[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-117377-en":3,"doc-seo-117377-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":4,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":27,"seo_description":14,"update_tm":28,"read_time":29},117377,7971461740886,"Theodore","https://ap-avatar.wpscdn.com/davatar_3d24733baf745e90a7e4bdd5f77d97b2",8,"Research & Report","Combining Fractional Derivatives and Machine Learning: A Review","Fractional calculus explains complex dynamics using non-integer order derivatives, reflecting spatiotemporal memory and a capacity to represent naturally occurring phenomena. Machine learning, which learns patterns from historical data, supports analysis, modeling, and prediction across many scientific fields. This review links the two areas by compiling and contextualizing prior combined approaches, focusing on how fractional derivatives enable preprocessing, feature augmentation, physically informed learning, and improved hyperparameter optimization. The article excludes neural networks due to existing literature and targets practitioners building data-driven models.","entropy   \nReview  \nCombining Fractional Derivatives and Machine Learning: A Review  \nSebastian Raubitzek 1, *, Kevin Mallinger 2 and Thomas Neubauer 2  \nCitation: Raubitzek, S.; Mallinger, K.; Neubauer, T. Combining Fractional Derivatives and Machine Learning: A Review. Entropy 2023, 25, 35 . [https://doi.org/10.3390/e25010035](https://doi.org/10.3390/e25010035)  \n1 Data Science Research Unit, TU Wien, Favoritenstrasse 9-11/194, 1040 Vienna, Austria  \n2 SBA Research gGmbh, Floragasse 7, 1040 Vienna, Austria  \n* Correspondence: [sebastian.raubitzek@tuwien.ac.at](sebastian.raubitzek@tuwien.ac.at)  \nAbstract: Fractional calculus has gained a lot of attention in the last couple of years. Researchers have discovered that processes in various fields follow fractional dynamics rather than ordinary integer-ordered dynamics, meaning that the corresponding differential equations feature non-integer valued derivatives. There are several arguments for why this is the case, one of which is that fractional derivatives inherit spatiotemporal memory and/or the ability to express complex naturally occurring phenomena. Another popular topic nowadays is machine learning, i.e., learning behavior and patterns from historical data. In our ever-changing world with ever-increasing amounts of data, machine learning is a powerful tool for data analysis, problem-solving, modeling, and prediction. It has provided many further insights and discoveries in various scientific disciplines. As these two modern-day topics hold a lot of potential for combined approaches in terms of describing complex dynamics, this article review combines approaches from fractional derivatives and machine learning from the past, puts them into context, and thus provides a list of possible combined approaches and the corresponding techniques. Note, however, that this article does not deal with neural networks, as there is already extensive literature on neural networks and fractional calculus. We sorted past combined approaches from the literature into three categories, i.e., preprocessing, machine learning and fractional dynamics, and optimization. The contributions of fractional derivatives to machine learning are manifold as they provide powerful preprocessing and feature augmentation techniques, can improve physically informed machine learning, and are capable of improving hyperparameter optimization. Thus, this article serves to motivate researchers dealing with data-based problems, tobe specific machine learning practitioners, to adopt new tools, and enhance their existing approaches.  \nKeywords: fractional derivative; fractional calculus; machine learning; artificial intelligence; complexity; regression analysis  \n1. Introduction  \nAcademic Editor: Ivanka Stamova  \nReceived: 7 December 2022  \nRevised: 20 December 2022  \nAccepted: 21 December 2022  \nPublished: 24 December 2022  \nCopyright: © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license ([https://](https://)[ ](https://)[creativecommons.org/licenses/by/](creativecommons.org/licenses/by/)[ ](creativecommons.org/licenses/by/)[4.0/](4.0/)) .  \nMachine learning and fractional calculus are tools capable of dealing with and describing complex real-life phenomena and its relation to its inherent nonlinear properties.  \nWhereas machine learning dynamically learns complex behavior from data in most cases, the framework of fractional calculus was previously used to describe complex phenomena by statically modeling them. The issue of a fractional derivative first came up in a letter from de l'Hopital to Leibniz in 1695, i.e., what one would obtain from a derivative of non-integer order n, e.g., n = ~~12~~. Leibniz's famous response to this day was: “It will lead to a paradox from which one day useful consequences will be drawn” [1] .  \nNowadays, both frameworks feature a multitude of applications ","cbCaik5unnKQLgzb","https://ap.wps.com/l/cbCaik5unnKQLgzb","pdf",411654,1,22,"English","en",105,"# Abstract\n# Introduction\n## Motivation and scope","[{\"question\":\"What problem does fractional calculus address in modeling real-world processes?\",\"answer\":\"Fractional calculus models dynamics using non-integer order derivatives, capturing memory effects and more complex behavior than standard integer-ordered equations.\"},{\"question\":\"How does machine learning contribute to data-based analysis and prediction?\",\"answer\":\"Machine learning learns behavior and patterns from historical data, enabling data analysis, problem-solving, modeling, and prediction across disciplines.\"},{\"question\":\"What combined approaches does the review organize and what do fractional derivatives add?\",\"answer\":\"The review categorizes past combined approaches into preprocessing, machine learning and fractional dynamics, and optimization, highlighting that fractional derivatives support preprocessing/feature augmentation, physically informed learning, and hyperparameter optimization.\"}]","Combining Fractional Derivatives and Machine Learning: A Review | 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problem does fractional calculus address in modeling real-world processes?","Question",{"text":75,"@type":76},"Fractional calculus models dynamics using non-integer order derivatives, capturing memory effects and more complex behavior than standard integer-ordered equations.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does machine learning contribute to data-based analysis and prediction?",{"text":80,"@type":76},"Machine learning learns behavior and patterns from historical data, enabling data analysis, problem-solving, modeling, and prediction across disciplines.",{"name":82,"@type":73,"acceptedAnswer":83},"What combined approaches does the review organize and what do fractional derivatives add?",{"text":84,"@type":76},"The review categorizes past combined approaches into preprocessing, machine learning and fractional dynamics, and optimization, highlighting that fractional derivatives support preprocessing/feature augmentation, physically informed learning, and 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