[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-123476-en":3,"doc-seo-123476-105":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":4,"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":27,"seo_description":14,"update_tm":28,"read_time":29},123476,1374391974585,"Genevieve","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","Kolmogorov n-Widths for Multitask Physics-Informed Machine Learning (PIML) Methods - Towards Robust Metrics","Physics-informed machine learning (PIML) is a prominent approach for solving partial differential equations (PDEs), often in multitask settings where a model learns from physical laws embedded into training instead of relying on large datasets. Despite strong practical results, comparing PIML architectures remains difficult due to limited analysis, unreliable benchmarking, and validation uncertainty from selective sampling and overfitting. Kolmogorov n-widths are introduced as an effectiveness metric to obtain lower accuracy bounds and study learned basis functions across multiple PDE problems. The metric also guides regularization to improve generalizability, while highlighting how activation choices can affect worst-case performance.","arXiv :2402 . 1 1 126v2 [ cs .LG] 4 Sep 2024  \nKolmogorov n-Widths for Multitask Physics-Informed Machine Learning (PIML) Methods: Towards Robust Metrics  \nMichael Penwarden 1 ,2 , Houman Owhadi3 , Robert M. Kirby 1 ,2  \n1 Scientific Computing and Imaging Institute, University of Utah, Salt Lake City, UT 84112, USA.  \n2 Kahlert School of Computing, University of Utah, Salt Lake City, UT 84112, USA.  \n3 Department of Computing and Mathematical Sciences, Caltech, Pasadena, CA 91125, USA  \nAbstract  \nPhysics-informed machine learning (PIML) as a means of solving partial differential equations (PDEs) has garnered much attention in the Computational Science and Engineering (CS&E) world. This topic encompasses a broad array of methods and models aimed at solving a single or a collection of PDE problems, called multitask learning. PIML is characterized by the incorporation of physical laws into the training process of machine learning models in lieu of large data when solving PDE problems. Despite the overall success of this collection of methods, it remains incredibly difficult to analyze, benchmark, and generally compare one approach to another. Using Kolmogorov n-widths as a measure of effectiveness of approximating functions, we judiciously apply this metric in the comparison of various multitask PIML architectures. We compute lower accuracy bounds and analyze the model’s learned basis functions on various PDE problems. This is the first objective metric for comparing multitask PIML architectures and helps remove uncertainty in model validation from selective sampling and overfitting. We also identify avenues of improvement for model architectures, such as the choice of activation function, which can drastically affect model generalization to“worst-case” scenarios, which is not observed when reporting task-specific errors. We also incorporate this metric into the optimization process through regularization, which improves the models’ generalizability over the multitask PDE problem.  \nKeywords: Physics-informed Neural Networks (PINNs), Neural Operators, Kolmogorov n-width, Multitask Learning  \n1. Introduction  \nPhysics-informed machine learning (PIML) has emerged as a popular framework for solving partial differential equations (PDEs) . PIML has been particularly effective and efficient in solving ill-posed and inverse problems over traditional methods [1] . The most ubiquitously used PIML implementation at present is the physics-informed neural network (PINN)[2] . This approach is often selected due to its flexibility in discretization and has been shown to be successful across a wide class of application domains [3, 4, 5, 6, 7, 8, 9] . Innumerable strategies to enhance PINN training have been proposed such as adaptive sampling [10, 11, 12], adaptive weighting [13, 14], adaptive activation functions [15, 16], additional loss terms [17], domain decomposition [18, 19, 20, 21], metalearning [22, 23, 24], and network architecture modification to obey characteristics [25, 26] . A thorough summary of PINN training challenges and their proposed solutions is provided in [25] . The physics-informed concept has also been incorporated into neural operator learning  \n∗ Email addresses: [mpenwarden@sci.utah.edu](mpenwarden@sci.utah.edu) (Michael Penwarden), [owhadi@caltech.edu](owhadi@caltech.edu) (Houman Owhadi), [kirby@cs.utah.edu](kirby@cs.utah.edu) (Robert M. Kirby)  \nPublished in the Journal of Neural Networks  \nin works such as physics-informed DeepONets [27, 28, 29] and physics-informed neural operators (PINO)[30, 31] . Comprehensive surveys of the state of PIML and future directions can be found in [32, 33] .  \nThe optimization process of PINNs and physics-informed methods, in general, limits the upper accuracy bound and can cause training challenges due to the interplay between minimizing the PDE residuals and the non-convex optimization of deep neural networks, which induces non-unique solutions [34] . Training challenges in PINNs c","cbCaicEUnGB7Q2lg","https://ap.wps.com/l/cbCaicEUnGB7Q2lg","pdf",44042209,1,25,"English","en",105,"# Introduction\n## Physics-informed machine learning for PDEs\n## Challenges in PINN optimization and benchmarking\n## Proposed Kolmogorov n-width methodology\n# Kolmogorov n-widths as a metric and regularizer\n## Numerical approximation via tri-optimization\n## Experimental comparison of multitask PIML architectures","[{\"question\":\"Why are Kolmogorov n-widths proposed for multitask PIML?\",\"answer\":\"They provide an objective effectiveness measure for approximating functions, enabling principled analysis and benchmarking of multitask PIML architectures beyond task-specific error reporting.\"},{\"question\":\"What problem does the paper address in evaluating PIML architectures?\",\"answer\":\"It targets the difficulty of analyzing, benchmarking, and comparing approaches reliably, since existing validation can be influenced by selective sampling and overfitting.\"},{\"question\":\"How is the proposed metric used beyond evaluation?\",\"answer\":\"Kolmogorov n-widths are incorporated into the optimization process through regularization, improving models’ generalizability across multitask PDE problems.\"}]","Kolmogorov n-Widths for Multitask Physics-Informed Machine Learning (PIML) Methods - Towards Robust Metrics | PDF",1785816724,63,{"code":4,"msg":31,"data":32},"ok",{"site_id":24,"language":23,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"kolmogorov-n-widths-for-multitask-physics-informed-machine-learning-piml-methods-towards-robust-metrics","",{"@graph":36,"@context":85},[37,54,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":20},"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/kolmogorov-n-widths-for-multitask-physics-informed-machine-learning-piml-methods-towards-robust-metrics/123476/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":23,"description":14,"dateModified":62,"datePublished":62,"encodingFormat":61,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-08-04",true,{"@type":65,"interactionType":66,"userInteractionCount":4},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"Why are Kolmogorov n-widths proposed for multitask PIML?","Question",{"text":75,"@type":76},"They provide an objective effectiveness measure for approximating functions, enabling principled analysis and benchmarking of multitask PIML architectures beyond task-specific error reporting.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What problem does the paper address in evaluating PIML architectures?",{"text":80,"@type":76},"It targets the difficulty of analyzing, benchmarking, and comparing approaches reliably, since existing validation can be influenced by selective sampling and overfitting.",{"name":82,"@type":73,"acceptedAnswer":83},"How is the proposed metric used beyond evaluation?",{"text":84,"@type":76},"Kolmogorov n-widths are incorporated into the optimization process through regularization, improving models’ generalizability across multitask PDE problems.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":89,"canonical":52},"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":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":53,"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"]