[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-seo-290994-105":3,"detail-sidebar-cat-0-en-105":81,"doc-detail-290994-en":131},{"code":4,"msg":5,"data":6},0,"ok",{"site_id":7,"language":8,"slug":9,"title":10,"keywords":11,"description":12,"schema_data":13,"social_meta":74,"head_meta":76,"extra_data":78,"updated_unix":80},105,"en","parametric-intrusive-reduced-order-models-enhanced-with-machine-learning-correction-terms","Parametric Intrusive Reduced Order Models enhanced with Machine Learning Correction Terms","","Equation-based parametric reduced order models are developed by augmenting reduced equations with data-driven correction terms to reintroduce contributions neglected in standard ROMs. Two contribution types are studied: turbulence modeling via a reduced-order eddy viscosity approximation and a correction model that restores discarded-mode effects. The model is extended to a parametric setting using machine learning, comparing dense and LSTM architectures, and tested on periodic turbulent cylinder flow and unsteady turbulent channel-driven cavity flow.",{"@graph":14,"@context":73},[15,34,56],{"@type":16,"itemListElement":17},"BreadcrumbList",[18,23,27,31],{"item":19,"name":20,"@type":21,"position":22},"https://docshare.wps.com","Home","ListItem",1,{"item":24,"name":25,"@type":21,"position":26},"https://docshare.wps.com/document/","Document",2,{"item":28,"name":29,"@type":21,"position":30},"https://docshare.wps.com/document/research-report/","Research & Report",3,{"item":32,"name":10,"@type":21,"position":33},"https://docshare.wps.com/document/parametric-intrusive-reduced-order-models-enhanced-with-machine-learning-correction-terms/290994/",4,{"url":32,"name":10,"@type":35,"image":36,"author":41,"headline":10,"publisher":44,"fileFormat":47,"inLanguage":8,"description":12,"dateModified":48,"datePublished":49,"encodingFormat":47,"isAccessibleForFree":50,"interactionStatistic":51},"DigitalDocument",{"url":37,"@type":38,"width":39,"height":40},"https://docshare.wps.com/thumbnails/parametric-intrusive-reduced-order-models-enhanced-with-machine-learning-correction-terms/290994.png","ImageObject",300,407,{"name":42,"@type":43},"Evangeline","Person",{"url":19,"name":45,"@type":46},"DocShare","Organization","application/pdf","2026-11-12","2026-09-17",true,{"@type":52,"interactionType":53,"userInteractionCount":55},"InteractionCounter",{"@type":54},"ViewAction",15,{"@type":57,"mainEntity":58},"FAQPage",[59,65,69],{"name":60,"@type":61,"acceptedAnswer":62},"What is the core idea of the proposed parametric ROM?","Question",{"text":63,"@type":64},"The paper proposes an equation-based parametric reduced order model whose accuracy is improved by adding data-driven terms into the reduced equations to reintroduce contributions ignored in standard ROMs.","Answer",{"name":66,"@type":61,"acceptedAnswer":67},"Which two types of contributions are investigated?",{"text":68,"@type":64},"It investigates turbulence modeling through a reduced-order approximation of the eddy viscosity field and a correction model designed to restore the contribution of discarded modes.",{"name":70,"@type":61,"acceptedAnswer":71},"How is machine learning used in the model extension?",{"text":72,"@type":64},"The approach extends the correction mechanism to a parametric setting using machine learning, testing neural network architectures from dense feed-forward networks to Long-Short Term Memory networks to find a suitable model.","https://schema.org",{"og:url":32,"og:type":75,"og:title":10,"og:site_name":45,"og:description":12},"article",{"robots":77,"canonical":32},"index,follow",{"doc_id":79,"site_id":7},290994,1789643153,{"code":4,"msg":82,"data":83},"success",[84,88,92,96,101,106,111,115,120,123,127],{"id":22,"doc_module":4,"doc_module_name":25,"category_name":85,"show_sort_weight":86,"slug":87},"Story & Novel",90,"story-novel",{"id":26,"doc_module":4,"doc_module_name":25,"category_name":89,"show_sort_weight":90,"slug":91},"Literature",80,"literature",{"id":33,"doc_module":4,"doc_module_name":25,"category_name":93,"show_sort_weight":94,"slug":95},"Exam",70,"exam",{"id":97,"doc_module":4,"doc_module_name":25,"category_name":98,"show_sort_weight":99,"slug":100},5,"Comic",60,"comic",{"id":102,"doc_module":4,"doc_module_name":25,"category_name":103,"show_sort_weight":104,"slug":105},6,"Technology",50,"technology",{"id":107,"doc_module":4,"doc_module_name":25,"category_name":108,"show_sort_weight":109,"slug":110},7,"Healthcare",40,"healthcare",{"id":112,"doc_module":4,"doc_module_name":25,"category_name":29,"show_sort_weight":113,"slug":114},8,30,"research-report",{"id":116,"doc_module":4,"doc_module_name":25,"category_name":117,"show_sort_weight":118,"slug":119},9,"Religion & Spirituality",20,"religion-spirituality",{"id":118,"doc_module":4,"doc_module_name":25,"category_name":121,"show_sort_weight":118,"slug":122},"World Cup","world-cup",{"id":124,"doc_module":4,"doc_module_name":25,"category_name":125,"show_sort_weight":124,"slug":126},10,"Lifestyle","lifestyle",{"id":128,"doc_module":4,"doc_module_name":25,"category_name":129,"show_sort_weight":97,"slug":130},19,"General","general",{"code":4,"msg":82,"data":132},{"doc_id":79,"user_id":133,"nickname":42,"user_avatar":134,"doc_module":4,"category_id":112,"category_name":29,"doc_title":10,"doc_description":12,"doc_content":135,"file_id":136,"file_url":137,"file_type":138,"file_size":139,"view_count":55,"is_deleted":4,"is_public":22,"is_downloadable":22,"audit_status":22,"page_count":140,"language":141,"language_code":8,"site_id":7,"html_lang":8,"table_of_contents":142,"faqs":143,"seo_title":144,"seo_description":12,"update_tm":80,"read_time":145,"preview_page_count":4},13056703019662,"https://ap-avatar.wpscdn.com/avatar/be000253a8e92610077?_k=1778726343310543188","arXiv :2406 .04169v1 [math .NA] 6 Jun 2024  \nParametric Intrusive Reduced Order Models enhanced with Machine Learning Correction Terms  \nAnna Ivagnes  \nSISSA, International School for Advanced Studies,  \nMathematics Area, mathLab, Trieste, Italy.  \n[aivagnes@sissa.it](aivagnes@sissa.it)  \nGiovanni Stabile  \nSant’Anna School of Advanced Studies  \nThe Biorobotics Institute, Pontedera, Pisa, Italy.  \n[giovanni.stabile@santannapisa.it](giovanni.stabile@santannapisa.it)  \nGianluigi Rozza  \nSISSA, International School for Advanced Studies,  \nMathematics Area, mathLab, Trieste, Italy.  \n[grozza@sissa.it](grozza@sissa.it)  \nAbstract  \nIn this paper, we propose an equation-based parametric Reduced Order Model (ROM), whose accuracy is improved with data-driven terms added into the reduced equations. These additions have the aim of reintroducing contributions that in standard ROMs are not taken into account.  \nIn particular, in this work we consider two types of contributions: the turbulence modeling, added through a reduced-order approximation of the eddy viscosity field, and the correction model, aimed to re-introduce the contribution of the discarded modes. Both approaches have been investigated in previous works such as [32, 19, 22, 21] and the goal of this paper is to extend the model to a parametric setting making use of ad-hoc machine learning procedures. More in detail, we investigate different neural networks’ architectures, from simple dense feed-forward to Long-Short Term Memory neural networks, in order to find the most suitable model for there-introduced contributions. We tested the methods on two test cases with different behaviors:  \nthe periodic turbulent flow past a circular cylinder and the unsteady turbulent flow in a channeldriven cavity. In both cases, the parameter considered is the Reynolds number and the machine learning-enhanced ROM considerably improved the pressure and velocity accuracy with respect to the standard ROM. A visual version of the abstract can be found in Figure 1 .  \nFigure 1: Visual abstract.  \n1. Introduction  \nReduced Order Models (ROMs) [9, 10, 11, 43, 42, 44] are a powerful tool used to reduce the computational effort of time-demanding simulations. One of the fields where this class of techniques is widespread is Computational Fluid Dynamics, where high-fidelity simulations may take days or weeks, even in the case of parallel computations on many cores. For this reason, a simplified model is necessary to efficiently compute the solutions for unseen configurations.  \nMost of the ROMs are built upon an offline-online paradigm [44] . The offline stage consists of the computation of a large number of expensive high-fidelity simulations, performed after setting the Full Order Model (FOM), which is usually a discretized version of complex PDEs, like the Navier– Stokes Equations (NSE) . The goal of this stage is the collection of the so-called snapshots, namely the solutions of the simulations. On the other hand, in the online stage the full-order manifold is projected into a space with reduced dimensionality, resulting in a reduced representation of the snapshots. This reduction step may be assessed both with linear or nonlinear approaches. In particular, we employ the Proper Orthogonal Decomposition (POD) [25, 13], a linear technique which may be considered equivalent to the Principal Component Analysis (PCA) [38, 55] and Singular Value Decomposition (SVD) [27, 17, 16] .  \nWhile the offline stage usually employs the numerical resolution of complex PDEs, in intrusive ROMs the online stage allows for the resolution of ODEs, which consists of a reduced and simplified version of the FOM.  \nIn particular, in this contribution, we focus on POD-Galerkin ROMs, that are based on a Galerkin projection [35, 12, 26, 7] . The main assumption of these models is that the solution maybe approximated as a convex combination of a reduced number of global basis (the modes), whose  \ncoefficients are the reduced repre","cbCaittRxwsNMcWn","https://ap.wps.com/l/cbCaittRxwsNMcWn","pdf",8602161,38,"English","# Abstract\n# 1. Introduction\n## Reduced order models and offline-online paradigm\n## Intrusive POD-Galerkin ROMs and limitations\n## Data-driven stabilization and closure approaches\n## Goal and scope of the manuscript","[{\"question\":\"What is the core idea of the proposed parametric ROM?\",\"answer\":\"The paper proposes an equation-based parametric reduced order model whose accuracy is improved by adding data-driven terms into the reduced equations to reintroduce contributions ignored in standard ROMs.\"},{\"question\":\"Which two types of contributions are investigated?\",\"answer\":\"It investigates turbulence modeling through a reduced-order approximation of the eddy viscosity field and a correction model designed to restore the contribution of discarded modes.\"},{\"question\":\"How is machine learning used in the model extension?\",\"answer\":\"The approach extends the correction mechanism to a parametric setting using machine learning, testing neural network architectures from dense feed-forward networks to Long-Short Term Memory networks to find a suitable model.\"}]","Parametric Intrusive Reduced Order Models enhanced with Machine Learning Correction Terms | PDF",96]