# Neural-Network Solutions to Real-Space Charge Density and Generalization

arXiv: https://arxiv.org/abs/2609.14906 · HTML: https://mlsanity.com/paper/2609.14906/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 46 references that are in the map.
- First posted on arXiv 14 Sep 2026.

## Builds on (most cited first, 13 of 13)

- [FiLM: Visual Reasoning with a General Conditioning Layer](https://mlsanity.com/paper/1709.07871/) (2017)
- [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)
- [MACE: Higher Order Equivariant Message Passing Neural Networks for Fast and Accurate Force Fields](https://mlsanity.com/paper/2206.07697/) (2022)
- [Reducing SO(3) Convolutions to SO(2) for Efficient Equivariant GNNs](https://mlsanity.com/paper/2302.03655/) (2023)
- [Equivariant graph neural networks for fast electron density estimation of molecules, liquids, and solids](https://mlsanity.com/paper/2112.00652/) (2021)
- [Higher-Order Equivariant Neural Networks for Charge Density Prediction in Materials](https://mlsanity.com/paper/2312.05388/) (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)
- [Spectral/Spatial Tensor Atomic Cluster Expansion with Universal Embeddings in Cartesian Space](https://mlsanity.com/paper/2509.14961/) (2025)
- [Edge Cluster Expansion with Radial Rotary Attention for Interatomic Potentials](https://mlsanity.com/paper/2607.10664/) (2026)
- [ED-DiT: Physics-Guided Diffusion Pretraining for Transferable Molecular Representations from Electron Density](https://mlsanity.com/paper/2608.03260/) (2026)

## Abstract (arXiv)

The Hohenberg-Kohn theorem establishes that, in principle, the ground state (GS) charge density contains all GS information of a many-electron system, such that all GS observables can be expressed as functionals of the GS charge density. Conventional Kohn-Sham density functional theory requires iterative solution of the self-consistent-field equations at substantial computational cost, motivating the development of deep learning surrogates for electronic structure calculations and, in turn, accelerating computer-aided materials design. Here, we propose AIDEN, an Atomic-Interaction Density Equivariant Network for solving real-space charge density. AIDEN separates the element-dependent one-center density from environment-induced density redistribution and represents the latter through complementary atom- and edge-centered tensor correlations. A continuous low-rank Gaussian decoder then reconstructs the density at arbitrary spatial coordinates while reusing atomic encodings independently of the evaluation grid. AIDEN achieves state-of-the-art accuracy on periodic crystal benchmarks while remaining competitive for molecular systems, and further demonstrates zero-shot transferability across several structurally distinct out-of-distribution case studies. Furthermore, AIDEN provides substantially faster inference than both baseline models and full SCF calculations, enabling efficient charge density reconstruction for large-scale electronic structure calculations.