[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-120036-en":3,"doc-seo-120036-105":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},120036,4398048950312,"Violet","https://ap-avatar.wpscdn.com/avatar/400002538284de19e3c?_k=1778320343897328908",8,"Research & Report","Solving variational problems by machine learning - Bachelor thesis","The bachelor thesis explores fundamental approaches to calculus of variations problems in physics through machine learning. It reviews methods on simple, specific benchmarks and contrasts them with traditional numerical techniques. After introducing variational calculus and neural networks, the thesis derives analytical solutions for core problems such as the brachistochrone, harmonic oscillator, and Laplace and Poisson equations. Machine learning methods—direct minimization of the action functional, neural-network-based minimization, and Neural ODEs—are implemented and tested in Julia, with results compared to analytical and numerical baselines.","BACHELOR THESIS  \nPˇremysl Kaska  \nSolving variational problems by machine learning  \nMathematical Institute of Charles University  \nSupervisor of the bachelor thesis: doc. RNDr. Michal Pavelka, Ph.D. Study programme: Mathematics  \nStudy branch: Mathematical modeling  \nI declare that I carried out this bachelor thesis independently, and only with the cited sources, literature and other professional sources. It has not been used to obtain another or the same degree.  \nI understand that my work relates to the rights and obligations under the Act No. 121/2000 Sb., the Copyright Act, as amended, in particular the fact that the Charles University has the right to conclude a license agreement on the use of this work as a school work pursuant to Section 60 subsection 1 of the Copyright Act.  \n[In . . . . . . . . . . . . . date . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .](In . . . . . . . . . . . . . date . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .) .  \nAuthor’s signature  \nI would like to thank my supervisor, doc. RNDr. Michal Pavelka, Ph.D. , for the introduction to this topic and for counseling. I would also like to thank my family for their never-ending support.  \nTitle: Solving variational problems by machine learning  \nAuthor: Pˇremysl Kaska  \nInstitute: Mathematical Institute of Charles University  \nSupervisor: doc. RNDr. Michal Pavelka, Ph.D. , Mathematical Institute of Charles University  \nAbstract: The bachelor thesis aims to explore various basic approaches to the calculus of variations problems in physics using machine learning. Our goal is to review these methods on simple, specific problems and compare them to more traditional numerical methods. The thesis begins with an introduction to the calculus of variations in physics and to neural networks. Next, I choose basic, variational problems, including the brachistochrone, harmonic oscillator, and the Laplace and Poisson equation, and provide their analytical solutions. Finally, I introduce machine learning methods such as direct minimization of the action functional, minimizing the functional with neural networks and Neural ODEs, and apply them to our problems. The experiments are run in the programming language Julia, and the results are then discussed and compared with the analytical solution and other numerical methods.  \nKeywords: neural networks, calculus of variations, machine learning  \nContents  \nIntroduction 2  \n1 Introduction to Calculus of Variations 4  \n1.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4  \n1.2 Functionals Represented by an Integral . . . . . . . . . . . . . . . 5  \n1.3 Applications in Physics ........................ 6  \n2 Introduction to Deep Learning 8  \n2. 1 Neural Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . 8  \n2.1. 1 Universal Approximation Theorem   10  \n2.1.2 Training Neural Networks . . . . . . . . . . . . . . . . . . 11  \n2.1.3 Gradient Descent-based Methods for Optimization   11  \n2.1.4 Backpropagation . . . . . . . . . . . . . . . . . . . . . . . 12  \n3 Analytical Solution of Classical Problems 14  \n3. 1 Brachistochrone . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14  \n3.2 Laplace and Poisson Equation   16  \n3.3 Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . 19  \n4 Machine Learning Methods for Solving Variational Problems 20  \n4. 1 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20  \n4.1.1 Directly Minimizing the Functional . . . . . . . . . . . . . 20  \n4.1.2 Minimizing the Functional with Neural Network . . . . . . 21  \n4.1.3 Neural Ordinary Differential Equations   22  \n4.2 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23  \nConclusion 28  \nBibliography 29  \nList of Figures 32  \nList of Tables 33  \nList of Abbreviations 34  \nA Attachments 35  \nA. 1 Code . . . . . . . . . . . . . . . . . . . . . ","cbCaiq7ZsqGPVwyM","https://ap.wps.com/l/cbCaiq7ZsqGPVwyM","pdf",1202320,1,46,"English","en",105,"# Introduction\n## Overview of machine learning in physics\n# Introduction to Calculus of Variations\n## Theory\n## Functionals represented by an integral\n## Applications in physics\n# Introduction to Deep Learning\n## Neural networks\n## Universal approximation theorem\n## Training neural networks\n## Optimization and backpropagation\n# Analytical Solution of Classical Problems\n## Brachistochrone\n## Laplace and Poisson equation\n## Harmonic oscillator\n# Machine Learning Methods for Solving Variational Problems\n## Directly minimizing the functional\n## Minimizing the functional with neural networks\n## Neural ordinary differential equations\n## Experiments\n# Conclusion","[{\"question\":\"What is the main objective of the thesis?\",\"answer\":\"The thesis aims to study basic calculus of variations methods in physics using machine learning, test them on simple problems, and compare the results with traditional numerical approaches.\"},{\"question\":\"Which variational problems are used as benchmarks?\",\"answer\":\"Benchmarks include the brachistochrone, harmonic oscillator, and the Laplace and Poisson equations, each with analytical solutions provided.\"},{\"question\":\"What machine learning methods are applied to solve the variational problems?\",\"answer\":\"The thesis applies direct minimization of the action functional, minimization using neural networks, and Neural ODEs, implemented and evaluated via experiments.\"}]","Solving variational problems by machine learning - Bachelor thesis | PDF",1785727831,116,{"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},"solving-variational-problems-by-machine-learning-bachelor-thesis","",{"@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/solving-variational-problems-by-machine-learning-bachelor-thesis/120036/",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 is the main objective of the thesis?","Question",{"text":75,"@type":76},"The thesis aims to study basic calculus of variations methods in physics using machine learning, test them on simple problems, and compare the results with traditional numerical approaches.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Which variational problems are used as benchmarks?",{"text":80,"@type":76},"Benchmarks include the brachistochrone, harmonic oscillator, and the Laplace and Poisson equations, each with analytical solutions provided.",{"name":82,"@type":73,"acceptedAnswer":83},"What machine learning methods are applied to solve the variational problems?",{"text":84,"@type":76},"The thesis applies direct minimization of the action functional, minimization using neural networks, and Neural ODEs, implemented and evaluated via experiments.","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"]