[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84696-en":3,"doc-seo-84696-105":29,"detail-sidebar-cat-0-en-105":90},{"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":13,"seo_description":14,"update_tm":27,"read_time":28},84696,4398048949847,"Eliana","https://ap-avatar.wpscdn.com/avatar/400002536579ef2da7f?_k=1778318612642679267",8,"Research & Report","Level-Set Physics-Informed Neural Networks for Domain Inverse Problems of Gravimetry","Proposes level-set physics-informed neural networks (PINNs) for domain inverse problems in gravimetry, targeting the ill-posedness and non-uniqueness of inverse density recovery from surface gravity measurements. The method encodes the unknown domain through a continuous level-set function whose zero level set represents sharp interfaces. To address exploding/vanishing gradients during training, an interface-aware backpropagation strategy redefines interface-evolution derivatives and enlarges the effective support region. Includes adaptive refinement of collocation points near interfaces, investigates architecture choices, and validates results with extensive 2D and 3D experiments.","arXiv :2607 .03772v1 [math .NA] 4 Jul 2026  \nLEVEL-SET PHYSICS-INFORMED NEURAL NETWORKS FOR DOMAIN INVERSE PROBLEMS OF GRAVIMETRY  \nJINGNAN YAO∗ , WENBIN LI∗†, AND JIANLIANG QIAN‡  \nAbstract. We propose level-set physics-informed neural networks (PINNs) for domain inverse problems of gravimetry. The domain inverse problem establishes a correctness class for ill-posed inverse gravimetry, which we solve within the PINNs framework. Directly representing the domain inverse problem via neural networks is problematic due to the discontinuous nature of interfaces. We consider a level-set formulation where the neural network represents a continuous level-set function, and its zero level-set depicts sharp interfaces. To overcome the challenges of exploding and vanishing gradients caused by sharp interfaces during training, we propose an interface-aware backpropagation strategy. By redefining the derivative associated with interface evolution, this strategy enables abroader support region to drive the evolution process. Detailed analysis is provided to justify the efficacy of this strategy. Additionally, we introduce a simple procedure for adaptive refinement of collocation points near interfaces. The selection of network architecture is investigated by studying the solution spaces and the approximation properties of neural networks. Finally, extensive 2D and 3D numerical examples demonstrate the effectiveness of the proposed method.  \nKey words. Level-set; Physics-informed neural networks; Domain inverse problems; Inverse gravimetry  \nMSC codes. 65N21; 49Q10; 68T07; 86A22  \n1. Introduction. Gravimetric surveys are valuable in the exploration of largescale and deep-seated structures. The primary objective of gravity inversion is toreconstruct the internal density distribution of the Earth using surface measurements of the gravity potential or related anomaly data [32] . It is also useful in deep-space exploration and planetary science, allowing scientists to probe the deep internal structures of celestial bodies [27, 34] . However, the inverse problem of gravimetry is notoriously ill-posed. As dictated by the equivalent source principle, an infinite number of subsurface mass configurations can produce identical gravity responses, leading to severe non-uniqueness in the Hadamard sense [13] .  \nTo tackle the ill-posedness of inverse gravimetry, classical methodologies have relied on constructing regularization terms based on prior geological or mathematical assumptions [8] . Prominent strategies include depth-weighting regularization, which is designed to counteract the natural decay of potential fields with depth [20], and sparsity-promoting regularization, which favors the recovery of compact anomalous bodies with sharp boundaries [6] . Driven by advancements in well-posedness theory, an alternative and highly effective paradigm has emerged: mitigating ill-posedness by explicitly restricting the mathematical form of the solution. A particularly successful framework parameterizes the anomalous mass as a volume distribution fχ D , where f denotes the density contrast and χD is the characteristic function of the domain D. Given a priori knowledge of f and certain geometric constraints on D, the domain inverse problem of gravimetry admits a unique solution [13] . This uniqueness result has lead to the use of level-set methods in inverse gravimetry [14, 15, 18, 19, 17], where the unknown domain D is recovered implicitly by tracking its boundary ∂D asthe zero-level-set of a level-set function.  \n∗ School of Science, Harbin Institute of Technology, Shenzhen, Shenzhen, 518055, China ([24B958002@stu.hit.edu.cn](24B958002@stu.hit.edu.cn), [liwenbin@hit.edu.cn](liwenbin@hit.edu.cn)).  \n†Corresponding author.  \n‡Department of Mathematics and Department of CMSE, Michigan State University, East Lansing, MI 48824, USA ([jqian@msu.edu](jqian@msu.edu)).  \n2 YAO, LI, AND QIAN  \nRecently, the application of neural networks and deep learning strategies has","cbCaibXeTF2oPJh4","https://ap.wps.com/l/cbCaibXeTF2oPJh4","pdf",2693602,1,27,"English","en",105,"# Introduction\n## Ill-posedness in inverse gravimetry and classical regularization\n## Domain parameterization and level-set methods\n# Physics-Informed Neural Networks in inverse gravimetry\n## Limitations of end-to-end learning\n## PINNs paradigm and loss construction","[{\"question\":\"What is the key challenge the proposed method addresses in inverse gravimetry?\",\"answer\":\"Inverse gravimetry is ill-posed and suffers from severe non-uniqueness, meaning many subsurface mass configurations can produce identical gravity responses. The method targets reliable recovery within a constrained correctness class while handling interface discontinuities.\"},{\"question\":\"How does the level-set formulation represent the unknown domain?\",\"answer\":\"A continuous level-set function is learned by the neural network, and its zero level set depicts the sharp interfaces that define the recovered domain boundary.\"},{\"question\":\"What is the purpose of interface-aware backpropagation?\",\"answer\":\"Sharp interfaces can cause exploding or vanishing gradients during training. The strategy redefines the derivative related to interface evolution, enabling a broader support region to drive the evolution and improving trainability.\"}]",1784197705,68,{"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":85,"head_meta":87,"extra_data":89,"updated_unix":27},"level-set-physics-informed-neural-networks-for-domain-inverse-problems-of-gravimetry","",{"@graph":35,"@context":84},[36,53,67],{"@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/level-set-physics-informed-neural-networks-for-domain-inverse-problems-of-gravimetry/84696/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":61,"encodingFormat":60,"isAccessibleForFree":62,"interactionStatistic":63},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-16",true,{"@type":64,"interactionType":65,"userInteractionCount":4},"InteractionCounter",{"@type":66},"ViewAction",{"@type":68,"mainEntity":69},"FAQPage",[70,76,80],{"name":71,"@type":72,"acceptedAnswer":73},"What is the key challenge the proposed method addresses in inverse gravimetry?","Question",{"text":74,"@type":75},"Inverse gravimetry is ill-posed and suffers from severe non-uniqueness, meaning many subsurface mass configurations can produce identical gravity responses. The method targets reliable recovery within a constrained correctness class while handling interface discontinuities.","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"How does the level-set formulation represent the unknown domain?",{"text":79,"@type":75},"A continuous level-set function is learned by the neural network, and its zero level set depicts the sharp interfaces that define the recovered domain boundary.",{"name":81,"@type":72,"acceptedAnswer":82},"What is the purpose of interface-aware backpropagation?",{"text":83,"@type":75},"Sharp interfaces can cause exploding or vanishing gradients during training. The strategy redefines the derivative related to interface evolution, enabling a broader support region to drive the evolution and improving trainability.","https://schema.org",{"og:url":51,"og:type":86,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":88,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":91},[92,96,100,104,109,114,119,122,127,130,134],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":93,"show_sort_weight":94,"slug":95},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":97,"show_sort_weight":98,"slug":99},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":101,"show_sort_weight":102,"slug":103},"Exam",70,"exam",{"id":105,"doc_module":4,"doc_module_name":45,"category_name":106,"show_sort_weight":107,"slug":108},5,"Comic",60,"comic",{"id":110,"doc_module":4,"doc_module_name":45,"category_name":111,"show_sort_weight":112,"slug":113},6,"Technology",50,"technology",{"id":115,"doc_module":4,"doc_module_name":45,"category_name":116,"show_sort_weight":117,"slug":118},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":120,"slug":121},30,"research-report",{"id":123,"doc_module":4,"doc_module_name":45,"category_name":124,"show_sort_weight":125,"slug":126},9,"Religion & Spirituality",20,"religion-spirituality",{"id":125,"doc_module":4,"doc_module_name":45,"category_name":128,"show_sort_weight":125,"slug":129},"World Cup","world-cup",{"id":131,"doc_module":4,"doc_module_name":45,"category_name":132,"show_sort_weight":131,"slug":133},10,"Lifestyle","lifestyle",{"id":135,"doc_module":4,"doc_module_name":45,"category_name":136,"show_sort_weight":105,"slug":137},19,"General","general"]