[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-125955-en":3,"doc-seo-125955-105":31,"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":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":28,"seo_description":14,"update_tm":29,"read_time":30},125955,137451207643,"Noah","https://ap-avatar.wpscdn.com/davatar_3d24733baf745e90a7e4bdd5f77d97b2",8,"Research & Report","Machine learning-based system reliability analysis with Gaussian Process Regression","Machine learning-based reliability analysis methods deliver improved computational efficiency and accuracy, yet existing work rarely addresses provably optimal learning strategies. This study proposes theoretical results to guide efficient sequential enrichment, explicitly contrasting designs that include or neglect correlations among candidate samples. It reformulates the U-learning function into an optimal learning form under assumptions of neglected Kriging correlation, and develops Bayesian estimates using corresponding loss functions. Simulations show that accounting for Kriging correlation reduces the required evaluations, at the cost of higher computational resources.","Machine learning-based system reliability analysis with Gaussian Process Regression  \nLisang Zhou1, Ziqian Luo2*, Xueting Pan3  \n1Bazaarvoice Inc., Austin, 78759, TX, United States, Email: [lzhou@berkeley.edu](lzhou@berkeley.edu)  \n2Oracle, Seattle, 98101, WA, United States, Corresponding author, Email: [luoziqian98@gmail.com](luoziqian98@gmail.com)  \n3Oracle, Seattle, 98101, WA, United States, [Email:](Email: xtpan8800@gmail.com)[ ](Email: xtpan8800@gmail.com)[xtpan8800@gmail.com](Email: xtpan8800@gmail.com)  \nABSTRACT  \nMachine learning-based reliability analysis methods have shown great advancements for their computational efficiency and accuracy. Recently, many efficient learning strategies have been proposed to enhance the computational performance. However, few of them explores the theoretical optimal learning strategy. In this article, we propose several theorems that facilitates such exploration. Specifically, cases that considering and neglecting the correlations among the candidate design samples are well elaborated. Moreover, we prove that the well-known U learning function can be reformulated to the optimal learning function for the case neglecting the Kriging correlation. In addition, the theoretical optimal learning strategy for sequential multiple training samples enrichment is also mathematically explored through the Bayesian estimate with the corresponding lost functions. Simulation results show that the optimal learning strategy considering the Kriging correlation works better than that neglecting the Kriging correlation and other stateof-the art learning functions from the literatures in terms of the reduction of number of evaluations of performance function. However, the implementation needs to investigate very large computational resource.  \nKey words: Reliability analysis; Risk analysis; Surrogate models; Kriging; Gaussian Process Regression; Active Learning;  \n1. Introduction  \nProbabilities of occurrence of risky events are critically decisive for engineers and researchers to quantify the risk of planed operations and conductions. These probabilities are typically characterized by conducting reliability analysis, of which the target is to estimate the probability of failure, denoted as 􀜲􀯙 . In reliability analysis, 􀜲􀯙 can be calculated as:  \n􀜲􀯙 = (􀢞)≤0􀟩 (􀢞)􀝀􀢞 =  􀜫􀯚 (􀢞)􀟩 (􀢞)􀝀􀢞 , (1)  \nwhere 􀢞 is the vector of random variables, 􀝃 (􀢞) is the so-called performance or limit state function, 􀟩(􀢞) is the joint probability density function (PDF) of 􀢞 and 􀜫􀯚 (􀢞) is a failure indicator function. Note that 􀜫􀯚 (􀢞) = 1 when 􀝃 (􀢞) ≤ 0, and 􀜫􀯚 (􀢞) = 0 when 􀝃 (􀢞) > 0. Modern reliability analysis methodologies aim to estimate 􀜲􀯙 with as less as prossible number of evluations to the computationally demanding numerical models. Those well-developed techniques primarily include simulation-based sampling techniques (e.g., the crude Monte-Carlo simulation (MCS) [1], [2], importance sampling (IS) [3], and subset simulation (SS)  \n[4]) and approximation-based approaches (e.g. first-or second-order reliability analysis methods) [5], [6]), which estimate 􀜲􀯙 by searching for the most probable point in the probabilistic space. Despite the first group can produce desirable estimate of 􀜲􀯙 , it is computationally expensive. On the contrary, approximation-based methods are often computationally fast, they lack the accuracy for problems with non-linear responses near the limit state. To address aforementioned limitations, surrogate model-based reliability analysis has emerged and shown advancement. Those surrogate models are mainly response surfaces [7], [8], artificial neutral networks [9], [10], [11], support vector machines [12], [13], polynomial chaos expansions [14], [15]  \nand the Gaussian Process Regression or Kriging model [16], [17] . Among these techniques, the Kriging model has been developing extremely fast out of its inherent advantages [18], [19], [20], [21] .  \nThe Kriging model or Gaussian process regression is a Bayesian","cbCaiiwGiGnA1BTP","https://ap.wps.com/l/cbCaiiwGiGnA1BTP","pdf",8446418,2,1,27,"English","en",105,"# Introduction\n## Reliability analysis problem formulation\n## Surrogate model-based reliability analysis\n## Kriging as a Bayesian surrogate model\n## Active learning strategies for sequential enrichment\n# Active learning in Kriging-based reliability\n## Role of learning functions\n## Existing learning functions and variants","[{\"question\":\"What problem does the document address in reliability analysis?\",\"answer\":\"It addresses how to estimate the probability of failure with as few evaluations as possible when reliability analysis relies on computationally demanding numerical models.\"},{\"question\":\"How does Gaussian Process Regression (Kriging) support active learning for reliability?\",\"answer\":\"Kriging provides a Bayesian regression model with predictive mean and variance, enabling adaptive enrichment of training samples so the surrogate can replace expensive evaluations during Monte Carlo simulation.\"},{\"question\":\"What is the main theoretical contribution regarding learning strategies?\",\"answer\":\"The document proposes several theorems for exploring theoretically optimal learning strategies, including reformulating the U-learning function for the case neglecting Kriging correlation and analyzing sequential multiple-sample enrichment via Bayesian estimates with loss functions.\"}]","Machine learning-based system reliability analysis with Gaussian Process Regression | 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problem does the document address in reliability analysis?","Question",{"text":76,"@type":77},"It addresses how to estimate the probability of failure with as few evaluations as possible when reliability analysis relies on computationally demanding numerical models.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"How does Gaussian Process Regression (Kriging) support active learning for reliability?",{"text":81,"@type":77},"Kriging provides a Bayesian regression model with predictive mean and variance, enabling adaptive enrichment of training samples so the surrogate can replace expensive evaluations during Monte Carlo simulation.",{"name":83,"@type":74,"acceptedAnswer":84},"What is the main theoretical contribution regarding learning strategies?",{"text":85,"@type":77},"The document proposes several theorems for exploring theoretically optimal learning strategies, including reformulating the U-learning function for the case neglecting Kriging correlation and analyzing sequential 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