[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82538-en":3,"doc-seo-82538-105":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},82538,549758146520,"Patrick","https://ap-avatar.wpscdn.com/avatar/80002397d8c0411e94?_k=1775819394049821470",8,"Research & Report","A Multilevel Stochastic-Gradient Neural Solver for Boundary Integral Equations","A multilevel stochastic-gradient neural solver is developed for second-kind boundary integral equations. The boundary density is modeled by a multilayer perceptron and trained by minimizing a Nyström-discretized residual across a hierarchy of refining quadrature grids, warm-started from the previous level. Each optimization step relies on dense matrix-vector products on collocation mini-batches and neural forward passes, mapping efficiently onto GPUs. Residual contraction is analyzed via the empirical neural tangent kernel and frequency-principle plateaus, with quadrature refinement acting as multigrid-style smoother, yielding uniform-work and rigorous a posteriori error control.","arXiv :2607 .00560v1 [math .NA] 1 Jul 2026  \nA MULTILEVEL STOCHASTIC-GRADIENT NEURAL SOLVER FOR  \nBOUNDARY INTEGRAL EQUATIONS  \nBing-Ze Lu ∗ and Richard Tsai †  \nABSTRACT  \nWe develop a multilevel stochastic-gradient neural solver for boundary integral equations of the second kind. The unknown density is represented by a multilayer perceptron, trained by minimizing the Nyström-discretized residual on a ladder of refining quadrature grids, each level warm-started from the parameters of the previous one. Each step requires only dense matrix-vector products on mini-batches of collocation rows and network passes, operations that map directly onto GPU hardware. The residual contraction is governed by the empirical neural tangent kernel (NTK), the discrete sample of a single continuum kernel.  \nOn a fixed grid, training stalls once the residual concentrates in modes the network contracts slowly, the plateau described by the frequency principle; a spectral analysis explains, and experiments confirm, how refining the quadrature resolves more of the continuum kernel’s spectrum and returns these modes to the optimizer’s reach. Spectral bias, elsewhere an obstruction to neural network solvers, thus serves as the smoother of a multigrid-type iteration, with quadrature refinement in place of coarse-grid correction. Under a uniform regularity bound on the network, the total work is a constant multiple of the work on the finest grid, and the uniform conditioning of the discrete second-kind operator leaves the NTK as the sole rate-determining spectrum while converting the training residual into ana posteriori error bound. Experiments on interior Dirichlet Laplace/Poisson problems and exterior Neumann Helmholtz problems, using both parametric and signed-distance surface representations, demonstrate the effectiveness and efficiency of the proposed method compared with GMRES at comparable tolerances.  \nKeywords Boundary integral equations, implicit boundary integral method, multilevel training, neural network solvers, stochastic optimization, neural tangent kernel  \n1 Introduction  \nThis paper develops a GPU-friendly multilevel algorithm for solving a class of dense linear systems that arise from the discretization of boundary integral equations (BIEs) . The unknown density on the boundary is represented by a multilayer perceptron (MLP), and the algorithm trains the network through residual minimization on a sequence of progressively refined quadrature grids.  \n∗Department of Mathematics, National Chung Cheng University, Minhsiung, Chiayi 100190, Taiwan. [bingzelu.math@gmail.com](bingzelu.math@gmail.com)  \n†Department of Mathematics and Oden Institute for Computational Engineering and Sciences, The University of Texas at Austin, Austin, Texas 78712, [U.S.A.](U.S.A. ytsai@math.utexas.edu)[ ytsai@math.utexas.edu](U.S.A. ytsai@math.utexas.edu)  \nLet Ω ⊂ Rd , d = 2 , 3, be a bounded domain with C2 boundary Γ = ∂Ω . We consider Fredholm integral equations of the second kind posed on Γ,  \nAρ := ~~1~~2ρ + Kρ = g, (Kρ)(x) := ZΓ k (x, y)ρ (y)dS (y), x ∈ Γ , (1)  \nwhere ρ : Γ → C is the unknown density, g ∈ H 1/2(Γ) is prescribed boundary data, and K is a compact integral operator on L2 (Γ) (or, under appropriate regularity, on C(Γ)) . The kernel k is determined by the underlying problem.  \nEquations of the form (1) can be derived from elliptic boundary value problems posed on Ω or on the exterior Ωc [14, 8] . A representative pair of examples is the Laplace equation with Dirichlet data on Ω and the exterior Helmholtz problem with Neumann data on Ωc. The ~~1~~2 in (1) is the canonical jump term for the interior-Dirichlet double-layer reformulation; the analysis below is invariant under the sign of the jump and applies equally to formulations in which − ~~1~~2ρ appears in place of + ~~1~~2ρ . In each case, the PDE solution admits a layer potential representation  \nu (x) = ZΓ ˜k(x, y)ρ (y)dS (y), x ∈ Rd \\ Γ , (2)  \nin which ˜k is a (possibly different) kern","cbCaiiXwTBgSi6bl","https://ap.wps.com/l/cbCaiiXwTBgSi6bl","pdf",19800437,2,1,33,"English","en",105,"# Abstract\n# Introduction","[{\"question\":\"How is the boundary density represented and trained in the proposed method?\",\"answer\":\"The unknown density on the boundary is represented by a multilayer perceptron. Training minimizes a Nyström-discretized residual on successively refining quadrature grids, using warm-start parameters inherited from the previous level.\"},{\"question\":\"What controls the rate of residual contraction in the algorithm?\",\"answer\":\"The contraction rate is governed by the empirical neural tangent kernel (NTK), interpreted as the discrete sample of an underlying continuum kernel.\"},{\"question\":\"How does quadrature refinement resolve neural network training stagnation?\",\"answer\":\"On a fixed grid, training stalls when residual concentrates in modes contracted slowly by the network. Refining the quadrature resolves more of the continuum kernel spectrum, bringing those modes back within the optimizer’s effective reach, functioning as a smoother in a multigrid-like iteration.\"}]",1784181396,83,{"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-multilevel-stochastic-gradient-neural-solver-for-boundary-integral-equations","",{"@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-multilevel-stochastic-gradient-neural-solver-for-boundary-integral-equations/82538/",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","2026-07-23","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},"How is the boundary density represented and trained in the proposed method?","Question",{"text":75,"@type":76},"The unknown density on the boundary is represented by a multilayer perceptron. Training minimizes a Nyström-discretized residual on successively refining quadrature grids, using warm-start parameters inherited from the previous level.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What controls the rate of residual contraction in the algorithm?",{"text":80,"@type":76},"The contraction rate is governed by the empirical neural tangent kernel (NTK), interpreted as the discrete sample of an underlying continuum kernel.",{"name":82,"@type":73,"acceptedAnswer":83},"How does quadrature refinement resolve neural network training stagnation?",{"text":84,"@type":76},"On a fixed grid, training stalls when residual concentrates in modes contracted slowly by the network. Refining the quadrature resolves more of the continuum kernel spectrum, bringing those modes back within the optimizer’s effective reach, functioning as a smoother in a multigrid-like iteration.","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":25},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]