[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-122575-en":3,"doc-seo-122575-105":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},122575,7971461741311,"Ophelia","https://ap-avatar.wpscdn.com/avatar/74000253aff267980c6?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779345379180704826",8,"Research & Report","Mini-Workshop - High-Dimensional Control Problems and Mean-Field Equations with Applications in Machine Learning","High-dimensional control problems and mean-field equations received renewed attention, with emphasis on numerical methods designed to overcome the curse of dimensionality. The workshop connected these optimization tasks to machine-learning settings such as data-driven optimal control and training deep neural networks. Participants exchanged ideas on synergies between control theory tools and learning models, covering high-dimensional approximation, PDE control, multi-agent mean-field strategies, and agent-based optimization in probability measure spaces.","Mathematisches Forschungsinstitut Oberwolfach  \nReport No. 56/2024  \nDOI: 10.4171/OWR/2024/56  \nMini-Workshop: High-Dimensional Control Problems and Mean-Field Equations with Applications in Machine  \nLearning  \nOrganized by  \nGiacomo Borghi, Edinburgh  \nElisa Iacomini, Ferrara  \nMathias Oster, Aachen  \nChiara Segala, Lugano  \n8 December – 13 December 2024  \nAbstract. High-dimensional control problems and mean 􀀌eld equations have been of increased interest in recent years and novel numerical tools tackling the curse of dimensionality have been developed. These optimization tasks are strongly related to learning problems such as data-driven optimal control and learning of deep neural networks. As a consequence, there is a huge potential to employ control theoretical techniques in Machine Learning. The Mini-Workshop was devoted to discuss possible synergies among the various  \ntools developed in these 􀀌elds.  \nMathematics Subject Classi􀀌cation (2020): 34H05, 68T07, 65D40, 49M41, 49N80 .  \nLicense: Unless otherwise noted, the content of this report is licensed under CC BY SA 4.0 .  \nIntroduction by the Organizers  \nThe workshop High-Dimensional Control Problems and Mean-Field Equations with Applications in Machine Learning brought together 16 mathematicians from Germany, Italy, UK, US, France, the Netherlands and Switzerland. Each participant contributed to the event by delivering a 45-minute seminar, presenting their latest research findings and theoretical advancements, and actively engaging in open discussion sessions held in the evenings.  \nThe workshop focused on several interconnected themes: high-dimensional control problems and mean-field equations, numerical tools to address the curse of  \n3212 Oberwolfach Report 56/2024  \ndimensionality, and data-driven optimal control techniques using deep neural networks. The theme of high-dimensional approximation tools addressed methods such as kernel methods, tensor decomposition techniques, and neural networks, which aim to circumvent the curse of dimensionality by exploiting structural properties in the data. These tools are essential for controlling semi-linear PDEs and designing Lyapunov and value functions with bounded complexity. Mean-field optimal control explored strategies for managing complex multi-agent systems, addressing challenges like non-locality, non-linearity, and non-convexity, with approaches such as sparse controls, model predictive control, and turnpike properties offering computational advantages. Agent-based methods in optimization highlighted the potential of interacting particle systems and mean-field approaches for tackling high-dimensional optimization tasks in spaces of probability measures, leveraging metrics like Wasserstein and Fisher–Rao distances. Lastly, the machine learning theme examined the interplay between mean-field PDEs, optimal control, and machine learning models such as ResNets and kernel methods, showing how these frameworks can provide both practical tools for solving control problems and deeper mathematical insights into machine learning processes.  \nThe first part of the week emphasized optimal control frameworks, featuring diverse perspectives on the topic. Chiara Segala explored controllability of continuous networks through kernel-based learning approximations, while Alessandro Scagliotti addressed strategies for managing uncertainty in control systems via averaged and uniform ensemble optimal control. Luca Saluzzi discussed the role of sparsity and low-rank structures in high-dimensional parametrized optimal control problems, and Mathias Oster examined adaptive ResNet architectures through insights from the mean-field limit. Tobias Breiten offered a mathematical perspective on optimal control for hypocoercive Fokker–Planck equations, while Giacomo Albi presented innovative approaches to controlling high-dimensional particle systems in magnetically confined fusion plasma. Finally, Susana Gomes investigated the dynamics of opinion formation ","cbCaioLiiOQnsMrZ","https://ap.wps.com/l/cbCaioLiiOQnsMrZ","pdf",816292,1,44,"English","en",105,"# Introduction\n## Participants and seminar structure\n## Core themes and technical approaches\n## Week schedule: optimal control focus\n## Cross-disciplinary joint session\n## Mid-week focus: optimization, gradient flows, and game theory\n## Spontaneous discussions and impromptu seminars","[{\"question\":\"What were the main topics of the mini-workshop?\",\"answer\":\"The mini-workshop focused on high-dimensional control problems and mean-field equations, including numerical tools to mitigate the curse of dimensionality and data-driven optimal control using deep neural networks.\"},{\"question\":\"How did the workshop link control theory with machine learning?\",\"answer\":\"It highlighted connections between mean-field PDEs, optimal control, and machine-learning models such as ResNets and kernel methods, showing how both practical solution tools and mathematical insights can emerge.\"},{\"question\":\"Which application areas and methods were emphasized for tackling high-dimensional problems?\",\"answer\":\"Discussions covered high-dimensional approximation tools (kernel methods, tensor decompositions, neural networks), mean-field optimal control for multi-agent systems (sparse controls, model predictive control, turnpike properties), and agent-based/particle and probability-measure approaches using metrics like Wasserstein and Fisher–Rao distances.\"}]","Mini-Workshop - High-Dimensional Control Problems and Mean-Field Equations with Applications in Machine Learning | PDF",1785811391,111,{"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},"mini-workshop-high-dimensional-control-problems-and-mean-field-equations-with-applications-in-machine-learning","",{"@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/mini-workshop-high-dimensional-control-problems-and-mean-field-equations-with-applications-in-machine-learning/122575/",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","2026-08-04",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 were the main topics of the mini-workshop?","Question",{"text":75,"@type":76},"The mini-workshop focused on high-dimensional control problems and mean-field equations, including numerical tools to mitigate the curse of dimensionality and data-driven optimal control using deep neural networks.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How did the workshop link control theory with machine learning?",{"text":80,"@type":76},"It highlighted connections between mean-field PDEs, optimal control, and machine-learning models such as ResNets and kernel methods, showing how both practical solution tools and mathematical insights can emerge.",{"name":82,"@type":73,"acceptedAnswer":83},"Which application areas and methods were emphasized for tackling high-dimensional problems?",{"text":84,"@type":76},"Discussions covered high-dimensional approximation tools (kernel methods, tensor decompositions, neural networks), mean-field optimal control for multi-agent systems (sparse controls, model predictive control, turnpike properties), and agent-based/particle and probability-measure approaches using metrics like Wasserstein and Fisher–Rao distances.","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"]