[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-121143-en":3,"doc-seo-121143-105":30,"detail-sidebar-cat-0-en-105":95},{"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},121143,1099514068035,"Ezra","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","Machine Learning for Chaotic Dynamical Systems - Dissertation","This dissertation studies machine learning methods for dynamical systems, with emphasis on chaotic dynamics. Chapter 1 introduces chaotic dynamical systems and machine learning, and summarizes later chapters. Chapter 2 develops a modified Physics Informed Neural Network for forecasting COVID-19 spread using the SICRD model, including identifiability checks, wavelet-based data processing, and a loss modification that stabilizes estimation for multiple unknowns and supports time-varying parameters. Chapter 3 proposes the Discrete-Temporal Sobolev Network (DTSN) to improve forecasting accuracy for discrete Lorenz-63 and Chua circuit models via temporal Sobolev norms. Chapter 4 introduces an EDMD variant for systems with symbolic representations, constructing dictionaries from cylinder set indicators and proving operator convergence with finite-step bounds, demonstrated on dyadic and logistic maps.","Machine Learning for Chaotic Dynamical Systems  \n\n| Item Type | Dissertation (Open Access) |\n| --- | --- |\n| Authors | Kennedy, Connor |\n| DOI | 10.7275/55152 |\n| Rights | Attribution-NonCommercial 4.0 International |\n| Download date | 2026-03-18 07:10:29 |\n| Item License | [http://creativecommons.org/licenses/by-nc/4.0/](http://creativecommons.org/licenses/by-nc/4.0/) |\n| Link to Item | [https://hdl.handle. net/20.500.14394/55152](https://hdl.handle. net/20.500.14394/55152) |\n\nMACHINE LEARNING FOR CHAOTIC DYNAMICAL SYSTEMS  \nA Dissertation Presented  \nby  \nCONNOR KENNEDY  \nSubmitted to the Graduate School of the University of Massachusetts Amherst in partial fulfillment of the requirements for the degree of  \nDOCTOR OF PHILOSOPHY  \nSeptember 2024  \nDepartment of Mathematics and Statistics  \n© Copyright by Connor Kennedy 2024 All Rights Reserved  \nMACHINE LEARNING FOR CHAOTIC DYNAMICAL SYSTEMS  \nA Dissertation Presented  \nby  \nCONNOR KENNEDY  \nApproved as to style and content by:  \n\n| Hongkun Zhang, Chair |\n| --- |\n| Panayotis Kevrekidis, Member |\n| Luc Rey-Bellet, Member |\n\nHui Guan, Member  \nNathaniel Whitaker, Department Head Mathematics and Statistics  \nACKNOWLEDGMENTS  \nI would like to thank the following people.  \nHongkun Zhang: For her consistent patience in working with me, even on the other side of the world.  \nPanayotis Kevrekidis: For offering some especially helpful grounding on my first publication.  \nJohn Kaushagen, Sankaran Vaidyanathan, Haoran Hu: For working hard alongside me even as each of you were undergoing your own struggles through graduate school.  \nAli: For commiserating with me as we both struggled with our graduate work Harry Rockland Miller: without whose grounding support I’m unsure I would have been able to finish this work.  \nAnd to so many more who have supported me emotionally over this incredibly long journey.  \nFinally, I would like to express my gratitude to the Mathematics Department atthe University of Massachusetts Amherst.  \nABSTRACT  \nMACHINE LEARNING METHODS FOR CHAOTIC DYNAMICAL SYSTEMS  \nSEPTEMBER 2024  \nCONNOR M. KENNEDY, B.S. , UNIVERSITY OF MASSACHUSETTS  \nLOWELL  \nPh.D., UNIVERSITY OF MASSACHUSETTS AMHERST Directed by: Professor Hongkun Zhang  \nThis dissertation is on the usage of machine learning for the study of dynamical systems, particularly chaotic dynamical systems. Chapter 1 provides a brief introduction to the fields of chaotic dynamical systems and machine learning as well asa small overview of chaptes 2-4  \nIn chapter 2 we study the usage of machine learning methods to forecast the spread of COVID-19 . We consider the Susceptible-Infected-Confirmed-RecoveredDeceased (SICRD) compartmental model, with the goal of estimating the unknown infected compartment I, and several unknown parameters. We apply a variation of a “Physics Informed Neural Network” (PINN), which uses knowledge of the system to aid learning. First, we ensure estimation is possible by verifying the model’s identifiability. Then, we propose a wavelet transform to process data for the network training. Finally, our central result is a novel modification of the PINN’s loss function to reduce the number of simultaneously considered unknowns. We find that our modified network is capable of stable, efficient, and accurate  \nestimation, while the unmodified network consistently yields incorrect values. The modified network is also shown to be efficient enough to be applied to a model with time-varying parameters. We present an application of our model results in ranking states by estimated relative testing efficiency. Our findings suggest the effectiveness of our modified PINN network, especially in this case of multiple unknown variables.  \nIn chapter 3 we introduce the Discrete-Temporal Sobolev Network (DTSN), a neural network loss function that assists dynamical system forecasting by minimizing variational differences between the network output and the training data via a temporal Sobolev norm. This approach i","cbCaiqr84WynbrBk","https://ap.wps.com/l/cbCaiqr84WynbrBk","pdf",18051751,1,170,"English","en",105,"# Acknowledgments\n# Abstract\n# Chapter 1: Introduction\n# Chapter 2: Machine Learning for COVID-19 Forecasting (SICRD)\n## Physics Informed Neural Network modification\n## Identifiability and wavelet preprocessing\n## Application: ranking states by estimated efficiency\n# Chapter 3: Discrete-Temporal Sobolev Network (DTSN)\n## Lorenz-63 and Chua circuit test cases\n## Comparison with MSE and PINN losses\n# Chapter 4: Extended Dynamic Mode Decomposition (EDMD) for Symbolic Systems\n## Dictionary construction from cylinder sets\n## Convergence and finite-step bounds\n## Numerical applications to dyadic and logistic maps","[{\"question\":\"What is the dissertation’s main focus in machine learning for chaotic dynamics?\",\"answer\":\"It investigates machine learning methods for dynamical systems, especially chaotic systems, developing approaches for forecasting, improving learning stability under sensitivity to noise, and connecting data-driven models with structured dynamical representations.\"},{\"question\":\"How does Chapter 2 use machine learning to forecast COVID-19 spread?\",\"answer\":\"It applies a modified Physics Informed Neural Network to a SICRD compartmental model, first checking identifiability and then using wavelet-transformed data. The modified loss function reduces the number of simultaneously treated unknowns, improving stable and accurate estimation compared with an unmodified network.\"},{\"question\":\"What is the purpose of the Discrete-Temporal Sobolev Network (DTSN) in Chapter 3?\",\"answer\":\"DTSN improves dynamical forecasting by replacing the usual loss with a temporal Sobolev norm that minimizes variational differences over time. The method is data-driven, architecture agnostic, and does not require derivative information from the underlying system.\"},{\"question\":\"What new EDMD method is introduced in Chapter 4 and how is accuracy supported?\",\"answer\":\"The chapter presents an EDMD variant for systems with symbolic representations by building a dictionary from indicator functions of cylinder sets. It proves strong operator topology convergence for the projection and the estimated Koopman operator, and derives finite-step estimation bounds, with numerical demonstrations on dyadic and logistic maps.\"}]","Machine Learning for Chaotic Dynamical Systems - Dissertation | PDF",1785734056,428,{"code":4,"msg":31,"data":32},"ok",{"site_id":24,"language":23,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":90,"head_meta":92,"extra_data":94,"updated_unix":28},"machine-learning-for-chaotic-dynamical-systems-dissertation","",{"@graph":36,"@context":89},[37,54,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/machine-learning-for-chaotic-dynamical-systems-dissertation/121143/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":23,"description":14,"dateModified":62,"datePublished":62,"encodingFormat":61,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-08-03",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81,85],{"name":72,"@type":73,"acceptedAnswer":74},"What is the dissertation’s main focus in machine learning for chaotic dynamics?","Question",{"text":75,"@type":76},"It investigates machine learning methods for dynamical systems, especially chaotic systems, developing approaches for forecasting, improving learning stability under sensitivity to noise, and connecting data-driven models with structured dynamical representations.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does Chapter 2 use machine learning to forecast COVID-19 spread?",{"text":80,"@type":76},"It applies a modified Physics Informed Neural Network to a SICRD compartmental model, first checking identifiability and then using wavelet-transformed data. The modified loss function reduces the number of simultaneously treated unknowns, improving stable and accurate estimation compared with an unmodified network.",{"name":82,"@type":73,"acceptedAnswer":83},"What is the purpose of the Discrete-Temporal Sobolev Network (DTSN) in Chapter 3?",{"text":84,"@type":76},"DTSN improves dynamical forecasting by replacing the usual loss with a temporal Sobolev norm that minimizes variational differences over time. The method is data-driven, architecture agnostic, and does not require derivative information from the underlying system.",{"name":86,"@type":73,"acceptedAnswer":87},"What new EDMD method is introduced in Chapter 4 and how is accuracy supported?",{"text":88,"@type":76},"The chapter presents an EDMD variant for systems with symbolic representations by building a dictionary from indicator functions of cylinder sets. It proves strong operator topology convergence for the projection and the estimated Koopman operator, and derives finite-step estimation bounds, with numerical demonstrations on dyadic and logistic maps.","https://schema.org",{"og:url":52,"og:type":91,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":93,"canonical":52},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":96},[97,101,105,109,114,119,124,127,132,135,139],{"id":20,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":106,"show_sort_weight":107,"slug":108},"Exam",70,"exam",{"id":110,"doc_module":4,"doc_module_name":46,"category_name":111,"show_sort_weight":112,"slug":113},5,"Comic",60,"comic",{"id":115,"doc_module":4,"doc_module_name":46,"category_name":116,"show_sort_weight":117,"slug":118},6,"Technology",50,"technology",{"id":120,"doc_module":4,"doc_module_name":46,"category_name":121,"show_sort_weight":122,"slug":123},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":125,"slug":126},30,"research-report",{"id":128,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":130,"slug":131},9,"Religion & Spirituality",20,"religion-spirituality",{"id":130,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":130,"slug":134},"World Cup","world-cup",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":136,"slug":138},10,"Lifestyle","lifestyle",{"id":140,"doc_module":4,"doc_module_name":46,"category_name":141,"show_sort_weight":110,"slug":142},19,"General","general"]