[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-122670-en":3,"doc-seo-122670-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":27,"seo_description":14,"update_tm":28,"read_time":29},122670,549758146520,"Patrick","https://ap-avatar.wpscdn.com/avatar/80002397d8c0411e94?_k=1775819394049821470",8,"Research & Report","Machine Learning Enhanced Hankel Dynamic-Mode Decomposition - Research Overview","Machine learning combined with dynamic mode decomposition (DMD) is used to construct more accurate dynamical models from time-series data, addressing the long-standing challenge of building dynamical models directly from observations. A deep learning DMD approach, called Deep Learning Hankel DMD (DLHDMD), leverages Takens' Embedding Theorem to adaptively approximate higher-dimensional and chaotic dynamics. The method is analyzed through its learned mappings and their effect on mutual information between dynamical dimensions. Quantitative information-theoretic studies clarify the behavior of the underlying machine-learning tools and support broader data-analysis and modeling potential in physical sciences.","arXiv :2303 .06289v 3 [ cs .LG] 18 Jul 2023  \nMachine Learning Enhanced Hankel Dynamic-Mode Decomposition  \nChristopher W. Curtis 1,* , D. Jay Alford-Lago 1,2 , Erik Bollt3, 4 , and  \nAndrew Tuma 1  \n1 Department of Mathematics and Statistics, San Diego State University, San Diego, CA, 92182, USA  \n2 Naval Information Warfare Center Paci􀀌c, San Diego, CA, 92152, USA  \n3 Department of Electrical and Computer Engineering, Clarkson University, 8 Clarkson Ave., Potsdam, NY, 13699, USA  \n4 Clarkson Center for Complex Systems Science, Clarkson University, 8 Clarkson Ave., Potsdam, NY, 13699, USA  \n* Corresponding author: Christopher W. Curtis, [ccurtis@sdsu.edu](ccurtis@sdsu.edu)  \nAbstract  \nWhile the acquisition of time series has become more straightforward, developing dynamical models from time series is still a challenging and evolving problem domain. Within the last several years, to address this problem, there has been a merging of machine learning tools with what is called the dynamic mode decomposition (DMD) . This general approach has been shown to be an especially promising avenue for accurate model development. Building on this prior body of work, we develop a deep learning DMD based method which makes use of the fundamental insight of Takens' Embedding Theorem to build an adaptive learning scheme that better approximates higher dimensional and chaotic dynamics. We call this method the Deep Learning Hankel DMD (DLHDMD) . We likewise explore how our method learns mappings which tend, after successful training, to signi􀀌cantly change the mutual information between dimensions in the dynamics. This appears to be a key feature in enhancing the DMD overall, and it should help provide further insight for developing other deep learning methods for time series analysis and model generation.  \nThis work uses machine learning to develop an accurate method for generating models of chaotic dynamical systems using measurements alone. A number of challenging examples are examined which show the broad utility of the method and point towards its potential impacts in advancing data analysis and modeling in the  \nphysical sciences. Finally, we present quantitative studies of the information theoretic behavior of the machine learning tools used in our work, thereby allowing for a more detailed understanding of what can otherwise be an inscrutable method.  \n1 Introduction  \nThe incorporation of modern machine learning methodology into dynamical systems is creating an ever expanding array of techniques pushing the boundaries of what is possible with regards to describing and predicting nonlinear multi-dimensional time series. Longstanding problems such as 􀀌nding optimal Takens' embeddings [1, 2] now have powerful and novel deep learning based algorithmic approaches [3] which would not have been feasible even ten years ago. Likewise, the 􀀌eld of equation free modeling using Koopman operator methods, broadly described by Dynamic Mode Decomposition (DMD), has seen several innovative deep learning based methods emerge over the last several years [4, 5, 6] which have been shown to greatly expand the accuracy and 􀀍exibility of DMD based approaches. There have also been related and signi􀀌cant advances in model identi􀀌cation and solving nonlinear partial di􀀋erential equations via deep learning techniques [7, 8, 9, 10] .  \nWith this background in mind, in this work we focus on extending the methods in [6] which were called Deep Learning DMD (DLDMD) . In that work, a relatively straightforward method merging auto-encoders with the extended DMD (EDMD) was developed. This was done by using an encoder to embed dynamics in a su􀀎ciently high enough dimensional space which then generated a su􀀎ciently large enough space of observables for the EDMD to generate accurate linear models of the embedded dynamics. Decoding then returned the embedded time series to the original variables in such a way as to guarantee the global stability of iterating the linear model","cbCail6ErbV9pMBh","https://ap.wps.com/l/cbCail6ErbV9pMBh","pdf",8758433,1,33,"English","en",105,"# Abstract\n# Introduction\n## Dynamic modeling from time series with machine learning\n## Deep Learning DMD and its limitations on chaotic dynamics\n## Motivation for DLHDMD using Takens' Embedding Theorem\n## Global EDMD and adaptive Hankel ordering","[{\"question\":\"What problem does DLHDMD address in time-series modeling?\",\"answer\":\"It targets the difficulty of developing dynamical models from time series, especially for higher-dimensional and chaotic dynamics, by combining deep learning with DMD principles.\"},{\"question\":\"How does DLHDMD differ from prior Deep Learning DMD (DLDMD)?\",\"answer\":\"DLHDMD expands the earlier framework by making the EDMD over embedded coordinates global rather than local, and by introducing an adaptive Hankel matrix-based ordering of embedded coordinates.\"},{\"question\":\"Why does the paper highlight mutual information changes in the learned mappings?\",\"answer\":\"It reports that after successful training, the learned mappings significantly change mutual information between dimensions, suggesting this as a key mechanism for enhancing DMD performance.\"}]","Machine Learning Enhanced Hankel Dynamic-Mode Decomposition - 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