[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-86260-en":3,"doc-seo-86260-105":30,"detail-sidebar-cat-0-en-105":92},{"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},86260,687197207919,"Theodora","https://ap-avatar.wpscdn.com/avatar/a000253d6f5f7c60be?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779446848396160552",8,"Research & Report","Machines that Predict Trajectories from Templates","Study trajectory prediction using libraries of stored output templates, aiming to forecast a system’s future from observed past behavior without identifying the underlying state-space model. Trajectory libraries generated by dynamical systems define behavioral spaces that act as prediction mechanisms. For linear systems, exact prediction is characterized via continuation maps, behavioral containment, and spectral conditions on output-visible eigenvalues. The work quantifies robustness to noisy observations and noisy libraries, provides error bounds for out-of-library trajectories, composes template libraries under interconnection constraints, and extends to nonlinear systems whose outputs lie in finite-dimensional linear behaviors.","Machines that Predict Trajectories from Templates  \nC. De Persis and P. Tesi  \narXiv :2607 . 11551v1 [ ee ss . SY] 13 Jul 2026  \nAbstract—We study trajectory prediction from libraries of stored output templates. Given the past of an unknown trajectory, the goal is to predict its future without identifying the state-space model that generated it. We show that libraries of trajectories generated by one or more dynamical systems define behavioral spaces that can be used as prediction mechanisms. For linear systems, we characterize exact prediction in terms of continuation maps, behavioral containment, and spectral conditions on output-visible eigenvalues. We also analyze robustness to noisy observations and noisy libraries, derive error bounds for out-of-library trajectories, and show how interconnection constraints can compose template libraries into new behavioral spaces with emergent modes. Finally, we extend the framework to nonlinear systems whose output trajectories are contained in, or immersed into, finite-dimensional linear behaviors. These results provide a theory of template-based prediction machines capable of generalizing beyond the stored trajectories and, in some cases, beyond the systems that generated them.  \nI. INTRODUCTION  \nSolving complex tasks using templates. Using stored examples, or templates, to solve complex tasks is an intuitive and powerful idea that appears in many areas of science and engineering. In classification and pattern recognition [1], [2], new objects are assigned labels by comparison with stored examples from known categories. In instance-based regression [3], predictions are obtained by exploiting similarities with previously observed samples. The appeal of this paradigm is that it can support fast inference and decision making: rather than constructing a full model of the phenomenon from scratch, one reuses representative examples stored in memory. In systems and control, related ideas appear in several contexts. For example, in fault detection, online measurements can be compared with stored nominal trajectories to detect abnormal behavior [4] . More generally, one may ask whether stored dynamical behaviors can be used to infer stability properties, predict future trajectories, or supervise control systems without first identifying a complete parametric model.  \nDespite the importance of these tasks, a general theory of control, estimation and prediction from trajectory templates is still largely undeveloped.  \nPredicting dynamical behaviors from templates. In this paper, we study the problem of predicting the behavior of an unknown dynamical system from templates. We assume that the portion of an output trajectory is observed, and we aim to predict its future evolution using a stored library of sample trajectories. Concretely, we assume access to collections of finite length output windows generated by a finite number of dynamical systems, and we address the task of predicting  \nC. De Persis is with ENTEG, University of Groningen, 9747 AG Groningen, The Netherlands. Email: [c.de.persis@rug.nl](c.de.persis@rug.nl).  \nP. Tesi is with DINFO, University of Florence, 50139 Firenze, Italy. E-mail: [pietro.tesi@unifi.it](pietro.tesi@unifi.it).  \nthe future of a new trajectory from its observed past without knowing which system is generating it.  \nThis viewpoint casts trajectory prediction as a dictionarybased inference problem. The library provides a repertoire of representative behaviors, and prediction amounts to completing a partially observed output sequence using this repertoire, rather than estimating the underlying state-space matrices. The use of multiple template systems is not only a way to describe multi-mode behavior. It also provides a mechanism for approximating [5], [6], [7], and in special cases exactly representing [8], [9], nonlinear behaviors: a finite collection of linear templates can approximate, or even span, the output trajectories of a nonlinear system when these traj","cbCaieTlEVTuQUx2","https://ap.wps.com/l/cbCaieTlEVTuQUx2","pdf",501831,5,1,25,"English","en",105,"# Introduction\n## Predicting dynamical behaviors from templates\n## From stored trajectories to generative behavioral spaces","[{\"question\":\"What does the template-based trajectory prediction framework assume and aim to do?\",\"answer\":\"It assumes access to finite-length output windows stored in a library from multiple known dynamical systems, and it observes part of an unknown trajectory’s output. The goal is to predict the future evolution using the library without knowing which system generated the trajectory.\"},{\"question\":\"How is exact prediction characterized for linear systems?\",\"answer\":\"For linear systems, exact prediction is characterized using continuation maps, behavioral containment, and spectral conditions tied to output-visible eigenvalues.\"},{\"question\":\"How does the framework handle real-world uncertainty and extend beyond linear systems?\",\"answer\":\"It analyzes robustness to noisy observations and noisy libraries, deriving error bounds for trajectories outside the library. It also extends the approach to nonlinear systems when their output trajectories are contained in, or immersed into, finite-dimensional linear behaviors.\"}]",1784209874,63,{"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":87,"head_meta":89,"extra_data":91,"updated_unix":28},"machines-that-predict-trajectories-from-templates","",{"@graph":36,"@context":86},[37,54,69],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"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":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/machines-that-predict-trajectories-from-templates/86260/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":24,"description":14,"dateModified":62,"datePublished":63,"encodingFormat":61,"isAccessibleForFree":64,"interactionStatistic":65},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-07-25","2026-07-16",true,{"@type":66,"interactionType":67,"userInteractionCount":20},"InteractionCounter",{"@type":68},"ViewAction",{"@type":70,"mainEntity":71},"FAQPage",[72,78,82],{"name":73,"@type":74,"acceptedAnswer":75},"What does the template-based trajectory prediction framework assume and aim to do?","Question",{"text":76,"@type":77},"It assumes access to finite-length output windows stored in a library from multiple known dynamical systems, and it observes part of an unknown trajectory’s output. The goal is to predict the future evolution using the library without knowing which system generated the trajectory.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"How is exact prediction characterized for linear systems?",{"text":81,"@type":77},"For linear systems, exact prediction is characterized using continuation maps, behavioral containment, and spectral conditions tied to output-visible eigenvalues.",{"name":83,"@type":74,"acceptedAnswer":84},"How does the framework handle real-world uncertainty and extend beyond linear systems?",{"text":85,"@type":77},"It analyzes robustness to noisy observations and noisy libraries, deriving error bounds for trajectories outside the library. It also extends the approach to nonlinear systems when their output trajectories are contained in, or immersed into, finite-dimensional linear behaviors.","https://schema.org",{"og:url":52,"og:type":88,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":90,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":93},[94,98,102,106,110,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":95,"show_sort_weight":96,"slug":97},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":99,"show_sort_weight":100,"slug":101},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":103,"show_sort_weight":104,"slug":105},"Exam",70,"exam",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},"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":20,"slug":138},19,"General","general"]