[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-121782-en":3,"doc-seo-121782-105":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},121782,962075006959,"Anda","https://ap-avatar.wpscdn.com/avatar/e0002397efbe92a78e?_k=1776741047341049297",8,"Research & Report","Machine Learning Discovery of Optimal Quadrature Rules for Isogeometric Analysis - Preprint","We propose machine learning to discover optimal quadrature rules for constructing stiffness and mass matrices in isogeometric analysis (IGA). The method starts from 1D spline spaces over uniform and non-uniform knot sequences and extends optimal rules to higher dimensions via tensor products. Quadrature search is formulated as a non-convex optimization problem solved with gradient descent, improved by dynamic programming using parameter initialization from lower-knot optimal solutions. Results cover up to 50 uniform elements and degree up to 8, including practical savings compared with Gaussian integration.","Machine Learning Discovery of Optimal Quadrature Rules for  \nIsogeometric Analysis  \nTomas Teijeiro  a,∗, Jamie M. Taylor  b , Ali Hashemian  a , David Pardo  c,a,d  \naBCAM – Basque Center for Applied Mathematics, Bilbao, Basque Country, Spain b Department of Quantitative Methods, CUNEF University, Madrid, Spain c University of the Basque Country (UPV/EHU), Leioa, Basque Country, Spain dIkerbasque – Basque Foundation for Sciences, Bilbao, Basque Country, Spain  \nAbstract  \nWe propose the use of machine learning techniques to find optimal quadrature rules for the construction of stiffness and mass matrices in isogeometric analysis (IGA) . We initially consider 1D spline spaces of arbitrary degree spanned over uniform and non-uniform knot sequences, and then the generated optimal rules are used for integration over higher-dimensional spaces using tensor product sense. The quadrature rule search is posed as an optimization problem and solved by a machine learning strategy based on gradientdescent. However, since the optimization space is highly non-convex, the success of the search strongly depends on the number of quadrature points and the parameter initialization. Thus, we use a dynamic programming strategy that initializes the parameters from the optimal solution over the spline space with a lower number of knots. With this method, we found optimal quadrature rules for spline spaces when using IGA discretizations with up to 50 uniform elements and polynomial degrees up to 8, showing the generality of the approach in this scenario. For non-uniform partitions, the method also finds an optimal rule in a reasonable number of test cases. We also assess the generated optimal rules in two practical case studies, namely, the eigenvalue problem of the Laplace operator and the eigenfrequency analysis of freeform curved beams, where the latter problem shows the applicability of the method to curved geometries. In particular, the proposed method results in savings with respect to traditional Gaussian integration of up to 44% in 1D, 68% in 2D, and 82% in 3D spaces.  \nKeywords: Numerical integration, optimal quadrature rules, machine learning, dynamic programming, isogeometric analysis  \n1. Introduction  \nDeveloping efficient and accurate integration methods plays a crucial role in many numerical analysis techniques. In the context of the isogeometric analysis (IGA) [1], a common practical approach for numerical integration is to use an element-wise Gaussian (EWG) quadrature rule when constructing system matrices in the sense of Galerkin discretizations. This follows the classical system construction technique of the finite element analysis (FEA) . However, it is known that there exist optimal quadrature rules for spline spaces of higher continuities, thus requiring a significantly fewer number of quadrature points than the classical EWG. For instance, one may consider the fast matrix formation technique by Calabr`o et al. [2], where each row of the system matrices is integrated by its own quadrature rule obtained by a linear system of equations. Bartoˇn et al. [3] propose weighted Gaussian quadrature rules for B-splines that require the minimum number of quadrature points while guaranteeing the exactness of integration with respect to the weight function. Other techniques by Bartoˇn and Calo [4, 5 , 6] use a polynomial homotopy continuation (PHC), a numerical scheme for solving polynomial systems of equations [7], to generate Gaussian quadrature rules for spline  \n∗ Corresponding author  \nEmail address: [tteijeiro@bcamath.org](tteijeiro@bcamath.org) (Tomas Teijeiro  )  \nPreprint submitted to Computer Methods in Applied Mechanics and Engineering April 3, 2023  \nspaces of higher continuities. To generate a Gaussian rule in a target spline space, they built an associated source space with known quadratures (e.g., a union of polynomial Gaussian rules) and transform the rule from the source space to the target space, while preserving the optima","cbCailpmmFK16KiX","https://ap.wps.com/l/cbCailpmmFK16KiX","pdf",3030135,1,18,"English","en",105,"# Introduction\n## Background in isogeometric analysis integration\n## Prior work on optimal quadrature rules\n## Machine learning for numerical integration and quadrature\n## Optimization formulation for discovering quadrature rules","[{\"question\":\"What problem does the document address in isogeometric analysis?\",\"answer\":\"It addresses how to find optimal quadrature rules that construct stiffness and mass matrices efficiently in IGA by minimizing the number of quadrature points while preserving exactness.\"},{\"question\":\"How are the optimal 1D quadrature rules extended to higher dimensions?\",\"answer\":\"The approach applies the generated 1D optimal rules for integration over higher-dimensional spaces using tensor product constructions.\"},{\"question\":\"Why is dynamic programming used in addition to gradient descent?\",\"answer\":\"Because the quadrature-rule search is highly non-convex, success depends strongly on initialization and the number of quadrature points; dynamic programming initializes parameters from optimal solutions in spline spaces with fewer knots.\"}]","Machine Learning Discovery of Optimal Quadrature Rules for Isogeometric Analysis - Preprint | PDF",1785806803,45,{"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-discovery-of-optimal-quadrature-rules-for-isogeometric-analysis-preprint","",{"@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-discovery-of-optimal-quadrature-rules-for-isogeometric-analysis-preprint/121782/",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 problem does the document address in isogeometric analysis?","Question",{"text":75,"@type":76},"It addresses how to find optimal quadrature rules that construct stiffness and mass matrices efficiently in IGA by minimizing the number of quadrature points while preserving exactness.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How are the optimal 1D quadrature rules extended to higher dimensions?",{"text":80,"@type":76},"The approach applies the generated 1D optimal rules for integration over higher-dimensional spaces using tensor product constructions.",{"name":82,"@type":73,"acceptedAnswer":83},"Why is dynamic programming used in addition to gradient descent?",{"text":84,"@type":76},"Because the quadrature-rule search is highly non-convex, success depends strongly on initialization and the number of quadrature points; 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