{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,13]],"date-time":"2026-02-13T13:06:06Z","timestamp":1770987966305,"version":"3.50.1"},"reference-count":44,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2022,10,7]],"date-time":"2022-10-07T00:00:00Z","timestamp":1665100800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2022,10,7]],"date-time":"2022-10-07T00:00:00Z","timestamp":1665100800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Neural Comput &amp; Applic"],"published-print":{"date-parts":[[2023,2]]},"DOI":"10.1007\/s00521-022-07728-x","type":"journal-article","created":{"date-parts":[[2022,10,7]],"date-time":"2022-10-07T14:03:54Z","timestamp":1665151434000},"page":"3101-3114","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":14,"title":["A novel fractional operator application for neural networks using proportional Caputo derivative"],"prefix":"10.1007","volume":"35","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7883-3131","authenticated-orcid":false,"given":"Gokhan","family":"Altan","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Sertan","family":"Alkan","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Dumitru","family":"Baleanu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2022,10,7]]},"reference":[{"key":"7728_CR1","doi-asserted-by":"publisher","first-page":"19","DOI":"10.1016\/j.neunet.2017.02.007","volume":"89","author":"J Wang","year":"2017","unstructured":"Wang J, Wen Y, Gou Y, Ye Z, Chen H (2017) Fractional-order gradient descent learning of BP neural networks with Caputo derivative. Neural Netw 89:19\u201330. https:\/\/doi.org\/10.1016\/j.neunet.2017.02.007","journal-title":"Neural Netw"},{"key":"7728_CR2","doi-asserted-by":"publisher","first-page":"15","DOI":"10.1016\/j.neucom.2014.12.031","volume":"154","author":"H Wang","year":"2015","unstructured":"Wang H, Yu Y, Wen G, Zhang S, Yu J (2015) Global stability analysis of fractional-order Hopfield neural networks with time delay. Neurocomputing 154:15\u201323. https:\/\/doi.org\/10.1016\/j.neucom.2014.12.031","journal-title":"Neurocomputing"},{"key":"7728_CR3","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1155\/2018\/7361628","volume":"2018","author":"C Bao","year":"2018","unstructured":"Bao C, Pu Y, Zhang Y (2018) Fractional-order deep backpropagation neural network. Comput Intell Neurosci 2018:1\u201310. https:\/\/doi.org\/10.1155\/2018\/7361628","journal-title":"Comput Intell Neurosci"},{"key":"7728_CR4","first-page":"337","volume":"9","author":"MA Marchaud","year":"1925","unstructured":"Marchaud MA (1925) Sur les d\u00e9riv\u00e9es et sur les diff\u00e9rences des fonctions de variables r\u00e9elles. J Math Pures Appl 9:337\u2013425","journal-title":"J Math Pures Appl"},{"issue":"5","key":"7728_CR5","doi-asserted-by":"publisher","first-page":"529","DOI":"10.1111\/j.1365-246X.1967.tb02303.x","volume":"13","author":"M Caputo","year":"1967","unstructured":"Caputo M (1967) Linear models of dissipation whose q is almost frequency independent-ii. Geophys J Int 13(5):529\u2013539","journal-title":"Geophys J Int"},{"issue":"21","key":"7728_CR6","first-page":"1","volume":"13","author":"J Liouville","year":"1832","unstructured":"Liouville J (1832) M\u00e9moire sur quelques qu\u00e9stions de g\u00e9ometrie et de m\u00e9canique. J l\u2019\u00e9cole Polytech 13(21):1\u201369","journal-title":"J l\u2019\u00e9cole Polytech"},{"key":"7728_CR7","doi-asserted-by":"publisher","unstructured":"Hilfer R (2000). Applications of fractional calculus in physics. https:\/\/doi.org\/10.1142\/3779","DOI":"10.1142\/3779"},{"key":"7728_CR8","doi-asserted-by":"publisher","unstructured":"Meerschaert MM, Sikorskii A (2011) Stochastic Models for Fractional Calculus. De Gruyter, Berlin. https:\/\/doi.org\/10.1515\/9783110258165","DOI":"10.1515\/9783110258165"},{"issue":"2","key":"7728_CR9","doi-asserted-by":"publisher","first-page":"73","DOI":"10.12785\/pfda\/010201","volume":"1","author":"M Caputo","year":"2015","unstructured":"Caputo M, Fabrizio M (2015) A new definition of fractional derivative without singular kernel. Prog Fract Differ Appl 1(2):73\u201385. https:\/\/doi.org\/10.12785\/pfda\/010201","journal-title":"Prog Fract Differ Appl"},{"issue":"2","key":"7728_CR10","doi-asserted-by":"publisher","first-page":"763","DOI":"10.2298\/TSCI160111018A","volume":"20","author":"A Atangana","year":"2016","unstructured":"Atangana A, Baleanu D (2016) New fractional derivatives with nonlocal and non-singular kernel:theory and application to heat transfer model. Therm Sci 20(2):763\u2013769. https:\/\/doi.org\/10.2298\/TSCI160111018AarXiv:1602.03408","journal-title":"Therm Sci"},{"issue":"3","key":"7728_CR11","doi-asserted-by":"publisher","first-page":"360","DOI":"10.3390\/math8030360","volume":"8","author":"D Baleanu","year":"2020","unstructured":"Baleanu D, Fernandez A, Akg\u00fcl A (2020) On a fractional operator combining proportional and classical differintegrals. Mathematics 8(3):360. https:\/\/doi.org\/10.3390\/math8030360","journal-title":"Mathematics"},{"key":"7728_CR12","doi-asserted-by":"publisher","DOI":"10.1140\/epjs\/s11734-022-00455-3","author":"E Viera-Martin","year":"2022","unstructured":"Viera-Martin E, G\u00f3mez-Aguilar JF, Sol\u00eds-P\u00e9rez JE, Hern\u00e1ndez-P\u00e9rez JA, Escobar-Jim\u00e9nez RF (2022) Artificial neural networks: a practical review of applications involving fractional calculus. Eur Phys J Spec Top. https:\/\/doi.org\/10.1140\/epjs\/s11734-022-00455-3","journal-title":"Eur Phys J Spec Top"},{"key":"7728_CR13","doi-asserted-by":"publisher","first-page":"14560","DOI":"10.1109\/ACCESS.2017.2679185","volume":"5","author":"J Wang","year":"2017","unstructured":"Wang J, Yang G, Zhang B, Sun Z, Liu Y, Wang J (2017) Convergence analysis of Caputo-type fractional order complex-valued neural networks. IEEE Access 5:14560\u201314571. https:\/\/doi.org\/10.1109\/ACCESS.2017.2679185","journal-title":"IEEE Access"},{"key":"7728_CR14","doi-asserted-by":"publisher","first-page":"260","DOI":"10.1016\/j.neucom.2018.10.090","volume":"391","author":"M-R Chen","year":"2020","unstructured":"Chen M-R, Chen B-P, Zeng G-Q, Lu K-D, Chu P (2020) An adaptive fractional-order BP neural network based on extremal optimization for handwritten digits recognition. Neurocomputing 391:260\u2013272. https:\/\/doi.org\/10.1016\/j.neucom.2018.10.090","journal-title":"Neurocomputing"},{"issue":"6","key":"7728_CR15","doi-asserted-by":"publisher","first-page":"1655","DOI":"10.1016\/j.camwa.2012.01.019","volume":"64","author":"GA Anastassiou","year":"2012","unstructured":"Anastassiou GA (2012) Fractional neural network approximation. Comput Math Appl 64(6):1655\u20131676. https:\/\/doi.org\/10.1016\/j.camwa.2012.01.019","journal-title":"Comput Math Appl"},{"issue":"2","key":"7728_CR16","doi-asserted-by":"publisher","first-page":"593","DOI":"10.1007\/s00034-017-0572-z","volume":"37","author":"S Khan","year":"2018","unstructured":"Khan S, Ahmad J, Naseem I, Moinuddin M (2018) A novel fractional gradient-based learning algorithm for recurrent neural networks. Circuits Syst Sig Process 37(2):593\u2013612. https:\/\/doi.org\/10.1007\/s00034-017-0572-z","journal-title":"Circuits Syst Sig Process"},{"key":"7728_CR17","doi-asserted-by":"publisher","unstructured":"Chen Y, Zhao G (2019) A Caputo-type fractional-order gradient descent learning of deep BP neural networks. In: 2019 IEEE 3rd advanced information management, communicates, electronic and automation control conference (IMCEC), pp 546\u2013550. IEEE. https:\/\/doi.org\/10.1109\/IMCEC46724.2019.8984089. https:\/\/ieeexplore.ieee.org\/document\/8984089\/","DOI":"10.1109\/IMCEC46724.2019.8984089"},{"key":"7728_CR18","doi-asserted-by":"publisher","DOI":"10.1016\/j.chaos.2019.109444","volume":"130","author":"CJ Zu\u00f1iga Aguilar","year":"2020","unstructured":"Zu\u00f1iga Aguilar CJ, G\u00f3mez-Aguilar JF, Alvarado-Mart\u00ednez VM, Romero-Ugalde HM (2020) Fractional order neural networks for system identification. Chaos Solitons Fractals 130:109444. https:\/\/doi.org\/10.1016\/j.chaos.2019.109444","journal-title":"Chaos Solitons Fractals"},{"issue":"10","key":"7728_CR19","doi-asserted-by":"publisher","first-page":"2319","DOI":"10.1109\/TNNLS.2016.2582512","volume":"28","author":"Y-F Pu","year":"2017","unstructured":"Pu Y-F, Yi Z, Zhou J-L (2017) Fractional Hopfield neural networks: fractional dynamic associative recurrent neural networks. IEEE Trans Neural Netw Learn Syst 28(10):2319\u20132333. https:\/\/doi.org\/10.1109\/TNNLS.2016.2582512","journal-title":"IEEE Trans Neural Netw Learn Syst"},{"key":"7728_CR20","doi-asserted-by":"crossref","unstructured":"PU Y-F, Wang J (2019) Fractional-order backpropagation neural networks: modified fractional-order steepest descent method for family of backpropagation neural networks arXiv:1906.09524","DOI":"10.1631\/FITEE.1900593"},{"issue":"4","key":"7728_CR21","doi-asserted-by":"publisher","first-page":"1201","DOI":"10.1007\/s00521-017-3153-y","volume":"31","author":"A Sharafian","year":"2019","unstructured":"Sharafian A, Ghasemi R (2019) Fractional neural observer design for a class of nonlinear fractional chaotic systems. Neural Comput Appl 31(4):1201\u20131213. https:\/\/doi.org\/10.1007\/s00521-017-3153-y","journal-title":"Neural Comput Appl"},{"key":"7728_CR22","doi-asserted-by":"publisher","first-page":"327","DOI":"10.1016\/j.neucom.2021.02.082","volume":"442","author":"Z Yang","year":"2021","unstructured":"Yang Z, Zhang J, Hu J, Mei J (2021) New results on finite-time stability for fractional-order neural networks with proportional delay. Neurocomputing 442:327\u2013336. https:\/\/doi.org\/10.1016\/j.neucom.2021.02.082","journal-title":"Neurocomputing"},{"key":"7728_CR23","doi-asserted-by":"publisher","DOI":"10.1007\/s00521-021-06765-2","author":"Z Yu","year":"2022","unstructured":"Yu Z, Sun G, Lv J (2022) A fractional-order momentum optimization approach of deep neural networks. Neural Comput Appl. https:\/\/doi.org\/10.1007\/s00521-021-06765-2","journal-title":"Neural Comput Appl"},{"key":"7728_CR24","doi-asserted-by":"publisher","DOI":"10.1007\/bf00993164","author":"AR Barron","year":"1994","unstructured":"Barron AR (1994) Approximation and estimation bounds for artificial neural networks. Mach Learn. https:\/\/doi.org\/10.1007\/bf00993164","journal-title":"Mach Learn"},{"issue":"3","key":"7728_CR25","doi-asserted-by":"publisher","first-page":"267","DOI":"10.1198\/tech.2003.s770","volume":"45","author":"ER Ziegel","year":"2003","unstructured":"Ziegel ER (2003) The elements of statistical learning. Technometrics 45(3):267\u2013268. https:\/\/doi.org\/10.1198\/tech.2003.s770","journal-title":"Technometrics"},{"key":"7728_CR26","unstructured":"Haykin S (1999) Neural networks: a comprehensive foundation (3rd Edn)"},{"key":"7728_CR27","doi-asserted-by":"publisher","DOI":"10.1006\/acha.1998.0248","author":"EJ Cand\u00e8s","year":"1999","unstructured":"Cand\u00e8s EJ (1999) Harmonic analysis of neural networks. Appl Comput Harmonic Anal. https:\/\/doi.org\/10.1006\/acha.1998.0248","journal-title":"Appl Comput Harmonic Anal"},{"issue":"31","key":"7728_CR28","doi-asserted-by":"publisher","first-page":"1","DOI":"10.17485\/ijst\/2016\/v9i31\/84278","volume":"9","author":"A Muayad","year":"2016","unstructured":"Muayad A, Irtefaa AN (2016) Ridge regression using artificial neural network. Indian J Sci Technol 9(31):1\u20135. https:\/\/doi.org\/10.17485\/ijst\/2016\/v9i31\/84278","journal-title":"Indian J Sci Technol"},{"key":"7728_CR29","first-page":"1929","volume":"15","author":"N Srivastava","year":"2014","unstructured":"Srivastava N, Hinton G, Krizhevsky A, Sutskever I, Salakhutdinov R (2014) Dropout: a simple way to prevent neural networks from overfitting. J Mach Learn Res 15:1929\u20131958","journal-title":"J Mach Learn Res"},{"key":"7728_CR30","unstructured":"Cavazza J, Haeffele BD, Lane C, Morerio P, Murino V, Vidal R (2018) Dropout as a low-rank regularizer for matrix factorization. In: international conference on artificial intelligence and statistics, AISTATS 2018, pp 435\u2013444"},{"key":"7728_CR31","unstructured":"Rumelhart DE, Hinton GE, Williams RJ (1986) Learning internal representations by error propagation. In: Rumenhart D, McCelland J (eds) Explorations in the micro-structure of cognition, vol 1, Foundations (Rumenhart edn), pp 318\u2013362. MIT Press, Cambridge"},{"issue":"1","key":"7728_CR32","doi-asserted-by":"publisher","first-page":"55","DOI":"10.1080\/00401706.1970.10488634","volume":"12","author":"AE Hoerl","year":"1970","unstructured":"Hoerl AE, Kennard RW (1970) Ridge regression: biased estimation for nonorthogonal problems. Technometrics 12(1):55\u201367. https:\/\/doi.org\/10.1080\/00401706.1970.10488634","journal-title":"Technometrics"},{"issue":"1","key":"7728_CR33","doi-asserted-by":"publisher","first-page":"247","DOI":"10.1186\/s13662-017-1306-z","volume":"2017","author":"F Jarad","year":"2017","unstructured":"Jarad F, U\u011furlu E, Abdeljawad T, Baleanu D (2017) On a new class of fractional operators. Adv Differ Equ 2017(1):247. https:\/\/doi.org\/10.1186\/s13662-017-1306-z","journal-title":"Adv Differ Equ"},{"key":"7728_CR34","doi-asserted-by":"publisher","first-page":"98","DOI":"10.1016\/j.neunet.2014.03.012","volume":"55","author":"H Wang","year":"2014","unstructured":"Wang H, Yu Y, Wen G (2014) Stability analysis of fractional-order Hopfield neural networks with time delays. Neural Netw 55:98\u2013109. https:\/\/doi.org\/10.1016\/j.neunet.2014.03.012","journal-title":"Neural Netw"},{"key":"7728_CR35","unstructured":"Verhulst P-F (2009) Notice sur la loi que la population poursuit dans son accroissement. Correspondance math\u00e9matique et physique 10: 113\u2013121. Technical report, retrieved on 09\/08"},{"key":"7728_CR36","unstructured":"Dugas C, Bengio Y, B\u00e9lisle F, Nadeau C, Garcia R (2001) Incorporating second-order functional knowledge for better option pricing. In: advances in neural information processing systems"},{"key":"7728_CR37","unstructured":"Ramachandran P, Zoph B, Le QV (2017) Searching for activation functions arXiv:1710.05941"},{"key":"7728_CR38","unstructured":"Mhaskar HN, Micchelli CA (1994) How to choose an activation function. In: Advances in Neural Information Processing Systems, pp 319\u2013326"},{"key":"7728_CR39","unstructured":"Nwankpa C, Ijomah W, Gachagan A, Marshall S (2018) Activation functions: comparison of trends in practice and research for deep learning arXiv:1811.03378"},{"key":"7728_CR40","unstructured":"LeCun Y, Cortes C (2010) MNIST handwritten digit database. AT &T Labs [Online]. Available: http:\/\/yann. lecun. com\/exdb\/mnist"},{"key":"7728_CR41","unstructured":"Fisher RA (2021) Iris data set, UCI repository of machine learning databases. https:\/\/archive.ics.uci.edu\/ml\/datasets\/iris"},{"issue":"3","key":"7728_CR42","doi-asserted-by":"publisher","first-page":"837","DOI":"10.2307\/2531595","volume":"44","author":"ER DeLong","year":"1988","unstructured":"DeLong ER, DeLong DM, Clarke-Pearson DL (1988) Comparing the areas under two or more correlated receiver operating characteristic curves: a nonparametric approach. Biometrics 44(3):837. https:\/\/doi.org\/10.2307\/2531595","journal-title":"Biometrics"},{"key":"7728_CR43","unstructured":"Krizhevsky A, Sutskever I, Hinton GE (2012) ImageNet classification with deep convolutional neural networks. In: advances in neural information processing systems"},{"key":"7728_CR44","unstructured":"Simonyan K, Zisserman A (2015) Very deep convolutional networks for large-scale image recognition. In: 3rd international conference on learning representations, ICLR 2015 - Conference track proceedings"}],"container-title":["Neural Computing and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00521-022-07728-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00521-022-07728-x\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00521-022-07728-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,1,27]],"date-time":"2023-01-27T05:02:00Z","timestamp":1674795720000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00521-022-07728-x"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,10,7]]},"references-count":44,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2023,2]]}},"alternative-id":["7728"],"URL":"https:\/\/doi.org\/10.1007\/s00521-022-07728-x","relation":{},"ISSN":["0941-0643","1433-3058"],"issn-type":[{"value":"0941-0643","type":"print"},{"value":"1433-3058","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,10,7]]},"assertion":[{"value":"23 September 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"12 August 2022","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"7 October 2022","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that they have no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}},{"value":"Not applicable.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Ethics approval"}},{"value":"Not applicable.","order":4,"name":"Ethics","group":{"name":"EthicsHeading","label":"Consent to participate"}},{"value":"Not applicable.","order":5,"name":"Ethics","group":{"name":"EthicsHeading","label":"Consent for publication"}}]}}