Introduction to Koopman Operator Theory VI
This is the sixth article in a series on Koopman Operator Theory. Koopman Operator Theory (KOT) is a theory and application of nonlinear dynamics based on the elegant theorem developed by Koopman and Von Neumann in the 1930’s as a precursor to the Feynman-Kac formula path integral.
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?It is known as the KvN Integral crucial to the development of Ergodic Theory and later Dynamic and Chaotic Systems Theory by M′ezic [3][5][7][15]. The linear Koopman Operator Theory includes a state space of infinite dimensions to control a finite dimensional nonlinear dynamic system using a dynamic mode decomposition of data snapshot eigenvalues and mode functions.
This data can be stochastic non-deterministic [17][19][23].
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Continuing from article five, we are defining the transformation from one K-space to the next in a time evolutionary process.
The time factor in K-space connects the KOT to the RG [20]. They are linked in K-space through the time evolution defined within as a fully dynamical nonlinear to linear system.
Since the Koopman Operator is used to define all possible states of measurement of the system, it can be applied to additional parameters such as the Wilson Loop path integrals of quantum particles discretized within a Schrodinger quantum well, the subject of a previous paper [13][17].
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We are defining the transformation from one K-space to the next in a time evolutionary process.
The Galerkin finite element method (FEM) projection was used to define the discretization of the energy field equations in the quantum well, bounded by the Wilson Loop with the Airy Function Ai.
The Galerkin FEM algorithm is used to calculate Koopman operators from time series datasets [18][21]. The magnetism is a function of the Hamiltonian. The magnetic field corresponds to the K-space.
The matrix can have a time delay for each observable.
References
[1] K. Becker, M. Becker, J. & Schwartz. String Theory and M-Theory: An Introduction. Cambridge University Press, New York. ISBN: 10-521-86069-5. 2007.
[2] N. Beisert et al., \Review of AdS/CFT Integrability: An Overview", Lett. Math. Phys. 99, 3 (2012), arXiv:112.3982.
[3] Petar Bevanda, Stefan Sosnowski, Sandra Hirche. Koopman Operator Dynamical Models: Learning, Analysis and Control. arXiv:2102.02522v2 [eess.SY] 22 Dec 2021. https://www.sciencedirect.com/science/article/abs/pii/S1367578821000729
[4] Brodsky, Stanley. (2008). Novel LHC Phenomena. 002. 10.22323/1.045.0002.
[5] Steven L. Brunton and Matthew J. Colbrook. 23/01/2023, 22:13 Resilient Data-driven Dynamical Systems with Koopman: An Infinite-dimensional Numerical Analysis Perspective https://www.damtp.cam.ac.uk/user/mjc249/pdfs/Koopman_NA.pdf
[6] Brunton, Steven & Kutz, J. & Fu, Xing & Grosek, Jacob. (2016). Dynamic Mode Decomposition for Robust PCA with Applications to Foreground/Background Subtraction in Video Streams and Multi-Resolution Analysis. https://www.researchgate.net/publication/301627453_Dynamic_Mode_Decomposition_for_Robust_PCA_with_Applications_to_ForegroundBackground_Subtraction_in_Video_Streams_and_Multi-Resolution_Analysis
[7] P. Carta, E. Gozzi, D. Mauro. Koopman-von Neumann Formulation of Classical Yang-Mills Theories: I. arXiv:hep-th/0508244v1 31 Aug 2005. https://arxiv.org/abs/hep-th/0508244
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[8] B. Duplantier et al. Schramm Loewner Evolution and Liouville Quantum Gravity. Phys.Rev.Lett. 107 (2011) 131305 arXiv:1012.4800 [math-ph].
[9] Razvan Ciuca, Oscar F. Hern′andez and Michael Wolman. A Convolutional Neural Network For Cosmic String Detection in CMB Temperature Maps. 14 Mar 2019.arXiv:1708.08878v3 [astro-ph.CO]
?
[10] Et. Al. Constraints on cosmic strings using data from the ?rst Advanced LIGO observing run. [Submitted on 11 Sep 2008]https://arxiv.org/abs/1712.01168v2
[11] Gell-Mann, M.; Low, F. E. (1954). "Quantum Electrodynamics at Small Distances" (PDF). Physical Review. 95 (5): 1300–1312. Bibcode:1954PhRv...95.1300G. doi:10.1103/PhysRev.95.1300.
[12] Isabel Fernandez-Nu~nez and Oleg Bulashenko. Wave propagation in metamaterials mimicking the topology of a cosmic string. 6 Mar 2018. arXiv:1711.02420v2 [physics.optics].
[13] Junkee Jeon and Ji-Hun Yoon. Discount Barrier Option Pricing with a Stochastic Interest Rate: Mellin Transform Techniques and Method of Images. Commun. Korean Math. Soc. 33 (2018), No. 1, pp. 345–360. https://doi.org/10.4134/CKMS.c170060. pISSN: 1225-1763 / eISSN: 2234-3024.
[14] E. Kiritsis. String Theory in a Nutshell. Princeton University Press. ISBN: 10:-0-691-12230-X. 19 March 2007.
[15] Milan Korda, Yoshihiko Susuki, Igor Mezi?, Power grid transient stabilization using Koopman model predictive control. IFAC-PapersOnLine, Volume 51, Issue 28, 2018,
Pages 297-302, ISSN 2405-8963, https://doi.org/10.1016/j.ifacol.2018.11.718
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[16] Vihar Kurama. Beginner's Guide to Boltzmann Machines in PyTorch. May 2021. https://blog.paperspace.com/beginners-guide-to-boltzmann-machines-pytorch/
[17] S. Little. AdS-CFT Stochastic Feynman-Kac Mellin Transform with Chaotic Boundaries. Academia.edu. December 28, 2021. https://www.academia.edu/66244508/AdS_CFT_Stochastic_Feynman_Kac_Mellin_Transform_with_Chaotic_Boundaries?source=swp_share.
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[18] Jorge Mallo. An introduction to Koopman operator theory and its applications. Universidad de Deusto. October 2019. https://cmc.deusto.eus/wp-content/uploads/2019/10/07-koopman-jorgeMallo.pdf
[19] Gerard McCaul and Denys. I. Bondar. How to Win Friends and In?uence Functionals: Deducing Stochasticity From Deterministic Dynamics. arXiv:1904.04918v2 [cond-mat.stat-mech] 1 Dec 2020. https://arxiv.org/abs/1904.04918
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[20] William T. Redman 9 June 2020. Renormalization Group as a Koopman Operator. arXiv:1912.13010v3 [cond-mat.stat-mech] [1912.13010v3] Renormalization Group as a Koopman Operator (arxiv.org)
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[21] Reinhold Schneider, L. Sallandt, M. Oster and P. Trunschke. (TU Berlin) (Deep Neural Networks: Niklas Nüsken U Potsdam, Lorenz. Richter ZIP Berlin - U Cottbus. FBSDE (forward-backward stochastic di?erential equations) and Hierarchical Tensors.
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[22] El-Showk, Sheer; Paulos, Miguel F.; Poland, David; Rychkov, Slava; Simmons-Duffin, David; Vichi, Alessandro (2014). "Solving the 3d Ising Model with the Conformal Bootstrap II. C -Minimization and Precise Critical Exponents" (PDF). Journal of Statistical Physics. 157 (4–5): 869–914. arXiv:1403.4545. https://inspirehep.net/literature/1286327
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[23] Favio Vázquez. Deep Learning made easy with Deep Cognition. Dec 21, 2017. https://becominghuman.ai/deep-learning-made-easy-with-deep-cognition-403fbe445351
[24] Williams, M.O., Kevrekidis, I.G. & Rowley, C.W. A Data–Driven Approximation of the Koopman Operator: Extending Dynamic Mode Decomposition. J Nonlinear Sci 25, 1307–1346 (2015). https://doi.org/10.1007/s00332-015-9258-5
[25] Git Hub Python boltzmannclean https://github.com/facultyai/boltzmannclean.