[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84589-en":3,"doc-seo-84589-105":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},84589,16904993612988,"Olivia Brown","https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd",8,"Research & Report","Convolutional Symmetric Autoencoders Enhancing Latent Stability Via Differential Geometry","Autoencoders are used for non-linear dimensionality reduction and, in reduced-order modelling (ROM), for learning low-dimensional representations of solution manifolds of parametric PDEs. Their high expressivity can fail to preserve the properties needed for accurate, robust ROMs, even when reconstruction error is minimized. The study extends representation consistency from fully connected AEs to convolutional layers by introducing symmetric convolutional autoencoders aligned with manifold parametrization mappings. Integrated into ROMs, the method improves predictive capability, producing more accurate latent trajectories, lower reconstruction errors, and greater robustness on advection, viscous Burgers, and Kuramoto–Sivashinsky test cases.","arXiv :2607 .00669v1 [math .NA] 1 Jul 2026  \nCONVOLUTIONAL SYMMETRIC AUTOENCODERS: ENHANCING LATENT STABILITY VIA DIFFERENTIAL GEOMETRY  \nGASPARE LI CAUSI 1 , NICCOLÒ TONICELLO 1 , LUCA MAGRI2 ,3 AND GIANLUIGI ROZZA 1  \n1 mathLab, Mathematics Area, SISSA, via Bonomea 265, I-34136 Trieste, Italy  \n2 Department of Aeronautics, Imperial College London, SW7 2AZ, London, UK  \n3 Politecnico di Torino, DIMEAS, Corso Duca degli Abruzzi, 24 10129 Torino, Italy  \nAbstract . Autoencoders (AEs) have emerged as powerful tools for non-linear dimensionality reduction, often surpassing traditional linear methods such as Proper Orthogonal Decomposition (POD) in scenarios characterized by slowly decaying Kolmogorov n-widths. In the realm of Reduced-Order Modelling (ROM), these models are increasingly utilized to learn low-dimensional representations of solution manifolds associated with parametric Partial Differential Equations (PDEs) . However, the high expressivity of AEs presents a challenge: although trained networks typically minimize reconstruction error, they often struggle to capture the essential properties necessary for building accurate and robust ROMs. Recent works by [33] and [6] have tackled this challenge in fully connected AEs by proposing representation-consistent architectures, which preserve some of the properties belonging to POD. This study builds upon that concept by extending representation consistency for convolutional layers. We introduce a novel class of symmetric Convolutional AutoEncoders (CAEs) designed to embody the primary properties of manifold parametrization mappings. When integrated into a ROM framework, this architecture demonstrates significantly improved predictive capabilities. Specifically, we compared the performance of the ROMs based on classical and symmetric CAEs on three one dimensional academic test cases, namely the Linear Advection, Viscous Burger and Kuramoto–Sivashinsky equation. Numerical results demonstrate that our proposed symmetric approach consistently yields more accurate latent trajectories, lower reconstruction errors, and enhanced model robustness.  \n1. Introduction  \nOver the past few decades, numerical simulations of physical systems have provided invaluable insight, significantly enhancing our understanding of complex phenomena. These systems are typically modeled using Partial Differential Equations (PDEs) for which closed-form solutions are rarely attainable. Instead, they must be solved through high-fidelity numerical frameworks such as Finite Difference (FD), Finite Element Methods (FEM), and Finite Volume (FV) techniques.  \nThe complexity of these systems does not translate directly into a more difficult implementation of those numerical schemes, but rather arises from the challenge of managing diverse temporal and spatial scales. These scales, often differing by several orders of magnitude, do not evolve separately; they interact dynamically due to the couplings arising from the nonlinear terms, leading to phenomena such as the energy cascade observed in viscous dissipative systems like turbulence. To achieve accurate solutions, a simulation must resolve the entire spectrum of scales, from the largest structures down to the smallest dissipative eddies. This requirement results in discretized systems with a massive number of Degrees of Freedom (DoF) .  \nDespite the continuous advancement of High-Performance Computing (HPC) facilities over the last thirty years [22, 45], certain realistic solutions remain virtually unattainable within reasonable time frames. A prominent example is the Direct Numerical Simulation (DNS) of an entire aircraft under cruise conditions, which remains computationally prohibitive even with modern HPC hardware. Even in scenarios where DNS is technically feasible, its application is often impractical for multi-query contexts such as structural parameter optimization or for real-time prediction models embedded in control systems. In this context Reduced Or","cbCailJ8zFmAQBwi","https://ap.wps.com/l/cbCailJ8zFmAQBwi","pdf",21133605,2,1,28,"English","en",105,"# Introduction\n# Symmetric CAEs: Enhancing Latent Stability via Differential Geometry","[{\"question\":\"What problem does the paper address in autoencoder-based ROMs?\",\"answer\":\"Autoencoders may minimize reconstruction error during training but still fail to capture essential properties needed for accurate and robust reduced-order models.\"},{\"question\":\"What is the key contribution of the proposed method?\",\"answer\":\"The paper introduces a novel class of symmetric convolutional autoencoders that extend representation-consistency ideas to convolutional layers and better reflect manifold parametrization mapping properties.\"},{\"question\":\"How was the method evaluated and what were the outcomes?\",\"answer\":\"The authors compared ROMs built on classical versus symmetric convolutional autoencoders on three 1D academic PDE test cases (Linear Advection, Viscous Burger, and Kuramoto–Sivashinsky). Results show improved latent trajectories, lower reconstruction errors, and enhanced robustness for the symmetric approach.\"}]",1784196962,71,{"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},"convolutional-symmetric-autoencoders-enhancing-latent-stability-via-differential-geometry","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,47,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":20},"https://docshare.wps.com/document/","Document",{"item":48,"name":12,"@type":43,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/convolutional-symmetric-autoencoders-enhancing-latent-stability-via-differential-geometry/84589/",4,{"url":51,"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","2026-07-22","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},"What problem does the paper address in autoencoder-based ROMs?","Question",{"text":75,"@type":76},"Autoencoders may minimize reconstruction error during training but still fail to capture essential properties needed for accurate and robust reduced-order models.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What is the key contribution of the proposed method?",{"text":80,"@type":76},"The paper introduces a novel class of symmetric convolutional autoencoders that extend representation-consistency ideas to convolutional layers and better reflect manifold parametrization mapping properties.",{"name":82,"@type":73,"acceptedAnswer":83},"How was the method evaluated and what were the outcomes?",{"text":84,"@type":76},"The authors compared ROMs built on classical versus symmetric convolutional autoencoders on three 1D academic PDE test cases (Linear Advection, Viscous Burger, and Kuramoto–Sivashinsky). Results show improved latent trajectories, lower reconstruction errors, and enhanced robustness for the symmetric approach.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]