[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81519-en":3,"doc-seo-81519-105":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},81519,1099513958762,"Logic","https://ap-avatar.wpscdn.com/avatar/1000023916a998db790?x-image-process=image/resize,m_fixed,w_180,h_180&k=1784791008015729253",8,"Research & Report","A Posteriori Certification of PDE Approximations with Particular Application to Neural Networks","Rigorous, efficiently computable a posteriori lower and upper error bounds are developed for given PDE approximations on potentially geometrically complex domains. The method embeds or envelopes the original domain into geometrically simpler subdomains, enabling avoidance of direct computations on the complex set. Residual functionals are extended and restricted via norm-preserving Hahn–Banach extensions, followed by Riesz representations on the simpler domains to obtain sharp bounds. The framework targets elliptic and parabolic problems, controlling error in the natural variational norm with minimal regularity assumptions and validated numerical performance. It motivates certification for physics-informed, mesh-free neural-network PDE solvers but applies to any approximation with evaluable variational residual.","arXiv :2502 .20336v5 [math .NA] 10 Jul 2026  \nA POSTERIORI CERTIFICATION OF PDE APPROXIMATIONS WITH PARTICULAR APPLICATION TO NEURAL NETWORKS  \nLEWIN ERNST, NIKOLAOS REKATSINAS, AND KARSTEN URBAN  \nAbstract. We propose rigorous and efficiently computable lower and upper aposteriori error bounds for given approximations to PDEs on a given domain, which might be geometrically complex. This is done by embedding or enveloping the original domain towards geometrically simpler domains, enabling the use of fast numerical solvers. To this end, we extend and restrict the residual and provide efficient methods to compute those Hahn-Banach extensions.  \nThen, we efficiently compute their Riesz representations on the geometrically simpler domains and obtain the desired a posteriori bounds for which we prove  \nthat they are sharp.  \nThe resulting bounds control the error in the natural norm induced by a well-posed variational formulation, require only minimal regularity assumptions, and thus remain applicable on complex geometries. The framework is detailed for elliptic as well as parabolic problems. Numerical experiments demonstrate the good quantitative behavior of the derived upper and lower error bounds.  \nA central motivation for this paper comes from physics-informed and related neural-network approximations of PDEs, which are naturally mesh-free and can be evaluated pointwise on complex or parameter-dependent geometries. Nevertheless, the framework applies to any approximation for which the variational residual can be evaluated.  \n1. Introduction  \nPartial differential equations (PDEs) are well-known to model a huge variety of processes. Accordingly, there is a huge literature of numerical methods for approximately solving PDEs. On the other hand, there are several black-box solvers which produce an approximation for the solution of a PDE without full access to the mathematical structure behind. Examples include commercial solvers or the increasing use of neural networks (NNs) for solving PDEs. The aim of this paper is to provide rigorous and efficiently computable a posteriori bounds for the error of some given approximation. In other words, the aim of this paper is to discuss the certification of given approximations to PDEs.  \nSuch black box solvers are particularly attractive in cases where the underlying domain Ω ⊂ Rd is geometrically complex so that one might want to avoid a triangulation e.g. for finite element or finite volume discretizations. Other scenarios include parameterized PDEs, where e.g. the domain is subject to parametric changes.  \n2020 Mathematics Subject Classification. 35J20, 65M15, 68T07 .  \nKey words and phrases. A Posteriori Error Bound; Parameterized Partial Differential Equations; Extension and Restriction of Functionals; Physics Informed Neural Networks.  \n2 LEWIN ERNST, NIKOLAOS REKATSINAS, AND KARSTEN URBAN  \nIn such cases, we do not want to perform any computations on the original domain. Instead, we suggest to embed or envelop Ω by geometrically simpler domains  \n(1 . 1) \\# ⊂ Ω ⊂ □ ⊂ Rd.  \nAssuming that the PDE is well-posed, it is well-known that the dual norm of the residual is a bound for the error. In order to avoid computing this norm on Ω, we extend and restrict it to □ and \\# for an upper and lower bound, respectively. This is done by efficiently computing norm-preserving Hahn-Banach extensions of the residual. Having them at hand, we efficiently compute the Riesz-representationson □ and \\#, so that their norms are the desired upper and lower bound. For that procedure to work, we only need access to point evaluations of the given approximation, which is typically provided by black box solvers as well by NN approximations including physics-informed neural networks (PINNs) .  \nThe broad and continuous interest on the usage of NNs for the numerical approximation of solutions to PDEs is in fact the main motivation for this paper, see e.g. [1–5], where this list is far from being complete. For inst","cbCaihFkeOkVnfgs","https://ap.wps.com/l/cbCaihFkeOkVnfgs","pdf",633246,2,1,34,"English","en",105,"# Introduction\n## Black-box solvers and complex domains\n## Residual-to-error bounds via dual norms\n## Neural networks and physics-informed approaches\n## Relation to generalization-based error control","[{\"question\":\"What does the paper provide for a given PDE approximation?\",\"answer\":\"It provides rigorous, efficiently computable a posteriori lower and upper error bounds for the approximation error on a given domain, even when the domain geometry is complex.\"},{\"question\":\"How are error bounds obtained without computing norms on the original complex domain?\",\"answer\":\"The approach embeds or envelopes the original domain into geometrically simpler domains, extends and restricts the residual using norm-preserving Hahn–Banach extensions, and then computes Riesz representations there to derive upper and lower bounds.\"},{\"question\":\"Why is the framework relevant to neural-network-based PDE solvers?\",\"answer\":\"Physics-informed and related neural-network approximations are often mesh-free and can be evaluated pointwise on complex or parameter-dependent geometries; the framework certifies approximations as long as the variational residual can be evaluated, including for such neural networks.\"}]",1784173961,86,{"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},"a-posteriori-certification-of-pde-approximations-with-particular-application-to-neural-networks","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,47,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":20},"https://docshare.wps.com/document/","Document",{"item":48,"name":12,"@type":43,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/a-posteriori-certification-of-pde-approximations-with-particular-application-to-neural-networks/81519/",4,{"url":51,"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 does the paper provide for a given PDE approximation?","Question",{"text":75,"@type":76},"It provides rigorous, efficiently computable a posteriori lower and upper error bounds for the approximation error on a given domain, even when the domain geometry is complex.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How are error bounds obtained without computing norms on the original complex domain?",{"text":80,"@type":76},"The approach embeds or envelopes the original domain into geometrically simpler domains, extends and restricts the residual using norm-preserving Hahn–Banach extensions, and then computes Riesz representations there to derive upper and lower bounds.",{"name":82,"@type":73,"acceptedAnswer":83},"Why is the framework relevant to neural-network-based PDE solvers?",{"text":84,"@type":76},"Physics-informed and related neural-network approximations are often mesh-free and can be evaluated pointwise on complex or parameter-dependent geometries; 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