[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81535-en":3,"doc-seo-81535-105":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},81535,687197100911,"Himbo","https://ap-avatar.wpscdn.com/avatar/a000239b6f1da00475?x-image-process=image/resize,m_fixed,w_180,h_180&k=1782698725881665579",8,"Research & Report","Adaptive Feature Capture Method for Solving Partial Differential Equations with Near Singular Solutions","Partial differential equations (PDEs) with near singular solutions are difficult for traditional numerical schemes, especially in complex geometries where adaptive refinement and mesh generation are costly. Deep-learning alternatives like Physics-Informed Neural Networks (PINNs) and the Random Feature Method (RFM) avoid meshing but often lack adaptive resolution in regions with steep gradients. This work introduces the Adaptive Feature Capture Method (AFCM), a machine-learning framework that redistributes neurons and collocation points in high-gradient areas using a gradient-norm monitor, extending RFM’s mesh-free efficiency while improving accuracy and efficiency.","arXiv :2507 . 12941v4 [math .NA] 10 Jul 2026  \nAdaptive feature capture method for solving partial differential equations with near singular solutions  \nYangtao Denga , Qiaolin Hea,∗, Xiaoping Wangb,c,∗  \na School of Mathematics, Sichuan University, Chengdu, 610065, China b School of Science and Engineering, The Chinese University of Hong Kong, Shenzhen,  \nGuangdong, 518172, China  \nc Shenzhen International Center for Industrial and Applied Mathematics, Shenzhen  \nResearch Institute of Big Data, Guangdong, 518172, China  \nAbstract  \nPartial differential equations (PDEs) with near singular solutions pose significant challenges for traditional numerical methods, particularly in complex geometries where mesh generation and adaptive refinement become computationally expensive. Although deep-learning-based approaches, such as Physics-Informed Neural Networks (PINNs) and the Random Feature Method (RFM), offer mesh-free alternatives, they often lack adaptive resolution in critical regions, limiting their accuracy for solutions with steep gradients or singularities. In this work, we propose the Adaptive Feature Capture Method (AFCM), a novel machine learning framework that adaptively redistributes neurons and collocation points in high-gradient regions to enhance local expressive power. Inspired by adaptive moving mesh techniques, AFCM uses the gradient norm of an approximate solution as a monitor function to guide the reinitialization of feature function parameters. This ensures that partition hyperplanes and collocation points cluster where they are most needed, achieving higher resolution without increasing computational overhead. The AFCM extends the capabilities of RFM to handle PDEs with near-singular solutions while preserving its mesh-free efficiency. Numerical experiments demonstrate the method’s effectiveness in accurately resolving near-singular problems with a performance that is better than that  \n∗ Corresponding author  \nEmail addresses: [ytdeng1998@foxmail.com](ytdeng1998@foxmail.com) (Yangtao Deng), [qlhejenny@scu.edu.cn](qlhejenny@scu.edu.cn) (Qiaolin He ), [wangxiaoping@cuhk.edu.cn](wangxiaoping@cuhk.edu.cn) (Xiaoping Wang )  \nof the traditional finite element method in terms of accuracy and efficiency. AFCM offers a robust and scalable approach to solving challenging PDEs in scientific and engineering applications.  \nKeywords:  \npartial differential equations, near singular, adaptive feature capture method, random feature method  \n1. Introduction  \nPartial differential equations (PDEs) are widely applied in diverse fields such as physics, engineering, economics, and biology [1, 2, 3] . Traditional numerical methods, including finite difference [4], finite volume[5], and finite element methods [6], have made significant theoretical and practical contributions to solving PDEs. However, these methods face notable challenges. For instance, complex geometries often lead to distorted mesh elements, which degrade computational accuracy and efficiency [4, 7, 8, 9] .  \nIn contrast, the success of deep learning in computer vision and natural language processing [10] has spurred interest in its application to scientific computing. Neural networks, with their universal approximation capabilities [11], have been explored for solving ordinary and partial differential equations (ODEs and PDEs) [12, 13, 14, 15, 16, 17, 18, 19] . Various deep-learning-based approaches have emerged, such as the Deep Ritz Method (DRM) [14], Deep Galerkin Method (DGM) [15], Physics-Informed Neural Networks (PINNs)  \n[17], and Weak Adversarial Networks (WAN) [16] . These methods offer meshfree alternatives, circumventing the need for computationally intensive mesh generation. However, a critical limitation of these approaches is the lack of reliable error estimation. Without knowledge of the exact solution, numerical approximations often fail to exhibit clear convergence trends, even as network parameters increase [20], raising concerns about their reli","cbCaic90hofOJ1RG","https://ap.wps.com/l/cbCaic90hofOJ1RG","pdf",6436212,4,1,38,"English","en",105,"# Introduction\n## Challenges of traditional numerical methods\n## Deep-learning and mesh-free PDE solvers\n## Random Feature Method (RFM) and its limitation\n## Adaptive moving-mesh ideas\n# Proposed method (AFCM)\n## Gradient-norm monitor and feature redistribution\n## Extension of RFM and expected benefits","[{\"question\":\"What problem does the Adaptive Feature Capture Method (AFCM) target?\",\"answer\":\"AFCM targets PDEs whose solutions are near singular, where traditional and non-adaptive mesh-free methods struggle to capture steep gradients accurately.\"},{\"question\":\"How does AFCM decide where to increase resolution?\",\"answer\":\"AFCM uses the gradient norm of an approximate solution as a monitor function to guide the reinitialization of feature parameters, clustering partition hyperplanes and collocation points in critical regions.\"},{\"question\":\"In what way does AFCM extend the Random Feature Method (RFM)?\",\"answer\":\"AFCM extends RFM by adding an adaptive feature/point redistribution mechanism that improves local expressive power for near-singular behavior while preserving mesh-free efficiency and avoiding additional computational overhead.\"}]",1784174095,96,{"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},"adaptive-feature-capture-method-for-solving-partial-differential-equations-with-near-singular-solutions","",{"@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/adaptive-feature-capture-method-for-solving-partial-differential-equations-with-near-singular-solutions/81535/",{"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","2026-07-24","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 the Adaptive Feature Capture Method (AFCM) target?","Question",{"text":75,"@type":76},"AFCM targets PDEs whose solutions are near singular, where traditional and non-adaptive mesh-free methods struggle to capture steep gradients accurately.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does AFCM decide where to increase resolution?",{"text":80,"@type":76},"AFCM uses the gradient norm of an approximate solution as a monitor function to guide the reinitialization of feature parameters, clustering partition hyperplanes and collocation points in critical regions.",{"name":82,"@type":73,"acceptedAnswer":83},"In what way does AFCM extend the Random Feature Method (RFM)?",{"text":84,"@type":76},"AFCM extends RFM by adding an adaptive feature/point redistribution mechanism that improves local expressive power for near-singular behavior while preserving mesh-free efficiency and avoiding additional computational overhead.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":21,"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":20,"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":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]