[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-125401-en":3,"doc-seo-125401-105":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":20,"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},125401,1374391974564,"Clementine","https://ap-avatar.wpscdn.com/avatar/14000253aa45c000a9e?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779874745381141002",8,"Research & Report","Finite-element Gaussian processes for the machine learning of steady-state linear partial differential equations - Research article overview","Introduces finite-element Gaussian processes (FEGPs) as a physics-informed machine learning framework for inverse problems governed by steady-state, linear PDEs. The method combines a Gaussian process prior for the unknown solution with a likelihood enforcing the PDE through its weak form, approximated using finite-element basis functions. By avoiding explicit kernel differentiation required by strong-form physics-informed GPs (PIGPs), the approach improves computational scalability. Numerical experiments on synthetic benchmarks show better results than PIGPs and competitiveness with physics-informed neural networks, with improved uncertainty quantification.","| Finite-element Gaussian processes for the machine learning of steady-state linear partial differential equations\u003Cbr>David Dalton∗, Hao Gao  , Dirk Husmeier \u003Cbr>University of Glasgow, G12 8QQ, United Kingdom |  |  |  |\n| --- | --- | --- | --- |\n| a r t i c l e i n f o |  | a b s t r a c t |  |\n| Keywords:\u003Cbr>Physics-informed machine learning Gaussian processes\u003Cbr>Finite-elements\u003Cbr>Inverse problems |  | We introduce finite-element Gaussian processes (FEGPs), a novel physics-informed machine learning approach for solving inverse problems involving steady-state, linear partial differential equations (PDEs). Our framework combines a Gaussian process prior for the unknown solution function with a likelihood that incorporates the PDE in its weak form, using a finite-element approximation. This approach offers significantly better scalability than physics-informed Gaussian processes (PIGPs), which rely on the strong form of the PDE. Through numerical experiments on a range of synthetic benchmark problems, we show that FEGPs offer results which outperform PIGPs, and are competitive with physics-informed neural networks (PINNs) with improved uncertainty quantification. |  |\n\n1. Introduction  \nOver the past generation, scientific research has witnessed substantial advances in the modelling of multiphysics phenomena, with applications spanning chemistry, molecular biology, geophysics, and climate forecasting [1]. These models are typically expressed in the form of partial differential equations (PDEs), which are often sufficiently complex to necessitate numerical solution techniques. Despite parallel progress in the development of numerical PDE solvers-such as the finite-element and finite-difference methods [2] -significant challenges persist in effectively integrating these solvers with observational data. These challenges are particularly acute in inverse problems, where the aim is to infer latent physical properties of a system, such as material stiffness or thermal conductivity. In such cases, conventional solvers can become computationally prohibitive [3], and can struggle to adapt to ill-posed settings, for instance where boundary condition information is incomplete.  \nPhysics-informed machine learning (PIML) has recently emerged as a promising alternative framework for modelling physical systems. PIML encompasses a variety of techniques that integrate data-driven machine learning models with physics-based mathematical descriptions, facilitating a coherent fusion of empirical observations with established physical laws [4]. These methods address several limitations of traditional numerical solvers, particularly when dealing with ill-posed problems [3].  \nAmong these approaches, physics-informed neural networks (PINNs) [5] have gained prominence due to their flexibility and computational efficiency [6,7]. PINNs approximate the unknown state of a physical system using a neural network, which is trained to minimise a composite loss function that penalises both deviations from observational data and violations of the governing physical laws, typically encoded as PDEs.  \nIn practical applications, accounting for predictive uncertainty-particularly that arising from incomplete or noisy data-is often essential. This need has led to the development of probabilistic PIML methods, among which Gaussian process (GP) regression has  \n∗ Corresponding author.  \nE-mail addresses: [david.dalton@glasgow.ac.uk](david.dalton@glasgow.ac.uk) (D. Dalton), [hao.gao@glasgow.ac.uk](hao.gao@glasgow.ac.uk) (H. Gao), [dirk.husmeier@glasgow.ac.uk](dirk.husmeier@glasgow.ac.uk) (D. Husmeier).  \n[https://doi.org/10.1016/j.cma.2025.118580](https://doi.org/10.1016/j.cma.2025.118580)  \nReceived 21 July 2025; Received in revised form 15 October 2025; Accepted 11 November 2025 Available online 20 December 2025  \n0045-7825/© 2025 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license  \n([http://creativecommons.org/licenses/by/4.0/](http://creativ","cbCaitxakGYCYwF6","https://ap.wps.com/l/cbCaitxakGYCYwF6","pdf",3360108,1,22,"English","en",105,"# Introduction\n## Physics-informed machine learning context\n## Physics-informed neural networks and uncertainty\n# Background\n## Governing equations","[{\"question\":\"What are finite-element Gaussian processes (FEGPs) in this work?\",\"answer\":\"FEGPs are a probabilistic physics-informed machine learning approach that uses a Gaussian process prior and a PDE-informed likelihood built from the weak form using finite-element basis functions.\"},{\"question\":\"How do FEGPs differ from physics-informed Gaussian processes (PIGPs)?\",\"answer\":\"FEGPs integrate the PDE via its weak form, avoiding the need to explicitly differentiate the GP kernel, whereas PIGPs rely on the strong form and require kernel differentiation.\"},{\"question\":\"How is model performance and uncertainty assessed?\",\"answer\":\"Performance and efficiency are evaluated theoretically and through synthetic numerical experiments, showing FEGPs outperform PIGPs and are competitive with physics-informed neural networks while providing improved uncertainty quantification.\"}]","Finite-element Gaussian processes for the machine learning of steady-state linear partial differential equations - Research article overview | PDF",1785898689,55,{"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},"finite-element-gaussian-processes-for-the-machine-learning-of-steady-state-linear-partial-differential-equations-research-article-overview","",{"@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/finite-element-gaussian-processes-for-the-machine-learning-of-steady-state-linear-partial-differential-equations-research-article-overview/125401/",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","2026-08-05",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What are finite-element Gaussian processes (FEGPs) in this work?","Question",{"text":75,"@type":76},"FEGPs are a probabilistic physics-informed machine learning approach that uses a Gaussian process prior and a PDE-informed likelihood built from the weak form using finite-element basis functions.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How do FEGPs differ from physics-informed Gaussian processes (PIGPs)?",{"text":80,"@type":76},"FEGPs integrate the PDE via its weak form, avoiding the need to explicitly differentiate the GP kernel, whereas PIGPs rely on the strong form and require kernel differentiation.",{"name":82,"@type":73,"acceptedAnswer":83},"How is model performance and uncertainty assessed?",{"text":84,"@type":76},"Performance and efficiency are evaluated theoretically and through synthetic numerical experiments, showing FEGPs outperform PIGPs and are competitive with physics-informed neural networks while providing improved uncertainty quantification.","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"]