[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-290825-105":59,"doc-detail-290825-en":131},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":124,"head_meta":126,"extra_data":128,"updated_unix":130},105,"en","machine-learning-gravity-compactifications-on-negatively-curved-manifolds-abstract-contents-overview","Machine Learning Gravity Compactifications on Negatively Curved Manifolds - Abstract & Contents Overview","","Machine Learning techniques are used to address the computational difficulty of constructing vacua for higher-dimensional gravity theories by solving low-energy (semi-)classical equations of motion. The work applies neural networks to solve Einstein PDEs on warped gravity compactifications built from non-trivial three-manifolds, including manifolds formed by filling cusps of hyperbolic manifolds. Results indicate scalable feasibility and suggest broader applicability to constructing negatively curved Einstein metrics and four-dimensional de Sitter compactifications in M-theory.",{"@graph":69,"@context":123},[70,84,106],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/machine-learning-gravity-compactifications-on-negatively-curved-manifolds-abstract-contents-overview/290825/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/machine-learning-gravity-compactifications-on-negatively-curved-manifolds-abstract-contents-overview/290825.png","ImageObject",300,407,{"name":92,"@type":93},"Skyler","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-11-03","2026-09-17",true,{"@type":102,"interactionType":103,"userInteractionCount":105},"InteractionCounter",{"@type":104},"ViewAction",13,{"@type":107,"mainEntity":108},"FAQPage",[109,115,119],{"name":110,"@type":111,"acceptedAnswer":112},"What is the main problem the document addresses?","Question",{"text":113,"@type":114},"Constructing the landscape of vacua for higher-dimensional gravity theories by directly solving low-energy (semi-)classical equations of motion is computationally challenging. The document targets this difficulty for warped gravity compactifications.","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"How does the work test the feasibility of machine learning in this setting?",{"text":118,"@type":114},"As a proof of concept, neural networks are used to solve Einstein PDEs on non-trivial three-manifolds generated by filling one or more cusps of hyperbolic manifolds.",{"name":120,"@type":111,"acceptedAnswer":121},"What broader applications does the document suggest for the developed methods?",{"text":122,"@type":114},"The approach could help numerically construct higher-dimensional negatively curved Einstein metrics and also construct four-dimensional de Sitter compactifications of M-theory on related manifolds.","https://schema.org",{"og:url":83,"og:type":125,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":127,"canonical":83},"index,follow",{"doc_id":129,"site_id":62},290825,1789642662,{"code":4,"msg":5,"data":132},{"doc_id":129,"user_id":133,"nickname":92,"user_avatar":134,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":135,"file_id":136,"file_url":137,"file_type":138,"file_size":139,"view_count":105,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":140,"language":141,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":142,"faqs":143,"seo_title":144,"seo_description":67,"update_tm":130,"read_time":145,"preview_page_count":4},2336464648746,"https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c","arXiv :2501 .00093v1 [hep-th] 30 Dec 2024  \nMachine Learning Gravity Compactifications on Negatively Curved Manifolds  \nG. Bruno De Luca  \n1 Stanford Institute for Theoretical Physics, Stanford University,  \n382 Via Pueblo Mall, Stanford, CA 94305, United States  \n[gbdeluca@stanford.edu](gbdeluca@stanford.edu)  \nAbstract  \nConstructing the landscape of vacua of higher-dimensional theories of gravity by directly solving the low-energy (semi-)classical equations of motion is notoriously difficult. In this work, we investigate the feasibility of Machine Learning techniques as tools for solving the equations of motion for general warped gravity compactifications. As a proof-of-concept we use Neural Networks to solve the Einstein PDEs on non-trivial three manifolds obtained by filling one or more cusps of hyperbolic manifolds. While in three dimensions an Einstein metric is also locally hyperbolic, the generality and scalability of Machine Learning methods, the availability of explicit families of hyperbolic manifolds in higher dimensions, and the universality of the filling procedure strongly suggest that the methods and code developed in this work can be of broader applicability. Specifically, they can be used to tackle both the geometric problem of numerically constructing novel higher-dimensional negatively curved Einstein metrics, as well asthe physical problem of constructing four-dimensional de Sitter compactifications of M-theory on the same manifolds.  \nContents  \n1 Introduction 2  \n2 Machine Learning methods for gravity compactifications 5  \n2.1 The equations of motion .......................... 6  \n2.2 Machine Learning methods for geometrical PDEs ............ 8  \n2.2.1 A method for compactifications .................. 10  \n2.3 de Sitter compactifications of M-theory on negatively curved manifolds 13  \n3 Machine Learning negatively curved Einstein metrics 14  \n3.1 Hyperbolic manifolds and Anderson-Dehn filling ............. 15  \n3.1.1 The three-dimensional case in detail ................ 17  \n3.2 An explicit 3d manifold ........................... 19  \n3.2.1 The UHS realization ........................ 20  \n3.2.2 The piece-wise metric ........................ 23  \n3.2.3 The boundary conditions ...................... 24  \n3.3 The Machine Learning method ....................... 25  \n3.3.1 The algorithm ............................ 25  \n3.3.2 Results ................................ 27  \n4 Conclusions and outlook 29  \nA Additional details on the Machine Learning method 31  \nB Additional numerical results 33  \nC Additional details on M3 33  \nC.1 Maps and Jacobians ............................ 35  \n1 Introduction  \nIn theories with extra dimensions, four-dimensional vacua are non-trivial even classically. In fact, even a configuration that looks like a vacuum in four dimensions can have (and often has) a very rich structure for the geometry and the matter fields in the extra dimensions. The study of these structures, which are simultaneously rich and highly constrained by the UV completion of the theory, is important to extract the fourdimensional physics, as well as for holographic approaches to quantum gravity.  \nAs we will discuss in detail below, the problem of directly solving the equations of motion for vacuum compactifications is computationally challenging, and a popular approach to bypass this challenge is to exploit supersymmetry in some form. This is the case, for example, of the famous KKLT proposal [1] for obtaining de Sitter vacua, which uses as starting point a supersymmetric Calabi-Yau compactification, on top of which supersymmetry-breaking effects are added. But this is true also for AdS: most of the explicitly known AdS compactifications of string/M-theory are either supersymmetric, obtained by starting from supersymmetric compactifications and turning on supersymmetry breaking effects, or as non-supersymmetric vacua of lower-dimensional supergravities obtained from reduction around supersymmetry vacua. Sup","cbCaipJctagGd7rC","https://ap.wps.com/l/cbCaipJctagGd7rC","pdf",1433435,43,"English","# Contents\n## Introduction\n## Machine Learning methods for gravity compactifications\n## Machine Learning negatively curved Einstein metrics\n## Conclusions and outlook\n## Additional details on the Machine Learning method\n## Additional numerical results\n## Additional details on M3","[{\"question\":\"What is the main problem the document addresses?\",\"answer\":\"Constructing the landscape of vacua for higher-dimensional gravity theories by directly solving low-energy (semi-)classical equations of motion is computationally challenging. The document targets this difficulty for warped gravity compactifications.\"},{\"question\":\"How does the work test the feasibility of machine learning in this setting?\",\"answer\":\"As a proof of concept, neural networks are used to solve Einstein PDEs on non-trivial three-manifolds generated by filling one or more cusps of hyperbolic manifolds.\"},{\"question\":\"What broader applications does the document suggest for the developed methods?\",\"answer\":\"The approach could help numerically construct higher-dimensional negatively curved Einstein metrics and also construct four-dimensional de Sitter compactifications of M-theory on related manifolds.\"}]","Machine Learning Gravity Compactifications on Negatively Curved Manifolds - Abstract & Contents Overview | PDF",108]