{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,15]],"date-time":"2026-06-15T06:39:57Z","timestamp":1781505597593,"version":"3.54.1"},"reference-count":52,"publisher":"Elsevier BV","issue":"8","license":[{"start":{"date-parts":[[2026,5,1]],"date-time":"2026-05-01T00:00:00Z","timestamp":1777593600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/tdm\/userlicense\/1.0\/"},{"start":{"date-parts":[[2026,5,1]],"date-time":"2026-05-01T00:00:00Z","timestamp":1777593600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/legal\/tdmrep-license"}],"funder":[{"DOI":"10.13039\/501100021171","name":"Basic and Applied Basic Research Foundation of Guangdong Province","doi-asserted-by":"publisher","award":["2024A1515011016"],"award-info":[{"award-number":["2024A1515011016"]}],"id":[{"id":"10.13039\/501100021171","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["62376290"],"award-info":[{"award-number":["62376290"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100017610","name":"Shenzhen Science and Technology Innovation Program","doi-asserted-by":"publisher","award":["JCYJ20250604175409013"],"award-info":[{"award-number":["JCYJ20250604175409013"]}],"id":[{"id":"10.13039\/501100017610","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["elsevier.com","sciencedirect.com"],"crossmark-restriction":true},"short-container-title":["Journal of the Franklin Institute"],"published-print":{"date-parts":[[2026,5]]},"DOI":"10.1016\/j.jfranklin.2026.108647","type":"journal-article","created":{"date-parts":[[2026,4,7]],"date-time":"2026-04-07T06:50:53Z","timestamp":1775544653000},"page":"108647","update-policy":"https:\/\/doi.org\/10.1016\/elsevier_cm_policy","source":"Crossref","is-referenced-by-count":1,"title":["Gradient-activation recurrent neural network applied to temporally-variant matrix inversion"],"prefix":"10.1016","volume":"363","author":[{"given":"Zhiguo","family":"Tan","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2228-0395","authenticated-orcid":false,"given":"Yunong","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"78","reference":[{"issue":"11","key":"10.1016\/j.jfranklin.2026.108647_bib0001","doi-asserted-by":"crossref","first-page":"11972","DOI":"10.1109\/TVT.2022.3194870","article-title":"Iterative matrix inversion methods for precoding in cell-free massive MIMO systems","volume":"71","author":"Ataeeshojai","year":"2022","journal-title":"IEEE Trans. Veh. Technol."},{"issue":"2","key":"10.1016\/j.jfranklin.2026.108647_bib0002","doi-asserted-by":"crossref","first-page":"122","DOI":"10.23919\/AISE.2025.000009","article-title":"Improved event-triggered adaptive neural network control for multi-agent systems under denial-of-service attacks","volume":"1","author":"Zhang","year":"2025","journal-title":"Artif. Intell. Sci. Eng."},{"issue":"17","key":"10.1016\/j.jfranklin.2026.108647_bib0003","doi-asserted-by":"crossref","first-page":"15915","DOI":"10.1109\/JIOT.2022.3150956","article-title":"Enabling privacy-preserving parallel outsourcing matrix inversion in IoT","volume":"9","author":"Gao","year":"2022","journal-title":"IEEE Internet Things J."},{"issue":"7","key":"10.1016\/j.jfranklin.2026.108647_bib0004","doi-asserted-by":"crossref","first-page":"6205","DOI":"10.1109\/TIE.2020.2994865","article-title":"High-throughput FPGA implementation of matrix inversion for control systems","volume":"68","author":"Zhang","year":"2021","journal-title":"IEEE Trans. Ind. Electron."},{"issue":"2","key":"10.1016\/j.jfranklin.2026.108647_bib0005","doi-asserted-by":"crossref","first-page":"134","DOI":"10.23919\/AISE.2025.000010","article-title":"Inverse reinforcement learning optimal control for Takagi-Sugeno fuzzy systems","volume":"1","author":"Song","year":"2025","journal-title":"Artif. Intell. Sci. Eng."},{"key":"10.1016\/j.jfranklin.2026.108647_bib0006","volume":"26","author":"Sowmya","year":"2022","journal-title":"A novel hybrid Zhang neural network model for time-varying matrix inversion"},{"issue":"12","key":"10.1016\/j.jfranklin.2026.108647_bib0007","doi-asserted-by":"crossref","first-page":"13532","DOI":"10.1109\/TIE.2025.3566731","article-title":"A momentum recurrent neural network for sparse motion planning of redundant manipulators with majorization-minimization","volume":"72","author":"Huang","year":"2025","journal-title":"IEEE Trans. Ind. Electron."},{"issue":"12","key":"10.1016\/j.jfranklin.2026.108647_bib0008","doi-asserted-by":"crossref","first-page":"11436","DOI":"10.1109\/TMC.2024.3397242","article-title":"A distributed competitive and collaborative coordination for multirobot systems","volume":"23","author":"Liu","year":"2024","journal-title":"IEEE Trans. Mobile Comput."},{"key":"10.1016\/j.jfranklin.2026.108647_bib0009","doi-asserted-by":"crossref","first-page":"3363","DOI":"10.1109\/TSP.2024.3395897","article-title":"Learning sparse high-dimensional matrix-valued graphical models from dependent data","volume":"72","author":"Tugnait","year":"2024","journal-title":"IEEE Trans. Signal Process."},{"key":"10.1016\/j.jfranklin.2026.108647_bib0010","series-title":"Coevolutionary neural dynamics considering multiple strategies for nonconvex optimization","author":"Fan","year":"2025"},{"issue":"7","key":"10.1016\/j.jfranklin.2026.108647_bib0011","doi-asserted-by":"crossref","first-page":"1885","DOI":"10.1109\/TMI.2022.3150853","article-title":"Sampling possible reconstructions of undersampled acquisitions in MR imaging with a deep learned prior","volume":"41","author":"Tezcan","year":"2022","journal-title":"IEEE Trans. Med. Imag."},{"key":"10.1016\/j.jfranklin.2026.108647_bib0012","series-title":"Fundamentals of computer graphics","author":"Marschner","year":"2021"},{"issue":"7","key":"10.1016\/j.jfranklin.2026.108647_bib0013","doi-asserted-by":"crossref","first-page":"1405","DOI":"10.1109\/TCSI.2008.2007065","article-title":"From Zhang neural network to Newton iteration for matrix inversion","volume":"56","author":"Zhang","year":"2009","journal-title":"IEEE Trans. Circuits Syst. I., Regul. Pap."},{"issue":"6","key":"10.1016\/j.jfranklin.2026.108647_bib0014","doi-asserted-by":"crossref","DOI":"10.1016\/j.jfranklin.2024.106710","article-title":"A pre-defined finite time neural solver for the time-variant matrix equation E(t)X(t)G(t)=D(t)","volume":"361","author":"Chen","year":"2024","journal-title":"J. Frankl. Inst."},{"issue":"4","key":"10.1016\/j.jfranklin.2026.108647_bib0015","doi-asserted-by":"crossref","first-page":"1535","DOI":"10.1109\/TNNLS.2020.3042761","article-title":"A variable-parameter noise-tolerant zeroing neural network for time-variant matrix inversion with guaranteed robustness","volume":"33","author":"Xiao","year":"2022","journal-title":"IEEE Trans. Neural Netw. Learn. Syst."},{"issue":"13","key":"10.1016\/j.jfranklin.2026.108647_bib0016","doi-asserted-by":"crossref","first-page":"9707","DOI":"10.1016\/j.jfranklin.2023.07.010","article-title":"Fixed-time convergence integral-enhanced ZNN for calculating complex-valued flow matrix Drazin inverse","volume":"360","author":"Xiao","year":"2023","journal-title":"J. Frankl. Inst."},{"key":"10.1016\/j.jfranklin.2026.108647_bib0017","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.matcom.2024.05.006","article-title":"New activation functions and Zhangians in zeroing neural network and applications to time-varying matrix pseudoinversion","volume":"225","author":"Gao","year":"2024","journal-title":"Math. Comput. Simul."},{"key":"10.1016\/j.jfranklin.2026.108647_bib0018","doi-asserted-by":"crossref","first-page":"483","DOI":"10.1016\/j.neucom.2018.11.071","article-title":"A novel recurrent neural network and its finite-time solution to time-varying complex matrix inversion","volume":"331","author":"Xiao","year":"2019","journal-title":"Neurocomputing"},{"key":"10.1016\/j.jfranklin.2026.108647_bib0019","doi-asserted-by":"crossref","first-page":"456","DOI":"10.1016\/j.matcom.2024.10.031","article-title":"A fuzzy activation function based zeroing neural network for dynamic Arnold map image cryptography","volume":"230","author":"Jin","year":"2025","journal-title":"Math. Comput. Simul."},{"issue":"8","key":"10.1016\/j.jfranklin.2026.108647_bib0020","doi-asserted-by":"crossref","first-page":"2980","DOI":"10.1109\/TNNLS.2019.2934734","article-title":"New varying-parameter ZNN models with finite-time convergence and noise suppression for time-varying matrix Moore-Penrose inversion","volume":"31","author":"Tan","year":"2020","journal-title":"IEEE Trans. Neural Netw. Learn. Syst."},{"issue":"1","key":"10.1016\/j.jfranklin.2026.108647_bib0021","doi-asserted-by":"crossref","first-page":"197","DOI":"10.1049\/cit2.12161","article-title":"Double integral-enhanced zeroing neural network with linear noise rejection for time-varying matrix inverse","volume":"9","author":"Liao","year":"2024","journal-title":"CAAI Trans. Intell. Technol."},{"key":"10.1016\/j.jfranklin.2026.108647_bib0022","first-page":"10907","volume":"359","author":"Fu","year":"2022","journal-title":"Gener. 9 Instant Discrete Time Zhang Neural Netw. Time Depend. Appl."},{"issue":"1","key":"10.1016\/j.jfranklin.2026.108647_bib0023","doi-asserted-by":"crossref","first-page":"620","DOI":"10.1109\/TAC.2022.3144135","article-title":"Gradient-based differential neural-solution to time-dependent nonlinear optimization","volume":"68","author":"Jin","year":"2023","journal-title":"IEEE Trans. Autom. Contr."},{"issue":"12","key":"10.1016\/j.jfranklin.2026.108647_bib0024","doi-asserted-by":"crossref","first-page":"2514","DOI":"10.1080\/00207160.2021.1902512","article-title":"A novel robust fixed-time convergent zeroing neural network for solving time-varying noise-polluted nonlinear equations","volume":"98","author":"Zhao","year":"2021","journal-title":"Int. J. Comput. Math."},{"key":"10.1016\/j.jfranklin.2026.108647_bib0025","doi-asserted-by":"crossref","first-page":"50","DOI":"10.1016\/j.tcs.2016.07.024","article-title":"A new design formula exploited for accelerating Zhang neural network and its application to time-varying matrix inversion","volume":"647","author":"Xiao","year":"2016","journal-title":"Theor. Comput. Sci."},{"issue":"9","key":"10.1016\/j.jfranklin.2026.108647_bib0026","doi-asserted-by":"crossref","DOI":"10.1016\/j.jfranklin.2024.106870","article-title":"Fixed-time solution of inequality constrained time-varying linear systems via zeroing neural networks","volume":"361","author":"Jin","year":"2024","journal-title":"J. Frankl. Inst."},{"key":"10.1016\/j.jfranklin.2026.108647_bib0027","first-page":"89","article-title":"A recurrent neural network for real-time matrix inversion","volume":"55","author":"Wang","year":"1993","journal-title":"Appl. Math. Comput."},{"key":"10.1016\/j.jfranklin.2026.108647_bib0028","doi-asserted-by":"crossref","first-page":"321","DOI":"10.1007\/s11063-019-10025-9","article-title":"Improved GNN models for constant matrix inversion","volume":"50","author":"Stanimirovi\u0107","year":"2019","journal-title":"Neural Process. Lett."},{"key":"10.1016\/j.jfranklin.2026.108647_bib0029","series-title":"2024 International Conference on Advanced Computational Intelligence","first-page":"177","article-title":"Nonlinear functions activated gradient-based neural dynamics for online matrix inversion","author":"Lv","year":"2024"},{"key":"10.1016\/j.jfranklin.2026.108647_bib0030","doi-asserted-by":"crossref","DOI":"10.1016\/j.ins.2023.119729","article-title":"A new recurrent neural network based on direct discretization method for solving discrete time-variant matrix inversion with application","volume":"652","author":"Shi","year":"2024","journal-title":"Inf. Sci."},{"issue":"5","key":"10.1016\/j.jfranklin.2026.108647_bib0031","doi-asserted-by":"crossref","first-page":"1053","DOI":"10.1109\/TNN.2002.1031938","article-title":"A recurrent neural network for solving Sylvester equation with time-varying coefficients","volume":"13","author":"Zhang","year":"2002","journal-title":"IEEE Trans. Neural Netw."},{"issue":"6","key":"10.1016\/j.jfranklin.2026.108647_bib0032","doi-asserted-by":"crossref","first-page":"1477","DOI":"10.1109\/TNN.2005.857946","article-title":"Design and analysis of a general recurrent neural network model for time-varying matrix inversion","volume":"16","author":"Zhang","year":"2005","journal-title":"IEEE Trans. Neural Netw."},{"key":"10.1016\/j.jfranklin.2026.108647_bib0033","doi-asserted-by":"crossref","first-page":"289","DOI":"10.1016\/j.matcom.2020.12.030","article-title":"Robust zeroing neural network for fixed-time kinematic control of wheeled mobile robot in noise-polluted environment","volume":"185","author":"Zhao","year":"2021","journal-title":"Math. Comput. Simul."},{"issue":"9","key":"10.1016\/j.jfranklin.2026.108647_bib0034","doi-asserted-by":"crossref","first-page":"16407","DOI":"10.1109\/TNNLS.2025.3563991","article-title":"Milne-Hamming method with zeroing neural network for time-varying nonlinear optimization and redundant manipulator application","volume":"36","author":"Kong","year":"2025","journal-title":"IEEE Trans. Neural Netw. Learn. Syst."},{"key":"10.1016\/j.jfranklin.2026.108647_bib0035","doi-asserted-by":"crossref","DOI":"10.1016\/j.measurement.2024.115989","article-title":"A novel adaptive parameter zeroing neural network for the synchronization of complex chaotic systems and its field programmable gate array implementation","volume":"242","author":"Zhao","year":"2025","journal-title":"Measurement"},{"issue":"22","key":"10.1016\/j.jfranklin.2026.108647_bib0036","doi-asserted-by":"crossref","first-page":"8218","DOI":"10.1002\/cpe.8218","article-title":"A noise-tolerant fuzzy-type zeroing neural network for robust synchronization of chaotic systems","volume":"36","author":"Liu","year":"2024","journal-title":"Concurr. Comput. Pract. Exp."},{"key":"10.1016\/j.jfranklin.2026.108647_bib0037","doi-asserted-by":"crossref","first-page":"126","DOI":"10.1016\/j.neucom.2019.01.058","article-title":"An exponential-enhanced-type varying-parameter RNN for solving time-varying matrix inversion","volume":"338","author":"Zhang","year":"2019","journal-title":"Neurocomputing"},{"key":"10.1016\/j.jfranklin.2026.108647_bib0038","doi-asserted-by":"crossref","first-page":"614","DOI":"10.1016\/j.matcom.2021.01.018","article-title":"Simulation of varying parameter recurrent neural network with application to matrix inversion","volume":"185","author":"Stanimirovi\u0107","year":"2021","journal-title":"Math. Comput. Simul."},{"issue":"11","key":"10.1016\/j.jfranklin.2026.108647_bib0039","doi-asserted-by":"crossref","first-page":"3974","DOI":"10.1109\/TFUZZ.2023.3272653","article-title":"A fuzzy adaptive zeroing neural network model with event-triggered control for time-varying matrix inversion","volume":"31","author":"Dai","year":"2023","journal-title":"IEEE Trans. Fuzzy Syst."},{"issue":"6","key":"10.1016\/j.jfranklin.2026.108647_bib0040","doi-asserted-by":"crossref","first-page":"3887","DOI":"10.1109\/TCYB.2022.3179312","article-title":"A robust predefined-time convergence zeroing neural network for dynamic matrix inversion","volume":"53","author":"Jin","year":"2023","journal-title":"IEEE Trans. Cybern."},{"issue":"15","key":"10.1016\/j.jfranklin.2026.108647_bib0041","doi-asserted-by":"crossref","DOI":"10.1016\/j.jfranklin.2024.107143","article-title":"A novel fuzzy-type zeroing neural network for dynamic matrix solving and its applications","volume":"361","author":"Zhao","year":"2024","journal-title":"J. Frankl. Inst."},{"key":"10.1016\/j.jfranklin.2026.108647_bib0042","doi-asserted-by":"crossref","DOI":"10.1016\/j.ins.2024.121217","article-title":"Design and analysis of finite-time convergent complex-valued zeroing neural networks with application to time-variant complex matrix inversion","volume":"681","author":"Xiao","year":"2024","journal-title":"Inf. Sci."},{"key":"10.1016\/j.jfranklin.2026.108647_bib0043","series-title":"2021 International Joint Conference on Neural Networks","first-page":"1","article-title":"Gradient-Zhang neural dynamics models computing pseudoinverses of time-varying matrices via ZeaD and extrapolation formulas","author":"Zhang","year":"2021"},{"key":"10.1016\/j.jfranklin.2026.108647_bib0044","doi-asserted-by":"crossref","DOI":"10.1016\/j.eswa.2023.120249","article-title":"Discrete gradient-zeroing neural dynamics for future Moore-Penrose inverse with application to tracking control of manipulator","volume":"227","author":"Wu","year":"2023","journal-title":"Expert Syst. Appl."},{"key":"10.1016\/j.jfranklin.2026.108647_bib0045","series-title":"2024 International Symposium on Neural Networks","first-page":"421","article-title":"Simplified gradient-zeroing neuronet for temporally-variant convex objective function minimization","author":"Yu","year":"2024"},{"key":"10.1016\/j.jfranklin.2026.108647_bib0046","doi-asserted-by":"crossref","first-page":"475","DOI":"10.1016\/j.matcom.2025.02.009","article-title":"Discrete gradient-zeroing neural network algorithm for solving future Sylvester equation aided with left-right four-step rule as well as robot arm inverse kinematics","volume":"233","author":"Guo","year":"2025","journal-title":"Math. Comput. Simul."},{"issue":"3","key":"10.1016\/j.jfranklin.2026.108647_bib0047","doi-asserted-by":"crossref","first-page":"763","DOI":"10.1007\/s11063-014-9397-y","article-title":"Finite-time stability and its application for solving time-varying Sylvester equation by recurrent neural network","volume":"42","author":"Shen","year":"2015","journal-title":"Neural Process. Lett."},{"key":"10.1016\/j.jfranklin.2026.108647_bib0048","doi-asserted-by":"crossref","DOI":"10.1016\/j.asoc.2025.112695","article-title":"Finite-time convergent gradient-zeroing neurodynamic system for solving temporally-variant linear simultaneous equation","volume":"170","author":"Tan","year":"2025","journal-title":"Appl. Soft Comput."},{"issue":"8","key":"10.1016\/j.jfranklin.2026.108647_bib0049","doi-asserted-by":"crossref","first-page":"1940","DOI":"10.1109\/TAC.2009.2023779","article-title":"Performance analysis of gradient neural network exploited for online time-varying matrix inversion","volume":"54","author":"Zhang","year":"2009","journal-title":"IEEE Trans. Autom. Contr."},{"key":"10.1016\/j.jfranklin.2026.108647_bib0050","doi-asserted-by":"crossref","first-page":"7021","DOI":"10.1016\/j.jfranklin.2023.05.007","article-title":"Nonlinear function activated GNN versus ZNN for online solution of general linear matrix equations","volume":"360","author":"Tan","year":"2023","journal-title":"J. Frankl. Inst."},{"key":"10.1016\/j.jfranklin.2026.108647_bib0051","doi-asserted-by":"crossref","DOI":"10.1016\/j.chaos.2023.113279","article-title":"A fixed-time robust controller based on zeroing neural network for generalized projective synchronization of chaotic systems","volume":"169","author":"Xiao","year":"2023","journal-title":"Chaos Solitons Fract."},{"key":"10.1016\/j.jfranklin.2026.108647_bib0052","doi-asserted-by":"crossref","first-page":"840","DOI":"10.1016\/j.asoc.2017.09.016","article-title":"Nonconvex projection activated zeroing neurodynamic models for time-varying matrix pseudoinversion with accelerated finite-time convergence","volume":"62","author":"Jin","year":"2018","journal-title":"Appl. Soft Comput."}],"container-title":["Journal of the Franklin Institute"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0016003226002474?httpAccept=text\/xml","content-type":"text\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0016003226002474?httpAccept=text\/plain","content-type":"text\/plain","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2026,5,16]],"date-time":"2026-05-16T01:10:24Z","timestamp":1778893824000},"score":1,"resource":{"primary":{"URL":"https:\/\/linkinghub.elsevier.com\/retrieve\/pii\/S0016003226002474"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,5]]},"references-count":52,"journal-issue":{"issue":"8","published-print":{"date-parts":[[2026,5]]}},"alternative-id":["S0016003226002474"],"URL":"https:\/\/doi.org\/10.1016\/j.jfranklin.2026.108647","relation":{},"ISSN":["0016-0032"],"issn-type":[{"value":"0016-0032","type":"print"}],"subject":[],"published":{"date-parts":[[2026,5]]},"assertion":[{"value":"Elsevier","name":"publisher","label":"This article is maintained by"},{"value":"Gradient-activation recurrent neural network applied to temporally-variant matrix inversion","name":"articletitle","label":"Article Title"},{"value":"Journal of the Franklin Institute","name":"journaltitle","label":"Journal Title"},{"value":"https:\/\/doi.org\/10.1016\/j.jfranklin.2026.108647","name":"articlelink","label":"CrossRef DOI link to publisher maintained version"},{"value":"article","name":"content_type","label":"Content Type"},{"value":"\u00a9 2026 The Franklin Institute. Published by Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.","name":"copyright","label":"Copyright"}],"article-number":"108647"}}