MIMO Volterra kernel recovery in the frequency domain using neural networks

Abstract

This paper presents a detailed analysis of a process for recovering multi-degree-of-freedom Volterra kernels using neural network weights, addressing the fact that such Higher-order Frequency Response Functions (HFRFs) have not previously been directly recovered from data. A novel method is proposed for HFRF recovery of Volterra kernels up to third order, and its effectiveness is demonstrated on a variety of systems using simulated data for validation. The harmonic-probing algorithms are derived from a general multi-input-multi-output NARX neural network model. These algorithms recover Volterra kernels for time-invariant systems with up to n degrees of freedom. The results demonstrate the accuracy of the method and suggest a promising direction for HFRF recovery in nonlinear time-invariant systems.

Metadata

Item Type:	Article
Authors/Creators:	Preston, O.H.L. Rogers, T.J. https://orcid.org/0000-0002-3433-3247 Worden, K. https://orcid.org/0000-0002-1035-238X
Copyright, Publisher and Additional Information:	© The Author(s) 2026. Open Access: This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
Keywords:	Higher-order frequency response functions; Multi-degree-of-freedom systems; Neural networks; Volterra series
Dates:	Accepted: 16 May 2026 Published (online): 3 July 2026 Published: July 2026
Institution:	The University of Sheffield
Academic Units:	The University of Sheffield > Faculty of Engineering (Sheffield) > School of Mechanical, Aerospace and Civil Engineering
Funding Information:	Funder Grant number ENGINEERING AND PHYSICAL SCIENCE RESEARCH COUNCIL EP/W005816/1
Date Deposited:	22 Jul 2026 15:46
Last Modified:	22 Jul 2026 15:46
Status:	Published
Publisher:	Springer Science and Business Media LLC
Refereed:	Yes
Identification Number:	10.1007/s11071-026-12672-9
Related URLs:	PubMed URL
Open Archives Initiative ID (OAI ID):	oai:eprints.whiterose.ac.uk:243723

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MIMO Volterra kernel recovery in the frequency domain using neural networks

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