[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82487-en":3,"doc-seo-82487-105":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},82487,1099513958762,"Logic","https://ap-avatar.wpscdn.com/avatar/1000023916a998db790?x-image-process=image/resize,m_fixed,w_180,h_180&k=1784791008015729253",8,"Research & Report","Learning Dynamical Systems from Noisy Data with Weak-form Kernel Ridge Regression","Accurate prediction of complex dynamical systems from noisy measurements remains difficult in scientific computing. Kernel ridge regression performs well on clean data but often degrades under observational noise, especially for chaotic dynamics. Building on the idea that weak formulations filter noise, the manuscript explains the filtering mechanism and derives a bias-variance error decomposition. Weak-form Kernel Ridge Regression (WKRR) combines weak formulation with kernel learning, is simple to implement, and outperforms baselines on chaotic benchmarks and high-dimensional fluid data.","arXiv :2607 .00257v1 [ cs .LG] 30 Jun 2026  \nLEARNING DYNAMICAL SYSTEMS FROM NOISY DATA WITH WEAK-FORM KERNEL RIDGE REGRESSION  \nA PREPRINT  \nMax Kreider  \nDepartment of Mathematics  \nThe Pennsylvania State University, University Park, PA 16802, USA  \n[mbk6295@psu.edu](mbk6295@psu.edu)  \nJohn Harlim  \nDepartment of Mathematics, Institute for Computational and Data Sciences The Pennsylvania State University, University Park, PA 16802, USA  \n[jharlim@psu.edu](jharlim@psu.edu)  \nDaning Huang  \nDepartment of Aerospace Engineering  \nThe Pennsylvania State University, University Park, PA 16802, USA  \n[daning@psu.edu](daning@psu.edu)  \nJuly 2, 2026  \nABSTRACT  \nAccurate prediction of complex dynamical systems from noisy measurements remains a significant challenge in scientific computing. Kernel ridge regression learning strategies are often effective when applied to clean data, but have limited success with noisy data. Recent work has observed that a weak formulation can act to filter noisy data, and different learning strategies have achieved increased noise robustness with a weak-form framework. In this manuscript, we give an overview of the filtering mechanism behind the weak formulation and provide a bias-variance error decomposition. Using these insights, we combine a weak formulation with a kernel learning strategy to propose Weak-form Kernel Ridge Regression (WKRR) for learning dynamical systems.  \nThe proposed framework is simple to implement, effective for both clean and noisy data, and outperforms several baseline methods. We demonstrate the performance ofWKRR on chaotic benchmark systems in up to 64 dimensions, as well as 15,000-dimensional real-world fluid data.  \n1 Introduction  \nMany problems in scientific computing and engineering disciplines involve modeling and prediction of dynamical systems, with applications including weather [17, 37, 57], environmental and ecological science [29, 49, 64, 82], biology [28, 60, 77], fluid dynamics [1, 23, 47, 62], finance [14, 73], and traffic [5, 6, 78] . However, many physically relevant problems remain challenging due to high-dimensionality, complex or chaotic dynamics, lack of known underlying dynamics, and noisy or low-fidelity observational data.  \nPurely data-driven methods have emerged as a strong option for learning dynamical systems [11, 27, 58, 74, 75] . Such methods circumvent the need to form dynamical equations and typically seek to represent unknown dynamics with a high-fidelity reduced-order model. A variety of popular approaches have proven to be competitive in this context. Dynamic mode decomposition (DMD) and variants leverage a Koopman framework that lifts finite-dimensional nonlinear data to an infinite-dimensional linear representation, often leading to a simplified low-dimensional surrogate model [20, 21, 41, 43, 55, 75] . Sparse identification of nonlinear dynamical systems (SINDy) and variants  \nA PREPRINT-JULY 2, 2026  \ndiscover dynamical equations from data by choosing a suitable, often sparse, linear combination of dictionary functions [12, 13, 38, 85] . Neural ordinary differential equations (NODEs) learning underlying dynamics by training a neural network to represent a continuous-time vector field [15, 33, 44, 59, 83, 86] . Various machine learning techniques such as Long Short-Term Memory [35, 45, 71, 84], reservoir computing [26, 56, 66, 79], and autoencoders [25] have also gained prominence. Kernel-based approaches such as kernel ridge regression (KRR) mitigate the curse of dimensionality with the so-called “kernel trick” [2, 4, 24, 65, 72] . KRR is especially attractive because it does not require a dictionary of functions, and is straightforward to implement. Recent work has shown that KRRoutperforms multiple baseline methods in data-driven dynamical system learning and forecasting problems over a wide range of data sets [65] . While these approaches typically perform well when applied to clean data, their performance often degrades significantl","cbCaiqNxEHc8AdfG","https://ap.wps.com/l/cbCaiqNxEHc8AdfG","pdf",2924190,1,32,"English","en",105,"# Abstract\n# Introduction\n## Data-driven learning of dynamical systems\n## Noise robustness via weak formulations","[{\"question\":\"Why is learning dynamical systems from noisy measurements challenging?\",\"answer\":\"Physically relevant systems can involve high dimensionality, complex or chaotic dynamics, unknown underlying equations, and low-fidelity noisy observations, which together make prediction difficult.\"},{\"question\":\"What problem does kernel ridge regression face when data are noisy?\",\"answer\":\"Kernel ridge regression is effective on clean data, but its performance can degrade significantly with measurement noise, particularly for chaotic systems where small errors grow.\"},{\"question\":\"How does the weak formulation improve noise robustness in WKRR?\",\"answer\":\"The manuscript interprets the weak formulation as a filtering mechanism: by using weak residual constraints (orthogonality to test functions), it effectively projects noisy data onto a subspace that suppresses noise fluctuations.\"}]",1784180865,81,{"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},"learning-dynamical-systems-from-noisy-data-with-weak-form-kernel-ridge-regression","",{"@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/learning-dynamical-systems-from-noisy-data-with-weak-form-kernel-ridge-regression/82487/",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 is learning dynamical systems from noisy measurements challenging?","Question",{"text":75,"@type":76},"Physically relevant systems can involve high dimensionality, complex or chaotic dynamics, unknown underlying equations, and low-fidelity noisy observations, which together make prediction difficult.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What problem does kernel ridge regression face when data are noisy?",{"text":80,"@type":76},"Kernel ridge regression is effective on clean data, but its performance can degrade significantly with measurement noise, particularly for chaotic systems where small errors grow.",{"name":82,"@type":73,"acceptedAnswer":83},"How does the weak formulation improve noise robustness in WKRR?",{"text":84,"@type":76},"The manuscript interprets the weak formulation as a filtering mechanism: by using weak residual constraints (orthogonality to test functions), it effectively projects noisy data onto a subspace that suppresses noise 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