[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85701-en":3,"doc-seo-85701-105":29,"detail-sidebar-cat-0-en-105":83},{"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":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":13,"seo_description":14,"update_tm":27,"read_time":28},85701,137441390410,"Hazel","https://ap-avatar.wpscdn.com/avatar/2000252f4ab5702993?_k=1776741390130283984",8,"Research & Report","Knowledge-Constrained Shape Optimization with a Mixture-of-Experts Neural Operator for High-Confidence Design","Engineering shape optimization faces practical and technical bottlenecks. Engineers must manually encode optimization settings—editable regions, deformation limits, and design-preservation constraints—using domain expertise, while surrogate-based methods can lose reliability on heterogeneous geometry databases or when search trajectories enter out-of-distribution regions. A knowledge-constrained framework is proposed to map knowledge-based constraints and user intent into quantifiable parameters for DFFD-based deformation operators. A Mixture-of-Experts Neural Operator improves surrogate modeling on heterogeneous aerodynamic datasets, and Mahalanobis-distance uncertainty detection triggers physics-based local enrichment. Tests on in-house MPV, SUV, and Sedan datasets report 1.16% MAPE and 94.34% trend accuracy, with CFD-validated drag reductions of ~4%–10%.","arXiv :2607 .09763v 1 [ cs .CV] 7 Jul 2026  \nKnowledge-Constrained Shape Optimization with a Mixture-of-Experts Neural Operator for High-Confidence  \nDesign  \nWenhao Fan 1,2,3 , Yuanwei Bin∗2,3, Jianghan Gu3 , Wenfa Luo4 , Jiao Xiang4 , Yuntian  \nChen2 , and Shiyi Chen2  \n1 School of Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, China  \n2 Eastern Institute of Technology, Ningbo, Ningbo 315200, Zhejiang, China  \n3 TenFong Technology Co. , Ltd. , Shenzhen 518000, Guangdong, China  \n4 IM Motors Technology Co. , Ltd. , Shanghai 201210, China  \nAbstract  \nEngineering shape optimization presents challenges at both the application and technical levels. At the application level, the specification of optimization settings, including editable regions, deformation ranges, and design-preservation constraints, relies heavily on domain expertise and is typically performed manually by experienced engineers. At the technical level, surrogate-based optimization methods often exhibit limited reliability when applied to heterogeneous geometry databases or when optimization trajectories enter out-of-distribution regions. To address these challenges, we propose a knowledge-constrained shape-optimization framework that translates knowledge-based constraints and user intent into quantifiable parameters of DFFD-based deformation operators, thereby enabling engineering-aware and controllable constrained optimization. We further develop a Mixture-of-Experts Neural Operator (MoE-NO) to enhance surrogate modeling performance over heterogeneous aerodynamic datasets. Compared with baseline models, including Transolver [1] and DragSolver  \n[2], MoE-NO achieves improved drag-prediction accuracy and enhanced consistency in trend prediction. Based on the MoE-NO encoder and the Mahalanobis distance, we introduce an uncertainty-estimation approach to identify out-of-distribution geometries during the optimization process. Design candidates associated with high uncertainty are selectively evaluated using a physics-based solver for local sample enrichment, thereby improving optimization reliability without incurring the cost of evaluating all candidates. Experiments conducted on in-house MPV, SUV, and Sedan datasets demonstrate that MoE-NO achievesa test-set MAPE of 1. 16%, outperforming the best baseline result of 1.52% . In addition, MoE-NO improves trend-prediction accuracy to 94.34%, compared with the best baseline accuracy of 90.34% . Vehicle shape-optimization experiments yield CFD-validated drag coefficient reductions in the range of approximately 4% to 10% .  \nKeywords: Shape optimization; Knowledge-constrained design; Surrogate-assisted optimization; Mixture-of-Experts Neural Operator  \n1 Introduction  \nPhysics-governed shape optimization seeks geometry modifications that improve a performance objective evaluated by a numerical physics model. It appears in aerodynamic design [3–6], thermal design [7], structural design  [8], and multidisciplinary design [9 , 10], and is usually  \n∗ Corresponding author. E-mail: [ybin@eitech.edu.cn](ybin@eitech.edu.cn)  \nformulated through geometry parameterization [9 , 11], high-fidelity simulation [12 , 13], adjointor derivative-free search [3 , 14 , 15], surrogate modeling [16 , 17], and multidisciplinary design optimization [10] . In a typical workflow, a baseline geometry is mapped to a finite-dimensional design space, candidate geometries are evaluated by a physics solver, and the design variables are updated until a performance criterion is improved. This formulation has become a standard computational route for inverse design [18 , 19], but its use in industrial geometry optimization remains limited by two practical issues.  \nThe first challenge is the construction of engineering constraints. In many design problems, admissible modifications are not specified only by analytic inequalities. They are also encoded in design standards [20 , 21], engineering guidelines [22–24], benchmark s","cbCaimGmcH6ky6nr","https://ap.wps.com/l/cbCaimGmcH6ky6nr","pdf",10544596,1,34,"English","en",105,"# Abstract\n# Introduction","[{\"question\":\"How is out-of-distribution geometry detected and handled during optimization?\",\"answer\":\"The method uses Mahalanobis-distance-based uncertainty estimation on the MoE-NO encoder to identify high-uncertainty candidates, then selectively evaluates them with a physics-based solver for local sample enrichment.\"}]",1784205693,86,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":78,"head_meta":80,"extra_data":82,"updated_unix":27},"knowledge-constrained-shape-optimization-with-a-mixture-of-experts-neural-operator-for-high-confidence-design","",{"@graph":35,"@context":77},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/knowledge-constrained-shape-optimization-with-a-mixture-of-experts-neural-operator-for-high-confidence-design/85701/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71],{"name":72,"@type":73,"acceptedAnswer":74},"How is out-of-distribution geometry detected and handled during optimization?","Question",{"text":75,"@type":76},"The method uses Mahalanobis-distance-based uncertainty estimation on the MoE-NO encoder to identify high-uncertainty candidates, then selectively evaluates them with a physics-based solver for local sample enrichment.","Answer","https://schema.org",{"og:url":51,"og:type":79,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":81,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":84},[85,89,93,97,102,107,112,115,120,123,127],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":86,"show_sort_weight":87,"slug":88},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":90,"show_sort_weight":91,"slug":92},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":94,"show_sort_weight":95,"slug":96},"Exam",70,"exam",{"id":98,"doc_module":4,"doc_module_name":45,"category_name":99,"show_sort_weight":100,"slug":101},5,"Comic",60,"comic",{"id":103,"doc_module":4,"doc_module_name":45,"category_name":104,"show_sort_weight":105,"slug":106},6,"Technology",50,"technology",{"id":108,"doc_module":4,"doc_module_name":45,"category_name":109,"show_sort_weight":110,"slug":111},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":113,"slug":114},30,"research-report",{"id":116,"doc_module":4,"doc_module_name":45,"category_name":117,"show_sort_weight":118,"slug":119},9,"Religion & Spirituality",20,"religion-spirituality",{"id":118,"doc_module":4,"doc_module_name":45,"category_name":121,"show_sort_weight":118,"slug":122},"World Cup","world-cup",{"id":124,"doc_module":4,"doc_module_name":45,"category_name":125,"show_sort_weight":124,"slug":126},10,"Lifestyle","lifestyle",{"id":128,"doc_module":4,"doc_module_name":45,"category_name":129,"show_sort_weight":98,"slug":130},19,"General","general"]