{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,30]],"date-time":"2026-04-30T00:53:17Z","timestamp":1777510397005,"version":"3.51.4"},"reference-count":42,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2025,2,14]],"date-time":"2025-02-14T00:00:00Z","timestamp":1739491200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Information"],"abstract":"<jats:p>The Hopfield Recurrent Neural Network (HRNN) is a single-point descent metaheuristic that uses a single potential solution to explore the search space of optimization problems, whose constraints and objective function are aggregated into a typical energy function. The initial point is usually randomly initialized, then moved by applying operators, characterizing the discrete dynamics of the HRNN, which modify its position or direction. Like all single-point metaheuristics, HRNN has certain drawbacks, such as being more likely to get stuck in local optima or miss global optima due to the use of a single point to explore the search space. Moreover, it is more sensitive to the initial point and operator, which can influence the quality and diversity of solutions. Moreover, it can have difficulty with dynamic or noisy environments, as it can lose track of the optimal region or be misled by random fluctuations. To overcome these shortcomings, this paper introduces a population-based fuzzy version of the HRNN, namely Gaussian Takagi\u2013Sugeno Hopfield Recurrent Neural Network (G-TS-HRNN). For each neuron, the G-TS-HRNN associates an input fuzzy variable of d values, described by an appropriate Gaussian membership function that covers the universe of discourse. To build an instance of G-TS-HRNN(s) of size s, we generate s n-uplets of fuzzy values that present the premise of the Takagi\u2013Sugeno system. The consequents are the differential equations governing the dynamics of the HRNN obtained by replacing each premise fuzzy value with the mean of different Gaussians. The steady points of all the rule premises are aggregated using the fuzzy center of gravity equation, considering the level of activity of each rule. G-TS-HRNN is used to solve the random optimization method based on the support vector model. Compared with HRNN, G-TS-HRNN performs better on well-known data sets.<\/jats:p>","DOI":"10.3390\/info16020141","type":"journal-article","created":{"date-parts":[[2025,2,14]],"date-time":"2025-02-14T06:52:01Z","timestamp":1739515921000},"page":"141","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["G-TS-HRNN: Gaussian Takagi\u2013Sugeno Hopfield Recurrent Neural Network"],"prefix":"10.3390","volume":"16","author":[{"given":"Omar","family":"Bahou","sequence":"first","affiliation":[{"name":"Laboratory of Mathematics and Data Science, Polydisciplinary Faculty of Taza, Sidi Mohamed Ben Abdellah of Fez, Taza P.O. Box 1223, Morocco"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mohammed","family":"Roudani","sequence":"additional","affiliation":[{"name":"Laboratory of Mathematics and Data Science, Polydisciplinary Faculty of Taza, Sidi Mohamed Ben Abdellah of Fez, Taza P.O. Box 1223, Morocco"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3922-5592","authenticated-orcid":false,"given":"Karim","family":"El Moutaouakil","sequence":"additional","affiliation":[{"name":"Laboratory of Mathematics and Data Science, Polydisciplinary Faculty of Taza, Sidi Mohamed Ben Abdellah of Fez, Taza P.O. Box 1223, Morocco"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,2,14]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"748","DOI":"10.1109\/72.846746","article-title":"Neuro-fuzzy rule generation: Survey in soft computing framework","volume":"11","author":"Mitra","year":"2000","journal-title":"IEEE Trans. Neural Netw."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"136","DOI":"10.1016\/j.knosys.2018.04.014","article-title":"Recent advances in neuro-fuzzy system: A survey","volume":"152","author":"Shihabudheen","year":"2018","journal-title":"Knowl.-Based Syst."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"530","DOI":"10.1109\/91.728447","article-title":"Improving the interpretability of TSK fuzzy models by combining global learning and local learning","volume":"6","author":"Yen","year":"1998","journal-title":"IEEE Trans. Fuzzy Syst."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"055103","DOI":"10.1063\/5.0171983","article-title":"Active oscillatory associative memory","volume":"160","author":"Du","year":"2024","journal-title":"J. Chem. Phys."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"5735","DOI":"10.1007\/s00521-023-09373-4","article-title":"Enhancing the analog to digital converter using proteretic hopfield neural network","volume":"36","author":"Abdulrahman","year":"2024","journal-title":"Neural Comput. Appl."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Roudani, M., El Moutaouakil, K., Palade, V., Ba\u00efzri, H., Chellak, S., and Cheggour, M. (2024). Fuzzy Clustering SMOTE and Fuzzy Classifiers for Hidden Disease Predictions, Lecture Notes in Networks and Systems; Springer.","DOI":"10.1007\/978-3-031-67426-6_10"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"El Moutaouakil, K., Bouhanch, Z., Ahourag, A., Aberqi, A., and Karite, T. (2024). OPT-FRAC-CHN: Optimal Fractional Continuous Hopfield Network. Symmetry, 16.","DOI":"10.3390\/sym16070921"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"3294","DOI":"10.1109\/TNNLS.2019.2940920","article-title":"On the Working Principle of the Hopfield Neural Networks and its Equivalence to the GADIA in Optimization","volume":"31","author":"Uykan","year":"2020","journal-title":"IEEE Trans. Neural Netw. Learn. Syst."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"416","DOI":"10.1016\/j.neunet.2005.10.008","article-title":"The generalized quadratic knapsack problem. A neuronal network approach","volume":"19","year":"2006","journal-title":"Neural Netw."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"2179","DOI":"10.1016\/j.cor.2004.02.008","article-title":"A continuous Hopfield network equilibrium points algorithm","volume":"32","year":"2005","journal-title":"Comput. Oper. Res."},{"key":"ref_11","first-page":"134","article-title":"Intelligent optimal control of nonlinear diabetic population dynamics system using a genetic algorithm","volume":"1","year":"2024","journal-title":"Syst. Res. Inf. Technol."},{"key":"ref_12","first-page":"149","article-title":"Clustering Methods for Network Data Analysis in Programming","volume":"15","author":"Kurdyukov","year":"2023","journal-title":"Int. J. Commun. Netw. Inf. Secur."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1007\/s11075-022-01412-w","article-title":"A simple yet efficient two-step fifth-order weighted-Newton method for nonlinear models","volume":"93","author":"Singh","year":"2023","journal-title":"Numer. Algorithms"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"111403","DOI":"10.1016\/j.asoc.2024.111403","article-title":"Fuzzy clustering-based neural network based on linear fitting residual-driven weighted fuzzy clustering and convolutional regularization strategy","volume":"154","author":"Bu","year":"2024","journal-title":"Appl. Soft Comput."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Sharma, R., Goel, T., Tanveer, M., and Al-Dhaifallah, M. (IEEE Trans. Emerg. Top. Comput. Intell., 2025). Alzheimer\u2019s Disease Diagnosis Using Ensemble of Random Weighted Features and Fuzzy Least Square Twin Support Vector Machine, IEEE Trans. Emerg. Top. Comput. Intell., early access.","DOI":"10.1109\/TETCI.2024.3523714"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1287\/ijoc.11.1.15","article-title":"Neural networks for combinatorial optimization: A review of more than a decade of research","volume":"11","author":"Smith","year":"1999","journal-title":"Informs J. Comput."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"219","DOI":"10.1016\/S0925-2312(01)00337-X","article-title":"Hopfield neural networks for optimization: Study of the different dynamics","volume":"43","author":"Joya","year":"2002","journal-title":"Neurocomputing"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"747","DOI":"10.1109\/82.686695","article-title":"On the dynamics of discrete-time, continuous-state Hopfield neural networks","volume":"45","author":"Wang","year":"1998","journal-title":"IEEE Trans. Circuits Syst. II Analog Digit. Signal Process."},{"key":"ref_19","unstructured":"Demidowitsch, B.P., Maron, I.A., and Schuwalowa, E.S. (1980). Metodos Numericos de Analisis, Paraninfo."},{"key":"ref_20","unstructured":"Bagherzadeh, N., Kerola, T., Leddy, B., and Brice, R. (1987, January 21\u201324). On parallel execution of the traveling salesman problem on a neural network model. Proceedings of the IEEE International Conference on Neural Networks, San Diego, CA, USA."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Brandt, R.D., Wang, Y., Laub, A.J., and Mitra, S.K. (1988, January 24\u201327). Alternative networks for solving the traveling salesman problem and the list-matching problem. Proceedings of the International Conference on Neural Network (ICNN\u201988), San Diego, CA, USA.","DOI":"10.1109\/ICNN.1988.23945"},{"key":"ref_22","unstructured":"Wu, J.K. (1994). Neural Networks and Simulation Methods, Marcel Dekker."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"246","DOI":"10.1109\/81.224297","article-title":"Global convergence and suppression of spurious states of the Hopfield neural networks","volume":"40","author":"Abe","year":"1993","journal-title":"IEEE Trans. Circuits Syst."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"716","DOI":"10.1109\/72.701185","article-title":"On chaotic simulated annealing","volume":"9","author":"Wang","year":"1998","journal-title":"IEEE Trans. Neural Netw."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"425","DOI":"10.1007\/s10107-022-01887-4","article-title":"Non-asymptotic superlinear convergence of standard quasi-Newton methods","volume":"200","author":"Jin","year":"2023","journal-title":"Math. Program."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"677","DOI":"10.1080\/02331934.2021.1981897","article-title":"Iterative algorithm with self-adaptive step size for approximating the common solution of variational inequality and fixed point problems","volume":"72","author":"Ogwo","year":"2023","journal-title":"Optimization"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"114765","DOI":"10.1016\/j.cam.2022.114765","article-title":"Semismooth and smoothing Newton methods for nonlinear systems with complementarity constraints: Adaptivity and inexact resolution","volume":"420","author":"Gharbia","year":"2023","journal-title":"J. Comput. Appl. Math."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1171","DOI":"10.1007\/s11075-022-01463-z","article-title":"Generalized conformable fractional Newton-type method for solving nonlinear systems","volume":"93","author":"Candelario","year":"2023","journal-title":"Numer. Algorithms"},{"key":"ref_29","unstructured":"Dua, D., and Graff, C. (2019). UCI Machine Learning Repository, University of California, School of Information and Computer Science. Available online: http:\/\/archive.ics.uci.edu\/ml."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1145\/3580277","article-title":"Certifying zeros of polynomial systems using interval arithmetic","volume":"49","author":"Breiding","year":"2023","journal-title":"ACM Trans. Math. Softw."},{"key":"ref_31","unstructured":"Bhuvaneswari, S., and Karthikeyan, S. (2023, January 28\u201329). A Comprehensive Research of Breast Cancer Detection Using Machine Learning, Clustering and Optimization Techniques. Proceedings of the 2023 IEEE International Conference on Data Science and Network Security (ICDSNS), Tiptur, India."},{"key":"ref_32","unstructured":"Zimmermann, H.-J. (2010). Fuzzy Set Theory\u2014And Its Applications, Springer. [4th ed.]."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Rajafillah, C., El Moutaouakil, K., Patriciu, A.M., Yahyaouy, A., and Riffi, J. (2024). INT-FUP: Intuitionistic Fuzzy Pooling. Mathematics, 12.","DOI":"10.3390\/math12111740"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1016\/0165-0114(88)90113-3","article-title":"Structure identification of fuzzy model","volume":"28","author":"Sugeno","year":"1988","journal-title":"Fuzzy Sets Syst."},{"key":"ref_35","unstructured":"Sugeno, M. (1988). Fuzzy Control, North-Holland, Publishing Co."},{"key":"ref_36","first-page":"1","article-title":"Takagi-sugeno fuzzy modeling for process control","volume":"262","author":"Mehran","year":"2008","journal-title":"Ind. Autom. Robot. Artif. Intell."},{"key":"ref_37","unstructured":"Kawamoto, S., Tada, K., Ishigame, A., and Taniguchi, T. (1992, January 8\u201312). An approach to stability analysis of second order fuzzy systems. Proceedings of the IEEE International Conference on Fuzzy Systems, San Diego, CA, USA."},{"key":"ref_38","unstructured":"Tanaka, K. (1994). A Theory of Advanced Fuzzy Control, Springer."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"415","DOI":"10.1098\/rsta.1909.0016","article-title":"Functions of positive and negative type, and their connection the theory of integral equations","volume":"209","author":"Mercer","year":"1909","journal-title":"Philos. Trans. R. Soc. Lond."},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Sch\u00f6lkopf, B., Smola, A.J., and Bach, F. (2002). Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond, MIT Press.","DOI":"10.7551\/mitpress\/4175.001.0001"},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Pandey, R.K., and Agrawal, O.P. (2015, January 2\u20135). Comparison of four numerical schemes for isoperimetric constraint fractional variational problems with A-operator. Proceedings of the International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Boston, MA, USA.","DOI":"10.1115\/DETC2015-46570"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"692","DOI":"10.1016\/j.jksus.2017.12.017","article-title":"Approximations of fractional integrals and Caputo derivatives with application in solving Abel\u2019s integral equations","volume":"31","author":"Kumar","year":"2019","journal-title":"J. King Saud Univ.-Sci."}],"container-title":["Information"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2078-2489\/16\/2\/141\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T16:34:24Z","timestamp":1760027664000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2078-2489\/16\/2\/141"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,2,14]]},"references-count":42,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2025,2]]}},"alternative-id":["info16020141"],"URL":"https:\/\/doi.org\/10.3390\/info16020141","relation":{},"ISSN":["2078-2489"],"issn-type":[{"value":"2078-2489","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,2,14]]}}}