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@article{Pan2024,
doi = {10.21105/joss.05881},
url = {https://doi.org/10.21105/joss.05881},
year = {2024},
publisher = {The Open Journal},
volume = {9},
number = {94},
pages = {5881},
author = {Pan, Shaowu and Kaiser, Eurika and de Silva, Brian M. and Kutz, J. Nathan and Brunton, Steven L.},
title = {{PyKoopman}: A {Python} Package for Data-Driven Approximation of the {Koopman} Operator},
journal = {Journal of Open Source Software}
}
@inproceedings{Dey2023_L4DC,
author = {Dey, Sourya and Davis, Eric William},
title = {{DLKoopman}: A deep learning software package for {Koopman} theory},
booktitle = {Proceedings of The 5th Annual Learning for Dynamics and Control Conference},
pages = {1467--1479},
volume = {211},
series = {Proceedings of Machine Learning Research},
publisher = {PMLR},
year = {2023},
month = {Jun},
url = {https://proceedings.mlr.press/v211/dey23a.html}
}
@inproceedings{Fey/Lenssen/2019,
title = {Fast Graph Representation Learning with {PyTorch Geometric}},
author = {Fey, Matthias and Lenssen, Jan E.},
booktitle = {ICLR Workshop on Representation Learning on Graphs and Manifolds},
year = {2019}
}
@article{koopman1931,
author = {Koopman, B. O.},
title = {Hamiltonian Systems and Transformation in {Hilbert} Space},
journal = {Proceedings of the National Academy of Sciences},
volume = {17},
number = {5},
pages = {315--318},
year = {1931},
doi = {10.1073/pnas.17.5.315}
}
@article{Lusch2018,
author = {Lusch, Bethany and Kutz, J. Nathan and Brunton, Steven L.},
title = {Deep learning for universal linear embeddings of nonlinear dynamics},
journal = {Nature Communications},
volume = {9},
pages = {4950},
year = {2018},
doi = {10.1038/s41467-018-07210-0}
}
@article{Mezic2021,
author = {Mezi{\'c}, Igor},
title = {Koopman operator, geometry, and learning of dynamical systems},
journal = {Notices of the American Mathematical Society},
volume = {68},
number = {7},
pages = {1087--1105},
year = {2021}
}
@article{Mezic2022,
author = {Mezi{\'c}, Igor},
title = {On Numerical Approximations of the {Koopman} Operator},
journal = {Mathematics},
volume = {10},
number = {7},
pages = {1180},
year = {2022},
doi = {10.3390/math10071180}
}
@inproceedings{Mukherjee2022,
author = {Mukherjee, Sayak and Nandanoori, Sai Pushpak and Guan, Sheng and Agarwal, Khushbu and Sinha, Subhrajit and Kundu, Soumya and Pal, Seemita and Wu, Yinghui and Vrabie, Draguna L. and Choudhury, Sutanay},
title = {Learning Distributed Geometric {Koopman} Operator for Sparse Networked Dynamical Systems},
booktitle = {Proceedings of the First Learning on Graphs Conference},
series = {Proceedings of Machine Learning Research},
volume = {198},
pages = {45:1--45:17},
year = {2022},
publisher = {PMLR},
url = {https://proceedings.mlr.press/v198/mukherjee22a.html}
}
@inproceedings{Li2020CompositionalKoopman,
author = {Li, Yunzhu and He, Hao and Wu, Jiajun and Katabi, Dina and Torralba, Antonio},
title = {Learning Compositional {Koopman} Operators for Model-Based Control},
booktitle = {International Conference on Learning Representations},
year = {2020},
url = {https://openreview.net/forum?id=B5bSpiRVFD}
}
@inproceedings{Turja2023,
author = {Turja, Md Asadullah and Styner, Martin and Wu, Guorong},
title = {{DeepGraphDMD}: Interpretable Spatio-Temporal Decomposition of Non-linear Functional Brain Network Dynamics},
booktitle = {Medical Image Computing and Computer Assisted Intervention -- MICCAI 2023},
series = {Lecture Notes in Computer Science},
volume = {14227},
pages = {358--368},
year = {2023},
publisher = {Springer},
url = {https://arxiv.org/abs/2306.03088}
}
@misc{Guerra2024,
author = {Guerra, Michele and Scardapane, Simone and Bianchi, Filippo Maria},
title = {Interpreting Temporal Graph Neural Networks with {Koopman} Theory},
year = {2024},
eprint = {2410.13469},
archivePrefix = {arXiv},
primaryClass = {cs.LG},
url = {https://arxiv.org/abs/2410.13469}
}
@misc{koopmangraph2026,
author = {Kessler, Travis},
title = {{KoopmanGraph}: Graph Neural Networks with {Koopman} Operator Theory},
year = {2026},
publisher = {GitHub},
journal = {GitHub repository},
url = {https://github.com/tjkessler/KoopmanGraph},
version = {0.3.0}
}
@inproceedings{Azencot2020,
author = {Azencot, Omri and Erichson, N. Benjamin and Lin, Vanessa and Mahoney, Michael W.},
title = {Forecasting Sequential Data Using Consistent {Koopman} Autoencoders},
booktitle = {Proceedings of the 37th International Conference on Machine Learning},
series = {Proceedings of Machine Learning Research},
volume = {119},
pages = {475--485},
year = {2020},
publisher = {PMLR},
url = {https://proceedings.mlr.press/v119/azencot20a.html}
}
@article{Li2017EDMD,
author = {Li, Qianxiao and Dietrich, Felix and Bollt, Erik M. and Kevrekidis, Ioannis G.},
title = {Extended Dynamic Mode Decomposition with Dictionary Learning: A Data-Driven Adaptive Spectral Decomposition of the {Koopman} Operator},
journal = {Chaos: An Interdisciplinary Journal of Nonlinear Science},
volume = {27},
number = {10},
pages = {103111},
year = {2017},
doi = {10.1063/1.4993854}
}
@article{Korda2018,
author = {Korda, Milan and Mezi{\'c}, Igor},
title = {Linear Predictors for Nonlinear Dynamical Systems: {Koopman} Operator Meets Model Predictive Control},
journal = {Automatica},
volume = {93},
pages = {149--160},
year = {2018},
doi = {10.1016/j.automatica.2018.03.046}
}
@article{Proctor2016DMDc,
author = {Proctor, Joshua L. and Brunton, Steven L. and Kutz, J. Nathan},
title = {Dynamic Mode Decomposition with Control},
journal = {SIAM Journal on Applied Dynamical Systems},
volume = {15},
number = {1},
pages = {142--161},
year = {2016},
doi = {10.1137/15M1013857}
}
@article{Bruder2021,
author = {Bruder, Daniel and Fu, Xun and Vasudevan, Ram},
title = {Advantages of Bilinear {Koopman} Realizations for the Modeling and Control of Systems With Unknown Dynamics},
journal = {IEEE Robotics and Automation Letters},
volume = {6},
number = {3},
pages = {4369--4376},
year = {2021},
doi = {10.1109/LRA.2021.3068117}
}
@article{Nandanoori2022,
author = {Nandanoori, Sai Pushpak and Guan, Sheng and Kundu, Soumya and Pal, Seemita and Agarwal, Khushbu and Wu, Yinghui and Choudhury, Sutanay},
title = {Graph Neural Network and {Koopman} Models for Learning Networked Dynamics: A Comparative Study on Power Grid Transients Prediction},
journal = {IEEE Access},
volume = {10},
pages = {32337--32349},
year = {2022},
doi = {10.1109/ACCESS.2022.3160710}
}
@article{Williams2015,
author = {Williams, Matthew O. and Kevrekidis, Ioannis G. and Rowley, Clarence W.},
title = {A Data--Driven Approximation of the {Koopman} Operator: Extending Dynamic Mode Decomposition},
journal = {Journal of Nonlinear Science},
volume = {25},
number = {6},
pages = {1307--1346},
year = {2015},
doi = {10.1007/s00332-015-9258-5}
}
@inproceedings{Yu2018STGCN,
author = {Yu, Bing and Yin, Haoteng and Zhu, Zhanxing},
title = {Spatio-Temporal Graph Convolutional Networks: A Deep Learning Framework for Traffic Forecasting},
booktitle = {Proceedings of the Twenty-Seventh International Joint Conference on Artificial Intelligence ({IJCAI-18})},
pages = {3634--3640},
year = {2018},
doi = {10.24963/ijcai.2018/505}
}
@inproceedings{Li2018DCRNN,
author = {Li, Yaguang and Yu, Rose and Shahabi, Cyrus and Liu, Yan},
title = {Diffusion Convolutional Recurrent Neural Network: Data-Driven Traffic Forecasting},
booktitle = {International Conference on Learning Representations ({ICLR})},
year = {2018},
url = {https://openreview.net/forum?id=SJiHXGWAZ}
}
@inproceedings{Wu2019WaveNet,
author = {Wu, Zonghan and Pan, Shirui and Long, Guodong and Jiang, Jing and Zhang, Chengqi},
title = {Graph {WaveNet} for Deep Spatial-Temporal Graph Modeling},
booktitle = {Proceedings of the Twenty-Eighth International Joint Conference on Artificial Intelligence ({IJCAI-19})},
pages = {1907--1913},
year = {2019},
doi = {10.24963/ijcai.2019/264}
}
@incollection{Takens1981,
author = {Takens, Floris},
title = {Detecting Strange Attractors in Turbulence},
booktitle = {Dynamical Systems and Turbulence, Warwick 1980},
series = {Lecture Notes in Mathematics},
volume = {898},
pages = {366--381},
publisher = {Springer},
year = {1981},
doi = {10.1007/BFb0091924}
}
@article{Brunton2017HAVOK,
author = {Brunton, Steven L. and Brunton, Bingni W. and Proctor, Joshua L. and Kaiser, Eurika and Kutz, J. Nathan},
title = {Chaos as an Intermittently Forced Linear System},
journal = {Nature Communications},
volume = {8},
pages = {19},
year = {2017},
doi = {10.1038/s41467-017-00030-8}
}
@article{Arbabi2017HankelDMD,
author = {Arbabi, Hassan and Mezi{\'c}, Igor},
title = {Ergodic Theory, Dynamic Mode Decomposition, and Computation of Spectral Properties of the {Koopman} Operator},
journal = {SIAM Journal on Applied Dynamical Systems},
volume = {16},
number = {4},
pages = {2096--2126},
year = {2017},
doi = {10.1137/17M1125236}
}
@inproceedings{Surana2016,
author = {Surana, Amit},
title = {Koopman Operator Based Observer Synthesis for Control-Affine Nonlinear Systems},
booktitle = {2016 IEEE 55th Conference on Decision and Control (CDC)},
pages = {6492--6499},
year = {2016},
doi = {10.1109/CDC.2016.7799268}
}
@inproceedings{Wu2025K2VAE,
author = {Wu, Xingjian and Qiu, Xiangfei and Gao, Hongfan and Hu, Jilin and Yang, Bin and Guo, Chenjuan},
title = {{$K^2$VAE}: A {Koopman}-{Kalman} Enhanced Variational AutoEncoder for Probabilistic Time Series Forecasting},
booktitle = {Proceedings of the 42nd International Conference on Machine Learning},
series = {Proceedings of Machine Learning Research},
volume = {267},
pages = {67562--67583},
year = {2025},
publisher = {PMLR}
}