# From Coupled Oscillators to Graph Neural Networks: Reducing Over-smoothing via a Kuramoto Model-based Approach

arXiv: https://arxiv.org/abs/2311.03260 · HTML: https://mlsanity.com/paper/2311.03260/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 57 references that are in the map.
- First posted on arXiv 6 Nov 2023.

## Builds on (most cited first, 23 of 23)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Graph Attention Networks](https://mlsanity.com/paper/1710.10903/) (2017)
- [Inductive Representation Learning on Large Graphs](https://mlsanity.com/paper/1706.02216/) (2017)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [Open Graph Benchmark: Datasets for Machine Learning on Graphs](https://mlsanity.com/paper/2005.00687/) (2020)
- [Pitfalls of Graph Neural Network Evaluation](https://mlsanity.com/paper/1811.05868/) (2018)
- [Learning Mesh-Based Simulation with Graph Networks](https://mlsanity.com/paper/2010.03409/) (2020)
- [Simple and Deep Graph Convolutional Networks](https://mlsanity.com/paper/2007.02133/) (2020)
- [Geom-GCN: Geometric Graph Convolutional Networks](https://mlsanity.com/paper/2002.05287/) (2020)
- [Graph Neural Networks for Social Recommendation](https://mlsanity.com/paper/1902.07243/) (2019)
- [Image-based Recommendations on Styles and Substitutes](https://mlsanity.com/paper/1506.04757/) (2015)
- [Neural Controlled Differential Equations for Irregular Time Series](https://mlsanity.com/paper/2005.08926/) (2020)
- [Augmented Neural ODEs](https://mlsanity.com/paper/1904.01681/) (2019)
- [Stable Architectures for Deep Neural Networks](https://mlsanity.com/paper/1705.03341/) (2017)
- [Revisiting Graph Neural Networks: All We Have is Low-Pass Filters](https://mlsanity.com/paper/1905.09550/) (2019)
- [GRAND: Graph Neural Diffusion](https://mlsanity.com/paper/2106.10934/) (2021)
- [Graph-Coupled Oscillator Networks](https://mlsanity.com/paper/2202.02296/) (2022)
- [Continuous Graph Neural Networks](https://mlsanity.com/paper/1912.00967/) (2019)
- [Neural SDE: Stabilizing Neural ODE Networks with Stochastic Noise](https://mlsanity.com/paper/1906.02355/) (2019)
- [Poisson Learning: Graph Based Semi-Supervised Learning At Very Low Label Rates](https://mlsanity.com/paper/2006.11184/) (2020)

## Built on by (most cited first, 15 of 15)

- [Artificial Kuramoto Oscillatory Neurons](https://mlsanity.com/paper/2410.13821/) (2024)
- [Are We Measuring Oversmoothing in Graph Neural Networks Correctly?](https://mlsanity.com/paper/2502.04591/) (2025)
- [Kuramoto Orientation Diffusion Models](https://mlsanity.com/paper/2509.15328/) (2025)
- [Transformer Meets Twicing: Harnessing Unattended Residual Information](https://mlsanity.com/paper/2503.00687/) (2025)
- [Explore Brain-Inspired Machine Intelligence for Connecting Dots on Graphs Through Holographic Blueprint of Oscillatory Synchronization](https://mlsanity.com/paper/2602.00057/) (2026)
- [Kuramoto Oscillatory Phase Encoding: Neuro-inspired Synchronization for Improved Learning Efficiency](https://mlsanity.com/paper/2604.07904/) (2026)
- [Resolving Oversmoothing with Opinion Dissensus](https://mlsanity.com/paper/2501.19089/) (2025)
- [Approximation properties of neural ODEs](https://mlsanity.com/paper/2503.15696/) (2025)
- [Hyperbolic-PDE GNN: Spectral Graph Neural Networks in the Perspective of A System of Hyperbolic Partial Differential Equations](https://mlsanity.com/paper/2505.23014/) (2025)
- [Beyond ReLU: Bifurcation, Oversmoothing, and Topological Priors](https://mlsanity.com/paper/2602.15634/) (2026)
- [Weight Decay Regimes in Grokking Transformers: Cheap Online Diagnostics](https://mlsanity.com/paper/2605.20441/) (2026)
- [Wavy Transformer](https://mlsanity.com/paper/2508.12787/) (2025)
- [Winfree Oscillatory Neural Network](https://mlsanity.com/paper/2605.20922/) (2026)
- [Swarm-Inspired Generation of Collective Behaviors in Graph Dynamical Systems](https://mlsanity.com/paper/2606.24958/) (2026)
- [From Cortical Synchronous Rhythm to Brain Inspired Learning Mechanism: An Oscillatory Spiking Neural Network with Time-Delayed Coordination](https://mlsanity.com/paper/2605.01656/) (2026)

## Abstract (arXiv)

We propose the Kuramoto Graph Neural Network (KuramotoGNN), a novel class of continuous-depth graph neural networks (GNNs) that employs the Kuramoto model to mitigate the over-smoothing phenomenon, in which node features in GNNs become indistinguishable as the number of layers increases. The Kuramoto model captures the synchronization behavior of non-linear coupled oscillators. Under the view of coupled oscillators, we first show the connection between Kuramoto model and basic GNN and then over-smoothing phenomenon in GNNs can be interpreted as phase synchronization in Kuramoto model. The KuramotoGNN replaces this phase synchronization with frequency synchronization to prevent the node features from converging into each other while allowing the system to reach a stable synchronized state. We experimentally verify the advantages of the KuramotoGNN over the baseline GNNs and existing methods in reducing over-smoothing on various graph deep learning benchmark tasks.