{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,10]],"date-time":"2026-04-10T08:08:38Z","timestamp":1775808518451,"version":"3.50.1"},"reference-count":40,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2026,3,4]],"date-time":"2026-03-04T00:00:00Z","timestamp":1772582400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by-nc-nd\/4.0"},{"start":{"date-parts":[[2026,3,11]],"date-time":"2026-03-11T00:00:00Z","timestamp":1773187200000},"content-version":"vor","delay-in-days":7,"URL":"https:\/\/creativecommons.org\/licenses\/by-nc-nd\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Complex Intell. Syst."],"published-print":{"date-parts":[[2026,4]]},"DOI":"10.1007\/s40747-026-02233-3","type":"journal-article","created":{"date-parts":[[2026,3,4]],"date-time":"2026-03-04T19:33:36Z","timestamp":1772652816000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["ET-KAN: an energy-based transformer model with Kolmogorov\u2013Arnold network for image reconstruction"],"prefix":"10.1007","volume":"12","author":[{"given":"Chiara","family":"Marullo","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0009-8427-292X","authenticated-orcid":false,"given":"Giuseppe","family":"Buonaiuto","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Francesco","family":"Gargiulo","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Massimo","family":"Esposito","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2026,3,4]]},"reference":[{"key":"2233_CR1","unstructured":"Kolmogorov AN (1956) On the representation of continuous functions of several variables as superpositions of continuous functions of a smaller number of variables. Dokl Akad Nauk 108(2):179\u2013182"},{"key":"2233_CR2","doi-asserted-by":"publisher","first-page":"653","DOI":"10.1007\/s00365-009-9054-2","volume":"30","author":"J Braun","year":"2009","unstructured":"Braun J, Griebel M (2009) On a constructive proof of Kolmogorov\u2019s superposition theorem. Constr Approx 30:653\u2013675","journal-title":"Constr Approx"},{"key":"2233_CR3","unstructured":"Liu Z, Wang Y, Vaidya S, Ruehle F, Halverson J, Solja\u010di\u0107 M, Hou TY, Tegmark M (2024) KAN: Kolmogorov\u2013Arnold networks. arXiv preprint arXiv:2404.19756"},{"key":"2233_CR4","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511623257","volume-title":"Modeling brain function: the world of attractor neural networks","author":"DJ Amit","year":"1989","unstructured":"Amit DJ, Amit DJ (1989) Modeling brain function: the world of attractor neural networks. Cambridge University Press, Cambridge"},{"key":"2233_CR5","doi-asserted-by":"publisher","DOI":"10.1088\/1751-8121\/abca75","volume":"53","author":"E Agliari","year":"2020","unstructured":"Agliari E, Barra A, Sollich P, Zdeborov\u00e1 L (2020) Machine learning and statistical physics: preface. J Phys A Math Theor 53:500401","journal-title":"J Phys A Math Theor"},{"key":"2233_CR6","volume-title":"Spin glass theory and beyond: an introduction to the replica method and its applications","author":"M M\u00e9zard","year":"1987","unstructured":"M\u00e9zard M, Parisi G, Virasoro MA (1987) Spin glass theory and beyond: an introduction to the replica method and its applications, vol 9. World Scientific Publishing Company, Singapore"},{"key":"2233_CR7","doi-asserted-by":"publisher","DOI":"10.1093\/oso\/9780198530237.001.0001","volume-title":"Theory of neural information processing systems","author":"ACC Coolen","year":"2005","unstructured":"Coolen ACC, K\u00fchn R, Sollich P (2005) Theory of neural information processing systems. Oxford University Press, Oxford"},{"issue":"1","key":"2233_CR8","doi-asserted-by":"publisher","first-page":"34","DOI":"10.3390\/e23010034","volume":"23","author":"C Marullo","year":"2020","unstructured":"Marullo C, Agliari E (2020) Boltzmann machines as generalized Hopfield networks: a review of recent results and outlooks. Entropy 23(1):34","journal-title":"Entropy"},{"issue":"8","key":"2233_CR9","doi-asserted-by":"publisher","first-page":"2554","DOI":"10.1073\/pnas.79.8.2554","volume":"79","author":"JJ Hopfield","year":"1982","unstructured":"Hopfield JJ (1982) Neural networks and physical systems with emergent collective computational abilities. Proc Natl Acad Sci 79(8):2554\u20132558","journal-title":"Proc Natl Acad Sci"},{"issue":"10","key":"2233_CR10","doi-asserted-by":"publisher","first-page":"2909","DOI":"10.1088\/0305-4470\/20\/10\/035","volume":"20","author":"A Bruce","year":"1987","unstructured":"Bruce A, Gardner E, Wallace D (1987) Dynamics and statistical mechanics of the Hopfield model. J Phys A Math Gen 20(10):2909","journal-title":"J Phys A Math Gen"},{"key":"2233_CR11","unstructured":"Ramsauer H, Sch\u00e4fl B, Lehner J, Seidl P, Widrich M, Adler T, Gruber L, Holzleitner M, Pavlovi\u0107 M, Sandve GK et al (2020) Hopfield networks is all you need. arXiv preprint arXiv:2008.02217"},{"key":"2233_CR12","first-page":"23549","volume":"36","author":"B Hoover","year":"2024","unstructured":"Hoover B, Liang Y, Pham B, Panda R, Strobelt H, Chau DH, Zaki M, Krotov D (2024) Energy transformer. Adv Neural Inf Process Syst 36:23549\u201323576","journal-title":"Adv Neural Inf Process Syst"},{"key":"2233_CR13","doi-asserted-by":"publisher","DOI":"10.1142\/S0129055X24300115","author":"C Marullo","year":"2024","unstructured":"Marullo C, Buonaiuto G, Ciani D, Gargiulo F, Esposito M (2024) Exploring research trends in modern machine learning through a statistical mechanics lens: an Ising-inspired modeling perspective. Rev Math Phys. https:\/\/doi.org\/10.1142\/S0129055X24300115","journal-title":"Rev Math Phys"},{"key":"2233_CR14","unstructured":"Vaswani A, Shazeer N, Parmar N, Uszkoreit J, Jones L, Gomez AN, Kaiser \u0141, Polosukhin I (2017) Attention is all you need. In: Advances in neural information processing systems, vol 30"},{"key":"2233_CR15","doi-asserted-by":"crossref","unstructured":"Gillioz A, Casas J, Mugellini E, Abou\u00a0Khaled O (2020) Overview of the transformer-based models for NLP tasks. In: 2020 15th conference on computer science and information systems (FedCSIS). IEEE, pp 179\u2013183","DOI":"10.15439\/2020F20"},{"key":"2233_CR16","unstructured":"Topal MO, Bas A, Heerden I (2021) Exploring transformers in natural language generation: GPT, BERT, and XLNET. arXiv preprint arXiv:2102.08036"},{"issue":"15","key":"2233_CR17","doi-asserted-by":"publisher","first-page":"2112","DOI":"10.1093\/bioinformatics\/btab083","volume":"37","author":"Y Ji","year":"2021","unstructured":"Ji Y, Zhou Z, Liu H, Davuluri RV (2021) DNABERT: pre-trained bidirectional encoder representations from transformers model for DNA-language in genome. Bioinformatics 37(15):2112\u20132120","journal-title":"Bioinformatics"},{"key":"2233_CR18","unstructured":"Zong M, Krishnamachari B (2022) A survey on GPT-3. arXiv preprint arXiv:2212.00857"},{"issue":"4","key":"2233_CR19","doi-asserted-by":"publisher","first-page":"80","DOI":"10.3390\/asi5040080","volume":"5","author":"A Chaudhari","year":"2022","unstructured":"Chaudhari A, Bhatt C, Krishna A, Mazzeo PL (2022) ViTFER: facial emotion recognition with vision transformers. Appl Syst Innov 5(4):80","journal-title":"Appl Syst Innov"},{"key":"2233_CR20","doi-asserted-by":"publisher","DOI":"10.1016\/j.bspc.2022.104376","volume":"80","author":"J Yang","year":"2023","unstructured":"Yang J, Tu J, Zhang X, Yu S, Zheng X (2023) TSE DeepLab: an efficient visual transformer for medical image segmentation. Biomed Signal Process Control 80:104376","journal-title":"Biomed Signal Process Control"},{"key":"2233_CR21","doi-asserted-by":"publisher","first-page":"33","DOI":"10.1007\/s41095-021-0247-3","volume":"8","author":"Y Xu","year":"2022","unstructured":"Xu Y, Wei H, Lin M, Deng Y, Sheng K, Zhang M, Tang F, Dong W, Huang F, Xu C (2022) Transformers in computational visual media: a survey. Comput Vis Media 8:33\u201362","journal-title":"Comput Vis Media"},{"key":"2233_CR22","unstructured":"Krotov D, Hopfield J (2020) Large associative memory problem in neurobiology and machine learning. arXiv preprint arXiv:2008.06996"},{"issue":"7","key":"2233_CR23","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.132.077301","volume":"132","author":"C Lucibello","year":"2024","unstructured":"Lucibello C, M\u00e9zard M (2024) Exponential capacity of dense associative memories. Phys Rev Lett 132(7):077301","journal-title":"Phys Rev Lett"},{"key":"2233_CR24","doi-asserted-by":"publisher","first-page":"8825","DOI":"10.1007\/s00521-025-11035-6","volume":"37","author":"M Abdelsattar","year":"2025","unstructured":"Abdelsattar M, AbdelMoety A, Emad-Eldeen A (2025) Advanced machine learning techniques for predicting power generation and fault detection in solar photovoltaic systems. Neural Comput Appl 37:8825\u20138844","journal-title":"Neural Comput Appl"},{"issue":"1","key":"2233_CR25","doi-asserted-by":"publisher","first-page":"0317619","DOI":"10.1371\/journal.pone.0317619","volume":"20","author":"M Abdelsattar","year":"2025","unstructured":"Abdelsattar M, Ismeil MA, Menoufi K, AbdelMoety A, Emad-Eldeen A (2025) Evaluating machine learning and deep learning models for predicting wind turbine power output from environmental factors. PLoS One 20(1):0317619","journal-title":"PLoS One"},{"key":"2233_CR26","doi-asserted-by":"publisher","first-page":"14236","DOI":"10.1038\/s41598-025-96945-0","volume":"15","author":"ZA Abdallah Rabee","year":"2025","unstructured":"Abdallah Rabee ZA, AbdelMoety A, Abdelsallam A, Ali M (2025) Comparative analysis of automated foul detection in football using deep learning architectures. Sci Rep 15:14236","journal-title":"Sci Rep"},{"key":"2233_CR27","doi-asserted-by":"publisher","first-page":"4136","DOI":"10.1109\/ACCESS.2024.3525183","volume":"13","author":"M Abdelsattar","year":"2025","unstructured":"Abdelsattar M, Abdelmoety A, Ismeil MA, Emad-Eldeen A (2025) Automated defect detection in solar cell images using deep learning algorithms. IEEE Access 13:4136\u20134157","journal-title":"IEEE Access"},{"key":"2233_CR28","volume-title":"Eureka: software review","author":"R Dub\u010d\u00e1kov\u00e1","year":"2011","unstructured":"Dub\u010d\u00e1kov\u00e1 R (2011) Eureka: software review. Springer, Berlin\/Heidelberg"},{"key":"2233_CR29","unstructured":"Dugan O, Dangovski R, Costa A, Kim S, Goyal P, Jacobson J, Solja\u010di\u0107 M (2020) OccamNet: a fast neural model for symbolic regression at scale. arXiv preprint arXiv:2007.10784"},{"key":"2233_CR30","unstructured":"Martius G, Lampert CH (2016) Extrapolation and learning equations. arXiv preprint arXiv:1610.02995"},{"key":"2233_CR31","first-page":"24912","volume":"34","author":"T Mundhenk","year":"2021","unstructured":"Mundhenk T, Landajuela M, Glatt R, Santiago CP, Petersen BK et al (2021) Symbolic regression via deep reinforcement learning enhanced genetic programming seeding. Adv Neural Inf Process Syst 34:24912\u201324923","journal-title":"Adv Neural Inf Process Syst"},{"key":"2233_CR32","unstructured":"Xu K, Chen L, Wang S (2024) Kolmogorov\u2013Arnold networks for time series: bridging predictive power and interpretability. arXiv preprint arXiv:2406.02496"},{"key":"2233_CR33","unstructured":"Malhotra P, Vig L, Shroff G, Agarwal P et al (2015) Long short term memory networks for anomaly detection in time series. In: ESANN, vol 2015, p 89"},{"key":"2233_CR34","doi-asserted-by":"crossref","unstructured":"Genet R, Inzirillo H (2024) TKAN: temporal Kolmogorov\u2013Arnold networks. arXiv preprint arXiv:2405.07344","DOI":"10.2139\/ssrn.4825654"},{"key":"2233_CR35","doi-asserted-by":"crossref","unstructured":"Wang Y, Sun J, Bai J, Anitescu C, Eshaghi MS, Zhuang X, Rabczuk T, Liu Y (2024) Kolmogorov\u2013Arnold informed neural network: a physics-informed deep learning framework for solving PDES based on Kolmogorov\u2013Arnold networks. arXiv preprint arXiv:2406.11045","DOI":"10.2139\/ssrn.4868150"},{"issue":"1","key":"2233_CR36","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1080\/00401706.1974.10489142","volume":"16","author":"S Wold","year":"1974","unstructured":"Wold S (1974) Spline functions in data analysis. Technometrics 16(1):1\u201311","journal-title":"Technometrics"},{"key":"2233_CR37","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-04919-8","volume-title":"B\u00e9zier and B-spline techniques","author":"H Prautzsch","year":"2002","unstructured":"Prautzsch H, Boehm W, Paluszny M (2002) B\u00e9zier and B-spline techniques, vol 6. Springer, Berlin\/Heidelberg"},{"key":"2233_CR38","volume-title":"Numerical mathematics","author":"A Quarteroni","year":"2006","unstructured":"Quarteroni A, Sacco R, Saleri F (2006) Numerical mathematics, vol 37. Springer, Berlin\/Heidelberg"},{"key":"2233_CR39","unstructured":"Krizhevsky A, Hinton G et al (2009) Learning multiple layers of features from tiny images"},{"key":"2233_CR40","doi-asserted-by":"crossref","unstructured":"Hore A, Ziou D (2010) Image quality metrics: PSNR VS. SSIM. In: 2010 20th International conference on pattern recognition. IEEE, pp 2366\u20132369","DOI":"10.1109\/ICPR.2010.579"}],"container-title":["Complex & Intelligent Systems"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s40747-026-02233-3","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40747-026-02233-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40747-026-02233-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,10]],"date-time":"2026-04-10T07:32:50Z","timestamp":1775806370000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s40747-026-02233-3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,3,4]]},"references-count":40,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2026,4]]}},"alternative-id":["2233"],"URL":"https:\/\/doi.org\/10.1007\/s40747-026-02233-3","relation":{},"ISSN":["2199-4536","2198-6053"],"issn-type":[{"value":"2199-4536","type":"print"},{"value":"2198-6053","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,3,4]]},"assertion":[{"value":"19 April 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"4 January 2026","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"4 March 2026","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 no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}},{"value":"NA.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Ethical approval"}},{"value":"Consent for publication.","order":4,"name":"Ethics","group":{"name":"EthicsHeading","label":"Consent for publication"}}],"article-number":"123"}}