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title = {Learning Koopman invariant subspaces for dynamic mode decomposition},
year = {2017},
isbn = {9781510860964},
publisher = {Curran Associates Inc.},
address = {Red Hook, NY, USA},
abstract = {Spectral decomposition of the Koopman operator is attracting attention as a tool for the analysis of nonlinear dynamical systems. Dynamic mode decomposition is a popular numerical algorithm for Koopman spectral analysis; however, we often need to prepare nonlinear observables manually according to the underlying dynamics, which is not always possible since we may not have any a priori knowledge about them. In this paper, we propose a fully data-driven method for Koopman spectral analysis based on the principle of learning Koopman invariant subspaces from observed data. To this end, we propose minimization of the residual sum of squares of linear least-squares regression to estimate a set of functions that transforms data into a form in which the linear regression fits well. We introduce an implementation with neural networks and evaluate performance empirically using nonlinear dynamical systems and applications.},
booktitle = {Proceedings of the 31st International Conference on Neural Information Processing Systems},
pages = {1130–1140},
numpages = {11},
location = {Long Beach, California, USA},
series = {NIPS'17}
}

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  journal={Nature Communications},
  year={2017},
  volume={9},
  url={https://api.semanticscholar.org/CorpusID:4854885}
}

@INPROCEEDINGS{8815339,
  author={Yeung, Enoch and Kundu, Soumya and Hodas, Nathan},
  booktitle={2019 American Control Conference (ACC)}, 
  title={Learning Deep Neural Network Representations for Koopman Operators of Nonlinear Dynamical Systems}, 
  year={2019},
  volume={},
  number={},
  pages={4832-4839},
  keywords={},
  doi={10.23919/ACC.2019.8815339}
}

@article{doi:10.1137/18M1177846,
author = {Otto, Samuel E. and Rowley, Clarence W.},
title = {Linearly Recurrent Autoencoder Networks for Learning Dynamics},
journal = {SIAM Journal on Applied Dynamical Systems},
volume = {18},
number = {1},
pages = {558-593},
year = {2019},
doi = {10.1137/18M1177846},

URL = { 
    
        https://doi.org/10.1137/18M1177846
    
    

},
eprint = { 
    
        https://doi.org/10.1137/18M1177846
    
    

}
,
    abstract = { This paper describes a method for learning low-dimensional approximations of nonlinear dynamical systems, based on neural network approximations of the underlying Koopman operator. Extended Dynamic Mode Decomposition (EDMD) provides a useful data-driven approximation of the Koopman operator for analyzing dynamical systems. This paper addresses a fundamental problem associated with EDMD: a trade-off between representational capacity of the dictionary and overfitting due to insufficient data. A new neural network architecture combining an autoencoder with linear recurrent dynamics in the encoded state is used to learn a low-dimensional and highly informative Koopman-invariant subspace of observables. A method is also presented for balanced model reduction of overspecified EDMD systems in feature space. Nonlinear reconstruction using partially linear multikernel regression aims to improve reconstruction accuracy from the low-dimensional state when the data has complex but intrinsically low-dimensional structure. The techniques demonstrate the ability to identify Koopman eigenfunctions of the unforced Duffing equation, create accurate low-dimensional models of an unstable cylinder wake flow, and make short-time predictions of the chaotic Kuramoto--Sivashinsky equation. }
}



@InProceedings{pmlr-v119-azencot20a,
  title = 	 {Forecasting Sequential Data Using Consistent Koopman Autoencoders},
  author =       {Azencot, Omri and Erichson, N. Benjamin and Lin, Vanessa and Mahoney, Michael},
  booktitle = 	 {Proceedings of the 37th International Conference on Machine Learning},
  pages = 	 {475--485},
  year = 	 {2020},
  editor = 	 {III, Hal Daumé and Singh, Aarti},
  volume = 	 {119},
  series = 	 {Proceedings of Machine Learning Research},
  month = 	 {13--18 Jul},
  publisher =    {PMLR},
  pdf = 	 {http://proceedings.mlr.press/v119/azencot20a/azencot20a.pdf},
  url = 	 {https://proceedings.mlr.press/v119/azencot20a.html},
  abstract = 	 {Recurrent neural networks are widely used on time series data, yet such models often ignore the underlying physical structures in such sequences. A new class of physics-based methods related to Koopman theory has been introduced, offering an alternative for processing nonlinear dynamical systems. In this work, we propose a novel Consistent Koopman Autoencoder model which, unlike the majority of existing work, leverages the forward and backward dynamics. Key to our approach is a new analysis which explores the interplay between consistent dynamics and their associated Koopman operators. Our network is directly related to the derived analysis, and its computational requirements are comparable to other baselines. We evaluate our method on a wide range of high-dimensional and short-term dependent problems, and it achieves accurate estimates for significant prediction horizons, while also being robust to noise.}
}

@article{wu2020variational,
  title={Variational approach for learning Markov processes from time series data},
  author={Wu, Hao and No{\'e}, Frank},
  journal={Journal of Nonlinear Science},
  volume={30},
  number={1},
  pages={23--66},
  year={2020},
  publisher={Springer}
}

@article{Iwata2020NeuralDM,
  title={Neural Dynamic Mode Decomposition for End-to-End Modeling of Nonlinear Dynamics},
  author={Tomoharu Iwata and Y. Kawahara},
  journal={ArXiv},
  year={2020},
  volume={abs/2012.06191},
  url={https://api.semanticscholar.org/CorpusID:228376379}
}

@article{TianWu+2021+635+659,
url = {https://doi.org/10.1515/cmam-2020-0130},
title = {Kernel Embedding Based Variational Approach for Low-Dimensional Approximation of Dynamical Systems},
title = {},
author = {Wenchong Tian and Hao Wu},
pages = {635--659},
volume = {21},
number = {3},
journal = {Computational Methods in Applied Mathematics},
doi = {doi:10.1515/cmam-2020-0130},
year = {2021},
lastchecked = {2024-11-25}
}