Densities in a Blink](https://mlsanity.com/paper/2601.19966/) (2026)
- [Learning the Kohn-Sham map with neural operators for quasi-linear scaling density functional theory](https://mlsanity.com/paper/2608.23895/) (2026)
- [Enhancing molecular dynamics with equivariant machine-learned densities](https://mlsanity.com/paper/2604.24563/) (2026)
- [Surprisingly High Redundancy in Electronic Structure Data Across Materials Explained by Low Intrinsic Dimensionality](https://mlsanity.com/paper/2507.09001/) (2025)
- [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)

## Abstract (arXiv)

Drawing inspiration from the domain of image super-resolution, we view the electron density as a 3D grayscale image and use a convolutional residual network to transform a crude and trivially generated guess of the molecular density into an accurate ground-state quantum mechanical density. We find that this model outperforms all prior density prediction approaches. Because the input is itself a real-space density, the predictions are equivariant to molecular symmetry transformations even though the model is not constructed to be. Due to its simplicity, the model is directly applicable to unseen molecular conformations and chemical elements. We show that fine-tuning on limited new data provides high accuracy even in challenging cases of exotic elements and charge states. Our work suggests new routes to learning real-space physical quantities drawing from the established ideas of image processing.