[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-86580-en":3,"doc-seo-86580-105":30,"detail-sidebar-cat-0-en-105":84},{"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},86580,7971461741311,"Ophelia","https://ap-avatar.wpscdn.com/avatar/74000253aff267980c6?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779345379180704826",8,"Research & Report","Inf-Sup Neural Networks for High Dimensional PDEs","Solving partial differential equations (PDEs) in high dimensions is hindered by the curse of dimensionality. A neural-network-based framework is proposed that converts a PDE into an inf–sup optimization problem by introducing a Lagrange multiplier. Two networks parameterize the primal solution and the multiplier, computed through iterative saddle-point optimization. The approach proves theoretical equivalence to the original PDE and provides rigorous error estimates covering network approximation, sampling, and optimization errors, with numerical results showing accuracy, stability, and efficiency.","arXiv :2607 . 11718v1 [math .NA] 13 Jul 2026  \nINF–SUP NEURAL NETWORKS FOR HIGH DIMENSIONAL PDES  \nZIREN CHEN AND HAILIANG LIU  \nAbstract. Solving partial differential equations (PDEs) in high dimensions remains challenging due to the curse of dimensionality. We propose a neural-network-based framework that reformulates PDEs as inf–sup optimization problems through the introduction of a Lagrange multiplier. The primal solution and the associated Lagrange multiplier are parameterized by two networks and are computed via an iterative saddle-point optimization procedure. We prove the theoretical equivalence between the proposed optimization formulation and the original PDE problem, and we derive rigorous error estimates that quantify the total approximation error in terms of the network approximation error, statistical (sampling) error, and optimization error. Numerical experiments demonstrate the accuracy, stability, and efficiency of the proposed method for solving high-dimensional PDEs.  \n1. Introduction  \nPartial differential equations (PDEs) play a central role in numerous areas of mathematics, physics, and engineering. They provide a rigorous framework for modeling systems governed by physical laws and have found applications in diverse domains including fluid dynamics, electromagnetism and material science. Traditional numerical methods for solving them have achieved high accuracy in low-dimensional settings but rely heavily on discretizing the state space. As the dimensionality increases, the number of grid points required for accurate approximations grows exponentially, leading to prohibitive computational costs [25, 50] .  \nRecent advances in neural network-based methods have emerged as promising alternatives for PDE solvers, leveraging the representational power of deep neural networks to approximate complex solutions in a mesh-free, data-efficient manner [49, 12, 44, 59, 27, 34, 32, 16] . Among these approaches, physics-informed neural networks (PINNs) [44] and their variants [21, 17, 58, 36, 10] have gained considerable attention for incorporating physical laws directly into the training objective. Nevertheless, PINNs may exhibit stability issues and limited generalization in certain applications [22, 53, 55] . To address these challenges, weak-residual formulations have been explored, see e.g., [21, 59, 16] . In particular, Inf-SupNet [16] reformulate the PDE as a constrained optimization problem, where the boundary data mismatch is minimized subject to the governing equation. This formulation is then recast as an inf–sup problem through the introduction of a Lagrange multiplier. Building on this perspective, the present work develops a more general inf–sup neural framework applicable to a broader class of PDEs.  \n2020 Mathematics Subject Classification. 65K10, 68Q25 .  \nKey words and phrases. High-dimensional PDEs, Inf–sup Formulation, Saddle-point Optimization, Deep Neural Networks.  \n2 ZIREN CHEN AND HAILIANG LIU  \nIn this work, we introduce InfsupNet, a neural framework, for solving a broad range of PDEs by reformulating the solution process as an inf-sup optimization problem. InfsupNet builds upon a variational characterization of PDEs, enabling the learning of solutions through a game-theoretic perspective. By expressing the PDE problem as a saddle-point optimization between primal and dual functions, this approach naturally leads to improved stability and convergence properties. Crucially, it avoids direct residual minimization, which can obscure the true dynamics of the solution process and introduce training pathologies.  \nFrom a theoretical standpoint, error analysis for inf–sup formulations is inherently more involved. For well-posed PDE problems, stability guarantees that the error is controlled by the residual. This connection has been explored in [40, 9] to derive generalization error bounds for PINNs. However, further relating the residual error to the saddle-point loss requires a different approach, ","cbCaikbiYOyKypy5","https://ap.wps.com/l/cbCaikbiYOyKypy5","pdf",1236733,5,1,35,"English","en",105,"# Introduction\n## Related work\n### Residual-minimization formulations\n### Variational/Ritz formulations","[{\"question\":\"What components are included in the rigorous error estimates?\",\"answer\":\"The total approximation error is decomposed into network approximation error, sampling (statistical) error, and optimization error, each bounded explicitly under the framework.\"}]",1784212759,88,{"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":79,"head_meta":81,"extra_data":83,"updated_unix":28},"inf-sup-neural-networks-for-high-dimensional-pdes","",{"@graph":36,"@context":78},[37,54,69],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/inf-sup-neural-networks-for-high-dimensional-pdes/86580/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":24,"description":14,"dateModified":62,"datePublished":63,"encodingFormat":61,"isAccessibleForFree":64,"interactionStatistic":65},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-07-27","2026-07-16",true,{"@type":66,"interactionType":67,"userInteractionCount":20},"InteractionCounter",{"@type":68},"ViewAction",{"@type":70,"mainEntity":71},"FAQPage",[72],{"name":73,"@type":74,"acceptedAnswer":75},"What components are included in the rigorous error estimates?","Question",{"text":76,"@type":77},"The total approximation error is decomposed into network approximation error, sampling (statistical) error, and optimization error, each bounded explicitly under the framework.","Answer","https://schema.org",{"og:url":52,"og:type":80,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":82,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":85},[86,90,94,98,102,107,112,115,120,123,127],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":87,"show_sort_weight":88,"slug":89},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":91,"show_sort_weight":92,"slug":93},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":95,"show_sort_weight":96,"slug":97},"Exam",70,"exam",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":99,"show_sort_weight":100,"slug":101},"Comic",60,"comic",{"id":103,"doc_module":4,"doc_module_name":46,"category_name":104,"show_sort_weight":105,"slug":106},6,"Technology",50,"technology",{"id":108,"doc_module":4,"doc_module_name":46,"category_name":109,"show_sort_weight":110,"slug":111},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":113,"slug":114},30,"research-report",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},9,"Religion & Spirituality",20,"religion-spirituality",{"id":118,"doc_module":4,"doc_module_name":46,"category_name":121,"show_sort_weight":118,"slug":122},"World Cup","world-cup",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":124,"slug":126},10,"Lifestyle","lifestyle",{"id":128,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":20,"slug":130},19,"General","general"]