[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-117633-en":3,"doc-seo-117633-105":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},117633,962075114101,"Seraphina","https://ap-avatar.wpscdn.com/avatar/e000253a75eb197efd?x-image-process=image/resize,m_fixed,w_180,h_180&k=1780044092746381165",8,"Research & Report","Scientific Machine Learning - An Overview of Methodological Paradigms and Applications","Scientific Machine Learning (SciML) integrates scientific computing with machine learning to tackle complex modeling, simulation, and inference tasks in natural and engineering sciences. The work outlines foundational paradigms, emphasizing how physics-based models and data-driven methods complement each other. It analyzes central challenges—data scarcity, physical consistency, high dimensionality, and computational cost—and presents key advances including physics-informed neural networks, operator learning, hybrid modeling, and probabilistic approaches. Applications in fluid dynamics, turbulence modeling, control, energy management, and sustainable mobility demonstrate gains in accuracy, efficiency, and interpretability.","Scientific machine learning  \nCitation for published version (APA):  \nDietrich, F. , & Schilders, W. (2025) . Scientific machine learning. Mathematische Semesterberichte , 72(2), 89- 115. [https://doi.org/10.1007/s00591-025-00399-4](https://doi.org/10.1007/s00591-025-00399-4)  \nDocument license:  \nCC BY  \nDOI:  \n10.1007/s00591-025-00399-4  \nDocument status and date:  \nPublished: 01/10/2025  \nDocument Version:  \nPublisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers)  \nPlease check the document version of this publication:  \n• A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. 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Apr. 2026  \nMathematische Semesterberichte (2025) 72:89–115 [https://doi.org/10.1007/s00591-025-00399-4](https://doi.org/10.1007/s00591-025-00399-4)  \nMATHEMATIK IN FORSCHUNG UND ANWENDUNG - MATHEMATICAL RESEARCH AND APPLICATIONS  \nScientific machine learning  \nFelix Dietrich · Wil Schilders  \nReceived: 10 August 2025 / Accepted: 12 August 2025 / Published online: 5 September 2025 © The Author(s) 2025  \nAbstract Scientific Machine Learning (SciML) is an emerging interdisciplinary field that integrates the strengths of scientific computing and machine learning to address complex modeling, simulation, and inference tasks in the natural and engineering sciences. This paper provides a concise overview of the foundational paradigms underlying SciML, where we highlight the complementary roles of physics-based models and data-driven methods. We discuss the core challenges, including data scarcity, physical consistency, high dimensionality, and computational cost and then introduce key methodological advances such as physics-informed neural networks, operator learning, hybrid modeling, and probabilistic approaches, each designed to address specific limitations of traditional methods. The techniques are illustrated through applications in fluid dynamics and turbulence modeling, automation and control, energy management, and sustainable mobility, demonstrating how SciML enables more accurate, efficient, and interpretable solutions. Finally, we outline open problems and future directions, and emphasize the need for theoretical understanding, scalable algorithms, and interdisciplinary collaboration.  \nKeywords Scientific computing · Machine learning · Physics-informed learning · Operator learning  \nFelix Dietrich  \nTUM School of Computation, Infor","cbCaiqlviTwWzntA","https://ap.wps.com/l/cbCaiqlviTwWzntA","pdf",1594421,1,28,"English","en",105,"# Introduction\n## From scientific computing to machine learning\n## The role of SciML at the intersection\n## Key challenges and methodological advances\n## Applications and future directions","[{\"question\":\"What is Scientific Machine Learning (SciML) in this paper?\",\"answer\":\"SciML is an emerging interdisciplinary field combining scientific computing and machine learning to address complex modeling, simulation, and inference tasks. It leverages the strengths of both physics-based models and data-driven methods.\"},{\"question\":\"Which core challenges does SciML address?\",\"answer\":\"The paper highlights data scarcity, physical consistency, high dimensionality, and computational cost. These issues motivate methodological advances beyond traditional approaches.\"},{\"question\":\"What methodological advances are introduced for SciML?\",\"answer\":\"The paper introduces physics-informed neural networks, operator learning, hybrid modeling, and probabilistic approaches. Each method targets specific limitations of conventional techniques.\"}]","Scientific Machine Learning - An Overview of Methodological Paradigms and Applications | PDF",1785677484,71,{"code":4,"msg":31,"data":32},"ok",{"site_id":24,"language":23,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"scientific-machine-learning-an-overview-of-methodological-paradigms-and-applications","",{"@graph":36,"@context":85},[37,54,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/scientific-machine-learning-an-overview-of-methodological-paradigms-and-applications/117633/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":23,"description":14,"dateModified":62,"datePublished":62,"encodingFormat":61,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-08-02",true,{"@type":65,"interactionType":66,"userInteractionCount":4},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What is Scientific Machine Learning (SciML) in this paper?","Question",{"text":75,"@type":76},"SciML is an emerging interdisciplinary field combining scientific computing and machine learning to address complex modeling, simulation, and inference tasks. 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