[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-122588-en":3,"doc-seo-122588-105":30,"detail-sidebar-cat-0-en-105":95},{"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},122588,7971461741311,"Ophelia","https://ap-avatar.wpscdn.com/avatar/74000253aff267980c6?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779345379180704826",8,"Research & Report","PDE-Constrained Machine Learning with Gaussian Processes towards Digital Twins","This dissertation investigates PDE-constrained machine learning for digital twins using Gaussian processes and deep kernel learning. It studies how to merge deep neural network components with Gaussian processes, while enforcing physics constraints through partial differential equations. The work develops robust modelling under input uncertainty, including uncertain data locations, and addresses parameter inference in high-dimensional PDE settings. Comparative deep learning with GPs, surrogate modelling for linear PDEs, numerical experiments, and likelihood derivations support the proposed methodologies and conclusions.","PDE-Constrained Machine Learning with Gaussian Processes  \ntowards Digital Twins  \nWeihao Yan  \nPDE-Constrained Machine Learning with Gaussian Processes towards Digital Twins  \nWeihao Yan  \nPDE-Constrained Machine Learning with Gaussian Processes towards Digital Twins  \nproefschrift  \nter verkrijging van  \nde graad van doctor aan de Universiteit Twente, op gezag van de rector magnificus,  \n[prof. dr. ir. A. Veldkamp](prof. dr. ir. A. Veldkamp), volgens besluit van het College voor Promoties in het openbaar te verdedigen op maandag 29 september 2025 om 12.45 uur  \ndoor  \nWeihao Yan  \ngeboren op 4 Mei 1995  \nThis dissertation has been approved by:  \nPromotors  \nprof. dr. C. Brune dr. M. Guo  \nThe work in this thesis was carried out at the chair: Mathematics of Imaging & AI,  \nFaculty of Electrical Engineering, Mathematics and Computer Science, University of Twente, Drienerlolaan 5, 7522NB, The Netherlands.  \nPrinted by: Ipskamp printing  \nISBN (print): 978-90-365-6881-4  \nISBN (digital): 978-90-365-6882-1  \nURL: [https://doi.org/10.3990/1.9789036568821](https://doi.org/10.3990/1.9789036568821)  \n[Copyright](Copyright) © [2025 by Weihao Yan](2025 by Weihao Yan), [Enschede](Enschede), [The Netherlands](The Netherlands).  \nAll rights reserved. No parts of this thesis may be reproduced, stored in a retrieval system or transmitted in any form or by any means without permission of the author. Alle rechten voorbehouden. Niets uit deze uitgave mag wordenvermenigvuldigd, in enige vorm of op enige wijze, zonder voorafgaandeschriftelijke toestemming van de auteur.  \nGraduation Committee:  \nChairman:  \nprof. dr. B.R.H.M Haverkort  \nPromotors:  \nprof. dr. C. Brune dr. M. Guo  \nCommittee Members: prof. dr. D. T. Crommelin prof. dr. T. van Leeuwen [prof. dr. ing. B. Rosic](prof. dr. ing. B. Rosic)[dr](dr). M. Schlottbom  \nprof. dr. K. P. Veroy-Grepl  \nUniversity of Twente  \nUniversity of Twente Lund University  \nUniversity of Amsterdam Utrecht University University of Twente  \nUniversity of Twente  \nEindhoven University of Technology  \nContents  \n1 Introduction 1  \n1.1 Motivation and background ......................... 1  \n1.2 Merging deep neural networks and Gaussian processes ........ 5  \n1.3 From data-driven to physics-constrained modelling ........... 7  \n1.4 Robust modelling under input uncertainty ................ 10  \n1.5 Parameter inference in high dimensions .................. 11  \n1.6 Outline and summary of contributions .................. 14  \n2 Deep learning with GPs: a comparative study 17  \n2.1 Introduction .................................. 17  \n2.2 Background .................................. 20  \n2.3 Deep architectures with Gaussian processes ............... 22  \n2.4 Numerical experiments ........................... 29  \n2.5 Discussion ................................... 35  \n2.6 Conclusion and outlook ........................... 36  \n3 PDE-constrained DKL in high dimensionality 39  \n3.1 Introduction .................................. 39  \n3.2 Problem formulation and solution methods ................ 43  \n3.3 Deep kernel learning with PDE constraints ................ 46  \n3.4 Numerical results ............................... 52  \n3.5 Concluding remarks ............................. 60  \n4 PDE-constrained GP with uncertain data locations 65  \n4.1 Introduction .................................. 66  \n4.2 Bayesian approach to GP regression with uncertain inputs ....... 69  \n4.3 Surrogate modelling for linear PDEs using GP with uncertain inputs . 72  \n4.4 Numerical experiments ........................... 75  \n4.5 Conclusion ................................... 86  \n5 PDE-DKL for parameter estimation in high-dimensional PDEs 87  \n5.1 Introduction .................................. 88  \n5.2 Background .................................. 90  \n5.3 Methodology ................................. 95  \n5.4 Numerical experiments ........................... 104  \n5.5 Discussion ................................... 109  \n5.6 Conclusion and outl","cbCaic5h0TcFtThT","https://ap.wps.com/l/cbCaic5h0TcFtThT","pdf",23262071,1,153,"English","en",105,"# Contents\n## 1 Introduction\n## 2 Deep learning with GPs: a comparative study\n## 3 PDE-constrained DKL in high dimensionality\n## 4 PDE-constrained GP with uncertain data locations\n## 5 PDE-DKL for parameter estimation in high-dimensional PDEs\n## Appendix: Derivation of the likelihood function\n## 6 Conclusion and Outlooks\n## References\n## Publications and conference presentations\n## Acknowledgements\n## Summary\n## Samenvatting","[{\"question\":\"What is the dissertation’s main objective?\",\"answer\":\"To develop PDE-constrained machine learning methods using Gaussian processes and deep kernel learning for digital twin modelling, including robust inference under uncertainty.\"},{\"question\":\"How does the work combine deep learning with Gaussian processes?\",\"answer\":\"It studies deep neural network architectures with Gaussian processes and introduces deep kernel learning components to integrate learning with uncertainty-aware GP modelling.\"},{\"question\":\"How are uncertainties handled in the proposed models?\",\"answer\":\"The dissertation treats input uncertainty in multiple ways, including Bayesian GP regression with uncertain inputs and surrogate modelling that accounts for uncertain data locations.\"},{\"question\":\"What problem does the dissertation address in high-dimensional PDEs?\",\"answer\":\"It focuses on parameter estimation and inference in high-dimensional PDE settings using PDE-constrained deep kernel learning and GP-based methods, supported by numerical experiments and discussion.\"}]","PDE-Constrained Machine Learning with Gaussian Processes towards Digital Twins | 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is the dissertation’s main objective?","Question",{"text":75,"@type":76},"To develop PDE-constrained machine learning methods using Gaussian processes and deep kernel learning for digital twin modelling, including robust inference under uncertainty.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the work combine deep learning with Gaussian processes?",{"text":80,"@type":76},"It studies deep neural network architectures with Gaussian processes and introduces deep kernel learning components to integrate learning with uncertainty-aware GP modelling.",{"name":82,"@type":73,"acceptedAnswer":83},"How are uncertainties handled in the proposed models?",{"text":84,"@type":76},"The dissertation treats input uncertainty in multiple ways, including Bayesian GP regression with uncertain inputs and surrogate modelling that accounts for uncertain data locations.",{"name":86,"@type":73,"acceptedAnswer":87},"What problem does the dissertation address in high-dimensional PDEs?",{"text":88,"@type":76},"It focuses on parameter estimation and inference in high-dimensional PDE settings using PDE-constrained deep kernel learning and GP-based methods, supported by numerical experiments and discussion.","https://schema.org",{"og:url":52,"og:type":91,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":93,"canonical":52},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":96},[97,101,105,109,114,119,124,127,132,135,139],{"id":20,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Story & 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