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Mathematical and numerical analysis of the time-dependent Maxwell-Schrödinger Equations in the Coulomb gauge

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arXiv 2017年

作者： Ma, Chupeng Cao, Liqun Huang, Jizu Lin, Yanping Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China Department of Applied Mathematics Hong Kong Polytechnic University Kowloon Hong Kong

In this paper, we consider the initial-boundary value problem for the time-dependent Maxwell-Schrödinger equations in the Coulomb gauge. We first prove the global existence of weak solutions to the equations. Next we propose an energy-conserving fully discrete finite element scheme for the system and prove the existence and uniqueness of solutions to the discrete system. The optimal error estimates for the numerical scheme without any time-step restrictions are then derived. Numerical results are provided to support our theoretical analysis. Copyright © 2017, The Authors. All rights reserved.

关键词： Initial value problems

An efficient global algorithm for single-group multicast beamforming

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arXiv 2017年

作者： Lu, Cheng Liu, Ya-Feng School of Economics and Management North China Electric Power University Beijing102206 China State Key Laboratory of Scientific and Engineering Computing Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China

Consider the single-group multicast beamforming problem, where multiple users receive the same data stream simultaneously from a single transmitter. The problem is NPhard and all existing algorithms for the problem either find suboptimal approximate or local stationary solutions. In this paper, we propose an efficient branch-and-bound algorithm for the problem that is guaranteed to find its global solution. To the best of our knowledge, our proposed algorithm is the first tailored global algorithm for the single-group multicast beamforming problem. Simulation results show that our proposed algorithm is computationally efficient (albeit its theoretical worstcase iteration complexity is exponential with respect to the number of receivers) and it significantly outperforms a state-ofthe- art general-purpose global optimization solver called Baron. Our proposed algorithm provides an important benchmark for performance evaluation of existing algorithms for the same problem. By using it as the benchmark, we show that two state-of-theart algorithms, semidefinite relaxation algorithm and successive linear approximation algorithm, work well when the problem dimension (i.e., the number of antennas at the transmitter and the number of receivers) is small but their performance deteriorates quickly as the problem dimension increases. Copyright © 2017, The Authors. All rights reserved.

关键词： Beamforming

Broadband multi-wavelength properties of M87 during the 2018 EHT campaign including a very high energy flaring episode

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arXiv 2024年

作者： Algaba, J.C. Baloković, M. Chandra, S. Cheong, W.-Y. Cui, Y.-Z. D’Ammando, F. Falcone, A.D. Ford, N.M. Giroletti, M. Goddi, C. Gurwell, M.A. Hada, K. Haggard, D. Jorstad, S. Kaur, A. Kawashima, T. Kerby, S. Kim, J.-Y. Kino, M. Kravchenko, E.V. Lee, S.-S. Lu, R.-S. Markoff, S. Michail, J. Neilsen, J. Nowak, M.A. Principe, G. Ramakrishnan, V. Ripperda, B. Sasada, M. Savchenko, S.S. Sheridan, C. Akiyama, K. Alberdi, A. Alef, W. Anantua, R. Asada, K. Azulay, R. Bach, U. Baczko, A.-K. Ball, D. Bandyopadhyay, B. Barrett, J. Bauböck, M. Benson, B.A. Bintley, D. Blackburn, L. Blundell, R. Bouman, K.L. Bower, G.C. Boyce, H. Bremer, M. Brissenden, R. Britzen, S. Broderick, A.E. Broguiere, D. Bronzwaer, T. Bustamante, S. Carlstrom, J.E. Chael, A. Chan, C.-K. Chang, D.O. Chatterjee, K. Chatterjee, S. Chen, M.-T. Chen, Y. Cheng, X. Cho, I. Christian, P. Conroy, N.S. Conway, J.E. Crawford, T.M. Crew, G.B. Cruz-Osorio, A. Dahale, R. Davelaar, J. De Laurentis, M. Deane, R. Dempsey, J. Desvignes, G. Dexter, J. Dhruv, V. Dihingia, I.K. Doeleman, S.S. Dzib, S.A. Eatough, R.P. Emami, R. Falcke, H. Farah, J. Fish, V.L. Fomalont, E. Ford, H.A. Foschi, M. Fraga-Encinas, R. Freeman, W.T. Friberg, P. Fromm, C.M. Fuentes, A. Galison, P. Gammie, C.F. García, R. Gentaz, O. Georgiev, B. Gold, R. Gómez-Ruiz, A.I. Gómez, J.L. Gu, M. Hesper, R. Heumann, D. Ho, L.C. Ho, P. Honma, M. Huang, C.-W.L. Huang, L. Hughes, D.H. Ikeda, S. Impellizzeri, C.M.V. Inoue, M. Issaoun, S. James, D.J. Jannuzi, B.T. Janssen, M. Jeter, B. Jiang, W. Jiménez-Rosales, A. Johnson, M.D. Jones, A.C. Joshi, A.V. Jung, T. Karuppusamy, R. Keating, G.K. Kettenis, M. Kim, D.-J. Kim, J. Kim, J. Koay, J.Y. Kocherlakota, P. Kofuji, Y. Koch, P.M. Koyama, S. Kramer, C. Kramer, J.A. Kramer, M. Krichbaum, T.P. Kuo, C.-Y. La Bella, N. Levis, A. Li, Z. Department of Physics Faculty of Science University of Malaya Kuala Lumpur50603 Malaysia Yale Center for Astronomy & Astrophysics 52 Hillhouse Avenue New HavenCT06511 United States Department of Physics Yale University P.O. Box 2018120 New HavenCT06520 United States Centre for Space Research North-West University Potchefstroom2520 South Africa Physical Research Laboratory Gujarat Ahmedabad380009 India Korea Astronomy and Space Science Institute Daedeok-daero 776 Yuseong-gu Daejeon34055 Korea Republic of University of Science and Technology Gajeong-ro 217 Yuseong-gu Daejeon34113 Korea Republic of Research Center for Astronomical Computing Zhejiang Lab Hangzhou311100 China INAF Istituto di Radioastronomia Via P. Gobetti 101 BolognaI-40129 Italy 525 Davey Laboratory Department of Astronomy and Astrophysics Pennsylvania State University University ParkPA16802 United States Department of Physics McGill University 3600 University Street MontréalQCH3A 2T8 Canada Trottier Space Institute at McGill 3550 University Street MontréalQCH3A 2A7 Canada Radboud University P.O. Box 9010 Nijmegen6500 GL Netherlands Leiden Observatory—Allegro Leiden University P.O. Box 9513 Leiden2300 RA Netherlands Center for Astrophysics Harvard & Smithsonian CambridgeMA02138 United States Mizusawa VLBI Observatory National Astronomical Observatory of Japan 2-12 Hoshigaoka Mizusawa Iwate Oshu023-0861 Japan 2-21-1 Osawa Mitaka Tokyo181-8588 Japan Graduate School of Science Nagoya City University Yamanohata 1 Mizuho-cho Mizuho-ku Aichi Nagoya467-8501 Japan McGill Space Institute McGill University 3550 University Street MontréalQCH3A 2A7 Canada Institute for Astrophysical Research Boston University 725 Commonwealth Ave. BostonMA02215 United States Department of Astronomy and Astrophysics Pennsylvania State University University ParkPA16802 United States Institute for Cosmic Ray Research The University of Tokyo 5-1-5 Kashiwanoha Chiba

Context. The nearby elliptical galaxy M87 contains one of only two supermassive black holes whose emission surrounding the event horizon has been imaged by the Event Horizon Telescope (EHT). In 2018, more than two dozen multi-wavelength (MWL) facilities (from radio to γ-ray energies) took part in the second M87 EHT campaign. Aims. The goal of this extensive MWL campaign was to better understand the physics of the accreting black hole M87*, the relationship between the inflow and inner jets, and the high-energy particle acceleration. Understanding the complex astrophysics is also a necessary first step towards performing further tests of general relativity. Methods. The MWL campaign took place in April 2018, overlapping with the EHT M87* observations. We present a new, contemporaneous spectral energy distribution (SED) ranging from radio to very high-energy (VHE) γ-rays as well as details of the individual observations and light curves. We also conducted phenomenological modelling to investigate the basic source properties. Results. We present the first VHE γ-ray flare from M87 detected since 2010. The flux above 350 GeV more than doubled within a period of ≈36 hours. We find that the X-ray flux is enhanced by about a factor of two compared to 2017, while the radio and millimetre core fluxes are consistent between 2017 and 2018. We detect evidence for a monotonically increasing jet position angle that corresponds to variations in the bright spot of the EHT image. Conclusions. Our results show the value of continued MWL monitoring together with precision imaging for addressing the origins of high-energy particle acceleration. While we cannot currently pinpoint the precise location where such acceleration takes place, the new VHE γ-ray flare already presents a challenge to simple one-zone leptonic emission model approaches, and it emphasises the need for combined image and spectral modelling. © 2024, CC BY.

关键词： Black holes

Error estimates of the Crank-Nicolson Galerkin method for the time-dependent Maxwell-Schrödinger equations under the Lorentz gauge

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arXiv 2017年

作者： Chupeng, M.A. Liqun, C.A.O. Yanping, L.I.N. Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China Department of Applied Mathematics Hong Kong Polytechnic University Hung Hom Kowloon Hong Kong

In this paper we study the numerical method and the convergence for solving the time-dependent Maxwell-Schrödinger equations under the Lorentz gauge. An alternating Crank-Nicolson finite element method for solving the problem is presented and the optimal error estimate for the numerical algorithm is obtained by a mathematical inductive method. Numerical examples are then carried out to confirm the theoretical results. Copyright © 2017, The Authors. All rights reserved.

关键词： Galerkin methods

On efficient second order stabilized semi-implicit schemes for the cahn-hilliard phase-field equation

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arXiv 2017年

作者： Wang, Lin Yu, Haijun School of Mathematical Sciences University of Chinese Academy of Sciences Beijing100049 China NCMIS LSEC Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Beijing100190 China

Efficient and energy stable high order time marching schemes are very important but not easy to construct for the study of nonlinear phase dynamics. In this paper, we propose and study two linearly stabilized second order semi-implicit schemes for the Cahn-Hilliard phase-field equation. One uses backward differentiation formula and the other uses Crank- Nicolson method to discretize linear terms. In both schemes, the nonlinear bulk forces are treated explicitly with two second-order stabilization terms. This treatment leads to linear elliptic systems with constant coefficients, for which lots of robust and efficient solvers are available. The discrete energy dissipation properties are proved for both schemes. Rigorous error analysis is carried out to show that, when the time step-size is small enough, second order accuracy in time is obtained with a prefactor controlled by a fixed power of 1/ϵ, where ϵ is the characteristic interface thickness. Numerical results are presented to verify the accuracy and efficiency of proposed *** Codes 65M12, 65M15, 65P40 Copyright © 2017, The Authors. All rights reserved.

关键词： Energy dissipation

Ultralight vector dark matter search using data from the KAGRA O3GK run

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Physical Review D 2024年第4期110卷 042001页

作者： A. G. Abac R. Abbott H. Abe I. Abouelfettouh F. Acernese K. Ackley C. Adamcewicz S. Adhicary N. Adhikari R. X. Adhikari V. K. Adkins V. B. Adya C. Affeldt D. Agarwal M. Agathos O. D. Aguiar I. Aguilar L. Aiello A. Ain P. Ajith T. Akutsu S. Albanesi R. A. Alfaidi A. Al-Jodah C. Alléné A. Allocca S. Al-Shammari P. A. Altin S. Alvarez-Lopez A. Amato L. Amez-Droz A. Amorosi C. Amra S. Anand A. Ananyeva S. B. Anderson W. G. Anderson M. Andia M. Ando T. Andrade N. Andres M. Andrés-Carcasona T. Andrić J. Anglin S. Ansoldi J. M. Antelis S. Antier M. Aoumi E. Z. Appavuravther S. Appert S. K. Apple K. Arai A. Araya M. C. Araya J. S. Areeda N. Aritomi F. Armato N. Arnaud M. Arogeti S. M. Aronson K. G. Arun G. Ashton Y. Aso M. Assiduo S. Assis de Souza Melo S. M. Aston P. Astone F. Aubin K. AultONeal G. Avallone S. Babak F. Badaracco C. Badger S. Bae S. Bagnasco E. Bagui Y. Bai J. G. Baier R. Bajpai T. Baka M. Ball G. Ballardin S. W. Ballmer S. Banagiri B. Banerjee D. Bankar P. Baral J. C. Barayoga B. C. Barish D. Barker P. Barneo F. Barone B. Barr L. Barsotti M. Barsuglia D. Barta S. D. Barthelmy M. A. Barton I. Bartos S. Basak A. Basalaev R. Bassiri A. Basti M. Bawaj P. Baxi J. C. Bayley A. C. Baylor M. Bazzan B. Bécsy V. M. Bedakihale F. Beirnaert M. Bejger D. Belardinelli A. S. Bell V. Benedetto D. Beniwal W. Benoit J. D. Bentley M. Ben Yaala S. Bera M. Berbel F. Bergamin B. K. Berger S. Bernuzzi M. Beroiz D. Bersanetti A. Bertolini J. Betzwieser D. Beveridge N. Bevins R. Bhandare U. Bhardwaj R. Bhatt D. Bhattacharjee S. Bhaumik S. Bhowmick A. Bianchi I. A. Bilenko G. Billingsley A. Binetti S. Bini O. Birnholtz S. Biscoveanu A. Bisht M. Bitossi M.-A. Bizouard J. K. Blackburn C. D. Blair D. G. Blair F. Bobba N. Bode G. Bogaert G. Boileau M. Boldrini G. N. Bolingbroke A. Bolliand L. D. Bonavena R. Bondarescu F. Bondu E. Bonilla M. S. Bonilla A. Bonino R. Bonnand P. Booker A. Borchers V. Boschi S. Bose V. Bossilkov V. Boudart A. Boumerdassi A. Bozzi C. Bradaschia P. R. Brady M. Braglia A. Branch M. Branchesi M. Breschi T. B Max Planck Institute for Gravitational Physics (Albert Einstein Institute) D-14476 Potsdam Germany LIGO Laboratory California Institute of Technology Pasadena California 91125 USA Graduate School of Science Tokyo Institute of Technology 2-12-1 Ookayama Meguro-ku Tokyo 152-8551 Japan LIGO Hanford Observatory Richland Washington 99352 USA Dipartimento di Farmacia Università di Salerno I-84084 Fisciano Salerno Italy INFN Sezione di Napoli I-80126 Napoli Italy University of Warwick Coventry CV4 7AL United Kingdom OzGrav School of Physics & Astronomy Monash University Clayton 3800 Victoria Australia The Pennsylvania State University University Park Pennsylvania 16802 USA University of Wisconsin-Milwaukee Milwaukee Wisconsin 53201 USA Louisiana State University Baton Rouge Louisiana 70803 USA OzGrav Australian National University Canberra Australian Capital Territory 0200 Australia Max Planck Institute for Gravitational Physics (Albert Einstein Institute) D-30167 Hannover Germany Leibniz Universität Hannover D-30167 Hannover Germany Inter-University Centre for Astronomy and Astrophysics Pune 411007 India University of Cambridge Cambridge CB2 1TN United Kingdom Instituto Nacional de Pesquisas Espaciais 12227-010 São José dos Campos São Paulo Brazil Stanford University Stanford California 94305 USA Cardiff University Cardiff CF24 3AA United Kingdom INFN Sezione di Pisa I-56127 Pisa Italy International Centre for Theoretical Sciences Tata Institute of Fundamental Research Bengaluru 560089 India Gravitational Wave Science Project National Astronomical Observatory of Japan 2-21-1 Osawa Mitaka City Tokyo 181-8588 Japan Advanced Technology Center National Astronomical Observatory of Japan 2-21-1 Osawa Mitaka City Tokyo 181-8588 Japan Dipartimento di Fisica Università degli Studi di Torino I-10125 Torino Italy INFN Sezione di Torino I-10125 Torino Italy SUPA University of Glasgow Glasgow G12 8QQ United Kingdom OzGrav University of Western Austral

Among the various candidates for dark matter (DM), ultralight vector DM can be probed by laser interferometric gravitational wave detectors through the measurement of oscillating length changes in the arm cavities. In this context, KAGRA has a unique feature due to differing compositions of its mirrors, enhancing the signal of vector DM in the length change in the auxiliary channels. Here we present the result of a search for U(1)B−L gauge boson DM using the KAGRA data from auxiliary length channels during the first joint observation run together with GEO600. By applying our search pipeline, which takes into account the stochastic nature of ultralight DM, upper bounds on the coupling strength between the U(1)B−L gauge boson and ordinary matter are obtained for a range of DM masses. While our constraints are less stringent than those derived from previous experiments, this study demonstrates the applicability of our method to the lower-mass vector DM search, which is made difficult in this measurement by the short observation time compared to the auto-correlation timescale of DM.

关键词： Dark matter Particle dark matter Gravitational wave detectors

Euclid: Searches for strong gravitational lenses using convolutional neural nets in Early Release Observations of the Perseus field

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arXiv 2024年

作者： Pearce-Casey, R. Nagam, B.C. Wilde, J. Busillo, V. Ulivi, L. Andika, I.T. Manjón-García, A. Leuzzi, L. Matavulj, P. Serjeant, S. Walmsley, M. Barroso, J.A. Acevedo O’Riordan, C.M. Clément, B. Tortora, C. Collett, T.E. Courbin, F. Gavazzi, R. Metcalf, R.B. Cabanac, R. Courtois, H.M. Crook-Mansour, J. Delchambre, L. Despali, G. Ecker, L.R. Franco, A. Holloway, P. Jahnke, K. Mahler, G. Marchetti, L. Melo, A. Meneghetti, M. Müller, O. Nucita, A.A. Pearson, J. Rojas, K. Scarlata, C. Schuldt, S. Sluse, D. Suyu, S.H. Vaccari, M. Vegetti, S. Verma, A. Vernardos, G. Bolzonella, M. Kluge, M. Saifollahi, T. Schirmer, M. Stone, C. Paulino-Afonso, A. Bazzanini, L. Hogg, N.B. Koopmans, L.V.E. Kruk, S. Mannucci, F. Bromley, J.M. Díaz-Sánchez, A. Dickinson, H.J. Powell, D.M. Bouy, H. Laureijs, R. Altieri, B. Amara, A. Andreon, S. Baccigalupi, C. Baldi, M. Balestra, A. Bardelli, S. Battaglia, P. Bonino, D. Branchini, E. Brescia, M. Brinchmann, J. Caillat, A. Camera, S. Capobianco, V. Carbone, C. Carretero, J. Casas, S. Castellano, M. Castignani, G. Cavuoti, S. Cimatti, A. Colodro-Conde, C. Congedo, G. Conselice, C.J. Conversi, L. Copin, Y. Cropper, M. Da Silva, A. Degaudenzi, H. De Lucia, G. Di Giorgio, A.M. Dinis, J. Dubath, F. Dupac, X. Dusini, S. Farina, M. Farrens, S. Faustini, F. Ferriol, S. Frailis, M. Franceschi, E. Galeotta, S. George, K. Gillard, W. Gillis, B. Giocoli, C. Gómez-Alvarez, P. Grazian, A. Grupp, F. Haugan, S.V.H. Holmes, W. Hook, I. Hormuth, F. Hornstrup, A. Hudelot, P. Jhabvala, M. Joachimi, B. Keihänen, E. Kermiche, S. Kiessling, A. Kilbinger, M. Kubik, B. Kümmel, M. Kunz, M. Kurki-Suonio, H. Le Mignant, D. Ligori, S. Lilje, P.B. Lindholm, V. Lloro, I. Maiorano, E. Mansutti, O. Marggraf, O. Markovic, K. Martinelli, M. Martinet, N. Marulli, F. Massey, R. Medinaceli, E. Mei, S. Melchior, M. Mellier, Y. Merlin, E. Meylan, G. Moresco, M. Moscardini, L. Nakajima, R. School of Physical Sciences The Open University Milton KeynesMK7 6AA United Kingdom Kapteyn Astronomical Institute University of Groningen PO Box 800 Groningen9700 AV Netherlands INAF-Osservatorio Astronomico di Capodimonte Via Moiariello 16 Napoli80131 Italy Department of Physics "E. Pancini" University Federico II Via Cinthia 6 Napoli80126 Italy INFN section of Naples Via Cinthia 6 Napoli80126 Italy University of Trento Via Sommarive 14 TrentoI-38123 Italy Dipartimento di Fisica e Astronomia Università di Firenze via G. Sansone 1 Sesto Fiorentino Firenze50019 Italy INAF-Osservatorio Astrofisico di Arcetri Largo E. Fermi 5 Firenze50125 Italy Technical University of Munich TUM School of Natural Sciences Physics Department James-Franck-Str. 1 Garching85748 Germany Max-Planck-Institut für Astrophysik Karl-Schwarzschild-Str. 1 Garching85748 Germany Departamento Física Aplicada Universidad Politécnica de Cartagena Campus Muralla del Mar Murcia Cartagena30202 Spain Dipartimento di Fisica e Astronomia "Augusto Righi" Alma Mater Studiorum Università di Bologna via Piero Gobetti 93/2 Bologna40129 Italy INAF-Osservatorio di Astrofisica e Scienza dello Spazio di Bologna Via Piero Gobetti 93/3 Bologna40129 Italy University of Applied Sciences and Arts of Northwestern Switzerland School of Engineering Windisch5210 Switzerland Department of Astronomy & Astrophysics University of Toronto 50 St George Street TorontoONM5S 3H4 Canada Jodrell Bank Centre for Astrophysics Department of Physics and Astronomy University of Manchester Oxford Road ManchesterM13 9PL United Kingdom Observatoire de Sauverny Versoix1290 Switzerland Lausanne1015 Switzerland Institute of Cosmology and Gravitation University of Portsmouth PortsmouthPO1 3FX United Kingdom Martí i Franquès 1 Barcelona08028 Spain Passeig de Lluís Companys 23 Barcelona08010 Spain Aix-Marseille Université CNRS CNES LAM Marseille France Institut d’Astrophysique de Paris UMR 709

The Euclid Wide Survey (EWS) is predicted to find approximately 170 000 galaxy-galaxy strong lenses from its lifetime observation of 14 000 deg2 of the sky. Detecting this many lenses by visual inspection with professional astronomers and citizen scientists alone is infeasible. As a result, machine learning algorithms, particularly convolutional neural networks (CNNs), have been used as an automated method of detecting strong lenses, and have proven fruitful in finding galaxy-galaxy strong lens candidates, such that the usage of CNNs in lens identification has increased. We identify the major challenge to be the automatic detection of galaxy-galaxy strong lenses while simultaneously maintaining a low false positive rate, thus producing a pure and complete sample of strong lens candidates from Euclid with a limited need for visual inspection. One aim of this research is to have a quantified starting point on the achieved purity and completeness with our current version of CNN-based detection pipelines for the VIS images of EWS. This work is vital in preparing our CNN-based detection pipelines to be able to produce a pure sample of the > 100 000 strong gravitational lensing systems widely predicted for Euclid. We select all sources with VIS IE 5 lensing systems in Euclid, this implies 106 objects for human classification, which while very large is not in principle intractable and not without precedent. © 2024, CC BY.

关键词： Convolutional neural networks

A Mathematical Aspect of Hohenberg-Kohn Theorem∗

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arXiv 2017年

作者： Zhou, Aihui LSEC Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China School of Mathematical Sciences University of Chinese Academy of Sciences Beijing100049 China

The Hohenberg-Kohn theorem plays a fundamental role in density functional theory, which has become a basic tool for the study of electronic structure of matter. In this article, we study the Hohenberg-Kohn theorem for a class of external potentials based on a unique continuation *** Codes 81V70 Copyright © 2017, The Authors. All rights reserved.

关键词： Electronic structure

Extended dynamic mode decomposition with dictionary learning: A data-driven adaptive spectral decomposition of the Koopman operator

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arXiv 2017年

作者： Li, Qianxiao Dietrich, Felix Bollt, Erik M. Kevrekidis, Ioannis G. Institute of High Performance Computing Singapore Singapore Faculty of Mathematics Technical University of Munich Germany Department of Mathematics Clarkson University United States Department of Chemical and Biological Engineering Program in Applied and Computational Mathematics Princeton University United States

Numerical approximation methods for the Koopman operator have advanced considerably in the last few years. In particular, data-driven approaches such as dynamic mode decomposition (DMD) and its generalization, the extended-DMD (EDMD), are becoming increasingly popular in practical applications. The EDMD improves upon the classical DMD by the inclusion of a flexible choice of dictionary of observables that spans a finite dimensional subspace on which the Koopman operator can be approximated. This enhances the accuracy of the solution reconstruction and broadens the applicability of the Koopman formalism. Although the convergence of the EDMD has been established, applying the method in practice requires a careful choice of the observables to improve convergence with just a finite number of terms. This is especially difficult for high dimensional and highly nonlinear systems. In this paper, we employ ideas from machine learning to improve upon the EDMD method. We develop an iterative approximation algorithm which couples the EDMD with a trainable dictionary represented by an artificial neural network. Using the Duffing oscillator and the Kuramoto Sivashinsky PDE as examples, we show that our algorithm can effectively and efficiently adapt the trainable dictionary to the problem at hand to achieve good reconstruction accuracy without the need to choose a fixed dictionary a priori. Furthermore, to obtain a given accuracy we require fewer dictionary terms than EDMD with fixed dictionaries. This alleviates an important shortcoming of the EDMD algorithm and enhances the applicability of the Koopman framework to practical problems. Copyright © 2017, The Authors. All rights reserved.

关键词： Machine learning