[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82890-en":3,"doc-seo-82890-105":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":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":13,"seo_description":14,"update_tm":28,"read_time":29},82890,8796095462418,"Noah","https://ap-avatar.wpscdn.com/avatar/80000253c1241d02b47?x-image-process=image/resize,m_fixed,w_180,h_180&k=1778826106357471780",8,"Research & Report","Uncertainty-aware damage identification in short-span bridges via physics-informed variational autoencoder","Vibration-based damage identification for civil infrastructure is an ill-posed inverse problem affected by measurement noise, sparse sensors, and environmental variability. Deterministic deep learning system identification often lacks reliable uncertainty quantification and can produce physically inconsistent outputs. This work presents a robust probabilistic SciML framework, the physics-informed Gaussian copula variational autoencoder (PI-GCVAE) for structural health monitoring, embedding physics constraints and correlated latent modeling to recover feasible damage posteriors under uncertainty.","arXiv :2607 .05025v2 [ cs .LG] 8 Jul 2026  \nUncertainty-aware damage identification in short-span bridges via physics-informed variational autoencoder  \nAna Fernandez-Navamuel1*, ´Angel J. Omella2 , Diego Zamora-Sanchez3 ,  \nDavid Pardo4  \n1 Basque Center for Applied Mathematics (BCAM), Bilbao, Spain.  \n2 Departamento de Matematica Aplicada, Universidad de Zaragoza/IUMA, Zaragoza, Spain.  \n3 TECNALIA, Basque Research and Technology Alliance (BRTA), Parque Cientifico y Tecnologico de Bizkaia, Astondo bidea, Edificio 700, Derio, E-48160, Spain.  \n4 Department of Mathematics, University of the Basque Country (UPV/EHU), Spain.  \n*Corresponding author(s). E-mail(s): [afernandez@bcamath.org](afernandez@bcamath.org) ; Contributing authors: [aomella@unizar.es](aomella@unizar.es) ; [diego.zamora@tecnalia.com](diego.zamora@tecnalia.com) ; [dzubiaur@gmail.com](dzubiaur@gmail.com) ;  \nKeywords: Scientific machine learning, Uncertainty quantification, Variational autoencoder, Structural health monitoring, Gaussian copulas, Bridge damage identification  \nAbstract  \nVibration-based damage identification in civil infrastructure is a challenging, ill-posed inverse problem due to measurement noise, sparse sensor arrays, and environmental variability. While deep learning is powerful for system identification, deterministic approaches lack reliable uncertainty quantification and can yield physically inconsistent results. This work proposes a robust probabilistic Scientific Machine Learning (SciML) framework: a physics-informed Gaussian copula variational autoencoder (PI-GCVAE) for structural health monitoring (SHM) .  \nOur methodology introduces three main contributions. First, we eliminate the need for data-driven surrogates by embedding a differentiable numerical eigenvalue solver directly into the VAE architecture. This ensures that latent space samples satisfy the governing equations of structural dynamics, reducing the trainable parameter space and improving generalization. Second, we replace the conventional independence assumption of latent variables with a Gaussian copula. This model captures complex, physics-dependent spatial cross-correlations between adjacent structural elements, defining feasible solutions while accounting for inherent system variability and measurement errors. Third, compared with alternatives such as Gaussian mixtures, our copula-based VAE provides an efficient distributional model for high-dimensional, strongly correlated latent spaces.  \nWe validate the approach using a synthetic dataset of a simply supported bridge subjected to various damage scenarios and corrupted with stochastic Gaussian noise (2 .5% on frequencies, 5% on mode shapes) . Synthetic data enables exhaustive validation against ground-truth stiffness values unavailable in practice. Results demonstrate that the PI-GCVAE accurately recovers the true posterior distribution, achieving 77.2% coverage. The proposed framework provides a reliable, scalable tool for early-stage damage diagnosis in operating bridges.  \nKeywords: Scientific machine learning · Physics-informed Neural networks · Uncertainty quantification · Gaussian copula · Variational autoencoder · Bridge damage identification.  \n1 Introduction  \nVibration-based structural health monitoring (SHM) is increasingly applied to assess the condition of largescale operative systems [1 , 2] . It addresses an inverse problem known as system identification [3], which aims to discover the presence of damage from measurements of a system’s dynamic response [1, 4 , 5] . In the context of civil engineering, structural damage is defined as any reduction in the system’s performance capabilities resulting from a local or distributed decrease in its stiffness properties [6] . The measured response often comes from acceleration sensors sparsely distributed along the structure. However, since raw acceleration signals are highly sensitive to environmental and operational conditions, a standard practice is to process them using operati","cbCaidipWlJi0051","https://ap.wps.com/l/cbCaidipWlJi0051","pdf",7553151,4,1,27,"English","en",105,"# Introduction\n# Methodology (PI-GCVAE)\n## Physics-informed differentiable eigenvalue solver\n## Gaussian copula latent modeling\n# Validation and Results\n## Synthetic simply supported bridge scenarios\n## Coverage and posterior recovery","[{\"question\":\"Why is vibration-based bridge damage identification considered an ill-posed inverse problem?\",\"answer\":\"It is ill-posed due to measurement noise, sparse sensor arrays, and environmental/operational variability, which allow multiple damage scenarios to produce similar observable responses.\"},{\"question\":\"What are the key elements of the proposed PI-GCVAE framework?\",\"answer\":\"The method embeds a differentiable numerical eigenvalue solver inside the VAE to enforce governing structural dynamics, and it replaces latent independence with a Gaussian copula to model physics-dependent cross-correlations and uncertainty.\"},{\"question\":\"How is the approach validated and what performance metric is reported?\",\"answer\":\"The model is tested on a synthetic simply supported bridge with multiple damage scenarios and stochastic Gaussian noise. Results report accurate posterior recovery with 77.2% coverage.\"}]",1784183728,68,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"uncertainty-aware-damage-identification-in-short-span-bridges-via-physics-informed-variational-autoencoder","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":20},"https://docshare.wps.com/document/uncertainty-aware-damage-identification-in-short-span-bridges-via-physics-informed-variational-autoencoder/82890/",{"url":52,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-24","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"Why is vibration-based bridge damage identification considered an ill-posed inverse problem?","Question",{"text":75,"@type":76},"It is ill-posed due to measurement noise, sparse sensor arrays, and environmental/operational variability, which allow multiple damage scenarios to produce similar observable responses.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What are the key elements of the proposed PI-GCVAE framework?",{"text":80,"@type":76},"The method embeds a differentiable numerical eigenvalue solver inside the VAE to enforce governing structural dynamics, and it replaces latent independence with a Gaussian copula to model physics-dependent cross-correlations and uncertainty.",{"name":82,"@type":73,"acceptedAnswer":83},"How is the approach validated and what performance metric is reported?",{"text":84,"@type":76},"The model is tested on a synthetic simply supported bridge with multiple damage scenarios and stochastic Gaussian noise. 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