[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-128653-en":3,"doc-seo-128653-105":30,"detail-sidebar-cat-0-en-105":92},{"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},128653,962084925782,"Ava Thompson","https://ap-avatar.wpscdn.com/davatar_9964176cb1d06d4a9deccf72a44ae3dc",8,"Research & Report","An Overview on Machine Learning Methods for Partial Differential Equations - from Physics Informed Neural Networks to Deep Operator Learning","The approximation of partial differential equation (PDE) solutions using numerical algorithms is a core topic in applied mathematics. Over decades, methods such as finite difference, finite element, and spectral schemes have been widely studied. In recent years, machine learning–based approaches have drawn significant interest, relying on training artificial neural networks with stochastic gradient descent–type optimization. This work introduces representative methods and the mathematical foundations behind them, focusing on physics-informed neural networks, deep BSDE methods, and multiple operator learning frameworks.","arXiv :2408 . 13222v1 [math .NA] 23 Aug 2024  \nAn Overview on Machine Learning Methods for Partial Differential Equations: from Physics Informed Neural Networks to Deep Operator Learning  \nLukas Gonon 1 , Arnulf Jentzen2 ,3 , Benno Kuckuck4 , Siyu Liang5 ,6 ,7 , Adrian Riekert8 , Philippe von Wurstemberger9 , 10  \n1 Department of Mathematics, Imperial College London,  \nUnited Kingdom; e-mail: [l.gonon@imperial.ac.uk](l.gonon@imperial.ac.uk)  \n2 School of Data Science and Shenzhen Research Institute  \nof Big Data, The Chinese University of Hong Kong, Shenzhen  \n(CUHK-Shenzhen), China; e-mail: [ajentzen@cuhk.edu.cn](ajentzen@cuhk.edu.cn)  \n3 Applied Mathematics: Institute for Analysis and Numerics,  \nFaculty of Mathematics and Computer Science, University of Münster, Germany; e-mail: [ajentzen@uni-muenster.de](ajentzen@uni-muenster.de)  \n4 Applied Mathematics: Institute for Analysis and Numerics,  \nFaculty of Mathematics and Computer Science, University of Münster, Germany; e-mail: [bkuckuck@uni-muenster.de](bkuckuck@uni-muenster.de)  \n5 School of Mathematics and Statistics,  \nNanjing University of Science and Technology,  \nNanjing, China; e-mail: [liangsiyu@njust.edu.cn](liangsiyu@njust.edu.cn)  \n6 School of Data Science, The Chinese University  \nof Hong Kong, Shenzhen (CUHK-Shenzhen), China  \n7 Mathematisches Institut, Ludwig-Maximilians-Universität München, Germany  \n8 Applied Mathematics: Institute for Analysis and Numerics,  \nFaculty of Mathematics and Computer Science, University of Münster, Germany; e-mail: [ariekert@uni-muenster.de](ariekert@uni-muenster.de)  \n9 School of Data Science, The Chinese University  \nof Hong Kong, Shenzhen (CUHK-Shenzhen),  \nChina; e-mail: [philippevw@cuhk.edu.cn](philippevw@cuhk.edu.cn)  \n10 Risklab, Department of Mathematics,  \nETH Zurich, Switzerland;  \ne-mail: [philippe.vonwurstemberger@math.ethz.ch](philippe.vonwurstemberger@math.ethz.ch)  \nAugust 26, 2024  \nAbstract  \nThe approximation of solutions of partial differential equations (PDEs) with numerical algorithms is a central topic in applied mathematics. For many decades, various types of methods for this purpose have been developed and extensively studied. One class of methods which has received a lot of attention in recent years are machine learning-based methods, which typically involve the training of artificial neural networks (ANNs) by means of stochastic gradient descent type optimization methods. While approximation methods for PDEs using ANNs have first been proposed in the 1990s they have only gained wide popularity in the last decade with the rise of deep learning. This article aims to provide an introduction to some of these methods and the mathematical theory on which they are based. We discuss methods such as physics-informed neural networks (PINNs) and deep BSDE methods and consider several operator learning approaches.  \nContents  \n1 Introduction 3  \n1.1 Fully-connected feedforward artificial neural networks (ANNs) ............. 4  \n2 Machine learning approximation methods for PDEs based on residual formulations 4  \n2.1 Basic reformulation result for machine learning methods for PDEs I .......... 4  \n2.2 Physics-informed neural networks (PINNs) ........................ 6  \n2.2.1 PINNs for general boundary value PDE problems ................ 6  \n2.2.2 PINNs for time-dependent initial value PDE problems ............. 9  \n2.2.3 PINNs for free boundary Stefan problems .................... 10  \n3 Machine learning approximation methods for PDEs based on stochastic FeynmanKac-type representations 13  \n3.1 Basic reformulation result for machine learning methods for PDEs II .......... 13  \n3.2 Stochastic representations (Feynman-Kac formulas) for PDEs ............. 16  \n3.3 Deep Kolmogorov methods ................................. 20  \n3.4 Deep BSDE methods .................................... 23  \n3.4.1 Uniqueness for solutions of BSDEs ........................ 23  \n3.4.2 Reformulating semilinear PDEs as infinite-dimensional sto","cbCaivEWOYTYSeMk","https://ap.wps.com/l/cbCaivEWOYTYSeMk","pdf",3354554,1,59,"English","en",105,"# Introduction\n## Fully-connected feedforward artificial neural networks (ANNs)\n## Machine learning approximation methods for PDEs based on residual formulations\n### Physics-informed neural networks (PINNs) - general boundary value PDE problems\n### Physics-informed neural networks (PINNs) - time-dependent initial value PDE problems\n### Physics-informed neural networks (PINNs) - free boundary Stefan problems\n## Machine learning approximation methods for PDEs based on stochastic Feynman-Kac-type representations\n### Stochastic representations (Feynman-Kac formulas) for PDEs\n### Deep Kolmogorov methods\n### Deep BSDE methods - Uniqueness for solutions of BSDEs\n### Deep BSDE methods - Reformulating semilinear PDEs as infinite-dimensional stochastic optimization problems\n## Operator learning methods\n### Neural operator architectures\n### Physics-informed neural operators\n### Other deep operator learning approaches\n### Numerical results","[{\"question\":\"What methods does the overview cover for approximating PDE solutions with machine learning?\",\"answer\":\"It introduces machine learning approximation methods based on residual formulations and stochastic Feynman–Kac-type representations, including physics-informed neural networks and deep BSDE methods, as well as operator learning approaches.\"},{\"question\":\"Why did neural-network approaches become more popular in the last decade?\",\"answer\":\"The article links the recent widespread popularity of ANN-based PDE approximation to the rise of deep learning, which enabled more effective training and broader application.\"},{\"question\":\"How do physics-informed neural networks (PINNs) fit into the framework discussed?\",\"answer\":\"PINNs are presented as a key residual-based approach, with variants for general boundary value problems, time-dependent initial value problems, and free boundary Stefan problems.\"}]","An Overview on Machine Learning Methods for Partial Differential Equations - 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