References

AMezic17

Hassan Arbabi and Igor Mezić. Ergodic Theory, Dynamic Mode Decomposition, and Computation of Spectral Properties of the Koopman Operator. SIAM Journal on Applied Dynamical Systems, 16(4):2096–2126, January 2017. URL: https://epubs.siam.org/doi/10.1137/17M1125236, doi:10.1137/17M1125236.

BN03

Mikhail Belkin and Partha Niyogi. Laplacian Eigenmaps for Dimensionality Reduction and Data Representation. Neural Computation, 15(6):1373–1396, June 2003. URL: http://www.mitpressjournals.org/doi/10.1162/089976603321780317, doi:10.1162/089976603321780317.

BDR+04

Yoshua Bengio, Olivier Delalleau, Nicolas Le Roux, Jean-François Paiement, Pascal Vincent, and Marie Ouimet. Learning Eigenfunctions Links Spectral Embedding and Kernel PCA. Neural Computation, 16(10):2197–2219, October 2004. URL: http://www.mitpressjournals.org/doi/10.1162/0899766041732396, doi:10.1162/0899766041732396.

BCGreguricFerenvcekS13

Tyrus Berry, John Robert Cressman, Z. Gregurić-Ferenček, and Timothy Sauer. Time-Scale Separation from Diffusion-Mapped Delay Coordinates. SIAM Journal on Applied Dynamical Systems, 12(2):618–649, January 2013. URL: http://epubs.siam.org/doi/10.1137/12088183X, doi:10.1137/12088183X.

BH15

Tyrus Berry and John Harlim. Nonparametric Uncertainty Quantification for Stochastic Gradient Flows. arXiv:1407.6972 [math], February 2015. arXiv: 1407.6972. URL: http://arxiv.org/abs/1407.6972.

BH16

Tyrus Berry and John Harlim. Variable bandwidth diffusion kernels. Applied and Computational Harmonic Analysis, 40(1):68–96, January 2016. URL: https://linkinghub.elsevier.com/retrieve/pii/S1063520315000020, doi:10.1016/j.acha.2015.01.001.

BS19

Tyrus Berry and Timothy Sauer. Consistent Manifold Representation for Topological Data Analysis. arXiv:1606.02353 [math], February 2019. arXiv: 1606.02353. URL: http://arxiv.org/abs/1606.02353.

Bis06

Christopher M. Bishop. Pattern recognition and machine learning. Information science and statistics. Springer, New York, 2006. ISBN 978-0-387-31073-2.

BPK16

Steven L. Brunton, Joshua L. Proctor, and J. Nathan Kutz. Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proceedings of the National Academy of Sciences, 113(15):3932–3937, April 2016. URL: http://www.pnas.org/lookup/doi/10.1073/pnas.1517384113, doi:10.1073/pnas.1517384113.

CBK19

Kathleen P. Champion, Steven L. Brunton, and J. Nathan Kutz. Discovery of Nonlinear Multiscale Systems: Sampling Strategies and Embeddings. SIAM Journal on Applied Dynamical Systems, 18(1):312–333, January 2019. URL: https://epubs.siam.org/doi/10.1137/18M1188227, doi:10.1137/18M1188227.

CGD+14

Eliodoro Chiavazzo, Charles Gear, Carmeline Dsilva, Neta Rabin, and Ioannis Kevrekidis. Reduced Models in Chemical Kinetics via Nonlinear Data-Mining. Processes, 2(1):112–140, January 2014. URL: http://www.mdpi.com/2227-9717/2/1/112, doi:10.3390/pr2010112.

CL06a

Ronald R. Coifman and Stéphane Lafon. Diffusion maps. Applied and Computational Harmonic Analysis, 21(1):5–30, July 2006. URL: https://linkinghub.elsevier.com/retrieve/pii/S1063520306000546, doi:10.1016/j.acha.2006.04.006.

CL06b

Ronald R. Coifman and Stéphane Lafon. Geometric harmonics: A novel tool for multiscale out-of-sample extension of empirical functions. Applied and Computational Harmonic Analysis, 21(1):31–52, July 2006. URL: https://linkinghub.elsevier.com/retrieve/pii/S1063520306000522, doi:10.1016/j.acha.2005.07.005.

DTR18

Nicola Demo, Marco Tezzele, and Gianluigi Rozza. PyDMD: Python Dynamic Mode Decomposition. The Journal of Open Source Software, 3(22):530, February 2018. URL: http://joss.theoj.org/papers/10.21105/joss.00530, doi:10.21105/joss.00530.

DS11

Ethan R. Deyle and George Sugihara. Generalized Theorems for Nonlinear State Space Reconstruction. PLoS ONE, 6(3):e18295, March 2011. URL: https://dx.plos.org/10.1371/journal.pone.0018295, doi:10.1371/journal.pone.0018295.

DTK19

Felix Dietrich, Thomas N. Thiem, and Ioannis G. Kevrekidis. On the Koopman operator of algorithms. arXiv:1907.10807 [cs, math], August 2019. arXiv: 1907.10807. URL: http://arxiv.org/abs/1907.10807.

DG03

D. L. Donoho and C. Grimes. Hessian eigenmaps: Locally linear embedding techniques for high-dimensional data. Proceedings of the National Academy of Sciences, 100(10):5591–5596, May 2003. URL: http://www.pnas.org/cgi/doi/10.1073/pnas.1031596100, doi:10.1073/pnas.1031596100.

DTCK18

Carmeline J. Dsilva, Ronen Talmon, Ronald R. Coifman, and Ioannis G. Kevrekidis. Parsimonious representation of nonlinear dynamical systems through manifold learning: A chemotaxis case study. Applied and Computational Harmonic Analysis, 44(3):759–773, May 2018. URL: https://linkinghub.elsevier.com/retrieve/pii/S1063520315000949, doi:10.1016/j.acha.2015.06.008.

FernandezGonzalezDiazD15

Ángela Fernández, Ana M. González, Julia Díaz, and José R. Dorronsoro. Diffusion Maps for dimensionality reduction and visualization of meteorological data. Neurocomputing, 163:25–37, September 2015. URL: https://linkinghub.elsevier.com/retrieve/pii/S0925231215004257, doi:10.1016/j.neucom.2014.08.090.

FernandezRFD14

Ángela Fernández, Neta Rabin, Dalia Fishelov, and José R. Dorronsoro. Auto-adaptative Laplacian Pyramids for High-dimensional Data Analysis. arXiv:1311.6594 [cs, stat], May 2014. arXiv: 1311.6594. URL: http://arxiv.org/abs/1311.6594.

Gia15

Dimitrios Giannakis. Dynamics-Adapted Cone Kernels. SIAM Journal on Applied Dynamical Systems, 14(2):556–608, January 2015. URL: http://epubs.siam.org/doi/10.1137/140954544, doi:10.1137/140954544.

Gia19

Dimitrios Giannakis. Data-driven spectral decomposition and forecasting of ergodic dynamical systems. Applied and Computational Harmonic Analysis, 47(2):338–396, September 2019. URL: https://linkinghub.elsevier.com/retrieve/pii/S1063520317300982, doi:10.1016/j.acha.2017.09.001.

KNuskeP+20

Stefan Klus, Feliks Nüske, Sebastian Peitz, Jan-Hendrik Niemann, Cecilia Clementi, and Christof Schütte. Data-driven approximation of the Koopman generator: Model reduction, system identification, and control. Physica D: Nonlinear Phenomena, 406:132416, May 2020. arXiv: 1909.10638. URL: http://arxiv.org/abs/1909.10638, doi:10.1016/j.physd.2020.132416.

KBBP16

J. Nathan Kutz, Steven L. Brunton, Bingni W. Brunton, and Joshua L. Proctor. Dynamic mode decomposition. Data-Driven modelling of complex systems. Society for Industrial and Applied Mathematics, 2016. ISBN 978-1-61197-450-8.

Laf04

Stéphane S Lafon. Diffusion Maps and Geometric Harmonics. PhD thesis, Yale University, 2004.

LCVS17

Soledad Le Clainche, José M. Vega, and Julio Soria. Higher order dynamic mode decomposition of noisy experimental data: The flow structure of a zero-net-mass-flux jet. Experimental Thermal and Fluid Science, 88:336–353, November 2017. URL: https://linkinghub.elsevier.com/retrieve/pii/S089417771730184X, doi:10.1016/j.expthermflusci.2017.06.011.

ManojlovicFM+20

Iva Manojlović, Maria Fonoberova, Ryan Mohr, Aleksandr Andrejčuk, Zlatko Drmač, Yannis Kevrekidis, and Igor Mezić. Applications of Koopman Mode Analysis to Neural Networks. arXiv:2006.11765 [cs, math, stat], June 2020. arXiv: 2006.11765. URL: http://arxiv.org/abs/2006.11765.

PVG+11

Fabian Pedregosa, Gael Varoquaux, Alexandre Gramfort, Vincent Michel, Bertrand Thirion, Olivier Grisel, Mathieu Blondel, Peter Prettenhofer, Ron Weiss, Vincent Dubourg, Jake Vanderplas, Alexandre Passos, and David Cournapeau. Scikit-learn: Machine Learning in Python. MACHINE LEARNING IN PYTHON, pages 6, 2011.

RC12

Neta Rabin and Ronald R. Coifman. Heterogeneous datasets representation and learning using diffusion maps and Laplacian pyramids. In Proceedings of the 2012 SIAM International Conference on Data Mining, 189–199. Society for Industrial and Applied Mathematics, April 2012. URL: https://epubs.siam.org/doi/10.1137/1.9781611972825.17, doi:10.1137/1.9781611972825.17.

RMezicB+09

Clarence W. Rowley, Igor Mezić, Shervin Bagheri, Philipp Schlatter, and Dan S. Henningson. Spectral analysis of nonlinear flows. Journal of Fluid Mechanics, 641:115–127, December 2009. URL: https://www.cambridge.org/core/product/identifier/S0022112009992059/type/journal_article, doi:10.1017/S0022112009992059.

Sch10

Peter J. Schmid. Dynamic mode decomposition of numerical and experimental data. Journal of Fluid Mechanics, 656:5–28, August 2010. URL: https://www.cambridge.org/core/product/identifier/S0022112010001217/type/journal_article, doi:10.1017/S0022112010001217.

Tak81

Floris Takens. Detecting strange attractors in turbulence. In Dynamical Systems and Turbulence, Warwick 1980, volume 898, pages 366–381. Springer Berlin Heidelberg, Berlin, Heidelberg, 1981. URL: http://link.springer.com/10.1007/BFb0091924, doi:10.1007/BFb0091924.

TRL+14

Jonathan H. Tu, Clarence W. Rowley, Dirk M. Luchtenburg, Steven L. Brunton, and J. Nathan Kutz. On Dynamic Mode Decomposition: Theory and Applications. Journal of Computational Dynamics, 1(2):391–421, December 2014. arXiv: 1312.0041. URL: http://arxiv.org/abs/1312.0041, doi:10.3934/jcd.2014.1.391.

WKR15

Matthew O. Williams, Ioannis G. Kevrekidis, and Clarence W. Rowley. A Data–Driven Approximation of the Koopman Operator: Extending Dynamic Mode Decomposition. Journal of Nonlinear Science, 25(6):1307–1346, December 2015. URL: http://link.springer.com/10.1007/s00332-015-9258-5, doi:10.1007/s00332-015-9258-5.