# Equivariant Neural Operator Learning with Graphon Convolution

arXiv: https://arxiv.org/abs/2311.10908 · HTML: https://mlsanity.com/paper/2311.10908/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 32 of its 56 references that are in the map.
- First posted on arXiv 17 Nov 2023.

## Builds on (most cited first, 30 of 32)

- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [3D U-Net: Learning Dense Volumetric Segmentation from Sparse Annotation](https://mlsanity.com/paper/1606.06650/) (2016)
- [E(n) Equivariant Graph Neural Networks](https://mlsanity.com/paper/2102.09844/) (2021)
- [Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds](https://mlsanity.com/paper/1802.08219/) (2018)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [SE(3)-Transformers: 3D Roto-Translation Equivariant Attention Networks](https://mlsanity.com/paper/2006.10503/) (2020)
- [SchNet: A continuous-filter convolutional neural network for modeling quantum interactions](https://mlsanity.com/paper/1706.08566/) (2017)
- [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)
- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- [Learning from Protein Structure with Geometric Vector Perceptrons](https://mlsanity.com/paper/2009.01411/) (2020)
- [Vector Neurons: A General Framework for SO(3)-Equivariant Networks](https://mlsanity.com/paper/2104.12229/) (2021)
- [Fast and Uncertainty-Aware Directional Message Passing for Non-Equilibrium Molecules](https://mlsanity.com/paper/2011.14115/) (2020)
- [U-NO: U-shaped Neural Operators](https://mlsanity.com/paper/2204.11127/) (2022)
- [PPFNet: Global Context Aware Local Features for Robust 3D Point Matching](https://mlsanity.com/paper/1802.02669/) (2018)
- [Graphon Neural Networks and the Transferability of Graph Neural Networks](https://mlsanity.com/paper/2006.03548/) (2020)
- [By-passing the Kohn-Sham equations with machine learning](https://mlsanity.com/paper/1609.02815/) (2016)
- [Generalization Analysis of Message Passing Neural Networks on Large Random Graphs](https://mlsanity.com/paper/2202.00645/) (2022)
- [Equivariant Graph Neural Networks for 3D Macromolecular Structure](https://mlsanity.com/paper/2106.03843/) (2021)
- [Equivariant Point Cloud Analysis via Learning Orientations for Message Passing](https://mlsanity.com/paper/2203.14486/) (2022)
- [Equivariant graph neural networks for fast electron density estimation of molecules, liquids, and solids](https://mlsanity.com/paper/2112.00652/) (2021)
- [Graphon Filters: Graph Signal Processing in the Limit](https://mlsanity.com/paper/2003.02099/) (2020)
- [Convolutional Neural Networks on Manifolds: From Graphs and Back](https://mlsanity.com/paper/2210.00376/) (2022)
- [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, 11 of 11)

- [Principled Approaches for Extending Neural Architectures to Function Spaces for Operator Learning](https://mlsanity.com/paper/2506.10973/) (2025)
- [ELECTRA: A Cartesian Network for 3D Charge Density Prediction with Floating Orbitals](https://mlsanity.com/paper/2503.08305/) (2025)
- [Gaussian Plane-Wave Neural Operator for Electron Density Estimation](https://mlsanity.com/paper/2402.04278/) (2024)
- [A Recipe for Charge Density Prediction](https://mlsanity.com/paper/2405.19276/) (2024)
- [Topological Neural Operators](https://mlsanity.com/paper/2606.09806/) (2026)
- [Global Plane Waves From Local Gaussians: Periodic Charge Densities in a Blink](https://mlsanity.com/paper/2601.19966/) (2026)
- [Physics-embedded Fourier Neural Network for Partial Differential Equations](https://mlsanity.com/paper/2407.11158/) (2024)
- [Sobek: Streaming Equivariant Tensor Product Convolutions](https://mlsanity.com/paper/2607.18074/) (2026)
- [E3STO: Orbital Inspired SE(3)-Equivariant Molecular Representation for Electron Density Prediction](https://mlsanity.com/paper/2410.06119/) (2024)
- [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)

We propose a general architecture that combines the coefficient learning scheme with a residual operator layer for learning mappings between continuous functions in the 3D Euclidean space. Our proposed model is guaranteed to achieve SE(3)-equivariance by design. From the graph spectrum view, our method can be interpreted as convolution on graphons (dense graphs with infinitely many nodes), which we term InfGCN. By leveraging both the continuous graphon structure and the discrete graph structure of the input data, our model can effectively capture the geometric information while preserving equivariance. Through extensive experiments on large-scale electron density datasets, we observed that our model significantly outperformed the current state-of-the-art architectures. Multiple ablation studies were also carried out to demonstrate the effectiveness of the proposed architecture.