{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,9]],"date-time":"2026-03-09T03:43:44Z","timestamp":1773027824120,"version":"3.50.1"},"reference-count":75,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2022,11,23]],"date-time":"2022-11-23T00:00:00Z","timestamp":1669161600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Deanship of Scientific Research, Vice Presidency for Graduate Studies and Scientific Research, King Faisal University, Saudi Arabia","award":["1850"],"award-info":[{"award-number":["1850"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Fractional polytropic gas sphere problems and electrical engineering models typically simulated with interconnected circuits have numerous applications in physical, astrophysical phenomena, and thermionic currents. Generally, most of these models are singular-nonlinear, symmetric, and include time delay, which has increased attention to them among researchers. In this work, we explored deep neural networks (DNNs) with an optimization algorithm to calculate the approximate solutions for nonlinear fractional differential equations (NFDEs). The target data-driven design of the DNN-LM algorithm was further implemented on the fractional models to study the rigorous impact and symmetry of different parameters on RL, RC circuits, and polytropic gas spheres. The targeted data generated from the analytical and numerical approaches in the literature for different cases were utilized by the deep neural networks to predict the numerical solutions by minimizing the differences in mean square error using the Levenberg\u2013Marquardt algorithm. The numerical solutions obtained by the designed technique were contrasted with the multi-step reproducing kernel Hilbert space method (MS-RKM), Laplace transformation method (LTM), and Pad\u00e9 approximations. The results demonstrate the accuracy of the design technique as the DNN-LM algorithm overlaps with the actual results with minimum percentage absolute errors that lie between 10\u22128 and 10\u221212. The extensive graphical and statistical analysis of the designed technique showed that the DNN-LM algorithm is dependable and facilitates the examination of higher-order nonlinear complex problems due to the flexibility of the DNN architecture and the effectiveness of the optimization procedure.<\/jats:p>","DOI":"10.3390\/sym14122482","type":"journal-article","created":{"date-parts":[[2022,11,23]],"date-time":"2022-11-23T02:55:51Z","timestamp":1669172151000},"page":"2482","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":17,"title":["Application of a Machine Learning Algorithm for Evaluation of Stiff Fractional Modeling of Polytropic Gas Spheres and Electric Circuits"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6598-6240","authenticated-orcid":false,"given":"Fawaz Khaled","family":"Alarfaj","sequence":"first","affiliation":[{"name":"Computer Sciences Department, College of Computer Sciences & Information Technology, King Faisal University, Al-Ahsa P.O. Box 31982, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8680-723X","authenticated-orcid":false,"given":"Naveed Ahmad","family":"Khan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Abdul Wali Khan University, Mardan 23200, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4040-6211","authenticated-orcid":false,"given":"Muhammad","family":"Sulaiman","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Abdul Wali Khan University, Mardan 23200, Pakistan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Abdullah M.","family":"Alomair","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, Faculty of Science, King Faisal University, Al-Ahsa P.O. Box 31982, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,11,23]]},"reference":[{"key":"ref_1","unstructured":"Oldham, K., and Spanier, J. (1974). The Fractional Calculus Theory and Applications of Differentiation and Integration to Arbitrary Order, Elsevier."},{"key":"ref_2","unstructured":"Miller, K.S., and Ross, B. (1993). An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley and Sons."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"108651","DOI":"10.1016\/j.ijepes.2022.108651","article-title":"Gated spatial-temporal graph neural network based short-term load forecasting for wide-area multiple buses","volume":"145","author":"Huang","year":"2023","journal-title":"Int. J. Electr. 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