# DiffGCN: Graph Convolutional Networks via Differential Operators and Algebraic Multigrid Pooling

arXiv: https://arxiv.org/abs/2006.04115 · HTML: https://mlsanity.com/paper/2006.04115/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Jun 2020.

## Built on by (most cited first, 17 of 17)

- [PDE-GCN: Novel Architectures for Graph Neural Networks Motivated by Partial Differential Equations](https://mlsanity.com/paper/2108.01938/) (2021)
- [Histopathology Whole Slide Image Analysis with Heterogeneous Graph Representation Learning](https://mlsanity.com/paper/2307.04189/) (2023)
- [Multigrid-augmented deep learning preconditioners for the Helmholtz equation](https://mlsanity.com/paper/2203.11025/) (2022)
- [Improving Graph Neural Networks with Learnable Propagation Operators](https://mlsanity.com/paper/2210.17224/) (2022)
- [pathGCN: Learning General Graph Spatial Operators from Paths](https://mlsanity.com/paper/2207.07408/) (2022)
- [DeltaConv: Anisotropic Operators for Geometric Deep Learning on Point Clouds](https://mlsanity.com/paper/2111.08799/) (2021)
- [Mimetic Neural Networks: A unified framework for Protein Design and Folding](https://mlsanity.com/paper/2102.03881/) (2021)
- [Haar Wavelet Feature Compression for Quantized Graph Convolutional Networks](https://mlsanity.com/paper/2110.04824/) (2021)
- [CurveCloudNet: Processing Point Clouds with 1D Structure](https://mlsanity.com/paper/2303.12050/) (2023)
- [DiffHybrid-UQ: Uncertainty Quantification for Differentiable Hybrid Neural Modeling](https://mlsanity.com/paper/2401.00161/) (2023)
- [Global-Local Graph Neural Networks for Node-Classification](https://mlsanity.com/paper/2406.10863/) (2024)
- [Quantized Convolutional Neural Networks Through the Lens of Partial Differential Equations](https://mlsanity.com/paper/2109.00095/) (2021)
- [Every Node Counts: Improving the Training of Graph Neural Networks on Node Classification](https://mlsanity.com/paper/2211.16631/) (2022)
- [Unsupervised Image Semantic Segmentation through Superpixels and Graph Neural Networks](https://mlsanity.com/paper/2210.11810/) (2022)
- [First-order PDES for Graph Neural Networks: Advection And Burgers Equation Models](https://mlsanity.com/paper/2404.03081/) (2024)
- [RAPNet: Accelerating Algebraic Multigrid with Learned Sparse Corrections](https://mlsanity.com/paper/2605.26854/) (2026)
- [Learnable Hodge-Laplacian Driven Mixture of Experts for Steady-PDE Surrogates on Coarse Unstructured Meshes](https://mlsanity.com/paper/2610.20585/) (2026)

## Abstract (arXiv)

Graph Convolutional Networks (GCNs) have shown to be effective in handling unordered data like point clouds and meshes. In this work we propose novel approaches for graph convolution, pooling and unpooling, inspired from finite differences and algebraic multigrid frameworks. We form a parameterized convolution kernel based on discretized differential operators, leveraging the graph mass, gradient and Laplacian. This way, the parameterization does not depend on the graph structure, only on the meaning of the network convolutions as differential operators. To allow hierarchical representations of the input, we propose pooling and unpooling operations that are based on algebraic multigrid methods, which are mainly used to solve partial differential equations on unstructured grids. To motivate and explain our method, we compare it to standard convolutional neural networks, and show their similarities and relations in the case of a regular grid. Our proposed method is demonstrated in various experiments like classification and part-segmentation, achieving on par or better than state of the art results. We also analyze the computational cost of our method compared to other GCNs.