mlsanityData to 7 Oct 2026

PDE-GCN: Novel Architectures for Graph Neural Networks Motivated by Partial Differential Equations

arXiv 2108.01938 ↗First posted 4 Aug 2021cs.LG cs.CV cs.NEcited by 106

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Abstract (arXiv)

Graph neural networks are increasingly becoming the go-to approach in various fields such as computer vision, computational biology and chemistry, where data are naturally explained by graphs. However, unlike traditional convolutional neural networks, deep graph networks do not necessarily yield better performance than shallow graph networks. This behavior usually stems from the over-smoothing phenomenon. In this work, we propose a family of architectures to control this behavior by design. Our networks are motivated by numerical methods for solving Partial Differential Equations (PDEs) on manifolds, and as such, their behavior can be explained by similar analysis. Moreover, as we demonstrate using an extensive set of experiments, our PDE-motivated networks can generalize and be effective for various types of problems from different fields. Our architectures obtain better or on par with the current state-of-the-art results for problems that are typically approached using different architectures.

Key facts. Cited by 106 papers in the map (data through 7 Oct 2026). Builds on 44 of its 65 references that are in the map. First posted on arXiv 4 Aug 2021.

Builds on

44 of 65 references.

References · 44

  1. 2015Deep Residual Learning for Image Recognitioncited by 51,809
  2. 2014Adam: A Method for Stochastic Optimizationcited by 36,960
  3. 2019PyTorch: An Imperative Style, High-Performance Deep Learning Librarycited by 20,160
  4. 2016Semi-Supervised Classification with Graph Convolutional Networkscited by 11,282
  5. 2017Graph Attention Networkscited by 8,854
  6. 2017Inductive Representation Learning on Large Graphscited by 6,284
  7. 2018Neural Ordinary Differential Equationscited by 3,733
  8. 2017Neural Message Passing for Quantum Chemistrycited by 3,460
  9. 2018Dynamic Graph CNN for Learning on Point Cloudscited by 2,220
  10. 2016Convolutional Neural Networks on Graphs with Fast Localized Spectral Filteringcited by 2,085
  11. 2018Graph Convolutional Neural Networks for Web-Scale Recommender Systemscited by 1,133
  12. 2013Spectral Networks and Locally Connected Networks on Graphscited by 1,097
32 more
  1. 2016Revisiting Semi-Supervised Learning with Graph Embeddingscited by 970
  2. 2020Simple and Deep Graph Convolutional Networkscited by 857
  3. 2020Geom-GCN: Geometric Graph Convolutional Networkscited by 821
  4. 2021How Attentive are Graph Attention Networks?cited by 818
  5. 2019DropEdge: Towards Deep Graph Convolutional Networks on Node Classificationcited by 679
  6. 2019Measuring and Relieving the Over-smoothing Problem for Graph Neural Networks from the Topological Viewcited by 544
  7. 2019Multi-scale Attributed Node Embeddingcited by 543
  8. 2019Cluster-GCN: An Efficient Algorithm for Training Deep and Large Graph Convolutional Networkscited by 540
  9. 2019DeepGCNs: Can GCNs Go as Deep as CNNs?cited by 527
  10. 2016Geometric deep learning on graphs and manifolds using mixture model CNNscited by 468
  11. 2020Multipole Graph Neural Operator for Parametric Partial Differential Equationscited by 343
  12. 2017Stable Architectures for Deep Neural Networkscited by 340
  13. 2017Dynamic Edge-Conditioned Filters in Convolutional Neural Networks on Graphscited by 307
  14. 2019PairNorm: Tackling Oversmoothing in GNNscited by 297
  15. 2021GRAND: Graph Neural Diffusioncited by 258
  16. 2018PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Networkcited by 208
  17. 2017Beyond Finite Layer Neural Networks: Bridging Deep Architectures and Numerical Differential Equationscited by 203
  18. 2018Deep Neural Networks Motivated by Partial Differential Equationscited by 187
  19. 2018GaAN: Gated Attention Networks for Learning on Large and Spatiotemporal Graphscited by 164
  20. 2017SplineCNN: Fast Geometric Deep Learning with Continuous B-Spline Kernelscited by 160
  21. 2020Combining Differentiable PDE Solvers and Graph Neural Networks for Fluid Flow Predictioncited by 137
  22. 2017Deep Functional Maps: Structured Prediction for Dense Shape Correspondencecited by 134
  23. 2018DeepInf: Social Influence Prediction with Deep Learningcited by 128
  24. 2016Learning shape correspondence with anisotropic convolutional neural networkscited by 95
  25. 2017Reversible Architectures for Arbitrarily Deep Residual Neural Networkscited by 69
  26. 2018GeniePath: Graph Neural Networks with Adaptive Receptive Pathscited by 66
  27. 2017Deep Relaxation: partial differential equations for optimizing deep neural networkscited by 27
  28. 2020Utilising Graph Machine Learning within Drug Discovery and Developmentcited by 26
  29. 2019IMEXnet: A Forward Stable Deep Neural Networkcited by 22
  30. 2020DiffGCN: Graph Convolutional Networks via Differential Operators and Algebraic Multigrid Poolingcited by 16
  31. 2021Mimetic Neural Networks: A unified framework for Protein Design and Foldingcited by 7
  32. 2019LeanConvNets: Low-cost Yet Effective Convolutional Neural Networkscited by 5

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2021: 2; 2022: 19; 2023: 27; 2024: 26; 2025: 23; 2026: 9.

106 papers.

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  1. 2023A Survey on Oversmoothing in Graph Neural Networkscited by 246
  2. 2022Graph-Coupled Oscillator Networkscited by 123
  3. 2022Convolutional Neural Networks on Graphs with Chebyshev Approximation, Revisitedcited by 87
  4. 2023DIFFormer: Scalable (Graph) Transformers Induced by Energy Constrained Diffusioncited by 85
  5. 2022A Survey on Graph Representation Learning Methodscited by 66
  6. 2022Understanding convolution on graphs via energiescited by 58
  7. 2022Anti-Symmetric DGN: a stable architecture for Deep Graph Networkscited by 58
  8. 2022Gradient Gating for Deep Multi-Rate Learning on Graphscited by 49
  9. 2023A Fractional Graph Laplacian Approach to Oversmoothingcited by 45
  10. 2025On Vanishing Gradients, Over-Smoothing, and Over-Squashing in GNNs: Bridging Recurrent and Graph Learningcited by 42
  11. 2022ACMP: Allen-Cahn Message Passing with Attractive and Repulsive Forces for Graph Neural Networkscited by 31
  12. 2022Optimization-Induced Graph Implicit Nonlinear Diffusioncited by 28
38 more
  1. 2023PC-Conv: Unifying Homophily and Heterophily with Two-fold Filteringcited by 28
  2. 2023From Continuous Dynamics to Graph Neural Networks: Neural Diffusion and Beyondcited by 26
  3. 2024On Oversquashing in Graph Neural Networks Through the Lens of Dynamical Systemscited by 26
  4. 2024Long Range Propagation on Continuous-Time Dynamic Graphscited by 26
  5. 2022Geometric Knowledge Distillation: Topology Compression for Graph Neural Networkscited by 22
  6. 2024BENO: Boundary-embedded Neural Operators for Elliptic PDEscited by 22
  7. 2025GeoTransolver: Learning Physics on Irregular Domains Using Multi-scale Geometry Aware Physics Attention Transformercited by 20
  8. 2022Improving Graph Neural Networks with Learnable Propagation Operatorscited by 19
  9. 2023Deep learning for dynamic graphs: models and benchmarkscited by 19
  10. 2022pathGCN: Learning General Graph Spatial Operators from Pathscited by 15
  11. 2023AbODE: Ab Initio Antibody Design using Conjoined ODEscited by 15
  12. 2023Feature Transportation Improves Graph Neural Networkscited by 15
  13. 2021Graph Kernel Neural Networkscited by 13
  14. 2022Modular Flows: Differential Molecular Generationcited by 11
  15. 2024Temporal Graph ODEs for Irregularly-Sampled Time Seriescited by 11
  16. 2022Generalized energy and gradient flow via graph frameletscited by 10
  17. 2023Modeling Graphs Beyond Hyperbolic: Graph Neural Networks in Symmetric Positive Definite Matricescited by 10
  18. 2022Restructuring Graph for Higher Homophily via Adaptive Spectral Clusteringcited by 9
  19. 2023Geometric Graph Filters and Neural Networks: Limit Properties and Discriminability Trade-offscited by 9
  20. 2023TransformerG2G: Adaptive time-stepping for learning temporal graph embeddings using transformerscited by 9
  21. 2023Are GATs Out of Balance?cited by 9
  22. 2024On The Temporal Domain of Differential Equation Inspired Graph Neural Networkscited by 9
  23. 2024Understanding Oversmoothing in Diffusion-Based GNNs From the Perspective of Operator Semigroup Theorycited by 8
  24. 2023DRIP: Deep Regularizers for Inverse Problemscited by 7
  25. 2023AGNN: Alternating Graph-Regularized Neural Networks to Alleviate Over-Smoothingcited by 7
  26. 2023Re-Think and Re-Design Graph Neural Networks in Spaces of Continuous Graph Diffusion Functionalscited by 7
  27. 2025Approximation theory for 1-Lipschitz ResNetscited by 7
  28. 2023Interpretable A-posteriori Error Indication for Graph Neural Network Surrogate Modelscited by 6
  29. 2024Graph Neural Reaction Diffusion Modelscited by 6
  30. 2025Improving Long-Range Interactions in Graph Neural Simulators via Hamiltonian Dynamicscited by 6
  31. 2023RobustMat: Neural Diffusion for Street Landmark Patch Matching under Challenging Environmentscited by 5
  32. 2024Advection Augmented Convolutional Neural Networkscited by 5
  33. 2023Exploring the Approximation Capabilities of Multiplicative Neural Networks for Smooth Functionscited by 4
  34. 2023QDC: Quantum Diffusion Convolution Kernels on Graphscited by 4
  35. 2024Global-Local Graph Neural Networks for Node-Classificationcited by 4
  36. 2021Quantized Convolutional Neural Networks Through the Lens of Partial Differential Equationscited by 3
  37. 2023Learning PDE Solution Operator for Continuous Modeling of Time-Seriescited by 3
  38. 2023Deep Graph Neural Networks via Posteriori-Sampling-based Node-Adaptive Residual Modulecited by 3

50 most cited of 106 All 106 (JSON)