[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-123468-en":3,"doc-seo-123468-105":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},123468,687197207919,"Theodora","https://ap-avatar.wpscdn.com/avatar/a000253d6f5f7c60be?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779446848396160552",8,"Research & Report","SCIENTIFIC MACHINE LEARNING FOR CLOSURE MODELS IN MULTISCALE PROBLEMS - A REVIEW","Closure problems arise broadly in multiscale simulations when certain quantities or processes cannot be specified directly, yet their influence strongly affects predictive accuracy. Scientific machine learning is reviewed as a strategy that blends physics-based modeling with data-driven components, often by augmenting differential equations with neural networks. The review organizes reduced-model forms by the extent of known physics, contrasts a priori and a posteriori learning objectives, and emphasizes physical-law constraints such as symmetries and conservation laws. It also surveys discretization effects and discretization-invariant trends, and links closure to inverse problems, Mori-Zwanzig theory, and multi-fidelity methods. Despite progress, generalizability and interpretability remain key challenges for learned models.","arXiv :2403 .02913v2 [math .NA] 12 Sep 2024  \nSCIENTIFIC MACHINE LEARNING FOR CLOSURE MODELS IN MULTISCALE PROBLEMS: A REVIEW  \nB. SANDERSE, P. STINIS, R. MAULIK, AND S. E. AHMED  \nAbstract. Closure problems are omnipresent when simulating multiscale systems, where some quantities and processes cannot be fully prescribed despite their effects on the simulation’s accuracy. Recently, scientific machine learning approaches have been proposed as a way to tackle the closure problem, combining traditional (physics-based) modeling with data-driven (machinelearned) techniques, typically through enriching differential equations with neural networks. This paper reviews the different reduced model forms, distinguished by the degree to which they include known physics, and the different objectives of a priori and a posteriori learning. The importance of adhering to physical laws (such as symmetries and conservation laws) in choosing the reduced model form and choosing the learning method is discussed. The effect of spatial and temporal discretization and recent trends toward discretization-invariant models are reviewed.  \nIn addition, we make the connections between closure problems and several other research disciplines: inverse problems, Mori-Zwanzig theory, and multi-fidelity methods. In conclusion, much progress has been made with scientific machine learning approaches for solving closure problems, but many challenges remain. In particular, the generalizability and interpretability of learned  \nmodels is a major issue that needs to be addressed further.  \nKeywords: closure model, scientific machine learning, multiscale problem, turbulence  \n1. Introduction and background  \nClosure problems are everywhere. They arise when a mathematical-physical model that describes certain quantities of interest is created, which depends on quantities that are not of prime interest but whose effect cannot be neglected. A prominent example is in numerical weather prediction, where the aim is to predict tomorrow’s weather given the physical laws that govern fluid flows and today’s conditions. In principle, these physical laws describe all relevant physics, but in practice certain processes like cloud formation take place on such small spatial and temporal scales that resolving (simulating) them on a computer is impossible, even though their effect on the weather is crucial [1] .  \nThe weather prediction problem is an example of a multiscale problem in which the quantities of interest are associated with large scales, but depend on quantities associated with small scales. If these small scales are not resolved, the large-scale equations are unclosed. In this article, we focus on multiscale problems that lead to a closure problem, and in particular we consider multiscale problems that involve a continuum of scales (fluid flows being an important example) . There is no clear separation of scales, and the closure model is needed everywhere in the domain to supplement the large-scale model (Type “B” problems in the terminology of E [2]) .  \nClosure models appear under different names in different domains [3] . The term “closure” is most common in the fluid dynamics community, designating the turbulence closure problem—closure models are known as subgrid or subgrid-scale (SGS) models. In kinetic equations and radiation transport, the term “moment closure” is used [4, 5, 6] . In numerical weather prediction and climate science, the problem is known as “parameterization”, and finding accurate and stable parameterizations is considered a key challenge [1, 7] . In general circulation models (including atmospheric and oceanic models), the same terminology is used [8, 9] . In the reservoir simulation community, the closure problem is commonly referred to as “homogenization” or “upscaling” [10] . In materials science, molecular dynamics, and computational biology, the term “coarse graining”is used [11, 12] .  \nFinding a closure model for multiscale problems is diffi","cbCaimba3tfRAJkZ","https://ap.wps.com/l/cbCaimba3tfRAJkZ","pdf",877101,1,32,"English","en",105,"# Introduction and background\n## Multiscale simulations and the closure problem\n## Terminology across research domains\n## Hybrid physics–machine learning approaches\n# Reduced model forms and learning objectives\n## Incorporating known physics\n## A priori vs a posteriori learning","[{\"question\":\"What causes closure problems in multiscale simulations?\",\"answer\":\"Closure problems occur when the quantities of interest depend on smaller-scale processes that cannot be fully resolved or prescribed, even though their effects are crucial for accuracy.\"},{\"question\":\"How do scientific machine learning methods address closure models?\",\"answer\":\"They combine physics-based differential-equation models with data-driven neural-network components, typically enriching differential equations to approximate the influence of unresolved scales.\"},{\"question\":\"Why are physical laws important when choosing a reduced model and learning method?\",\"answer\":\"Adhering to physical constraints such as symmetries and conservation laws helps ensure the reduced model form and learning approach remain physically consistent and reliable.\"}]","SCIENTIFIC MACHINE LEARNING FOR CLOSURE MODELS IN MULTISCALE PROBLEMS - A REVIEW | PDF",1785816688,81,{"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-for-closure-models-in-multiscale-problems-a-review","",{"@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-for-closure-models-in-multiscale-problems-a-review/123468/",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","2026-08-04",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 causes closure problems in multiscale simulations?","Question",{"text":75,"@type":76},"Closure problems occur when the quantities of interest depend on smaller-scale processes that cannot be fully resolved or prescribed, even though their effects are crucial for accuracy.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How do scientific machine learning methods address closure models?",{"text":80,"@type":76},"They combine physics-based differential-equation models with data-driven neural-network components, typically enriching differential equations to approximate the influence of unresolved scales.",{"name":82,"@type":73,"acceptedAnswer":83},"Why are physical laws important when choosing a reduced model and learning method?",{"text":84,"@type":76},"Adhering to physical constraints such as symmetries and conservation laws helps ensure the reduced model form and learning approach remain physically consistent and reliable.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":89,"canonical":52},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]