[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84052-en":3,"doc-seo-84052-105":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":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},84052,1374391975076,"Riley","https://ap-avatar.wpscdn.com/avatar/14000253ca4ec9f6853?x-image-process=image/resize,m_fixed,w_180,h_180&k=1783305029341752051",8,"Research & Report","SplineNet An Isogeometric Deep Learning Method for Complex Shells","SplineNet introduces a seamless isogeometric deep learning workflow for shell structures with complex geometries, unifying CAD-accurate spline representations with neural network computation. The method uses watertight spline models such as analysis-suitable unstructured T-splines and exact CAD geometry descriptions inside the network. Bézier extraction structures the architecture and Bernstein polynomials act as nonlinear activations. It supports data-free formulations with energy-based losses using the Kirchhoff-Love model, and data-driven operation as a DeepONet trunk for rapid inference and interpretability on unseen inputs.","arXiv:2607.06026v1][cs.LG]7 Jul 2026  \n# SplineNet:An Isogeometric Deep Learning Method forComplex Shells\n\nShizhou Luo¹and Xiaodong Wei¹*  \n¹Global College,Shanghai Jiao Tong University,Shanghai,200240,China.  \n*Corresponding author(s).E-mail(s):xiaodong.wei@sjtu.edu.cn;  \nContributing authors:LuoShizhou@sjtu.edu.cn;  \n## Abstract\n\nWe present a novel isogeometric deep learning method,termed Spline Net,for the seamless design andanalysis of shell structures with complex geometries.The proposed approach is built upon watertightspline representations,e.g.,analysis-suitable unstructured T-splines,and features exact geometricdescriptions of Computer-Aided Design(CAD)models in neural networks.Bézier extraction is usedto build the network architecture,where Bernstein polynomials serve as the nonlinear activationfunctions.Spline Net can be applied in a data-free or data-driven way.In the data-free case,energy-based formulations can be naturally incorporated as loss terms,which fulfill the need of Computer-Aided Engineering(CAE)and can be accurately calculated.In particular,the Kirchhoff-Love(KL)model is adopted to solve for the mechanical behaviors of shell structures.This way,CAD and CAEcan be tightly integrated in a deep neural network without the time-consuming model/data exchangeprocess.In the data-driven case,SplineNet can be used as the trunk net of Deep Operator Networks(DeepONet)to provide interpretability.Given such a trained network and unseen input data,resultscan be immediately obtained without retraining the network or repeatedly performing the traditionalworkflow for analysis.In the end,a variety of numerical examples are studied to demonstrate theeffectiveness of the proposed method,especially when real-world complex geometries are involved.  \nKeywords:Deep Operator Networks,Interpretability,Isogeometric Analysis,Analysis-suitableUnstructured Splines,Bézier Extraction  \n## 1 Introduction\n\nIsogeometric Analysis(IGA)was proposed tobridge the gap between Computer-Aided Design(CAD)and Computer-Aided Engineering(CAE)by employing spline-based representations used inCAD as basis functions for numerical analysis [1]By directly utilizing exact geometric descrip-tions,IGA enables higher geometric fidelity andimproved numerical performance compared with  \nthe finite element method(FEM)[2].In partic-ular,spline-based discretizations possess higher-order continuity,making them especially suit-able for high-order partial differential equations(PDEs),such as the Kirchhoff-Love(KL)shell for-mulation [3].Despite these advantages,traditionalIGA methods become computationally expensivefor large-scale problems and require repeated sim-ulations in design optimization [4]and uncertaintyquantification [5].  \nTo overcome these drawbacks,leveraging deeplearning to enhance the performance of traditional  \nPDE solvers has attracted significant attention [6].Based on learning objectives,these approachescan be broadly classified into two categories:solu-tion learning and operator learning.  \nIn the solution learning paradigm,the neu-ral network acts as a function approximator tosolve a specific problem.Many methods havebeen studied accordingly based on both strongand energy/weak forms of PDEs,such as physics-informed neural networks(PINNs)[7],the deepRitz method (DRM)[8],the deep energy method(DEM)[9],and variational physics-informed neu-ral networks(VPINNs)[10].PINNs utilize strongform residuals to define loss functions.They pro-vide a flexible,mesh-free approach for solving for-ward and inverse problems.However,they requirehigh-order derivatives and lead to training difi-culties [11,12].DRM,DEM,and VPINNs employthe energy/weak form of PDEs as loss functions,which can help reduce the order of derivatives [13and improve computational efficiency [14].Despitethese improvements,in many implementationsof these methods,essential boundary conditionsare imposed through penalty terms that causeconflicting gradients during training [15,16].  \nAnother line o","cbCaifvQZpAn6rMo","https://ap.wps.com/l/cbCaifvQZpAn6rMo","pdf",8750042,4,1,25,"English","en",105,"# Abstract\n# 1 Introduction","[{\"question\":\"What problem does SplineNet target in shell analysis?\",\"answer\":\"SplineNet targets seamless design and analysis of shell structures with complex geometries by integrating exact CAD geometry representations into deep learning for efficient mechanical behavior prediction.\"},{\"question\":\"How does SplineNet connect CAD and CAE inside a neural network?\",\"answer\":\"It builds watertight spline representations (e.g., analysis-suitable unstructured T-splines) and embeds exact CAD geometric descriptions directly in the neural network, avoiding time-consuming model/data exchange.\"},{\"question\":\"What are the two usage modes of SplineNet?\",\"answer\":\"SplineNet can work in a data-free mode by incorporating energy-based loss terms (using the Kirchhoff-Love model), or in a data-driven mode as the trunk net of Deep Operator Networks for instant inference on unseen inputs without retraining.\"}]",1784192254,63,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"splinenet-an-isogeometric-deep-learning-method-for-complex-shells","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":21},"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":20},"https://docshare.wps.com/document/splinenet-an-isogeometric-deep-learning-method-for-complex-shells/84052/",{"url":52,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-27","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What problem does SplineNet target in shell analysis?","Question",{"text":75,"@type":76},"SplineNet targets seamless design and analysis of shell structures with complex geometries by integrating exact CAD geometry representations into deep learning for efficient mechanical behavior prediction.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does SplineNet connect CAD and CAE inside a neural network?",{"text":80,"@type":76},"It builds watertight spline representations (e.g., analysis-suitable unstructured T-splines) and embeds exact CAD geometric descriptions directly in the neural network, avoiding time-consuming model/data exchange.",{"name":82,"@type":73,"acceptedAnswer":83},"What are the two usage modes of SplineNet?",{"text":84,"@type":76},"SplineNet can work in a data-free mode by incorporating energy-based loss terms (using the Kirchhoff-Love model), or in a data-driven mode as the trunk net of Deep Operator Networks for instant inference on unseen inputs without 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