[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-119883-en":3,"doc-seo-119883-105":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},119883,1649267921044,"Ava Thompson","https://us-avatar.wpscdn.com/avatar/1800007509477c92dfb?_k=1782875107921204101",8,"Research & Report","Machine Learning Regularization for the Minimum Volume Formula of Toric Calabi-Yau 3-folds","提出一组用于Sasaki-Einstein 5维流形最小体积的显式公式。其锥空间为扭量子对称的三维扭（toric）Calabi-Yau 3-fold，并与一类4d N=1超对称规范理论相关，作为D3膜探测这些几何奇点的世界体理论。借助AdS/CFT对应，最小体积与相应4d N=1超共形场论的中心电荷（由a函数决定）成反比。公式以toric Calabi-Yau 3-fold的几何不变量表示，通过机器学习正则化得到，可在大规模数据集上以高精度近似最小体积，并提供可解释的表达式。","arXiv :2310 . 19276v1 [hep-th] 30 Oct 2023  \nUNIST-MTH-23-RS-05  \nMachine Learning Regularization  \nfor the Minimum Volume Formula of Toric Calabi-Yau 3-folds  \nEugene Choia and Rak-Kyeong Seonga,b ∗  \na Department of Mathematical Sciences, and  \nb Department of Physics,  \nUlsan National Institute of Science and Technology,  \n50 UNIST-gil, Ulsan 44919, South Korea  \nWe present a collection of explicit formulas for the minimum volume of Sasaki-Einstein 5-manifolds. The cone over these 5-manifolds is a toric Calabi-Yau 3-fold. These toric Calabi-Yau 3-folds are associated with an infinite class of 4d N = 1 supersymmetric gauge theories, which are realized as worldvolume theories of D3-branes probing the toric Calabi-Yau 3-folds. Under the AdS/CFT correspondence, the minimum volume of the Sasaki-Einstein base is inversely proportional to the central charge of the corresponding 4d N = 1 superconformal field theories. The presented formulas for the minimum volume are in terms of geometric invariants of the toric Calabi-Yau 3-folds. These explicit results are derived by implementing machine learning regularization techniques that advance beyond previous applications of machine learning for determining the minimum volume. Moreover, the use of machine learning regularization allows us to present interpretable and explainable formulas for the minimum volume. Our work confirms that, even for extensive sets of toric Calabi-Yau 3-folds, the proposed formulas approximate the minimum volume with remarkable accuracy.  \nI. INTRODUCTION  \nSince the introduction of machine learning techniques in [1–13] for studying problems that occur in the context of string theory, machine learning – both supervised [14– 20] and unsupervised [21–24] – has led to a variety of applications in string theory. A problem that appeared particularly suited for machine learning in 2017 [2] was the problem of identifying a formula for the minimum volume of Sasaki-Einstein 5-manifolds [25, 26] . The cone over these Sasaki-Einstein 5-manifolds is a toric Calabi-Yau 3-fold [27, 28] . Given that there are infinitely many toric Calabi-Yau 3-folds with corresponding Sasaki-Einstein 5-manifolds and that there is an infinite class of 4d N = 1 supersymmetric gauge theories associated to them via string theory [29–36], this beautiful correspondence between geometry and gauge theory was identified in [2] as an ideal testbed for introducing machine learning for string theory.  \nThese 4d N = 1 supersymmetric gauge theories corresponding to toric Calabi-Yau 3-folds are realized as worldvolume theories of D3-branes probing the CalabiYau singularities. Via the AdS/CFT correspondence [37–39], the minimum volume of the Sasaki-Einstein 5-manifolds is related to the maximized a-function [40–42] that gives the central charges of the corresponding 4d N = 1 superconformal field theories [43, 44] . The pro-  \n∗ Electronic address: [xeugenechoi@gmail.com](xeugenechoi@gmail.com), [seong@unist.ac.kr](seong@unist.ac.kr)  \nposal in [2] was that machine learning techniques can be used to give a formula of the minimum volume in terms of features taken from the toric diagram of the corresponding toric Calabi-Yau 3-folds. Such a formula would significantly simplify the computation of the minimum volume, which conventionally is computed by minimizing the volume function obtained from the equivariant index [25, 26] or Hilbert series of the toric Calabi-Yau 3-fold [45, 46] .  \nIn [2], we made use of multiple linear regression [47–51] and a combination of a regression model and a convolutional neural network (CNN) [52–55] to learn the minimum volume for toric Calabi-Yau 3-folds. As it is often the case for supervised machine learning [56, 57], the models lacked interpretability and explainability, achieving high accuracies in estimating the minimum volume with giving only little insight into the mathematical structure and physical origin of the estimating formula.  \n\n|  | 0 1 2 3 | 4 5 6 7 | 8 9 |\n| --- |","cbCairK7jg4VTfBS","https://ap.wps.com/l/cbCairK7jg4VTfBS","pdf",12865617,1,15,"English","en",105,"# Introduction\n## Motivation from machine learning in string theory\n## AdS/CFT relation and need for computable formulas\n## Limits of prior supervised ML approaches and role of regularization\n# Calabi-Yau 3-folds and Quiver Gauge Theories","[{\"question\":\"该文研究的核心问题是什么？\",\"answer\":\"如何为Sasaki-Einstein 5维流形给出最小体积的可计算显式公式，并将其表示为toric Calabi-Yau 3-fold的几何不变量。\"},{\"question\":\"最小体积与哪些物理量建立了联系？\",\"answer\":\"在AdS/CFT对应下，Sasaki-Einstein底的最小体积与对应4d N=1超共形场论的中心电荷相关，表现为与a函数最大化结果的反比关系。\"},{\"question\":\"作者如何用机器学习正则化改进以往方法？\",\"answer\":\"通过引入Lasso正则化等正则化机器学习模型，选择适合的多项式与对数回归结构，从而克服监督学习中可解释性不足的问题，得到可重复使用且具可解释性的候选公式。\"}]","Machine Learning Regularization for the Minimum Volume Formula of Toric Calabi-Yau 3-folds | PDF",1785726827,38,{"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-regularization-for-the-minimum-volume-formula-of-toric-calabi-yau-3-folds","",{"@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-regularization-for-the-minimum-volume-formula-of-toric-calabi-yau-3-folds/119883/",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},"该文研究的核心问题是什么？","Question",{"text":75,"@type":76},"如何为Sasaki-Einstein 5维流形给出最小体积的可计算显式公式，并将其表示为toric Calabi-Yau 3-fold的几何不变量。","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"最小体积与哪些物理量建立了联系？",{"text":80,"@type":76},"在AdS/CFT对应下，Sasaki-Einstein底的最小体积与对应4d N=1超共形场论的中心电荷相关，表现为与a函数最大化结果的反比关系。",{"name":82,"@type":73,"acceptedAnswer":83},"作者如何用机器学习正则化改进以往方法？",{"text":84,"@type":76},"通过引入Lasso正则化等正则化机器学习模型，选择适合的多项式与对数回归结构，从而克服监督学习中可解释性不足的问题，得到可重复使用且具可解释性的候选公式。","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"]