[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83791-en":3,"doc-seo-83791-105":29,"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":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},83791,4398048950312,"Violet","https://ap-avatar.wpscdn.com/avatar/400002538284de19e3c?_k=1778320343897328908",8,"Research & Report","Domain Decomposition Methods with Physics-informed Neural Networks for Elliptic Equations on Manifolds","提出两种用于紧致黎曼流形上椭圆方程的数值域分解方法（DDMs），其核心思想是将物理信息神经网络（PINNs）与流形上的域分解技术结合。该方案把流形问题转化为多个欧氏域上的子问题，并利用神经网络在高维情形下的优势提升求解效率与可行性。通过对多种流形（含边界与不含边界）进行数值实验，在5到10维范围内验证了方法有效性。","arXiv :2607 .04285v1 [math .NA] 5 Jul 2026  \nDOMAIN DECOMPOSITION METHODS WITH PHYSICS-INFORMED NEURAL NETWORKS FOR ELLIPTIC EQUATIONS ON MANIFOLDS  \nYUFANG JIANG ∗ , LIZHEN QIN †, AND FENG WANG ‡  \nAbstract. We propose two numerical domain decomposition methods (DDMs) for elliptic equations on compact Riemannian manifolds, based on physics-informed neural networks (PINNs) . Our approach incorporates the DDM technique for manifolds with the advantages of neural networks in high-dimensional settings. The proposed methods are validated through numerical experiments on various manifolds, both with and without boundary, in dimensions ranging from 5 to 10 .  \nAMS subject classifications. Primary 65N55; Secondary 58J05, 68T07 .  \nKey words. Riemannian manifolds, elliptic problems, domain decomposition methods, physicsinformed neural networks  \n1. Introduction. Elliptic equations on Riemannian manifolds are important in both analysis and geometry (see e.g. [3, 41 , 67]) . These equations appear in many areas, such as image processing, multifluid dynamics, micromagnetics, and theoretical physics (see e.g. [4, 5 , 8 , 18 , 35 , 36 , 37 , 39 , 40 , 57 , 66]) . A simple and important example is  \n(1.1) − ∆u + bu = f.  \nHere ∆ is the Laplace-Beltrami operator, or Laplacian for brevity, defined on a ddimensional Riemannian manifold M. Many manifolds of interest arise naturally as submanifolds of Euclidean spaces, where the “dimension” of a submanifold is understood as the topological dimension of the manifold, not that of the ambient Euclidean space. For instance, the unit sphere  \nSn−1 = {x ∈ Rn | ∥x∥ = 1}  \nin Rn has dimension n − 1.  \nWhen the manifold M is a two-dimensional Riemannian submanifold in R3 , i.e. a surface, the numerical methods to solve differential equations, particularly (1.1), on M have been extensively studied over many years (see e.g. [7, 22 , 23 , 56 , 58]) . A conventional and popular approach is to solve the equations by finite element methods (FEMs) based on a global grid or triangulation of M. Such a grid can be obtained by polyhedron approximation in R3 . This approach has been highly developed and widely applied (see e.g. [9, 17 , 24] for surveys and bibliographies) .  \nHowever, as pointed out in [12] and [62], the above approach on surfaces faces substantial obstacles when extended to general high-dimensional manifolds, due to their great topological and geometrical complexity. Moreover, many important manifolds are even not defined as submanifolds of Euclidean spaces. A case in point is the complex projective space CPk which is foundational to algebraic geometry. Such manifolds pose formidable challenges in numerical computation, particularly in the construction of global triangulations.  \n∗ School of Mathematical Sciences, Nanjing Normal University, Nanjing Jiangsu, China ([240902008@njnu.edu.cn](240902008@njnu.edu.cn)).  \n†School of Mathematics, Nanjing University, Nanjing, Jiangsu, China ([qinlz@nju.edu.cn](qinlz@nju.edu.cn)) .  \n‡School of Mathematical Sciences, Nanjing Normal University, Nanjing Jiangsu, China ([fwang@njnu.edu.cn](fwang@njnu.edu.cn)).  \n2 Y. JIANG, L. QIN, AND F. WANG  \nTo circumvent this difficulty, Qin-Zhang-Zhang in [63] proposed an idea to solve elliptic equations on manifolds using FEMs without global triangulations. Since ad-dimensional manifold M has local coordinate charts by definition, M can be decomposed into finitely many subdomains that carry local coordinates. Consequently, an elliptic equation on each subdomain can be transformed to one on a domain in Rd. Thus an elliptic problem on M can be solved by domain decomposition methods (DDMs) with subproblems posed on Euclidean domains, where grids are much easier to generate.  \nThe idea of [63] was further developed by Cao-Qin in [12] and by Qin-WangWang in [62] . Those works merge the philosophy in [63] and the seminal works of P. L. Lions. To solve elliptic differential equations in Euclidean domains, Lions proposed a sequen","cbCaiiMY1xbyoqdF","https://ap.wps.com/l/cbCaiiMY1xbyoqdF","pdf",861181,1,26,"English","en",105,"# Introduction\n## Elliptic equations on manifolds and existing FEM approaches\n## Domain decomposition without global triangulations\n## Continuous and numerical DDMs on manifolds\n## PINNs for high-dimensional elliptic problems","[{\"question\":\"文中提出的两种域分解方法基于什么框架？\",\"answer\":\"提出两种重叠式域分解方法，将流形上的椭圆问题按子域分解，并在数值层面用物理信息神经网络（PINNs）求解各子问题。\"},{\"question\":\"为什么需要把流形问题转化为欧氏域上的子问题？\",\"answer\":\"流形（尤其高维或非欧嵌入情形）难以构造全局网格/三角剖分；转化后子问题在欧氏域上设置，网格生成更容易。\"},{\"question\":\"数值实验覆盖哪些流形与维度范围？\",\"answer\":\"实验覆盖多种流形，既包括有边界也包括无边界情形；维度范围为5到10，并通过结果表明方法表现良好。\"}]",1784190429,66,{"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":86,"head_meta":88,"extra_data":90,"updated_unix":27},"domain-decomposition-methods-with-physics-informed-neural-networks-for-elliptic-equations-on-manifolds","",{"@graph":35,"@context":85},[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/domain-decomposition-methods-with-physics-informed-neural-networks-for-elliptic-equations-on-manifolds/83791/",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",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"文中提出的两种域分解方法基于什么框架？","Question",{"text":75,"@type":76},"提出两种重叠式域分解方法，将流形上的椭圆问题按子域分解，并在数值层面用物理信息神经网络（PINNs）求解各子问题。","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"为什么需要把流形问题转化为欧氏域上的子问题？",{"text":80,"@type":76},"流形（尤其高维或非欧嵌入情形）难以构造全局网格/三角剖分；转化后子问题在欧氏域上设置，网格生成更容易。",{"name":82,"@type":73,"acceptedAnswer":83},"数值实验覆盖哪些流形与维度范围？",{"text":84,"@type":76},"实验覆盖多种流形，既包括有边界也包括无边界情形；维度范围为5到10，并通过结果表明方法表现良好。","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":45,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":45,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":45,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":45,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":45,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":45,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":45,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]