[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82749-en":3,"doc-seo-82749-105":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},82749,4398048949847,"Eliana","https://ap-avatar.wpscdn.com/avatar/400002536579ef2da7f?_k=1778318612642679267",8,"Research & Report","Pointwise Error Estimates for Numerical Physics-Informed Neural Networks","Physics-informed neural networks are often evaluated using residual losses sampled at finitely many points, which do not certify pointwise values of a PDE solution. This work constructs deterministic pointwise error intervals for mesh-based, piecewise-linear numerical PINNs via a compatible finite-element reconstruction. A certifying residual is obtained by applying the finite-dimensional numerical system to this compatible field, yielding exact adjoint Green representations in square linear settings. Intractable correction is handled by computable variants, and discrete-to-continuous transfer uses comparison estimates with fully computable residual-based estimators and Taylor remainder extensions.","3 Jul 2026  \nPointwise Error Estimates for Numerical Physics-Informed Neural Networks  \nNivar Anwera,1 , Marien Chenaudb,c,1 , José Alvesc , Frédéric Magoulèsb  \na School of Computer Science, Georgia Institute of Technology, Atlanta, GA, USA, bMICS, CentraleSupélec, Paris Saclay University, 9 Rue Joliot-Curie, Gif sur Yvette, 91400, France, c Transvalor SA, 955 Avenue Roumanille, Biot, 06904, France,  \nAbstract  \nPhysics-informed neural networks are often evaluated by residual losses sampled at finitely many points, which do not by themselves certify pointwise values of a partial differential equation solution. In this work, deterministic pointwise error intervals are developed for mesh-based, piecewise-linear numerical physics-informed neural networks. The proposed error estimation is given for a compatible field, which is the finite-element reconstruction of an admissible prediction on a mesh. The certifying residual is then obtained by applying the finite-dimensional numerical system to this compatible field. For compatible square linear systems, the pointwise error relative to the discrete target has an exact adjoint Green representation, and the computed signed error recovers the finite element solution exactly. Norm-based, inexact, localized, and randomized variants provide computable intervals when the exact correction computation is impractical. The extension from the discrete target to the continuous solution is supplied by  \ncomparison estimates. For a one-dimensional coercive reaction-diffusion class, this transfer layer is made fully computable by an explicit residual-based a posteriori estimator with querywise constants. The error bound derivation is extended to nonlinear residual systems with explicit Taylor remainders. Numerical experiments assess compatibility and calibration on manufactured examples, on a large-scale public three-dimensional elasticity benchmark, and on projected neural load families on the same benchmark.  \n1. Introduction coefficient vector in a finite-dimensional space. The resulting continuous field is the finite-element interpolation of that vector.  \n[math .NA]  \narXiv :2607 .03431v1  \nPhysics-informed neural networks (PINNs) are commonly trained using sampled PDE residuals, boundary terms, and data constraints [1, 2] . However, the numerical values of the corresponding loss terms do not constitute a certificate for the predicted spatial field. In this work, we provide deterministic interval certifications for values of the exact PDE solution at specified query points. Such deterministic certificate must guarantee that the exact solution value is contained in the reported interval. A small empirical residual loss at finitely many samples does not provide such a guarantee; rigorous error control for PINN-type methods depends on stability, sampling, approximation, optimization, and loss consistency assumptions rather than on the sampled residual value alone [3, 4] .  \nThe inability to construct such certificate from loss values computed at fixed collocation points is structural. For an unprojected collocation PINN trained using finitely many automaticdifferentiation residual samples, the field may be altered away from sampled locations without changing the sampled loss, while an unsampled point value may be changed arbitrarily. Finite collocation sampling therefore does not control unsampled point values deterministically, unless additional structure is imposed. Proposition 3.2 formalizes this limitation, and the compatibility condition introduced next provides the finite-dimensional structure needed for certification.  \nThe certified model class is a mesh-based, piecewise-linear numerical PINN. The trainable model outputs an admissible  \n1These authors contributed equally to this work.  \nThe certifying residual that is used to derive the error bounds is the residual of the finite-dimensional numerical system applied to the same vector. This alignment is called compatibility. Compatibility","cbCaiifo1q0hriOc","https://ap.wps.com/l/cbCaiifo1q0hriOc","pdf",25043241,1,31,"English","en",105,"# Introduction\n## Motivation and certification gap\n## Structural limitation of collocation PINNs\n## Compatibility condition and certified PINN class\n## Error estimation workflow and extensions\n## Main contributions","[{\"question\":\"Why do sampled residual losses in PINNs fail to guarantee pointwise correctness?\",\"answer\":\"Loss values computed at finitely many collocation points do not certify the exact PDE solution at unsampled spatial locations. The sampled loss can remain unchanged while the field is modified away from sampled points, leaving pointwise values uncontrolled.\"},{\"question\":\"What is the compatibility condition and why is it central?\",\"answer\":\"Compatibility aligns the certifying residual with the same admissible discrete state by using a compatible finite-element reconstruction of an admissible prediction on a mesh. This alignment ensures the field, residual, and query act on one coherent discrete object.\"},{\"question\":\"How are pointwise error intervals computed when exact correction is impractical?\",\"answer\":\"The paper provides norm-based, inexact, localized, and randomized variants that yield computable interval bounds when full correction or correction computations cannot be performed exactly. Discrete-to-continuous transfer is then handled via rigorous comparison estimates, supported by residual-based a posteriori estimators in a coercive one-dimensional setting.\"}]",1784182677,78,{"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},"pointwise-error-estimates-for-numerical-physics-informed-neural-networks","",{"@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/pointwise-error-estimates-for-numerical-physics-informed-neural-networks/82749/",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 do sampled residual losses in PINNs fail to guarantee pointwise correctness?","Question",{"text":75,"@type":76},"Loss values computed at finitely many collocation points do not certify the exact PDE solution at unsampled spatial locations. The sampled loss can remain unchanged while the field is modified away from sampled points, leaving pointwise values uncontrolled.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What is the compatibility condition and why is it central?",{"text":80,"@type":76},"Compatibility aligns the certifying residual with the same admissible discrete state by using a compatible finite-element reconstruction of an admissible prediction on a mesh. This alignment ensures the field, residual, and query act on one coherent discrete object.",{"name":82,"@type":73,"acceptedAnswer":83},"How are pointwise error intervals computed when exact correction is impractical?",{"text":84,"@type":76},"The paper provides norm-based, inexact, localized, and randomized variants that yield computable interval bounds when full correction or correction computations cannot be performed exactly. Discrete-to-continuous transfer is then handled via rigorous comparison estimates, supported by residual-based a posteriori estimators in a coercive one-dimensional setting.","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"]