[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-122053-en":3,"doc-seo-122053-105":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},122053,8796095462418,"Noah","https://ap-avatar.wpscdn.com/avatar/80000253c1241d02b47?x-image-process=image/resize,m_fixed,w_180,h_180&k=1778826106357471780",8,"Research & Report","Physics-informed machine learning as a kernel method - Paper abstract","Physics-informed machine learning integrates physical modeling structure with data-driven prediction to improve performance in tasks governed by physical mechanisms. The work studies a regression setting where the empirical risk is regularized by a partial differential equation term measuring physical inconsistency. For linear differential priors, the regularized problem is reformulated as kernel regression. Convergence rates for the regularized minimizer are proved, reaching at least Sobolev minimax rates, with faster rates possible depending on the physical error.","arXiv :2402 .075 14v2 [ cs .AI] 19 Jun 2024  \nPhysics-informed machine learning as a kernel method  \nNathan Doumche NATHAN . DOUMECHE @ SORBONNE-UNIVERSITE . FR  \nLaboratory of Probability, Statistics, and Modeling, Sorbonne University, France  \nFrancis Bach FRANCIS . BACH @INRIA . FR  \nInria, Ecole Normale Sup e´ rieure, PSL Research University, France  \nGrard Biau GERARD . BIAU @ SORBONNE-UNIVERSITE . FR  \nLaboratory of Probability, Statistics, and Modeling, Sorbonne University, France  \nClaire Boyer CLAIRE . BOYER @ SORBONNE-UNIVERSITE . FR  \nIUF, Laboratory of Probability, Statistics, and Modeling, Sorbonne University, France  \nAbstract  \nPhysics-informed machine learning combines the expressiveness of data-based approaches with the interpretability of physical models. In this context, we consider a general regression problem where the empirical risk is regularized by a partial differential equation that quantifies the physical inconsistency. We prove that for linear differential priors, the problem can be formulated as a kernel regression task. Taking advantage of kernel theory, we derive convergence rates for the minimizer ˆfn of the regularized risk and show that ˆfn converges at least at the Sobolev minimax rate. However, faster rates can be achieved, depending on the physical error. This principle is illustrated with a one-dimensional example, supporting the claim that regularizing the empirical risk with physical information can be beneficial to the statistical performance of estimators.  \nKeywords: Physics-informed machine learning, Kernel methods, Rates of convergence, Physical regularization  \n1. Introduction  \nPhysics-informed machine learning. Physics-informed machine learning (PIML) refers to a subdomain of machine learning that combines physical knowledge and empirical data to enhance performance of tasks involving a physical mechanism. Following the influential work of Raissi et al.(2019), the field has experienced a notable surge in popularity, largely driven by scientific computing and engineering applications. We refer the reader to the surveys by Rai and Sahu (2020), Karniadakis et al. (2021), Cuomo et al. (2022), and Hao et al. (2022) . In a nutshell, the success of PIML relies on the smart interaction between machine learning and physics. In its most standard form, this achievement is realized by integrating physical equations into the loss function. Three common use cases include solving systems of partial differential equations (PDEs), addressing inverse problems (e.g., learning the PDE governing an observed phenomenon), and further improving the statistical performance of empirical risk minimization. This article focuses on the latter approach, known as hybrid modeling (e.g., Rai and Sahu, 2020) .  \n© N. Doumche, F. Bach, G. Biau & C. Boyer.  \nDOUMCHE BACH BIAU BOYER  \nHybrid modeling. Consider the classical regression model Y = f⋆(X) + ε, where the function f⋆ : Rd → R is unknown. The random variable Y ∈ R is the target, the random variable X ∈ Ω ⊆[−L, L]d the vector of features, and ε a random noise. Given a sample {(X1 , Y1 ) , ... ,(Xn, Yn)} ofi.i.d. copies of (X, Y ), the goal is to construct an estimator ˆfn of f⋆ based on these n observations. The distinctive element of PIML is the inclusion of a prior on f⋆, asserting its compliance with a known PDE. Therefore, it is assumed that f⋆ is at least weakly differentiable, belonging to the Sobolev space Hs (Ω) for some integer s > d/2, and that there is a known differential operator D such that D (f⋆ ) ≃ 0. For instance, if the desired solution f⋆ is intended to conform to the wave equation, then D (f)(x, t) = ∂2t,tf(x, t) − ∂2x,xf(x, t) for (x, t) ∈ Ω . Overall, we are interested in the minimizer of the empirical risk function  \nRn(f) = 1n Xi1 |f(Xi) − Yi|2 + λn∥f∥2Hsper([−2L,2L]d ) + µn∥D(f)∥2L2 (Ω) (1)  \nover the class F = Hsper([−2L,2L]d ) of candidate functions, where λn > 0 and µn ⩾ 0 are hyperparameters that weigh the relative importance of ","cbCaimmV8IK2dGxA","https://ap.wps.com/l/cbCaimmV8IK2dGxA","pdf",2511254,1,52,"English","en",105,"# Introduction\n## Hybrid modeling formulation\n# Contributions","[{\"question\":\"What does physics-informed machine learning combine to improve prediction tasks?\",\"answer\":\"It combines physical knowledge (expressed through differential equations) with empirical data to enhance learning tasks involving physical mechanisms.\"},{\"question\":\"How is the physical prior incorporated into the regression objective in this work?\",\"answer\":\"The empirical risk is regularized by a term that penalizes the mismatch with a known differential operator, quantified via an L2 norm of D(f) over the domain.\"},{\"question\":\"What result is shown when the differential prior is linear?\",\"answer\":\"The regularized regression problem can be formulated as a kernel regression task, enabling derived convergence rates for the minimizer.\"}]","Physics-informed machine learning as a kernel method - Paper abstract | PDF",1785808583,131,{"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},"physics-informed-machine-learning-as-a-kernel-method-paper-abstract","",{"@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/physics-informed-machine-learning-as-a-kernel-method-paper-abstract/122053/",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 does physics-informed machine learning combine to improve prediction tasks?","Question",{"text":75,"@type":76},"It combines physical knowledge (expressed through differential equations) with empirical data to enhance learning tasks involving physical mechanisms.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How is the physical prior incorporated into the regression objective in this work?",{"text":80,"@type":76},"The empirical risk is regularized by a term that penalizes the mismatch with a known differential operator, quantified via an L2 norm of D(f) over the domain.",{"name":82,"@type":73,"acceptedAnswer":83},"What result is shown when the differential prior is linear?",{"text":84,"@type":76},"The regularized regression problem can be formulated as a kernel regression task, enabling derived convergence rates for the minimizer.","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"]