[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-124934-en":3,"doc-seo-124934-105":30,"detail-sidebar-cat-0-en-105":83},{"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},124934,5909877438554,"Maeve","https://ap-avatar.wpscdn.com/avatar/5600025385ad2bf12a7?_k=1778553567797529272",8,"Research & Report","Extrapolating tipping points and simulating non-stationary dynamics of complex systems using efficient machine learning","Data-driven prediction of tipping point transitions in nonlinear dynamical systems remains difficult, especially when only observational samples are available and governing equations are unknown. This work introduces an efficient reservoir-computing-based machine learning method that extrapolates bifurcation behavior from stationary training data. The trained next-generation reservoir computing model further predicts non-stationary dynamics under time-varying bifurcation parameters, enabling simulation of post-tipping behavior in previously unseen parameter regions.","[www. nature.com/scientificreports](www. nature.com/scientificreports)  \nOPEN  \nExtrapolating tipping points and simulating non‑stationary dynamics of complex systems using efficient machine learning  \nDaniel Köglmayr * & Christoph Räth  \nModel‑free and data‑driven prediction of tipping point transitions in nonlinear dynamical systems is a challenging and outstanding task in complex systems science. We propose a novel, fully data‑ driven machine learning algorithm based on next‑generation reservoir computing to extrapolate the bifurcation behavior of nonlinear dynamical systems using stationary training data samples. We show that this method can extrapolate tipping point transitions. Furthermore, it is demonstrated that the trained next‑generation reservoir computing architecture can be used to predict non‑stationary dynamics with time‑varying bifurcation parameters. In doing so, post‑tipping point dynamics of unseen parameter regions can be simulated.  \nSmall perturbations in a complex system can dramatically change its evolution1. A lack of precision in determining the exact state of the system can lead to an amplified lack of certainty about the future behavior of the system. This is the case even when we know its true governing equations and the exact boundary conditions. How can we then deal with complex systems where, in addition, we do not know the governing equations and must rely solely on observational data?  \nIn recent years promising and remarkably efficient machine learning methods were proposed that use observational data as training data to autonomously generate a model that can explain the data2–4. One prominent example is a recurrent neural network method called reservoir computing5,6 (RC) . A reservoir computer creates a high-dimensional nonlinear representation of the observed dynamical system and synchronizes it with the corresponding input data. The synchronized representation is then trained on the desired output target so that the reservoir computer becomes an autonomous dynamical system whose output dynamics resemble that of the analyzed system. This way, it can achieve cutting-edge performances in predicting short-and long-term behavior of chaotic systems and outperforms other machine learning approaches like LSTMs or DNNs7,8. In September 2021, Gauthier et al. published the next-generation reservoir computing architecture (NG-RC), highlighting its lack of randomness, the fewer hyperparameters, the smaller amount of required training data, and its performance gain in speed compared to the traditional approach9. In traditional reservoir computing, randomly initialized matrices are used to feed the input variables ofthe dynamical system into a high-dimensional state space that is nonlinearized by applying a nonlinear activation function. The NG-RC uses a library of unique polynomials of time-shifted input variables to achieve a nonlinear dimensionality expansion. In both cases, the resulting state space is consistently trained on the desired output target using ridge regression to become an autonomous dynamical system. Both methods can generally be deployed with small state spaces, which, combined with the computational cheap regression, lead to highly efficient algorithms.  \nSo far, these algorithms have been used mainly for analyzing stationary dynamical systems, where the boundary conditions of the system are assumed to be fixed, i.e., time-independent. In this case, the qualitative behavior of the system, such as periodicity or chaoticity, remains the same over time. However, in most real-world systems, the boundary conditions can change over time, possibly leading to a qualitative change in the behavior of the system, e.g., from stable periodicity to chaos or from chaos to system collapse. These systems are called nonstationary dynamical systems, and the boundary condition under which the system undergoes such a critical transition is termed tipping point. Extrapolating tipping points is of great in","cbCaiqvBFRDlCzEV","https://ap.wps.com/l/cbCaiqvBFRDlCzEV","pdf",5526091,1,12,"English","en",105,"# Abstract\n# Problem Setting: tipping points in nonlinear dynamical systems\n# Background: reservoir computing and next-generation RC\n## Training on stationary data vs. non-stationary time series\n# Proposed Method and Capabilities\n## Extrapolating bifurcation behavior\n## Predicting non-stationary dynamics with time-varying parameters\n# Related Work and Motivation","[{\"question\":\"Can the trained reservoir model handle time-varying (non-stationary) dynamics?\",\"answer\":\"Yes. After training, the next-generation reservoir computing architecture can predict non-stationary dynamics with time-varying bifurcation parameters and simulate post-tipping point dynamics in unseen parameter regions.\"}]","Extrapolating tipping points and simulating non-stationary dynamics of complex systems using efficient machine learning | PDF",1785895465,30,{"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":78,"head_meta":80,"extra_data":82,"updated_unix":28},"extrapolating-tipping-points-and-simulating-non-stationary-dynamics-of-complex-systems-using-efficient-machine-learning","",{"@graph":36,"@context":77},[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/extrapolating-tipping-points-and-simulating-non-stationary-dynamics-of-complex-systems-using-efficient-machine-learning/124934/",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","2026-08-05",true,{"@type":65,"interactionType":66,"userInteractionCount":4},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71],{"name":72,"@type":73,"acceptedAnswer":74},"Can the trained reservoir model handle time-varying (non-stationary) dynamics?","Question",{"text":75,"@type":76},"Yes. After training, the next-generation reservoir computing architecture can predict non-stationary dynamics with time-varying bifurcation parameters and simulate post-tipping point dynamics in unseen parameter regions.","Answer","https://schema.org",{"og:url":52,"og:type":79,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":81,"canonical":52},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":84},[85,89,93,97,102,107,112,114,119,122,126],{"id":20,"doc_module":4,"doc_module_name":46,"category_name":86,"show_sort_weight":87,"slug":88},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":90,"show_sort_weight":91,"slug":92},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Exam",70,"exam",{"id":98,"doc_module":4,"doc_module_name":46,"category_name":99,"show_sort_weight":100,"slug":101},5,"Comic",60,"comic",{"id":103,"doc_module":4,"doc_module_name":46,"category_name":104,"show_sort_weight":105,"slug":106},6,"Technology",50,"technology",{"id":108,"doc_module":4,"doc_module_name":46,"category_name":109,"show_sort_weight":110,"slug":111},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":29,"slug":113},"research-report",{"id":115,"doc_module":4,"doc_module_name":46,"category_name":116,"show_sort_weight":117,"slug":118},9,"Religion & Spirituality",20,"religion-spirituality",{"id":117,"doc_module":4,"doc_module_name":46,"category_name":120,"show_sort_weight":117,"slug":121},"World Cup","world-cup",{"id":123,"doc_module":4,"doc_module_name":46,"category_name":124,"show_sort_weight":123,"slug":125},10,"Lifestyle","lifestyle",{"id":127,"doc_module":4,"doc_module_name":46,"category_name":128,"show_sort_weight":98,"slug":129},19,"General","general"]