{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,18]],"date-time":"2026-04-18T18:46:09Z","timestamp":1776537969453,"version":"3.51.2"},"reference-count":26,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2024,5,20]],"date-time":"2024-05-20T00:00:00Z","timestamp":1716163200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100000923","name":"Australian Research Council linkage grant","doi-asserted-by":"publisher","award":["LP160100528"],"award-info":[{"award-number":["LP160100528"]}],"id":[{"id":"10.13039\/501100000923","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000923","name":"Australian Research Council linkage grant","doi-asserted-by":"publisher","award":["ICG002517"],"award-info":[{"award-number":["ICG002517"]}],"id":[{"id":"10.13039\/501100000923","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000923","name":"Australian Research Council linkage grant","doi-asserted-by":"publisher","award":["SE67504"],"award-info":[{"award-number":["SE67504"]}],"id":[{"id":"10.13039\/501100000923","id-type":"DOI","asserted-by":"publisher"}]},{"name":"innovative connection grant","award":["LP160100528"],"award-info":[{"award-number":["LP160100528"]}]},{"name":"innovative connection grant","award":["ICG002517"],"award-info":[{"award-number":["ICG002517"]}]},{"name":"innovative connection grant","award":["SE67504"],"award-info":[{"award-number":["SE67504"]}]},{"name":"industry grant","award":["LP160100528"],"award-info":[{"award-number":["LP160100528"]}]},{"name":"industry grant","award":["ICG002517"],"award-info":[{"award-number":["ICG002517"]}]},{"name":"industry grant","award":["SE67504"],"award-info":[{"award-number":["SE67504"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>This paper proposes a new optimization algorithm for backpropagation (BP) neural networks by fusing integer-order differentiation and fractional-order differentiation, while fractional-order differentiation has significant advantages in describing complex phenomena with long-term memory effects and nonlocality, its application in neural networks is often limited by a lack of physical interpretability and inconsistencies with traditional models. To address these challenges, we propose a mixed integer-fractional (MIF) gradient descent algorithm for the training of neural networks. Furthermore, a detailed convergence analysis of the proposed algorithm is provided. Finally, numerical experiments illustrate that the new gradient descent algorithm not only speeds up the convergence of the BP neural networks but also increases their classification accuracy.<\/jats:p>","DOI":"10.3390\/a17050220","type":"journal-article","created":{"date-parts":[[2024,5,20]],"date-time":"2024-05-20T11:06:41Z","timestamp":1716203201000},"page":"220","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["An Integer-Fractional Gradient Algorithm for Back Propagation Neural Networks"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7717-547X","authenticated-orcid":false,"given":"Yiqun","family":"Zhang","sequence":"first","affiliation":[{"name":"School of Electrical Engineering, Computation and Mathematical Sciences, Curtin University, Kent Street, Perth, WA 6102, Australia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3212-2080","authenticated-orcid":false,"given":"Honglei","family":"Xu","sequence":"additional","affiliation":[{"name":"School of Electrical Engineering, Computation and Mathematical Sciences, Curtin University, Kent Street, Perth, WA 6102, Australia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yang","family":"Li","sequence":"additional","affiliation":[{"name":"School of Electrical Engineering, Computation and Mathematical Sciences, Curtin University, Kent Street, Perth, WA 6102, Australia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2905-4363","authenticated-orcid":false,"given":"Gang","family":"Lin","sequence":"additional","affiliation":[{"name":"School of Electrical Engineering, Computation and Mathematical Sciences, Curtin University, Kent Street, Perth, WA 6102, Australia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Liyuan","family":"Zhang","sequence":"additional","affiliation":[{"name":"School of Electrical Engineering, Computation and Mathematical Sciences, Curtin University, Kent Street, Perth, WA 6102, Australia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Chaoyang","family":"Tao","sequence":"additional","affiliation":[{"name":"School of Electrical Engineering, Computation and Mathematical Sciences, Curtin University, Kent Street, Perth, WA 6102, Australia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1028-1785","authenticated-orcid":false,"given":"Yonghong","family":"Wu","sequence":"additional","affiliation":[{"name":"School of Electrical Engineering, Computation and Mathematical Sciences, Curtin University, Kent Street, Perth, WA 6102, Australia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,5,20]]},"reference":[{"key":"ref_1","unstructured":"Edwards, C.J. 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