[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-117204-en":3,"doc-seo-117204-105":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},117204,5909877438554,"Maeve","https://ap-avatar.wpscdn.com/avatar/5600025385ad2bf12a7?_k=1778553567797529272",8,"Research & Report","Boundary Constrained Gaussian Processes for Robust Physics-Informed Machine Learning of Linear Partial Differential Equations","A framework is proposed to construct boundary constrained Gaussian process (BCGP) priors that enforce linear boundary conditions exactly, and to apply them to learning linear (initial) boundary value problems for partial differential equations. Boundary constrained mean and kernel functions are designed for Dirichlet, Neumann, Robin, and mixed conditions, supporting both forward and inverse problem settings. The BCGP kernel is shown to have universal representational capacity under Dirichlet conditions, with a formal equivalence to boundary-constrained neural networks of infinite width. Extensive experiments on multiple linear PDEs demonstrate robust inference under sparse, noisy data.","Boundary constrained Gaussian processes for robust physics-informed machine learning of linear partial  \ndi􀀋erential equations  \nDavid Dalton Alan Lazarus Hao Gao Dirk Husmeier  \nSchool of Mathematics and Statistics University of Glasgow  \nGlasgow G12 8QQ, UK  \n[david.dalton@glasgow.ac.uk](david.dalton@glasgow.ac.uk)[ ](david.dalton@glasgow.ac.uk)[alan.lazarus@glasgow.ac.uk](alan.lazarus@glasgow.ac.uk)[ ](alan.lazarus@glasgow.ac.uk)[hao.gao@glasgow.ac.uk](hao.gao@glasgow.ac.uk)  \n[dirk.husmeier@glasgow.ac.uk](dirk.husmeier@glasgow.ac.uk)  \nEditor: Jean-Philippe Vert  \nAbstract  \nWe introduce a framework for designing boundary constrained Gaussian process (BCGP) priors for exact enforcement of linear boundary conditions, and apply it to the machine learning of (initial) boundary value problems involving linear partial di􀀋erential equations (PDEs) . In contrast to existing work, we illustrate how to design boundary constrained mean and kernel functions for all classes of boundary conditions typically used in PDE modelling, namely Dirichlet, Neumann, Robin and mixed conditions. Importantly, this is done in a manner which allows for both forward and inverse problems to be naturally accommodated. We prove that the BCGP kernel has a universal representational capacity under Dirichlet conditions, and establish a formal equivalence between BCGPsand boundary-constrained neural networks (BCNNs) of in􀀌nite width. Finally, extensive numerical experiments are performed involving several linear PDEs, the results of which demonstrate the e􀀋ectiveness and robustness of BCGP inference in the presence of sparse, noisy data.  \nKeywords: Physics-informed machine learning, Gaussian processes, partial di􀀋erential equations, boundary-value problems, inverse problems  \n1. Introduction  \nPhysics-informed machine learning (PIML) is a rapidly developing 􀀌eld which integrates data-based machine learning approaches with physics-based mathematical methods (Karniadakis et al., 2021) . A PIML model of a physical system leverages observational data with known physical principles, which can include, for example, boundary constraints, conservation laws and partial di􀀋erential equations (PDEs) . Physics-informed approaches can o􀀋er more robust and interpretable predictions than purely data-based approaches, in addition to delivering insights and inference about the system of interest that would not be possible without accounting for domain-speci􀀌c information. Consequently, PIML has become oneof the most topical research areas in computational physics and machine learning, with  \n􀀍c2024 David Dalton, Alan Lazarus, Hao Gao and Dirk Husmeier.  \nLicense: CC-BY 4.0, see [https://creativecommons.org/licenses/by/4.0/](https://creativecommons.org/licenses/by/4.0/. Attribution)[. Attribution](https://creativecommons.org/licenses/by/4.0/. Attribution) requirements are provided  \nat [http://jmlr.org/papers/v25/23-1508.html](http://jmlr.org/papers/v25/23-1508.html).  \nDalton, Lazarus, Gao and Husmeier  \napplications in a wide range of disciplines. This includes quantum chemistry (Pun et al. , 2019), solid mechanics (Nguyen-Thanh et al., 2020), 􀀍uid dynamics (Cai et al., 2021), softtissue mechanics (Dalton et al., 2023), and climate modelling (L􀁿utjens et al., 2021), to give some examples.  \nGaussian process regression (GPR) is one machine learning framework that has found application in the context of PIML (Raissi et al., 2017) . GPR can be especially e􀀋ective for data that is limited or expensive to obtain, and it o􀀋ers well-calibrated predictive uncertainty estimates, which may be essential for scienti􀀌c applications. In this work, we will consider the application of GPR to physical systems subject to linear boundary and linear PDE constraints. GPR is particularly e􀀋ective in this case, as it allows for seamless integration of observational data with linear PDE information, which enables e􀀎cient joint inference of any unknown PDE parameters together with the solution fun","cbCaiokjggK8n4Eh","https://ap.wps.com/l/cbCaiokjggK8n4Eh","pdf",2430214,1,61,"English","en",105,"# Introduction\n## Related Work","[{\"question\":\"What is the core idea behind boundary constrained Gaussian processes (BCGP) in this work?\",\"answer\":\"BCGP priors are constructed so that linear boundary conditions are enforced exactly within the Gaussian process framework, enabling consistent learning for PDE boundary value problems.\"},{\"question\":\"How does the method handle different boundary condition types?\",\"answer\":\"It designs boundary constrained mean and kernel functions for Dirichlet, Neumann, Robin, and mixed boundary conditions commonly used in PDE modelling.\"},{\"question\":\"What evidence supports the robustness of the approach?\",\"answer\":\"The paper reports extensive numerical experiments on several linear PDEs, showing effective and robust BCGP inference even when data are sparse and noisy.\"}]","Boundary Constrained Gaussian Processes for Robust Physics-Informed Machine Learning of Linear Partial Differential Equations | 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is the core idea behind boundary constrained Gaussian processes (BCGP) in this work?","Question",{"text":75,"@type":76},"BCGP priors are constructed so that linear boundary conditions are enforced exactly within the Gaussian process framework, enabling consistent learning for PDE boundary value problems.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the method handle different boundary condition types?",{"text":80,"@type":76},"It designs boundary constrained mean and kernel functions for Dirichlet, Neumann, Robin, and mixed boundary conditions commonly used in PDE modelling.",{"name":82,"@type":73,"acceptedAnswer":83},"What evidence supports the robustness of the approach?",{"text":84,"@type":76},"The paper reports extensive numerical experiments on several linear PDEs, showing effective and robust BCGP inference even when data are sparse and 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