[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-116973-en":3,"doc-seo-116973-105":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},116973,34359740700684,"Finn","https://ap-avatar.wpscdn.com/avatar/1f400023980c374ae676?_k=1777273430885731487",8,"Research & Report","Challenges and Opportunities in Machine Learning for Geometry","Over recent decades, advances in machine learning have enabled the mathematical community to analyze large bodies of pure mathematical data using supervised, semi-supervised, and unsupervised techniques. Such approaches can support numerical conjecturing by leveraging models such as neural networks, support vector machines, and principal component analysis. The article reviews current machine-learning applications in geometry and anticipates significant growth. It also introduces a method for extracting geometric information from point clouds and reconstructing 2D or 3D models using generalized asymptotes.","mathematics  \nArticle  \nChallenges and Opportunities in Machine Learning for Geometry  \nRafael Magdalena-Benedicto 1, *,†, Sonia Pérez-Díaz 2,† and Adrià Costa-Roig 3,†  \nCitation: Magdalena-Benedicto, R.; Pérez-Díaz, S.; Costa-Roig, A. Challenges and Opportunities in Machine Learning for Geometry. Mathematics 2023, 11, 2576. [https://](https://)[ ](https://)[doi.org/10.3390/math11112576](doi.org/10.3390/math11112576)  \nAcademic Editors: Ioannis G. Tsoulos, Ivan Lorencin and Faheim Sufi  \nReceived: 12 April 2023  \nRevised: 15 May 2023  \nAccepted: 1 June 2023  \nPublished: 4 June 2023  \nCopyright: © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license ([https://](https://)[ ](https://)[creativecommons.org/licenses/by/](creativecommons.org/licenses/by/)[ ](creativecommons.org/licenses/by/)[4.0/](4.0/)) .  \n1 Department of Electronic Engineering, University of Valencia, 46010 Valencia, Spain  \n2 University of Alcalá, Department of Physics and Mathematics, 28871 Alcalá de Henares, Spain; [sonia.perez@uah.es](sonia.perez@uah.es)  \n3 Department of Pediatric Surgery, La Fe University and Polytechnic Hospital, 46026 Valencia, Spain; [adriacostaroig@gmail.com](adriacostaroig@gmail.com)  \n* Correspondence: [rafael.magdalena@uv.es](rafael.magdalena@uv.es)[ ](rafael.magdalena@uv.es)† These authors contributed equally to this work.  \nAbstract: Over the past few decades, the mathematical community has accumulated a significant amount of pure mathematical data, which has been analyzed through supervised, semi-supervised, and unsupervised machine learning techniques with remarkable results, e.g., artificial neural networks, support vector machines, and principal component analysis. Therefore, we consider as disruptive the use of machine learning algorithms to study mathematical structures, enabling the formulation of conjectures via numerical algorithms. In this paper, we review the latest applications of machine learning in the field of geometry. Artificial intelligence can help in mathematical problem solving, and we predict a blossoming of machine learning applications during the next years in the field of geometry. As a contribution, we propose a new method for extracting geometric information from the point cloud and reconstruct a 2D or a 3D model, based on the novel concept of generalized asymptotes.  \nKeywords: algebraic geometry; machine learning; generalized asymptotes  \nMSC: 14Q20; 14Q05; 68T01  \n1. Introduction  \nThe use of machine learning (ML) is gaining popularity in the scientific community, especially in domains such as data analysis, optimization, and statistics. ML algorithms are used to detect patterns in data and can be leveraged to solve a broad range of mathematical problems. Mathematics has benefited significantly from ML, particularly in the realm of data analysis. Through ML algorithms, mathematicians can analyze vast datasets and uncover underlying relationships and patterns that may be elusive using conventional statistical techniques. The discovery of these patterns and relationships have provided novel insights in various fields such as engineering, biology, and finance [1] .  \nMachine learning also plays an essential role in optimization, which refers to finding the most optimal solution for a given problem. Machine learning algorithms are particularly useful in searching for the best solution in high-dimensional spaces. Furthermore, through the application of machine learning techniques to analyze vast datasets and uncover patterns, mathematicians can develop new mathematical models that are better equipped to address the complexity of real-world systems. Machine learning can help in the creation of novel mathematical models and algorithms.  \nThe structure of the paper is as follows (see Figure 1): In Section 2, we discuss the outbreak of machine learning in certain fields closely related to","cbCaibaiEvzerPGh","https://ap.wps.com/l/cbCaibaiEvzerPGh","pdf",1335001,1,24,"English","en",105,"# Introduction\n## Scope and growth of machine learning in mathematics\n# Machine learning applications and trends in geometry\n# Problems and technical challenges\n# Asymptote-based point cloud reconstruction method","[{\"question\":\"What motivates using machine learning to study geometric structures?\",\"answer\":\"The work is motivated by the availability of substantial mathematical data and the ability of machine learning to uncover patterns and enable numerical formulation of conjectures. The authors emphasize disruptive potential in analyzing mathematical structures using modern ML techniques.\"},{\"question\":\"Which types of machine learning approaches are discussed for mathematical data?\",\"answer\":\"The article highlights supervised, semi-supervised, and unsupervised machine learning. It also cites examples such as artificial neural networks, support vector machines, and principal component analysis.\"},{\"question\":\"What is the proposed contribution for point cloud reconstruction?\",\"answer\":\"The paper proposes extracting geometric information from point clouds to reconstruct 2D or 3D models by fitting a set of asymptotes. It introduces generalized (g-) asymptotes rather than restricting to lines, enabling curves or surfaces that follow the asymptote structure.\"}]","Challenges and Opportunities in Machine Learning for Geometry | PDF",1785672923,60,{"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},"challenges-and-opportunities-in-machine-learning-for-geometry","",{"@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/challenges-and-opportunities-in-machine-learning-for-geometry/116973/",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","2026-08-02",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 motivates using machine learning to study geometric structures?","Question",{"text":75,"@type":76},"The work is motivated by the availability of substantial mathematical data and the ability of machine learning to uncover patterns and enable numerical formulation of conjectures. The authors emphasize disruptive potential in analyzing mathematical structures using modern ML techniques.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Which types of machine learning approaches are discussed for mathematical data?",{"text":80,"@type":76},"The article highlights supervised, semi-supervised, and unsupervised machine learning. It also cites examples such as artificial neural networks, support vector machines, and principal component analysis.",{"name":82,"@type":73,"acceptedAnswer":83},"What is the proposed contribution for point cloud reconstruction?",{"text":84,"@type":76},"The paper proposes extracting geometric information from point clouds to reconstruct 2D or 3D models by fitting a set of asymptotes. 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