[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85604-en":3,"doc-seo-85604-105":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":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},85604,3848291630094,"Emma Wilson","https://eur-avatar.wpscdn.com/davatar_085a072bc5b1113ac321206ff7593b45",8,"Research & Report","Branched Signature Kernel Solvers for ODEs with Rough Single-Trajectory Signals","Develops a branched signature kernel solver for linear and nonlinear ODEs driven by a single observed, potentially rough forcing trajectory, reflecting situations where only one forcing realization is available. Uses count-sampling to convert the single observation into a hierarchy of N+1 nested training paths, enabling branched signature kernel evaluation on one trajectory. Builds a kernel-collocation framework with ansatz placement on either the highest-order derivative or the solution, proves a universal approximation theorem via the Hairer–Kelly morphism, and extends the solver to streaming Test/Train/Retrain with optional online updates; numerical studies on six benchmarks show accurate, stable predictions.","arXiv :2605 .25826v2 [math .NA] 13 Jul 2026  \nBRANCHED SIGNATURE KERNEL SOLVERS FOR ODES WITH ROUGH  \nSINGLE-TRAJECTORY SIGNALS  \nMUNAWAR ALI∗ QI FENG† CHARLIE PYLE♯ GEORGE XU♯♯  \nAbstract . We develop a branched signature kernel solver for linear and nonlinear ordinary differential equations driven by a single observed trajectory of a possibly rough forcing signal—a setting common within earthquake engineering, finance, biology, and structural health monitoring, where only one forcing realization is available, and the solver must respect the underlying physical law without an ensemble of realizations. We first introduce a count-sampling construction method to turn the single observation into a hierarchical family of N + 1 nested training paths on which the branched signature kernel can be evaluated; this allows the signature kernel machinery, originally designed for multi-realization regression problems, to operate on a single-trajectory observation.  \nThen we build a kernel-collocation framework, which places the ansatz either on the highest-order derivative of the solution or on the solution itself. We prove a universal approximation theorem for the branched signature kernel, leveraging the Hairer–Kelly morphism to express branched signature evaluations through geometric signatures of time-extended paths. The offline solver is extended to a streaming Test/Train/Retrain protocol with optional closed-form online updates in both linear and nonlinear cases. Numerical experiments on six benchmarks show accurate, stable predictions across all regimes.  \nKeywords: Branched Signature; Branched Signature Kernel; Universal approximation theorem; fractional Brownian motion; Hopf algebra; ODE solver; count sampling; streaming kernel methods. 2000 AMS Mathematics subject classification: 60L10, 60L20, 46E22, 60G17, 65C20, 65C30, 60H10, 91B70 .  \n1. Introduction  \nThe numerical solution of ordinary differential equations driven by a single observed forcing trajectory arises in many engineering and scientific settings. A structure responding to a recorded ground acceleration during an earthquake, a financial state variable evolving under a single realized market signal, a biological process excited by a particular environmental input, and a coupled oscillator network perturbed by one realization of noise are all instances of the same problem: solve Nu(t) = f(t) on [0, T] given a single discrete sample of f. Two features distinguish this regime from the classical setting. First, no ensemble of independent realizations of f is available, so ensemble-averaged calibration is impossible. Second, the forcing is typically rough: high-frequency seismic accelerations, fractional Brownian motion type volatility, and noisy biological signals all exhibit Hölder regularity well below the bounded-variation assumption of classical numerical analysis. A solver that respects the physical law governing the ODE while learning a representation of the solution from a single rough trajectory is therefore needed.  \nDate: July 14, 2026 .  \n∗ : Department of Mathematics, Florida State University, Tallahassee, FL 32306; e-mail: [ma22bm@fsu.edu](ma22bm@fsu.edu).  \n† : Department of Mathematics, Florida State University, Tallahassee, FL 32306; e-mail: [qfeng2@fsu.edu](qfeng2@fsu.edu. This)[. This](qfeng2@fsu.edu. This)[ ](qfeng2@fsu.edu. This)[author is partially supported by the National Science Foundation under grant \\#DMS-2420029.](author is partially supported by the National Science Foundation under grant #DMS-2420029.)  \n♯: Department of Mathematics, Texas A&M University, College Station, TX 77843; e-mail: [charliepyle@tamu.edu](charliepyle@tamu.edu). This author is partially supported by the National Science Foundation under grant \\#DMS-2420029 through the Research Experience for Undergraduates (REU) program at Florida State University.  \n♯♯ : Department of Mathematics, Rutgers University, Piscataway, NJ 08854; e-mail: [gtx1@scarletmail.rutgers.edu](gtx1@scarl","cbCaijZhiXoIwZhF","https://ap.wps.com/l/cbCaijZhiXoIwZhF","pdf",2312161,4,1,36,"English","en",105,"# Introduction\n## Problem setting and motivation\n## Rough path and signature kernel background\n# Main contribution overview","[{\"question\":\"What problem does the branched signature kernel solver address?\",\"answer\":\"It solves linear and nonlinear ODEs when only a single observed forcing trajectory is available, and the forcing may be rough rather than smooth.\"},{\"question\":\"How is the single trajectory used to construct training data?\",\"answer\":\"Through a count-sampling construction that transforms the single observation into a hierarchical family of N+1 nested training paths for branched signature evaluation.\"},{\"question\":\"What theoretical result is proven for the proposed kernel method?\",\"answer\":\"A universal approximation theorem for the branched signature kernel, using the Hairer–Kelly morphism to relate branched signature evaluations to geometric signatures of time-extended paths.\"}]",1784204866,91,{"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":86,"head_meta":88,"extra_data":90,"updated_unix":28},"branched-signature-kernel-solvers-for-odes-with-rough-single-trajectory-signals","",{"@graph":36,"@context":85},[37,53,68],{"@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":20},"https://docshare.wps.com/document/branched-signature-kernel-solvers-for-odes-with-rough-single-trajectory-signals/85604/",{"url":52,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-26","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},"What problem does the branched signature kernel solver address?","Question",{"text":75,"@type":76},"It solves linear and nonlinear ODEs when only a single observed forcing trajectory is available, and the forcing may be rough rather than smooth.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How is the single trajectory used to construct training data?",{"text":80,"@type":76},"Through a count-sampling construction that transforms the single observation into a hierarchical family of N+1 nested training paths for branched signature evaluation.",{"name":82,"@type":73,"acceptedAnswer":83},"What theoretical result is proven for the proposed kernel method?",{"text":84,"@type":76},"A universal approximation theorem for the branched signature kernel, using the Hairer–Kelly morphism to relate branched signature evaluations to geometric signatures of time-extended 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