# When Graph Neural Networks Meet Dynamic Mode Decomposition

arXiv: https://arxiv.org/abs/2410.05593 · HTML: https://mlsanity.com/paper/2410.05593/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 48 of its 77 references that are in the map.
- First posted on arXiv 8 Oct 2024.

## Builds on (most cited first, 30 of 48)

- [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 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)
- [Convolutional Neural Networks on Graphs with Fast Localized Spectral Filtering](https://mlsanity.com/paper/1606.09375/) (2016)
- [Simplifying Graph Convolutional Networks](https://mlsanity.com/paper/1902.07153/) (2019)
- [Variational Graph Auto-Encoders](https://mlsanity.com/paper/1611.07308/) (2016)
- [On the Bottleneck of Graph Neural Networks and its Practical Implications](https://mlsanity.com/paper/2006.05205/) (2020)
- [Adaptive Universal Generalized PageRank Graph Neural Network](https://mlsanity.com/paper/2006.07988/) (2020)
- [Multi-scale Attributed Node Embedding](https://mlsanity.com/paper/1909.13021/) (2019)
- [Understanding over-squashing and bottlenecks on graphs via curvature](https://mlsanity.com/paper/2111.14522/) (2021)
- [Modern Koopman Theory for Dynamical Systems](https://mlsanity.com/paper/2102.12086/) (2021)
- [Long Range Graph Benchmark](https://mlsanity.com/paper/2206.08164/) (2022)
- [GRAND: Graph Neural Diffusion](https://mlsanity.com/paper/2106.10934/) (2021)
- [A Survey on Oversmoothing in Graph Neural Networks](https://mlsanity.com/paper/2303.10993/) (2023)
- [Graph Wavelet Neural Network](https://mlsanity.com/paper/1904.07785/) (2019)
- [Graph-Coupled Oscillator Networks](https://mlsanity.com/paper/2202.02296/) (2022)
- [Forecasting Sequential Data using Consistent Koopman Autoencoders](https://mlsanity.com/paper/2003.02236/) (2020)
- [Koopa: Learning Non-stationary Time Series Dynamics with Koopman Predictors](https://mlsanity.com/paper/2305.18803/) (2023)
- [PDE-GCN: Novel Architectures for Graph Neural Networks Motivated by Partial Differential Equations](https://mlsanity.com/paper/2108.01938/) (2021)
- [Understanding Oversquashing in GNNs through the Lens of Effective Resistance](https://mlsanity.com/paper/2302.06835/) (2023)
- [DIFFormer: Scalable (Graph) Transformers Induced by Energy Constrained Diffusion](https://mlsanity.com/paper/2301.09474/) (2023)
- [Beltrami Flow and Neural Diffusion on Graphs](https://mlsanity.com/paper/2110.09443/) (2021)
- [Expander Graph Propagation](https://mlsanity.com/paper/2210.02997/) (2022)
- [Diffusion Models for Time Series Applications: A Survey](https://mlsanity.com/paper/2305.00624/) (2023)
- [Quantum Graph Neural Networks](https://mlsanity.com/paper/1909.12264/) (2019)
- [On the Trade-off between Over-smoothing and Over-squashing in Deep Graph Neural Networks](https://mlsanity.com/paper/2212.02374/) (2022)
- [Understanding convolution on graphs via energies](https://mlsanity.com/paper/2206.10991/) (2022)
- [GREAD: Graph Neural Reaction-Diffusion Networks](https://mlsanity.com/paper/2211.14208/) (2022)

## Built on by (most cited first, 3 of 3)

- [Interpreting Temporal Graph Neural Networks with Koopman Theory](https://mlsanity.com/paper/2410.13469/) (2024)
- [PAINET: A Principled Efficient Transformer for 3D Dynamics Modeling](https://mlsanity.com/paper/2510.04233/) (2025)
- [Differentiable Koopman Operator for Contrastive Learning on Dynamic Graphs](https://mlsanity.com/paper/2610.02990/) (2026)

## Abstract (arXiv)

Graph Neural Networks (GNNs) have emerged as fundamental tools for a wide range of prediction tasks on graph-structured data. Recent studies have drawn analogies between GNN feature propagation and diffusion processes, which can be interpreted as dynamical systems. In this paper, we delve deeper into this perspective by connecting the dynamics in GNNs to modern Koopman theory and its numerical method, Dynamic Mode Decomposition (DMD). We illustrate how DMD can estimate a low-rank, finite-dimensional linear operator based on multiple states of the system, effectively approximating potential nonlinear interactions between nodes in the graph. This approach allows us to capture complex dynamics within the graph accurately and efficiently. We theoretically establish a connection between the DMD-estimated operator and the original dynamic operator between system states. Building upon this foundation, we introduce a family of DMD-GNN models that effectively leverage the low-rank eigenfunctions provided by the DMD algorithm. We further discuss the potential of enhancing our approach by incorporating domain-specific constraints such as symmetry into the DMD computation, allowing the corresponding GNN models to respect known physical properties of the underlying system. Our work paves the path for applying advanced dynamical system analysis tools via GNNs. We validate our approach through extensive experiments on various learning tasks, including directed graphs, large-scale graphs, long-range interactions, and spatial-temporal graphs. We also empirically verify that our proposed models can serve as powerful encoders for link prediction tasks. The results demonstrate that our DMD-enhanced GNNs achieve state-of-the-art performance, highlighting the effectiveness of integrating DMD into GNN frameworks.