[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-120892-en":3,"doc-seo-120892-105":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":4,"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},120892,4810365810221,"Aurora","https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d",8,"Research & Report","Machine learning based partial differential equation (PDE) recovery - Dissertation","The dissertation develops automated methods for recovering governing partial differential equations from data, emphasizing sparsity-driven identification and physics-informed neural networks. It presents techniques for sparse modeling to identify PDE terms, dictionary construction, and synthetic validation on canonical equations such as wave, Helmholtz, and Burgers forms. It extends recovery to spatially varying acoustical properties using denoising by integration, and introduces SD-PINN for deep learning based spatially dependent PDE reconstruction via low-rank coefficient assumptions and matrix completion-style learning.","UC San Diego  \nUC San Diego Electronic Theses and Dissertations  \nTitle  \nMachine learning based partial differential equation (PDE) recovery  \nPermalink  \n[https://escholarship.org/uc/item/7ds343sj](https://escholarship.org/uc/item/7ds343sj)  \nAuthor  \nLiu, Ruixian  \nPublication Date  \n2023  \nPeer reviewed|Thesis/dissertation  \n[eScholarship.org](eScholarship.org) Powered by the California Digital Library  \nUniversity of California  \nUNIVERSITY OF CALIFORNIA SAN DIEGO  \nMachine learning based partial differential equation (PDE) recovery  \nA dissertation submitted in partial satisfaction of the requirements for the degree  \nDoctor of Philosophy  \nin  \nElectrical Engineering (Signal and Image Processing)  \nby  \nRuixian Liu  \nCommittee in charge:  \nPeter Gerstoft, Chair  \nMichael Bianco  \nMichael J. Buckingham  \nWenyuan Fan  \nWilliam S. Hodgkiss  \nTruong Quang Nguyen  \nBhaskar D. Rao  \nCopyright Ruixian Liu, 2023 All rights reserved.  \nThe Dissertation of Ruixian Liu is approved, and it is acceptable in quality and form for publication on microfilm and electronically.  \nUniversity of California San Diego  \n2023  \nDEDICATION  \nTO MY FAMILY, MY FRIENDS AND COLLABORATORS  \nTABLE OF CONTENTS  \nDissertation Approval Page .................................................... iii  \nDedication .................................................................. iv  \nTable of Contents ............................................................ v  \nList of Figures ............................................................... vii  \nList of Tables ................................................................ xi  \nAcknowledgements ........................................................... xiii  \nVita ........................................................................ xv  \nAbstract of the Dissertation .................................................... xvi  \nChapter 1 Introduction ..................................................... 1  \n1.1 Basics of PDEs recovery ............................................... 2  \n1.2 PDE identification using sparse modeling ................................ 3  \n1.3 Physics informed neural network ........................................ 4  \n1.4 Dissertation overview ................................................. 5  \nChapter 2 Automated Partial Differential Equation Identification .................. 7  \n2.1 Introduction ......................................................... 7  \n2.2 Theory .............................................................. 8  \n2.2.1 Background ................................................... 8  \n2.2.2 Building a dictionary ........................................... 9  \n2.2.3 Identifying PDE terms .......................................... 11  \n2.3 Synthetic experiments ................................................. 14  \n2.3.1 Wave equation ................................................. 14  \n2.3.2 Helmholtz equation ............................................ 19  \n2.3.3 Burgers equation .............................................. 21  \n2.4 Application to real video ............................................... 23  \n2.5 Conclusion .......................................................... 25  \n2.6 Supplemental materials ................................................ 26  \n2.6.1 Wavenumber determined by wave equations with attenuation ......... 26  \n2.6.2 Threshold least squares (TLS) ................................... 28  \n2.6.3 Comparison with SINDy ........................................ 29  \n2.7 Acknowledgments .................................................... 30  \nChapter 3 Recovery of Spatially Varying Acoustical Properties via Automated PDE Identification .................................................... 31  \n3.1 Introduction ......................................................... 31  \n3.2 Theory .............................................................. 35  \n3.2.1 PDE identification ............................................. 35","cbCaik5to6uLxpvH","https://ap.wps.com/l/cbCaik5to6uLxpvH","pdf",11258528,1,120,"English","en",105,"# Chapter 1 Introduction\n## Basics of PDEs recovery\n## PDE identification using sparse modeling\n## Physics informed neural network\n## Dissertation overview\n# Chapter 2 Automated Partial Differential Equation Identification\n## Introduction\n## Theory\n## Synthetic experiments\n## Application to real video\n## Conclusion\n## Supplemental materials\n# Chapter 3 Recovery of Spatially Varying Acoustical Properties via Automated PDE Identification\n## Introduction\n## Theory\n## Numerical experiments\n## extracting PDEs for a vibrating plate\n## Efficiency\n## Conclusion\n# Chapter 4 SD-PINN: Deep Learning based Spatially Dependent PDEs Recovery\n## Introduction\n## Theory of SD-PINN\n## Experiments for SD-PINN\n## Comparison with two baseline methods\n## Conclusion\n# Chapter 5 Conclusion","[{\"question\":\"What core problem does the dissertation address?\",\"answer\":\"Recovering governing partial differential equations from observational data using automated identification methods and learning-based models.\"},{\"question\":\"How does the work perform automated PDE identification?\",\"answer\":\"It leverages sparse modeling with dictionary building to identify PDE terms, and evaluates approaches through synthetic experiments and additional supplemental techniques.\"},{\"question\":\"What is SD-PINN and how does it improve PDE recovery?\",\"answer\":\"SD-PINN is a deep learning framework for spatially dependent PDE recovery that uses low-rank assumptions for spatial coefficient variation and formulates coefficient recovery as a matrix completion problem.\"}]","Machine learning based partial differential equation (PDE) recovery - Dissertation | PDF",1785732524,302,{"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":86,"head_meta":88,"extra_data":90,"updated_unix":28},"machine-learning-based-partial-differential-equation-pde-recovery-dissertation","",{"@graph":36,"@context":85},[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-based-partial-differential-equation-pde-recovery-dissertation/120892/",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":4},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What core problem does the dissertation address?","Question",{"text":75,"@type":76},"Recovering governing partial differential equations from observational data using automated identification methods and learning-based models.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the work perform automated PDE identification?",{"text":80,"@type":76},"It leverages sparse modeling with dictionary building to identify PDE terms, and evaluates approaches through synthetic experiments and additional supplemental techniques.",{"name":82,"@type":73,"acceptedAnswer":83},"What is SD-PINN and how does it improve PDE recovery?",{"text":84,"@type":76},"SD-PINN is a deep learning framework for spatially dependent PDE recovery that uses low-rank assumptions for spatial coefficient variation and formulates coefficient recovery as a matrix completion problem.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":89,"canonical":52},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]