[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-120591-en":3,"doc-seo-120591-105":30,"detail-sidebar-cat-0-en-105":95},{"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},120591,1099514068365,"Aurelia","https://ap-avatar.wpscdn.com/avatar/10000253d8d9f28188e?_k=1776742907772140068",8,"Research & Report","Machine-learning perspectives on Volterra system identification","Machine-learning perspectives on Volterra system identification reviews how the Volterra series supports nonlinear system identification and how its frequency-domain counterpart generalizes resonance curves via higher-order frequency-response functions (HFRFs). The paper addresses the challenge of estimating Volterra terms and links recent advances in engineering dynamics to machine-learning methods. It surveys neural networks, Gaussian processes, and reproducing kernel Hilbert spaces, and introduces new neural-network results for multi-input multi-output (MIMO) systems.","[eprints@whiterose.ac.uk](eprints@whiterose.ac.uk)[ ](eprints@whiterose.ac.uk)[https://eprints.whiterose.ac.uk](https://eprints.whiterose.ac.uk)  \nUniversities of Leeds, Sheffield and York  \nDeposited via The University of Sheffield.  \nWhite Rose Research Online URL for this paper:  \n[https://eprints.whiterose.ac.uk/id/eprint/237715/](https://eprints.whiterose.ac.uk/id/eprint/237715/)  \nVersion: Published Version  \nArticle:  \nWorden, K. , Rogers, T. and Preston, O. (2025) Machine-learning perspectives on Volterra system identification. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 383 (2305) . 20240053. ISSN: 1364-503X  \n[https://doi.org/10.1098/rsta.2024.0053](https://doi.org/10.1098/rsta.2024.0053)  \nReuse  \nThis article is distributed under the terms of the Creative Commons Attribution (CC BY) licence. This licence allows you to distribute, remix, tweak, and build upon the work, even commercially, as long as you credit the authors for the original work. More information and the full terms of the licence here: [https://creativecommons.org/licenses/](https://creativecommons.org/licenses/)  \nTakedown  \nIf you consider content in White Rose Research Online to be in breach of UK law, please notify us by  \nemailing [eprints@whiterose.ac.uk](eprints@whiterose.ac.uk) including the URL of the record and the reason for the withdrawal request.  \n[royalsocietypublishing.org/journal/rsta](royalsocietypublishing.org/journal/rsta)  \nCite this article: Worden K, Rogers T, Preston O. 2025 Machine-learning perspectives on Volterra system identification. Phil. Trans. R. Soc. A 383: 20240053 .  \n[https://doi.org/10.1098/rsta.2024.0053](https://doi.org/10.1098/rsta.2024.0053)  \nReceived: 11 February 2025  \nAccepted: 20 May 2025  \nOne contribution of 14 to a theme issue‘Frontiers of applied inverse problems in science and engineering’.  \nSubject Areas:  \nmechanical engineering  \nKeywords:  \nVolterra series, nonlinear dynamics, machine learning  \nAuthor for correspondence:  \nKeith Worden  \n[e-mail: k.worden@sheffield.ac.uk](e-mail: k.worden@sheffield.ac.uk)  \nSupplementary material is available online at [https://doi.org/10.6084/m9.figshare.c](https://doi.org/10.6084/m9.figshare.c). 8039942.  \nMachine-learning perspectiveson Volterra  \nsystem identification  \nKeith Worden, Timothy Rogers and Oliver Preston  \nDynamics Research Group, School of Mechanical, Aerospace and Civil Engineering, The University of Sheffield, Sheffield S1 3JD, UK  \n KW, 0000-0002-1035-238X; TR, 0000-0002-3433-3247  \nThe Volterra series has been used in nonlinear system identification (NLSI) for decades; its frequency‑ domain counterpart allows a generalization of ’reso‑ nance curves’for nonlinear systems—so‑called higher‑ order frequency‑response functions (HFRFs) . Estimat‑ ing the terms in the series has often proved to be a challenge; however, the (comparatively) recent uptake of machine‑learning technology into engineering dy‑ namics has led to advances in the identification of the series—both for the Volterra kernels themselves and for the HFRFs. The current paper provides an overview of a number of approaches based on neural networks, Gaussian processes (GPs) and reproducing kernel Hilbert spaces (RKHSs), and presents new results for multi‑input multi‑output (MIMO) systems based on neural networks.  \nThis article is part of the theme issue ‘Frontiers of applied inverse problems in science and engineering’.  \n1. Introduction  \nOne of the most important inverse problems in structural dynamics is system identification (SI); this is the problem of fitting a mathematical model of a system to measured data. Even for linear systems, this is an inverse problem of the second kind and is very often ill‑posed [1] . For nonlinear systems, the problem is much more difficult, as there are essentially an infinity of possible model forms to choose from. In general, there is no ‘one‑size‑fits‑all’solution to problems in nonlinear S","cbCaiadZirq5XSZL","https://ap.wps.com/l/cbCaiadZirq5XSZL","pdf",1406974,1,32,"English","en",105,"# Introduction\n## Volterra series and system identification\n## Frequency-domain view and HFRFs\n## Machine-learning approaches (NNs, GPs, RKHS)\n## New results for MIMO neural-network identification","[{\"question\":\"What role does the Volterra series play in nonlinear system identification?\",\"answer\":\"It provides a structured representation for nonlinear systems used for nonlinear system identification, including estimation of Volterra series terms.\"},{\"question\":\"How do higher-order frequency-response functions (HFRFs) relate to resonance curves?\",\"answer\":\"In the frequency domain, HFRFs generalize resonance curves for nonlinear systems, offering a higher-order extension beyond linear frequency response.\"},{\"question\":\"Which machine-learning methods are surveyed for Volterra system identification?\",\"answer\":\"The paper reviews approaches based on neural networks, Gaussian processes (GPs), and reproducing kernel Hilbert spaces (RKHSs).\"},{\"question\":\"What new results does the paper present for multi-input multi-output systems?\",\"answer\":\"It presents new neural-network-based identification results specifically for multi-input multi-output (MIMO) systems.\"}]","Machine-learning perspectives on Volterra system identification | 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role does the Volterra series play in nonlinear system identification?","Question",{"text":75,"@type":76},"It provides a structured representation for nonlinear systems used for nonlinear system identification, including estimation of Volterra series terms.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How do higher-order frequency-response functions (HFRFs) relate to resonance curves?",{"text":80,"@type":76},"In the frequency domain, HFRFs generalize resonance curves for nonlinear systems, offering a higher-order extension beyond linear frequency response.",{"name":82,"@type":73,"acceptedAnswer":83},"Which machine-learning methods are surveyed for Volterra system identification?",{"text":84,"@type":76},"The paper reviews approaches based on neural networks, Gaussian processes (GPs), and reproducing kernel Hilbert spaces (RKHSs).",{"name":86,"@type":73,"acceptedAnswer":87},"What new results does the paper present for multi-input multi-output systems?",{"text":88,"@type":76},"It 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