[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83064-en":3,"doc-seo-83064-105":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},83064,1374391975076,"Riley","https://ap-avatar.wpscdn.com/avatar/14000253ca4ec9f6853?x-image-process=image/resize,m_fixed,w_180,h_180&k=1783305029341752051",8,"Research & Report","PhyRes MDNF Physics Coupled Residual GNN Correction for Multilevel Discrete Neural Field Inversion","Coefficient inversion under PDE constraints is ill conditioned because sparse observations weakly constrain fine-scale parameters, while single-resolution optimization must recover state and coefficient fields across all scales at once. This yields slow, initialization-sensitive convergence and risks approximation errors from learned transfer models that rely on offline datasets. PhyRes-MDNF introduces a fixed-physics multilevel discrete neural field framework with joint optimization across levels. Under the same fine-grid update budget, it cuts coefficient and state errors by about 85% and 90%, and improves KTC2023 EIT mean Otsu mIoU by ~3.4% over linearized CEM and 16.9% over direct single-level DNF.","Highlights  \nPhyRes-MDNF: Physics-Coupled Residual GNN Correction for Multilevel Discrete Neural Field Inversion  \nZheng Lu, Jiwei Jia, Young Ju Lee  \n• Multilevel DNF improves Darcy inversion under matched fine-grid updates.  \n• PhyRes-GNN directly prolongs and corrects across levels without replacing physics.  \n• LUNDIsim and KTC2023 demonstrate generality across inverse problems.  \narXiv :2607 .06237v1 [math .NA] 7 Jul 2026  \nPhyRes-MDNF: Physics-Coupled Residual GNN Correction for Multilevel Discrete Neural Field Inversion  \nZheng Lua , Jiwei Jiaa,c,∗ and Young Ju Leeb,∗∗  \na School of Mathematics, Jilin University, Changchun, Jilin, 130012, China  \nb Department of Mathematics, Texas State University, San Marcos, TX, 78666, USA  \ncAI for Science and Engineering Center, Shenzhen Loop Area Institute, Shenzhen, 518048, China  \nARTICLE INFO  \nKeywords:  \nPDE-constrained inverse problems discrete neural fields  \nmultilevel optimization  \ngraph neural networks  \nDarcy flow  \nelectrical impedance tomography  \nAB STRACT  \nCoefficient inversion on fine grids under PDE constraints is ill conditioned: sparse observations weakly constrain fine-scale parameters, and direct single-resolution optimization must recover state and coefficient fields across all scales simultaneously. This causes slow, initializationsensitive convergence; learned transfer models require offline data and can introduce approximation error into the numerical physics. We propose PhyRes-MDNF, a fixed-physics multilevel discrete neural field framework. On each level, a single-level DNF represents the inverse unknowns directly as trainable fields and optimizes the discrete objective. In the full-space Darcy realization, state fields 􀁕 and their shared coefficient field 􀁋 are optimized jointly in one fixed-physics inverse process. Between levels, one zero-initialized PhyRes-GNN jointly performs fixed-stencil prolongation and bounded residual correction to construct an incoming target representation, which a fixed initialization map converts to the next DNF variables. It is fitted anew from the observations and unchanged numerical model, without offline pretraining or fine-grid truth. Coarse levels therefore resolve large-scale structure before refined degrees of freedom are introduced, shortening the fine-grid optimization path while retaining the original discrete operator. Under the same final-grid update budget, the multilevel Darcy realization reduces coefficient and state errors by approximately 85% and 90%, respectively, demonstrating improved accuracy and final-grid iteration efficiency. On measured KTC2023 EIT data, the full-􀁗 pipeline improves mean Otsu mIoU by approximately 3.4% over the official linearized CEM reconstruction and 16.9% over direct single-level DNF.  \n1. Introduction  \nCoefficient inversion for elliptic PDEs seeks a spatially varying material field from indirect measurements of states, voltages, or fluxes. The problem is intrinsically ill conditioned: observations are sparse relative to the number of coefficient degrees of freedom, fine-scale modes are only weakly visible, and errors in the reconstructed state can be absorbed by the coefficient. On a fine grid, direct single-resolution optimization must recover large-and smallscale structures simultaneously. This produces a long, initialization-sensitive optimization path and can cause the coefficient and state variables to settle into an inaccurate local solution before the large-scale structure has been identified [13, 30, 31] .  \nClassical finite-difference and finite-element inverse methods retain a trusted discrete forward model, but optimizing all fine-grid inverse variables from the start remains expensive and poorly conditioned. Physics-informed neural networks (PINNs) instead represent states and coefficients with coordinate networks and fit their weights from observation and residual losses [16, 25] . Although flexible, this representation can introduce spectral bias, loss imbalanc","cbCaikiLZIFoX7JP","https://ap.wps.com/l/cbCaikiLZIFoX7JP","pdf",1342349,1,27,"English","en",105,"# Introduction\n## Challenges in coefficient inversion under PDE constraints\n## Existing approaches: finite methods, PINNs, and learned surrogates\n# PhyRes-MDNF for inverse problems\n## Discrete neural fields and fixed-physics optimization\n## Multilevel DNF backbone and level-to-level correction","[{\"question\":\"Why is coefficient inversion for elliptic PDEs ill conditioned on fine grids?\",\"answer\":\"Sparse observations provide weak constraints on fine-scale coefficient modes, and state reconstruction errors can be absorbed into the coefficient. As a result, direct single-resolution optimization must recover large- and small-scale structures simultaneously, making convergence long and sensitive to initialization.\"},{\"question\":\"What is PhyRes-MDNF and how does it differ from transfer or offline pretrained models?\",\"answer\":\"PhyRes-MDNF is a fixed-physics multilevel discrete neural field framework that keeps the prescribed numerical operator unchanged. It is fitted anew from observations without offline pretraining or fine-grid truth, avoiding surrogate approximation errors being misinterpreted as physically plausible coefficient changes.\"},{\"question\":\"How does the method perform multilevel correction across resolutions?\",\"answer\":\"Within the multilevel DNF backbone, one zero-initialized PhyRes-GNN performs fixed-stencil prolongation and bounded residual correction to build an incoming target representation for the next level. A fixed initialization map converts it to the next DNF variables, allowing coarse levels to resolve large-scale structure before introducing refined degrees of freedom.\"}]",1784184964,68,{"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":86,"head_meta":88,"extra_data":90,"updated_unix":27},"phyres-mdnf-physics-coupled-residual-gnn-correction-for-multilevel-discrete-neural-field-inversion","",{"@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/phyres-mdnf-physics-coupled-residual-gnn-correction-for-multilevel-discrete-neural-field-inversion/83064/",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},"Why is coefficient inversion for elliptic PDEs ill conditioned on fine grids?","Question",{"text":75,"@type":76},"Sparse observations provide weak constraints on fine-scale coefficient modes, and state reconstruction errors can be absorbed into the coefficient. As a result, direct single-resolution optimization must recover large- and small-scale structures simultaneously, making convergence long and sensitive to initialization.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What is PhyRes-MDNF and how does it differ from transfer or offline pretrained models?",{"text":80,"@type":76},"PhyRes-MDNF is a fixed-physics multilevel discrete neural field framework that keeps the prescribed numerical operator unchanged. It is fitted anew from observations without offline pretraining or fine-grid truth, avoiding surrogate approximation errors being misinterpreted as physically plausible coefficient changes.",{"name":82,"@type":73,"acceptedAnswer":83},"How does the method perform multilevel correction across resolutions?",{"text":84,"@type":76},"Within the multilevel DNF backbone, one zero-initialized PhyRes-GNN performs fixed-stencil prolongation and bounded residual correction to build an incoming target representation for the next level. A fixed initialization map converts it to the next DNF variables, allowing coarse levels to resolve large-scale structure before introducing refined degrees of freedom.","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"]