[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-125441-en":3,"doc-seo-125441-105":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},125441,687197207919,"Theodora","https://ap-avatar.wpscdn.com/avatar/a000253d6f5f7c60be?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779446848396160552",8,"Research & Report","Automated Machine Learning Exact Dirichlet Boundary Physics-Informed Neural Networks for Solid Mechanics","Physics-informed neural networks (PINN) are applied to partial differential equations, but conventional formulations can suffer from convergence instability, loss-balancing difficulties, and a large number of trainable parameters. Exact Dirichlet boundary condition PINN (EPINN) addresses forward solid-mechanics problems using tensor decomposition, a distance-function approximation, and the principle of least work, delivering over 127× speedup. This work introduces mesh-free 3D BO-TPE Automated Machine Learning EPINN (AEPINN) to optimize hyperparameters without labeled solution-field data, achieving further speedups and accurate GPU-accelerated displacement predictions, while analyzing effects of training precision and hyperparameter sensitivity.","Engineering Structures 330 (2025) 119884  \nContents lists available at ScienceDirect  \nEngineering Structures  \njournal [homepage:](homepage: www.elsevier.com/locate/engstruct)[ www.elsevier.com/locate/engstruct](homepage: www.elsevier.com/locate/engstruct)  \n| Automated machine learning exact dirichlet boundary physics-informed neural networks for solid mechanics\u003Cbr>Xiaoge Tiana , Jiaji Wang a,* , Chul-Woo Kim b , Xiaowei Denga , Yingjie Zhu c \u003Cbr>a Department of Civil Engineering, The University of Hong Kong, China\u003Cbr>b Department of Civil and Earth Resources Engineering, Kyoto University, Kyoto, Japan c School of Civil Engineering, North China University of Technology, China |  |\n| --- | --- |\n| A R T I C L E I N F O\u003Cbr>Keywords:\u003Cbr>Solid mechanics\u003Cbr>Physics-informed neural network (PINN) Exact Dirichlet boundary PINN (EPINN) Bayesian-optimization tree-structured parzen estimator\u003Cbr>AutoML EPINN (AEPINN) | A B S T R A C T\u003Cbr>While Physics-informed neural networks (PINN) have made significant progress in solving partial differential equations (PDE), conventional PINN may have convergence issues due to spectral bias, the requirement of loss balancing, and a significant number of trainable weights. Exact Dirichlet boundary condition Physics-informed Neural Networks (EPINN) was developed to solve forward problems in solid mechanics by applying tensor decomposition, approximating distance function, and the principle of least work, achieving more than 127 times speedup compared to PINN. However, the sensitivity of hyperparameters of the PINN framework is less reported. To merge the gap, this study develops the mesh-free 3D Bayesian-Optimization Tree-Structured Parzen Estimator (BO-TPE) Automated Machine Learning EPINN to solve solid mechanics problems without labelled data of the solution field. Developed based on Nvidia modulus platform, the Automated Machine Learning EPINN (AEPINN) can achieve more than 20 times speedup for 2D plane stress problems and four times speedup for 3D bracket problems compared with the EPINN architecture. Compared with conventional PINN, AEPINN model achieved more than 200 times speedup for a plane stress problem and 400 times speed up for a bracket problem. For a twospan three-story frame composed of beams, columns, and slabs, the AEPINN model can simulate the frame displacement deformations comparable to ABAQUS results with adequate accuracy and speed with GPU accelerated. Optimized hyperparameters AEPINN can approach a hyperelastic cube rubber case within 60 s compared with Abaqus results of Mooney-Rivlin constitutive law. The comparison between single-precision and doubleprecision training is also illustrated. The influences of hyperparameters in the adopted EPINN framework are examined accordingly. |\n\n1. Introduction  \nAs one of the numerical solution methods to obtain approximated solution field Partial Differential Equations (PDE), Finite Element Methods (FEM) can mesh complex objects into simple elements achieving numerical results approximation [1]. Despite the performance ofFEM, it may be hard to obtain the derivatives ofthe solution field with respect to input parameters, especially for complicated structures. Hence, solving parametric simulation problems, inverse problems and design optimization problems may be challenging due to the numerical approximation in FEM [2,3]. GPU-accelerated differentiable solvers for solid mechanics problems have been rapidly developed to achieve significant speedup in parametric analysis, design optimization problems, and inverse problems. Deep learning has recently contributed to  \nintelligent computation research with substantial efficiencies and affordable costs in simulation and experimental parts [4–7]. Researchers added loss terms of governing equations (PDE loss) to the data loss terms and developed a physics-informed neural network framework (PINN) to solve scientific PDE computation problems by utilizing the Universal Approximation Theory of Deep Ne","cbCaijJsXksSPMDW","https://ap.wps.com/l/cbCaijJsXksSPMDW","pdf",26986498,1,20,"English","en",105,"# Introduction\n# Background and Motivation\n# Physics-Informed Neural Networks and Challenges\n# Exact Dirichlet Boundary PINN (EPINN)\n# Automated Machine Learning EPINN (AEPINN)\n## Bayesian-optimization BO-TPE framework\n## Implementation on Nvidia Modulus and GPU acceleration\n# Numerical Experiments and Results\n## Speedup and accuracy comparisons\n## Hyperparameter and precision effects\n# Conclusion","[{\"question\":\"What problem does the study address in conventional PINN methods for solid mechanics?\",\"answer\":\"Conventional PINN can show convergence issues due to spectral bias, loss balancing requirements, and many trainable weights, and extending to complex solid problems is limited by solver efficiency.\"},{\"question\":\"How does EPINN enforce exact Dirichlet boundary conditions in solid mechanics?\",\"answer\":\"EPINN applies tensor decomposition, approximates the distance function, and uses the principle of least work to solve forward solid-mechanics PDEs with exact Dirichlet boundaries.\"},{\"question\":\"What is the role of BO-TPE AutoML in the proposed AEPINN framework?\",\"answer\":\"BO-TPE automates hyperparameter optimization for EPINN without requiring labeled solution-field data, improving speed and maintaining accuracy across benchmark solid mechanics cases.\"}]","Automated Machine Learning Exact Dirichlet Boundary Physics-Informed Neural Networks for Solid Mechanics | 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problem does the study address in conventional PINN methods for solid mechanics?","Question",{"text":75,"@type":76},"Conventional PINN can show convergence issues due to spectral bias, loss balancing requirements, and many trainable weights, and extending to complex solid problems is limited by solver efficiency.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does EPINN enforce exact Dirichlet boundary conditions in solid mechanics?",{"text":80,"@type":76},"EPINN applies tensor decomposition, approximates the distance function, and uses the principle of least work to solve forward solid-mechanics PDEs with exact Dirichlet boundaries.",{"name":82,"@type":73,"acceptedAnswer":83},"What is the role of BO-TPE AutoML in the proposed AEPINN framework?",{"text":84,"@type":76},"BO-TPE automates hyperparameter optimization for EPINN without requiring labeled solution-field data, improving speed and maintaining accuracy across benchmark solid mechanics 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