[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-119742-en":3,"doc-seo-119742-105":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},119742,549758146520,"Patrick","https://ap-avatar.wpscdn.com/avatar/80002397d8c0411e94?_k=1775819394049821470",8,"Research & Report","Machine learning discovers invariants of braids and flat braids","Machine learning is used to classify braids (and flat braids) as trivial or non-trivial via supervised neural-network models based on multilayer perceptrons. When the classifiers achieve strong results, their internal structure is interpreted as mathematical conjectures, which are then rigorously proven as theorems. The study produces new, convenient invariants of braids and establishes a complete invariant for flat braids. Experimental choices emphasize interpretable network weights and the subsequent translation from learned patterns to precise results.","arXiv :2307 . 12185v1 [math .GT] 22 Jul 2023  \nMachine learning discovers invariants of braids and 􀀍at braids  \nAlexei Lisitsa, Mateo Salles and Alexei Vernitski  \nAbstract. We use machine learning to classify examples of braids (or 􀀍at braids) as trivial or non-trivial. Our ML takes form of supervised learning using neural networks (multilayer perceptrons) . When they achieve good results in classi􀀌cation, we are able to interpret their structure as mathematical conjectures and then prove these conjectures as theorems. As a result, we 􀀌nd new convenient invariants of braids, including a complete invariant of 􀀍at braids.  \n1. Introduction and AI background  \nAutomated discovery of theorems or, in other words, AI-assisted conjecturesis an increasingly important direction of research in recent years, see, for example,[3, 4, 16] . In this paper we report on our experiments with applying neural networks to braids and 􀀍at braids; this has led to forming conjectures, which we then were able to prove as Theorems 1, 2, 3 .  \nWe are satis􀀌ed with the results of this study, and are pleasantly surprised by them. Indeed, normally, our research concentrates on explainable AI; for instance, if we are speaking about a trivial braid, we are speaking of it in the context of being able to 􀀌nd an untangling sequence of Reidemeister moves for this braid [10, 11] . In this study we considered the problem of classifying braids as trivial or non-trivial using neural networks; this approach feels woolly and imprecise in comparison with what we normally do. As expected, neural networks were able to produce only a partial solution. Nevertheless, in some of the experiments the classi􀀌cation produced by the neural network was unexpectedly good, and it made us think that there must be a theorem there, and the entries in the trained neural network formed a distinctive pattern which we were able to re-formulate as a theorem. As a result, imprecise experiments with supervised learning and neural networks have led us to proving exact and unambiguous mathematical results. The lessons that we draw from this study are an inspiration to use supervised  \n2 A. Lisitsa, M. Salles and A. Vernitski  \nFigure 1 . An example of a braid to illustrate encodings.  \nlearning when looking for conjectures and useful skills that will help us to spot potential conjectures.  \nAn arti􀀌cial neural network is, in the simplest case, a perceptron, that is, the process of producing the output from the input by multiplying the input by a matrix (the entries in this matrix are called weights) and then applying a monotonic non-linear function (for example, 0 if x 􀀔 0 and 1 if x > 0) to the numbers in the output. If several perceptrons are applied consecutively one after another, this construction is called a multilayer perceptron (MLP); by layers one means the input, the output and the hidden layers, that is, the intermediate outputs that serve as inputs for other perceptrons. A slightly more general term feedforward neural network is also frequently used in practice with the same meaning as MLP. If one wants to stress that the desired behaviour of a neural network cannot be approximated by a neural network with a small number of layers, one speaks of a deep neural network. In this paper we do not use deep neural networks; instead, we use multilayer perceptrons with one hidden layer; this gives us an opportunity to inspect the weights and generalize them in the form of a mathematical conjecture.  \nLet us provide more machine learning details. In this paper we consider a problem of supervised learning of the binary classi􀀌ers of some properties of braids and 􀀍at braids. We use the classical model of multilayer perceptron (MLP) [13],[14] and its implementation as software called WEKA Workbench for Data Mining [18] . In terms of machine learning, a MLP is a kind of afeedforward neural network models which supports supervised learning using backpropagation [14] . It is known to be an universa","cbCaibTFl5UCuHyH","https://ap.wps.com/l/cbCaibTFl5UCuHyH","pdf",542530,1,24,"English","en",105,"# Introduction and AI background\n## Braids: definitions and encodings","[{\"question\":\"How does the paper use machine learning for braids and flat braids?\",\"answer\":\"It trains supervised neural networks to classify braid (or flat braid) examples as trivial or non-trivial.\"},{\"question\":\"What turns the neural-network results into mathematical results?\",\"answer\":\"When classification performs well, the learned structure is interpreted as mathematical conjectures, which are then proven as theorems.\"},{\"question\":\"What mathematical outcomes does the study report?\",\"answer\":\"It derives new convenient invariants of braids and obtains a complete invariant for flat braids.\"}]","Machine learning discovers invariants of braids and flat braids | PDF",1785726064,60,{"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},"machine-learning-discovers-invariants-of-braids-and-flat-braids","",{"@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-discovers-invariants-of-braids-and-flat-braids/119742/",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},"How does the paper use machine learning for braids and flat braids?","Question",{"text":75,"@type":76},"It trains supervised neural networks to classify braid (or flat braid) examples as trivial or non-trivial.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What turns the neural-network results into mathematical results?",{"text":80,"@type":76},"When classification performs well, the learned structure is interpreted as mathematical conjectures, which are then proven as theorems.",{"name":82,"@type":73,"acceptedAnswer":83},"What mathematical outcomes does the study report?",{"text":84,"@type":76},"It derives new convenient invariants of braids and obtains a complete invariant for flat braids.","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},[93,97,101,105,109,114,119,122,127,130,134],{"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":29,"slug":108},5,"Comic","comic",{"id":110,"doc_module":4,"doc_module_name":46,"category_name":111,"show_sort_weight":112,"slug":113},6,"Technology",50,"technology",{"id":115,"doc_module":4,"doc_module_name":46,"category_name":116,"show_sort_weight":117,"slug":118},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":120,"slug":121},30,"research-report",{"id":123,"doc_module":4,"doc_module_name":46,"category_name":124,"show_sort_weight":125,"slug":126},9,"Religion & Spirituality",20,"religion-spirituality",{"id":125,"doc_module":4,"doc_module_name":46,"category_name":128,"show_sort_weight":125,"slug":129},"World Cup","world-cup",{"id":131,"doc_module":4,"doc_module_name":46,"category_name":132,"show_sort_weight":131,"slug":133},10,"Lifestyle","lifestyle",{"id":135,"doc_module":4,"doc_module_name":46,"category_name":136,"show_sort_weight":106,"slug":137},19,"General","general"]