{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T00:59:38Z","timestamp":1760057978090,"version":"build-2065373602"},"reference-count":35,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2025,3,8]],"date-time":"2025-03-08T00:00:00Z","timestamp":1741392000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>The Euler-Type Universal Numerical Integrator (E-TUNI) is a discrete numerical structure that couples a first-order Euler-type numerical integrator with some feed-forward neural network architecture. Thus, E-TUNI can be used to model non-linear dynamic systems when the real-world plant\u2019s analytical model is unknown. From the discrete solution provided by E-TUNI, the integration process can be either forward or backward. Thus, in this article, we intend to use E-TUNI in a backward integration framework to model autonomous non-linear dynamic systems. Three case studies, including the dynamics of the non-linear inverted pendulum, were developed to verify the computational and numerical validation of the proposed model.<\/jats:p>","DOI":"10.3390\/a18030153","type":"journal-article","created":{"date-parts":[[2025,3,10]],"date-time":"2025-03-10T05:46:52Z","timestamp":1741585612000},"page":"153","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["The Euler-Type Universal Numerical Integrator (E-TUNI) with Backward Integration"],"prefix":"10.3390","volume":"18","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7069-8430","authenticated-orcid":false,"given":"Paulo M.","family":"Tasinaffo","sequence":"first","affiliation":[{"name":"Instituto Tecnol\u00f3gico de Aeron\u00e1utica (ITA), S\u00e3o Jos\u00e9 dos Campos 12228-900, SP, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0932-9171","authenticated-orcid":false,"given":"Gild\u00e1rcio S.","family":"Gon\u00e7alves","sequence":"additional","affiliation":[{"name":"Instituto Tecnol\u00f3gico de Aeron\u00e1utica (ITA), S\u00e3o Jos\u00e9 dos Campos 12228-900, SP, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1551-435X","authenticated-orcid":false,"given":"Johnny C.","family":"Marques","sequence":"additional","affiliation":[{"name":"Instituto Tecnol\u00f3gico de Aeron\u00e1utica (ITA), S\u00e3o Jos\u00e9 dos Campos 12228-900, SP, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5958-8011","authenticated-orcid":false,"given":"Luiz A. V.","family":"Dias","sequence":"additional","affiliation":[{"name":"Instituto Tecnol\u00f3gico de Aeron\u00e1utica (ITA), S\u00e3o Jos\u00e9 dos Campos 12228-900, SP, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2399-5066","authenticated-orcid":false,"given":"Adilson M.","family":"da Cunha","sequence":"additional","affiliation":[{"name":"Instituto Tecnol\u00f3gico de Aeron\u00e1utica (ITA), S\u00e3o Jos\u00e9 dos Campos 12228-900, SP, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,3,8]]},"reference":[{"key":"ref_1","first-page":"115","article-title":"A logical calculus of the ideas immanent in nervous activity","volume":"5","author":"McCulloch","year":"1943","journal-title":"Bull. Math. Biol."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"303","DOI":"10.1007\/BF02551274","article-title":"Approximation by superpositions of a sigmoidal function","volume":"2","author":"Cybenko","year":"1989","journal-title":"Math. Control Signals Syst."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"359","DOI":"10.1016\/0893-6080(89)90020-8","article-title":"Multilayer feedforward networks are universal approximators","volume":"2","author":"Hornik","year":"1989","journal-title":"Neural Netw."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"290","DOI":"10.1109\/72.80265","article-title":"The Stone-Weierstrass and its application to neural networks","volume":"1","author":"Cotter","year":"1990","journal-title":"IEEE Trans. Neural Netw."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Hanin, B. (2019). Universal function approximation by deep neural nets with bounded width and ReLU activations. Mathematics, 7.","DOI":"10.3390\/math7100992"},{"key":"ref_6","unstructured":"Haykin, S. (1999). Neural Networks: A Comprehensive Foundation, Prentice-Hall, Inc."},{"key":"ref_7","unstructured":"Goodfellow, I., Bengio, Y., and Courville, A. (2017). Deep Learning, MIT Press."},{"key":"ref_8","unstructured":"Henrici, P. (1964). Elements of Numerical Analysis, John Wiley and Sons."},{"key":"ref_9","unstructured":"Lapidus, L., and Seinfeld, J.H. (1971). Numerical Solution of Ordinary Differential Equations, Academic Press."},{"key":"ref_10","unstructured":"Lambert, J.D. (1973). Computational Methods in Ordinary Differential Equations, John Wiley and Sons."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Vidyasagar, M. (1978). Nonlinear Systems Analysis, Prentice-Hall Inc.","DOI":"10.1115\/1.3426360"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"199","DOI":"10.5556\/j.tkjm.39.2008.12","article-title":"Higher order composition Runge-Kutta methods","volume":"39","author":"Chen","year":"2008","journal-title":"Tamkang J. Math."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"425","DOI":"10.1007\/s11075-010-9437-2","article-title":"Multiplicative Adams Bashforth-Moulton methods","volume":"57","author":"Misirli","year":"2011","journal-title":"Numer. Algor."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"867948","DOI":"10.1155\/2012\/867948","article-title":"An optimized Runge-Kutta method for the numerical solution of the radial Schr\u00f6dinger equation","volume":"2012","author":"Ming","year":"2012","journal-title":"Math. Probl. Eng."},{"key":"ref_15","first-page":"5115","article-title":"Comparing accuracy of differential equation results between Runge-Kutta Fehlberg methods and Adams-Moulton methods","volume":"7","author":"Polla","year":"2013","journal-title":"Appl. Math. Sci."},{"key":"ref_16","first-page":"383","article-title":"An introduction to universal numerical integrators","volume":"15","author":"Tasinaffo","year":"2019","journal-title":"Int. J. Innov. Comput. Inf. Control"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1013","DOI":"10.1080\/00207178908559683","article-title":"Representations of nonlinear systems: The NARMAX model","volume":"49","author":"Chen","year":"1989","journal-title":"Int. J. Control"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1327","DOI":"10.1080\/00207179008953599","article-title":"Practical identification of NARMAX models using radial basis functions","volume":"52","author":"Chen","year":"1990","journal-title":"Int. J. Control"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"4","DOI":"10.1109\/72.80202","article-title":"Identification and control of dynamical systems using neural networks","volume":"1","author":"Narendra","year":"1990","journal-title":"IEEE Trans. Neural Netw."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1083","DOI":"10.1016\/0005-1098(92)90053-I","article-title":"Neural networks for control systems\u2014A survey","volume":"28","author":"Hunt","year":"1992","journal-title":"Automatica"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Norgaard, M., Ravn, O., Poulsen, N.K., and Hansen, L.K. (2000). Neural Networks for Modelling and Control of Dynamic Systems, Springer.","DOI":"10.1007\/978-1-4471-0453-7"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"294","DOI":"10.1109\/72.661124","article-title":"Runge-Kutta neural network for identification of dynamical systems in high accuracy","volume":"9","author":"Wang","year":"1998","journal-title":"IEEE Trans. Neural Netw."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"7769","DOI":"10.1007\/s00500-018-3405-5","article-title":"A Runge-Kutta neural network-based control method for nonlinear MIMO systems","volume":"23","year":"2019","journal-title":"Soft Comput."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"U\u00e7ak, K., and G\u00fcnel, G.O. (2018, January 25\u201327). An adaptive state feedback controller based on SVR for nonlinear systems. Proceedings of the 6th International Conference on Control Engineering and Information Technology (CEIT), Istanbul, Turkey. Available online: https:\/\/api.semanticscholar.org\/CorpusID:195775331.","DOI":"10.1109\/CEIT.2018.8751888"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1789","DOI":"10.1007\/s11063-019-10167-w","article-title":"A novel model predictive Runge-Kutta neural network controller for nonlinear MIMO systems","volume":"51","year":"2020","journal-title":"Neural Process. Lett."},{"key":"ref_26","first-page":"445","article-title":"Adams-Bashforth neural networks applied in a predictive control structure with only one horizon","volume":"15","author":"Tasinaffo","year":"2019","journal-title":"Int. J. Innov. Comput. Inf. Control"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"98","DOI":"10.21528\/LNLM-vol3-no2-art5","article-title":"Mean derivatives based neural Euler integrator for nonlinear dynamic systems modeling","volume":"3","author":"Tasinaffo","year":"2005","journal-title":"Learn. Nonlinear Model."},{"key":"ref_28","first-page":"1721","article-title":"Modeling autonomous nonlinear dynamic systems using mean derivatives, fuzzy logic and genetic algorithms","volume":"12","author":"Tasinaffo","year":"2016","journal-title":"Int. J. Innov. Comput. Inf. Control"},{"key":"ref_29","unstructured":"Munem, M.A., and Foulis, D.J. (1978). Calculus with Analytic Geometry (Volumes I and II), Worth Publishers, Inc."},{"key":"ref_30","unstructured":"Wilson, E. (1958). Advanced Calculus, Dover Publication."},{"key":"ref_31","first-page":"1","article-title":"A Survey About Universal Numerical Integrators (UNIs): Part II or Quantitative Approach","volume":"4","author":"Tasinaffo","year":"2024","journal-title":"Hum.-Centric Intell. Syst. (Preprint)"},{"key":"ref_32","unstructured":"Ames, W.F. (1988). Numerical Methods for Partial Differential Equations, Academic Press. [2nd ed.]."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"989","DOI":"10.1109\/72.329697","article-title":"Training feedforward networks with the Marquardt algorithm","volume":"5","author":"Hagan","year":"1994","journal-title":"IEEE Trans. Neural Netw."},{"key":"ref_34","unstructured":"Burden, R.L., and Faires, J.D. (2011). Numerical Analysis, Brooks\/Cole Inc.. [9th ed.]."},{"key":"ref_35","unstructured":"Cheney, W., and Kincaid, D. (2008). Numerical Mathematics and Computing, Thonson Brooks\/Cole. [6th ed.]."}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/18\/3\/153\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T16:49:24Z","timestamp":1760028564000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/18\/3\/153"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,3,8]]},"references-count":35,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2025,3]]}},"alternative-id":["a18030153"],"URL":"https:\/\/doi.org\/10.3390\/a18030153","relation":{},"ISSN":["1999-4893"],"issn-type":[{"type":"electronic","value":"1999-4893"}],"subject":[],"published":{"date-parts":[[2025,3,8]]}}}