mlsanityData to 7 Oct 2026

Beltrami Flow and Neural Diffusion on Graphs

arXiv 2110.09443 ↗First posted 18 Oct 2021cs.LG cs.AI stat.MLcited by 84

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Abstract (arXiv)

We propose a novel class of graph neural networks based on the discretised Beltrami flow, a non-Euclidean diffusion PDE. In our model, node features are supplemented with positional encodings derived from the graph topology and jointly evolved by the Beltrami flow, producing simultaneously continuous feature learning and topology evolution. The resulting model generalises many popular graph neural networks and achieves state-of-the-art results on several benchmarks.

Key facts. Cited by 84 papers in the map (data through 7 Oct 2026). Builds on 59 of its 100 references that are in the map. First posted on arXiv 18 Oct 2021.

Builds on

59 of 100 references.

References · 59

  1. 2017Attention Is All You Needcited by 59,932
  2. 2016Semi-Supervised Classification with Graph Convolutional Networkscited by 11,282
  3. 2017Graph Attention Networkscited by 8,854
  4. 2017Inductive Representation Learning on Large Graphscited by 6,284
  5. 2016PointNet: Deep Learning on Point Sets for 3D Classification and Segmentationcited by 5,223
  6. 2018How Powerful are Graph Neural Networks?cited by 4,355
  7. 2018Neural Ordinary Differential Equationscited by 3,733
  8. 2017Neural Message Passing for Quantum Chemistrycited by 3,460
  9. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,806
  10. 2018Dynamic Graph CNN for Learning on Point Cloudscited by 2,220
  11. 2020Open Graph Benchmark: Datasets for Machine Learning on Graphscited by 2,148
  12. 2016Convolutional Neural Networks on Graphs with Fast Localized Spectral Filteringcited by 2,085
38 more
  1. 2014DeepWalk: Online Learning of Social Representationscited by 1,543
  2. 2019Simplifying Graph Convolutional Networkscited by 1,470
  3. 2017Deep Setscited by 1,394
  4. 2021Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gaugescited by 1,131
  5. 2018Pitfalls of Graph Neural Network Evaluationcited by 994
  6. 2016Revisiting Semi-Supervised Learning with Graph Embeddingscited by 970
  7. 2015Image-based Recommendations on Styles and Substitutescited by 702
  8. 2020A Generalization of Transformer Networks to Graphscited by 653
  9. 2020On the Bottleneck of Graph Neural Networks and its Practical Implicationscited by 624
  10. 2016Geometric deep learning on graphs and manifolds using mixture model CNNscited by 468
  11. 2019Diffusion Improves Graph Learningcited by 414
  12. 2019Hyperbolic Graph Convolutional Neural Networkscited by 400
  13. 2019Augmented Neural ODEscited by 396
  14. 2018Tune: A Research Platform for Distributed Model Selection and Trainingcited by 395
  15. 2020Lagrangian Neural Networkscited by 388
  16. 2019Provably Powerful Graph Networkscited by 385
  17. 2020Multipole Graph Neural Operator for Parametric Partial Differential Equationscited by 343
  18. 2018Neural Relational Inference for Interacting Systemscited by 339
  19. 2020Improving Graph Neural Network Expressivity via Subgraph Isomorphism Countingcited by 312
  20. 2020SIGN: Scalable Inception Graph Neural Networkscited by 283
  21. 2020Scalable Gradients for Stochastic Differential Equationscited by 264
  22. 2018Learning Continuous Hierarchies in the Lorentz Model of Hyperbolic Geometrycited by 261
  23. 2019Revisiting Graph Neural Networks: All We Have is Low-Pass Filterscited by 260
  24. 2021GRAND: Graph Neural Diffusioncited by 258
  25. 2015Deep Convolutional Networks on Graph-Structured Datacited by 246
  26. 2019Hyperbolic Graph Neural Networkscited by 244
  27. 2019Learning Discrete Structures for Graph Neural Networkscited by 228
  28. 2020How to train your neural ODE: the world of Jacobian and kinetic regularizationcited by 228
  29. 2021Weisfeiler and Lehman Go Topological: Message Passing Simplicial Networkscited by 219
  30. 2017CayleyNets: Graph Convolutional Neural Networks with Complex Rational Spectral Filterscited by 203
  31. 2020Graph Information Bottleneckcited by 168
  32. 2017Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equationscited by 158
  33. 2019Graph Neural Ordinary Differential Equationscited by 153
  34. 2020Combining Differentiable PDE Solvers and Graph Neural Networks for Fluid Flow Predictioncited by 137
  35. 2020Directional Graph Networkscited by 132
  36. 2019Continuous Graph Neural Networkscited by 114
  37. 2019Hamiltonian Graph Networks with ODE Integratorscited by 107
  38. 2019Neural SDE: Stabilizing Neural ODE Networks with Stochastic Noisecited by 98

50 most cited of 59 All 59 (JSON)

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2022: 15; 2023: 23; 2024: 24; 2025: 14; 2026: 8.

84 papers.

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  1. 2023A Survey on Oversmoothing in Graph Neural Networkscited by 246
  2. 2022Neural Sheaf Diffusion: A Topological Perspective on Heterophily and Oversmoothing in GNNscited by 208
  3. 2022Graph-Coupled Oscillator Networkscited by 123
  4. 2023DIFFormer: Scalable (Graph) Transformers Induced by Energy Constrained Diffusioncited by 85
  5. 2022Understanding convolution on graphs via energiescited by 58
  6. 2022GREAD: Graph Neural Reaction-Diffusion Networkscited by 54
  7. 2022Gradient Gating for Deep Multi-Rate Learning on Graphscited by 49
  8. 2025On Vanishing Gradients, Over-Smoothing, and Over-Squashing in GNNs: Bridging Recurrent and Graph Learningcited by 42
  9. 2022Generalised Implicit Neural Representationscited by 31
  10. 2022On the Robustness of Graph Neural Diffusion to Topology Perturbationscited by 30
  11. 2023Graph Neural Convection-Diffusion with Heterophilycited by 26
  12. 2023From Continuous Dynamics to Graph Neural Networks: Neural Diffusion and Beyondcited by 26
38 more
  1. 2024On Oversquashing in Graph Neural Networks Through the Lens of Dynamical Systemscited by 26
  2. 2023Adversarial Robustness in Graph Neural Networks: A Hamiltonian Approachcited by 24
  3. 2022Heterogeneous manifolds for curvature-aware graph embeddingcited by 23
  4. 2022Geometric Knowledge Distillation: Topology Compression for Graph Neural Networkscited by 22
  5. 2023Exposition on over-squashing problem on GNNs: Current Methods, Benchmarks and Challengescited by 22
  6. 2022RobustLoc: Robust Camera Pose Regression in Challenging Driving Environmentscited by 18
  7. 2023Neural Latent Geometry Search: Product Manifold Inference via Gromov-Hausdorff-Informed Bayesian Optimizationcited by 18
  8. 2022TIDE: Time Derivative Diffusion for Deep Learning on Graphscited by 17
  9. 2024Unleashing the Potential of Fractional Calculus in Graph Neural Networks with FRONDcited by 16
  10. 2025Graph ODEs and Beyond: A Comprehensive Survey on Integrating Differential Equations with Graph Neural Networkscited by 14
  11. 2023Unifying over-smoothing and over-squashing in graph neural networks: A physics informed approach and beyondcited by 13
  12. 2024Bundle Neural Networks for message diffusion on graphscited by 13
  13. 2022Capturing Graphs with Hypo-Elliptic Diffusionscited by 12
  14. 2023Node Embedding from Neural Hamiltonian Orbits in Graph Neural Networkscited by 12
  15. 2022Generalized energy and gradient flow via graph frameletscited by 10
  16. 2023Projections of Model Spaces for Latent Graph Inferencecited by 9
  17. 2024Coupling Graph Neural Networks with Fractional Order Continuous Dynamics: A Robustness Studycited by 9
  18. 2024Understanding Oversmoothing in Diffusion-Based GNNs From the Perspective of Operator Semigroup Theorycited by 8
  19. 2023Supercharging Graph Transformers with Advective Diffusioncited by 7
  20. 2024Distributed-Order Fractional Graph Operating Networkcited by 7
  21. 2024PointDifformer: Robust Point Cloud Registration With Neural Diffusion and Transformercited by 6
  22. 2025Improving the Effective Receptive Field of Message-Passing Neural Networkscited by 6
  23. 2023QDC: Quantum Diffusion Convolution Kernels on Graphscited by 4
  24. 2023Self-Contrastive Graph Diffusion Networkcited by 4
  25. 2023Learning to Decouple Complex Systemscited by 3
  26. 2023DYMAG: Rethinking Message Passing Using Dynamical-systems-based Waveformscited by 3
  27. 2024Foundations and Frontiers of Graph Learning Theorycited by 3
  28. 2022Spectral Maps for Learning on Subgraphscited by 2
  29. 2023Node-Specific Space Selection via Localized Geometric Hyperbolicity in Graph Neural Networkscited by 2
  30. 2023Implicit Graph Neural Diffusion Networks: Convergence, Generalization, and Over-Smoothingcited by 2
  31. 2024PosDiffNet: Positional Neural Diffusion for Point Cloud Registration in a Large Field of View with Perturbationscited by 2
  32. 2024Graph Neural Aggregation-diffusion with Metastabilitycited by 2
  33. 2024Continuous Geometry-Aware Graph Diffusion via Hyperbolic Neural PDEcited by 2
  34. 2024Learning Divergence Fields for Shift-Robust Graph Representationscited by 2
  35. 2024G-Adaptivity: optimised graph-based mesh relocation for finite element methodscited by 2
  36. 2024PRFusion: Toward Effective and Robust Multi-Modal Place Recognition with Image and Point Cloud Fusioncited by 2
  37. 2024When Graph Neural Networks Meet Dynamic Mode Decompositioncited by 2
  38. 2024Faster Local Solvers for Graph Diffusion Equationscited by 2

50 most cited of 84 All 84 (JSON)