[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-121657-en":3,"doc-seo-121657-105":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},121657,1099513958762,"Logic","https://ap-avatar.wpscdn.com/avatar/1000023916a998db790?x-image-process=image/resize,m_fixed,w_180,h_180&k=1784791008015729253",8,"Research & Report","Supervised machine learning to estimate instabilities in chaotic systems - estimation of local Lyapunov exponents","In chaotic dynamical systems such as weather, prediction errors can grow at different rates depending on the system state and time window. Local Lyapunov exponents quantify how rapidly small errors diverge over finite intervals, but computing them is costly due to tangent linear model evolution, orthogonalisation, and large matrix storage. This feasibility study evaluates supervised machine learning to estimate current local Lyapunov exponents using current and recent trajectory time steps, nonintrusively as an alternative to the classical approach.","Supervised machine learning to estimate instabilities in chaotic systems: estimation of local Lyapunov exponents  \nArticle  \nPublished Version  \nCreative Commons: Attribution 4.0 (CC-BY)  \nOpen Access  \nAyers, D. ORCID: [https://orcid.org/0000-0002-5667-8174](https://orcid.org/0000-0002-5667-8174) , Lau, J., Amezcua, J. ORCID: [https://orcid.org/0000-0002-4952-](https://orcid.org/0000-0002-4952-)[ ](https://orcid.org/0000-0002-4952-)8354, Carrassi, A. ORCID: [https://orcid.org/0000-0003-0722-](https://orcid.org/0000-0003-0722-)[ ](https://orcid.org/0000-0003-0722-)5600 and Ojha, V. ORCID: [https://orcid.org/0000-0002-9256-](https://orcid.org/0000-0002-9256-)[ ](https://orcid.org/0000-0002-9256-)1192 (2023) Supervised machine learning to estimate instabilities in chaotic systems: estimation of local Lyapunov exponents. Quarterly Journal of the Royal Meteorological  \nSociety, 149 (753) . pp. 1236-1262. ISSN 1477-870X doi: 10.1002/qj.4450 Available at [https://centaur. reading.ac. uk/1](https://centaur. reading.ac. uk/1) 11511/  \nIt is advisable to refer to the publisher’s version if you intend to cite from the work. See Guidance on citing.  \nTo link to this article DOI: [http://dx.doi.org/10.1002/qj.4450](http://dx.doi.org/10.1002/qj.4450)  \n[Publisher: Wiley](Publisher: Wiley)  \nAll outputs in CentAUR are protected by Intellectual Property Rights law, including copyright law. Copyright and IPR is retained by the creators or other copyright holders . Terms and conditions for use of this material are defined in  \nthe End User Agreement  .  \n[www. reading.ac. uk/centaur](www. reading.ac. uk/centaur)  \nCentAUR  \nCentral Archive at the University of Reading  \nReading’s research outputs online  \nDOI: 10.1002/qj.4450  \nRESEAR CH ARTICLE  \nSupervised machine learning to estimate instabilities  \nin chaotic systems: Estimation of local Lyapunov exponents  \nDaniel Ayers1,2  Jack Lau3  Javier Amezcua1,4  Alberto Carrassi1,5  Varun Ojha3,6  \n1 Department of Meteorology, University of Reading, Reading, UK  \n2National Centre for Earth Observation, Reading, UK  \n3 Department of Computer Science, University of Reading, Reading, UK  \n4 School of Science and Engineering, Tecnológico de Monterrey, Mexico City, Mexico  \n5 Department of Physics and Astronomy“Augusto Righi”, University of Bologna, Bologna, Italy  \n6 School of Computing, Newcastle University, Newcastle upon Tyne, UK  \nCorrespondence  \nDaniel Ayers, Department of Meteorology, University of Reading, Reading, UK. [Email: d.ayers@pgr.reading.ac.uk](Email: d.ayers@pgr.reading.ac.uk)  \nFunding information  \nEngineering and Physical Sciences  \nResearch Council, Grant/Award Number: EP/N509723/1; National Centre for Earth Observation, Grant/Award Number:  \nNCEO02004; Schmidt Futures, Grant/Award Number: 353  \nAbstract  \nIn chaotic dynamical systems such as the weather, prediction errors grow faster in some situations than in others. Real-time knowledge about the error growth could enable strategies to adjust the modelling and forecasting infrastructure on the fly to increase accuracy and/or reduce computation time. For example, one could change the ensemble size, the distribution and type of target observations, and so forth. Local Lyapunov exponents are known indicators of the rate at which very small prediction errors grow over a finite time interval. However, their computation is very expensive: it requires maintaining and evolving a tangent linear model, orthogonalisation algorithms and storing large matrices. In this feasibility study, we investigate the accuracy of supervised machine learning in estimating the current local Lyapunov exponents, from input of current and recent time steps of the system trajectory, as an alternative to the classical method. Thus machine learning is not used here to emulate a physical model or some of its components, but “nonintrusively” as a complementary tool. We test four popular supervised learning algorithms: regression trees, multilayer perceptrons, co","cbCaisIDEDc4vPXM","https://ap.wps.com/l/cbCaisIDEDc4vPXM","pdf",22981277,1,29,"English","en",105,"# Abstract\n## Problem context: state-dependent error growth in chaotic systems\n## Local Lyapunov exponents and computational cost\n## Proposed approach: nonintrusive supervised learning\n## Methods and models tested\n## Main findings","[{\"question\":\"What problem does the study address in chaotic dynamical systems?\",\"answer\":\"It addresses how prediction errors grow at different rates depending on system state, and how to estimate that error-growth behavior more efficiently in real time.\"},{\"question\":\"Why are local Lyapunov exponents important, and what makes them expensive to compute?\",\"answer\":\"Local Lyapunov exponents indicate the rate at which very small prediction errors grow over finite intervals. Their computation is expensive because it requires evolving a tangent linear model, orthogonalisation algorithms, and storing large matrices.\"},{\"question\":\"Which supervised learning models are evaluated, and on what systems?\",\"answer\":\"The study tests regression trees, multilayer perceptrons, convolutional neural networks, and long short-term memory networks. Experiments are conducted on the Rössler and Lorenz 63 low-dimensional chaotic systems.\"}]","Supervised machine learning to estimate instabilities in chaotic systems - estimation of local Lyapunov exponents | PDF",1785805995,73,{"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},"supervised-machine-learning-to-estimate-instabilities-in-chaotic-systems-estimation-of-local-lyapunov-exponents","",{"@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/supervised-machine-learning-to-estimate-instabilities-in-chaotic-systems-estimation-of-local-lyapunov-exponents/121657/",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},"What problem does the study address in chaotic dynamical systems?","Question",{"text":75,"@type":76},"It addresses how prediction errors grow at different rates depending on system state, and how to estimate that error-growth behavior more efficiently in real time.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Why are local Lyapunov exponents important, and what makes them expensive to compute?",{"text":80,"@type":76},"Local Lyapunov exponents indicate the rate at which very small prediction errors grow over finite intervals. Their computation is expensive because it requires evolving a tangent linear model, orthogonalisation algorithms, and storing large matrices.",{"name":82,"@type":73,"acceptedAnswer":83},"Which supervised learning models are evaluated, and on what systems?",{"text":84,"@type":76},"The study tests regression trees, multilayer perceptrons, convolutional neural networks, and long short-term memory networks. Experiments are conducted on the Rössler and Lorenz 63 low-dimensional chaotic systems.","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"]