[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84902-en":3,"doc-seo-84902-105":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},84902,1099514068035,"Ezra","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","Learning Adaptive Coarse Spaces Using Transferable Neural Network Models for Linear and Nonlinear Overlapping Domain Decomposition Methods","Domain decomposition methods serve as efficient, parallelizable iterative solvers and preconditioners for large systems from PDE discretization, with overlapping Schwarz schemes working for both linear and nonlinear settings. For highly heterogeneous coefficients, standard methods lose convergence due to coefficient-contrast sensitivity, which is mitigated by adaptively enriching the coarse space via local generalized eigenvalue constraints. The adaptive coarse-space construction can dominate runtime, motivating a machine-learning framework that predicts coarse bases and their required count, avoiding online eigen-solves.","arXiv :2607 .06261v1 [math .NA] 7 Jul 2026  \nLEARNING ADAPTIVE COARSE SPACES USING TRANSFERABLE NEURAL NETWORK MODELS FOR LINEAR AND NONLINEAR OVERLAPPING DOMAIN DECOMPOSITION METHODS  \nAXEL KLAWONN†‡, MARTIN LANSER†‡, AND JANINE WEBER-HAMACHER†‡  \nJuly 8, 2026  \nAbstract. Domain decomposition methods have been established as efficient and parallel scalable iterative solvers and preconditioners for the solution of large-scale systems arising from the discretization of partial differential equations. In particular, overlapping Schwarz methods have been successfully applied to a wide range of linear and nonlinear problems. However, for problems with highly heterogeneous coefficients, standard domain decomposition methods typically suffer from deteriorating convergence rates. Robustness with respect to the coefficient contrast can be achieved by enriching the coarse space with adaptively selected constraints obtained from local generalized eigenvalue problems. The construction of these adaptive coarse spaces, however, can account for a significant part of the overall computing time.  \nIn the present work, machine learning techniques are employed to reduce this part of the computing time in the context of the adaptive Generalized Dryja-Smith-Widlund (AGDSW) coarse space. A two-stage approach is proposed in which regression neural networks are used to predict the adaptive coarse basis functions, while a classification neural network is employed to predict the number of basis functions required to ensure robustness. As a consequence, adaptive coarse spaces can beset up in the online phase without solving any eigenvalue problem. Particular attention is paid to problem-specific aspects, including sign-invariant loss functions and post-processing strategies to significantly improve the predicted constraints. The proposed approach is first investigated for scalar diffusion problems with high coefficient contrasts and is subsequently transferred, without retraining, to problems of linear elasticity and to nonlinear p-Laplace problems, also within a nonlinear Schwarz framework.  \nKey words. Machine Learning, Domain Decomposition, Schwarz method, GDSW, Adaptive Coarse Spaces, Scientific Machine Learning, nonlinear Schwarz  \nAMS subject classifications. 65F10, 65N30, 65N55, 68T05, 68T07  \n1. Introduction. Discretizing second-order elliptic partial differential equations (PDEs), for example by using finite elements, leads to linear or nonlinear systems of equations which are often very large and thus have to be solved on parallel computers. Parallel scalable, preconditioned iterative methods have been proven to bea good choice for the solution of these kind of problems. Prominent examples belong to the family of domain decomposition methods (DDMs) . In this work, we will focus on a specific problem that occurs when solving such discretized elliptic PDEs iteratively. If the contrast of the maximum and the minimum of certain coefficients of the PDEs becomes too large, usually the convergence rate of standard DDMs will deteriorate. There exist extensions of standard two-level DDMs where the secondlevel coarse space is enriched with additional information on the coefficient contrast such that the convergence rate becomes robust and the iterative methods converge independently of it. In the present work, we will focus on one member of the family of these robust DDMs, namely the adaptive Generalized Dryja-Smith-Widlund (AGDSW) overlapping Schwarz method. In the following, we will describe in a broad picture the main ingredient which makes this method robust with regard to a high  \n†Department of Mathematics and Computer Science, University of Cologne, Weyertal 86-90, 50931 Köln, Germany, {axel.klawonn, martin.lanser, [janine.weber}@uni-koeln.de](janine.weber}@uni-koeln.de), url: [http://www](http://www). [numerik.uni-koeln.de](numerik.uni-koeln.de)  \n‡Center for Data and Simulation Science, University of Cologne, Germany, url: [http://www.cds](htt","cbCaimesWthnwNd0","https://ap.wps.com/l/cbCaimesWthnwNd0","pdf",12272917,3,1,38,"English","en",105,"# Introduction\n## Problem setting and motivation\n## Adaptive AGDSW coarse spaces\n## Reducing online eigen-solves with neural networks","[{\"question\":\"Why do standard overlapping Schwarz domain decomposition methods struggle with highly heterogeneous coefficients?\",\"answer\":\"When the coefficient contrast becomes large, the convergence rate of standard domain decomposition methods deteriorates, making the iterative solver less robust.\"},{\"question\":\"How does AGDSW achieve robustness for coefficient-contrast problems?\",\"answer\":\"AGDSW enriches the second-level coarse space by adaptively selecting constraints derived from local generalized eigenvalue problems on interfaces between subdomains.\"},{\"question\":\"How does the proposed machine-learning approach reduce the cost of building adaptive coarse spaces?\",\"answer\":\"Regression neural networks predict adaptive coarse basis functions, while a classification neural network predicts how many basis functions are needed for robustness, enabling online setup without solving eigenvalue problems.\"}]",1784199264,96,{"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},"learning-adaptive-coarse-spaces-using-transferable-neural-network-models-for-linear-and-nonlinear-overlapping-domain-decomposition-methods","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,50],{"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":20},"https://docshare.wps.com/document/research-report/",{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/learning-adaptive-coarse-spaces-using-transferable-neural-network-models-for-linear-and-nonlinear-overlapping-domain-decomposition-methods/84902/",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},"Why do standard overlapping Schwarz domain decomposition methods struggle with highly heterogeneous coefficients?","Question",{"text":75,"@type":76},"When the coefficient contrast becomes large, the convergence rate of standard domain decomposition methods deteriorates, making the iterative solver less robust.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does AGDSW achieve robustness for coefficient-contrast problems?",{"text":80,"@type":76},"AGDSW enriches the second-level coarse space by adaptively selecting constraints derived from local generalized eigenvalue problems on interfaces between subdomains.",{"name":82,"@type":73,"acceptedAnswer":83},"How does the proposed machine-learning approach reduce the cost of building adaptive coarse spaces?",{"text":84,"@type":76},"Regression neural networks predict adaptive coarse basis functions, while a classification neural network predicts how many basis functions are needed for robustness, enabling online setup without solving 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