{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,19]],"date-time":"2025-12-19T22:13:17Z","timestamp":1766182397015,"version":"build-2065373602"},"reference-count":19,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2024,9,10]],"date-time":"2024-09-10T00:00:00Z","timestamp":1725926400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Portuguese Foundation for Science and Technology","award":["UIDB\/00013\/2020","UIDP\/00013\/2020","CPCA-IAC\/AV\/589164\/2023","CPCA-IAC\/AF\/589140\/2023","2021.05201.BD","2022.06672.PTDC\u2014iMAD"],"award-info":[{"award-number":["UIDB\/00013\/2020","UIDP\/00013\/2020","CPCA-IAC\/AV\/589164\/2023","CPCA-IAC\/AF\/589140\/2023","2021.05201.BD","2022.06672.PTDC\u2014iMAD"]}]},{"name":"FCT","award":["UIDB\/00013\/2020","UIDP\/00013\/2020","CPCA-IAC\/AV\/589164\/2023","CPCA-IAC\/AF\/589140\/2023","2021.05201.BD","2022.06672.PTDC\u2014iMAD"],"award-info":[{"award-number":["UIDB\/00013\/2020","UIDP\/00013\/2020","CPCA-IAC\/AV\/589164\/2023","CPCA-IAC\/AF\/589140\/2023","2021.05201.BD","2022.06672.PTDC\u2014iMAD"]}]},{"name":"FCT","award":["UIDB\/00013\/2020","UIDP\/00013\/2020","CPCA-IAC\/AV\/589164\/2023","CPCA-IAC\/AF\/589140\/2023","2021.05201.BD","2022.06672.PTDC\u2014iMAD"],"award-info":[{"award-number":["UIDB\/00013\/2020","UIDP\/00013\/2020","CPCA-IAC\/AV\/589164\/2023","CPCA-IAC\/AF\/589140\/2023","2021.05201.BD","2022.06672.PTDC\u2014iMAD"]}]},{"name":"FCT\/MCTES (PIDDAC)","award":["UIDB\/00013\/2020","UIDP\/00013\/2020","CPCA-IAC\/AV\/589164\/2023","CPCA-IAC\/AF\/589140\/2023","2021.05201.BD","2022.06672.PTDC\u2014iMAD"],"award-info":[{"award-number":["UIDB\/00013\/2020","UIDP\/00013\/2020","CPCA-IAC\/AV\/589164\/2023","CPCA-IAC\/AF\/589140\/2023","2021.05201.BD","2022.06672.PTDC\u2014iMAD"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Fractal Fract"],"abstract":"<jats:p>Neural Fractional Differential Equations (Neural FDEs) represent a neural network architecture specifically designed to fit the solution of a fractional differential equation to given data. This architecture combines an analytical component, represented by a fractional derivative, with a neural network component, forming an initial value problem. During the learning process, both the order of the derivative and the parameters of the neural network must be optimised. In this work, we investigate the non-uniqueness of the optimal order of the derivative and its interaction with the neural network component. Based on our findings, we perform a numerical analysis to examine how different initialisations and values of the order of the derivative (in the optimisation process) impact its final optimal value. Results show that the neural network on the right-hand side of the Neural FDE struggles to adjust its parameters to fit the FDE to the data dynamics for any given order of the fractional derivative. Consequently, Neural FDEs do not require a unique \u03b1 value; instead, they can use a wide range of \u03b1 values to fit data. This flexibility is beneficial when fitting to given data is required and the underlying physics is not known.<\/jats:p>","DOI":"10.3390\/fractalfract8090529","type":"journal-article","created":{"date-parts":[[2024,9,10]],"date-time":"2024-09-10T11:01:42Z","timestamp":1725966102000},"page":"529","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Neural Fractional Differential Equations: Optimising the Order of the Fractional Derivative"],"prefix":"10.3390","volume":"8","author":[{"ORCID":"https:\/\/orcid.org\/0009-0009-4502-937X","authenticated-orcid":false,"given":"Cec\u00edlia","family":"Coelho","sequence":"first","affiliation":[{"name":"Centre of Mathematics (CMAT), University of Minho, 4710-057 Braga, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6235-286X","authenticated-orcid":false,"given":"M. Fernanda P.","family":"Costa","sequence":"additional","affiliation":[{"name":"Centre of Mathematics (CMAT), University of Minho, 4710-057 Braga, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5477-3226","authenticated-orcid":false,"given":"Lu\u00eds L.","family":"Ferr\u00e1s","sequence":"additional","affiliation":[{"name":"Centre of Mathematics (CMAT), University of Minho, 4710-057 Braga, Portugal"},{"name":"CEFT\u2014Centro de Estudos de Fen\u00f3menos de Transporte, Department of Mechanical Engineering (Section of Mathematics), FEUP, University of Porto, 4200-465 Porto, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2024,9,10]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"686","DOI":"10.1016\/j.jcp.2018.10.045","article-title":"Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations","volume":"378","author":"Raissi","year":"2019","journal-title":"J. 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Adam: A method for stochastic optimization. arXiv."}],"container-title":["Fractal and Fractional"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2504-3110\/8\/9\/529\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T15:53:18Z","timestamp":1760111598000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2504-3110\/8\/9\/529"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,9,10]]},"references-count":19,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2024,9]]}},"alternative-id":["fractalfract8090529"],"URL":"https:\/\/doi.org\/10.3390\/fractalfract8090529","relation":{},"ISSN":["2504-3110"],"issn-type":[{"type":"electronic","value":"2504-3110"}],"subject":[],"published":{"date-parts":[[2024,9,10]]}}}