[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83606-en":3,"doc-seo-83606-105":29,"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":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":13,"seo_description":14,"update_tm":27,"read_time":28},83606,16904993612988,"Olivia Brown","https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd",8,"Research & Report","Structure-Preserving Reduced-Order Modeling via Low-Rank Transport Signatures","Parametrized PDEs with density-valued solutions are difficult to approximate with classical linear reduced-order models, particularly in transport-dominated regimes. The work proposes an optimal-transport-based ROM that represents each density via the Kantorovich potential mapping a fixed reference density to the target density, then converts potentials into transport signatures using a weighted Laplacian. These signatures embed the solution map in a Hilbert space with controlled induced transport and Wasserstein error, support low-rank skeleton decomposition and neural non-intrusive online evaluation, and include mass preservation by construction. The paper proves mean-squared Wasserstein error bounds separating low-rank, discretization, sampling, and learning contributions, and validates results on a 2D continuity equation where signatures yield lower-rank structure than raw snapshots.","arXiv :2607 .01696v1 [math .NA] 2 Jul 2026  \nStructure-Preserving Reduced-Order Modeling via Low-Rank Transport Signatures  \nJiajia Yu∗  \nJingwei Hu† Yunan Yang¶  \nFengyan Li‡ Shanyin Tong§ Zhaiming Shen ‖  \nAbstract  \nParametrized PDEs with density-valued solutions are often difficult to approximate with classical linear reduced-order models, especially in transport-dominated regimes. We introduce an optimal-transport-based reduced-order modeling that represents each density by the Kantorovich potential transporting a fixed reference density to the target density, and then maps these potentials to transport signatures using a weighted Laplacian associated with the reference measure. This embeds the density-valued solution map in a Hilbert space while preserving control of the induced transport maps and Wasserstein error. We treat the signature map as a continuous matrix indexed by parameters and space, construct a low-rank skeleton decomposition using a maximalvolume criterion, and learn the parameter-to-coefficient map with a neural network for efficient non-intrusive online evaluation. The reconstructed solution is obtained by pushing forward the reference density, so mass preservation is built into the method. We prove a mean-squared Wasserstein error bound separating low-rank approximation, discretization, sampling, and learning errors, and demonstrate the method on a twodimensional continuity equation, where transport signatures yield substantially lowerrank structure than the original density snapshots.  \n1 Introduction  \nParametrized partial differential equations arise in many-query and real-time settings where a high-fidelity model must be evaluated repeatedly for many parameter values. Reducedorder modeling (ROM) seeks to replace such high-dimensional simulations by inexpensive surrogate models that retain the dominant dependence of the solution on the parameters. Classical approaches, including proper orthogonal decomposition (POD) and reduced-basis  \n∗[jiajia.yu@duke.edu. Department](jiajia.yu@duke.edu. Department) of Mathematics, Duke University, Durham, NC 27710  \n†[hujw@uw.edu. Department](hujw@uw.edu. Department) of Applied Mathematics, University of Washington, Seattle, WA 98195 ‡[lif@rpi.edu. Department](lif@rpi.edu. Department) of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, NY 12180 §[tong3@sas.upenn.edu. Department](tong3@sas.upenn.edu. Department) of Mathematics, University of Pennsylvania, Philadelphia, PA 19104 ¶[yunan.yang@cornell.edu. Department](yunan.yang@cornell.edu. Department) of Mathematics, Cornell University, Ithaca, NY 14850 ‖[zshen49@gatech.edu. School](zshen49@gatech.edu. School) of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332  \nmethods, construct a low-dimensional linear approximation space from representative snapshots and then approximate new solutions by projection or regression in this space [4, 16 , 18] . These methods have been highly successful for many elliptic, parabolic, and mildly nonlinear problems, especially when the associated solution manifold has rapidly decaying Kolmogorov widths [9, 12 , 17 , 26] .  \nHowever, transport-dominated problems pose a persistent challenge for linear reducedorder models. When coherent structures translate, deform, or concentrate, the solution manifold may be intrinsically low-dimensional but poorly approximated by a fixed linear subspace. A simple traveling wave, for instance, can require many linear modes despite being described by only a few physical parameters. This phenomenon is often referred to as the Kolmogorov-width barrier for transport-dominated problems. It has motivated a broad class of nonlinear ROM techniques, including shifted bases, registration methods, Lagrangian formulations, transported subspaces, adaptive bases, and neural-network-based nonlinear manifolds [6, 25 , 28] .  \nThe difficulty is even more pronounced when the state variable is a probability distribution or density. Linear combinat","cbCaisWggm0ZQhxj","https://ap.wps.com/l/cbCaisWggm0ZQhxj","pdf",2489082,1,33,"English","en",105,"# Introduction\n## Challenges of classical linear ROM for transport-dominated PDEs\n## Density-valued problems and Wasserstein geometry\n## Related work on optimal-transport-based ROM and embeddings","[{\"question\":\"What problem does the proposed method target in reduced-order modeling?\",\"answer\":\"It addresses reduced-order modeling for parametrized PDEs whose solutions are density-valued and where transport-dominated dynamics make classical linear ROMs ineffective.\"},{\"question\":\"How does the method represent each density?\",\"answer\":\"Each density is represented using the Kantorovich potential that transports a fixed reference density to the target density, and then that potential is mapped to a transport signature.\"},{\"question\":\"How is the online evaluation made efficient?\",\"answer\":\"The transport-signature mapping is treated as a parameter-indexed continuous matrix, decomposed via a low-rank skeleton (maximal volume) approach, and the parameter-to-coefficient map is learned with a neural network for non-intrusive online evaluation.\"}]",1784189219,83,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"structure-preserving-reduced-order-modeling-via-low-rank-transport-signatures","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/structure-preserving-reduced-order-modeling-via-low-rank-transport-signatures/83606/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","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 proposed method target in reduced-order modeling?","Question",{"text":75,"@type":76},"It addresses reduced-order modeling for parametrized PDEs whose solutions are density-valued and where transport-dominated dynamics make classical linear ROMs ineffective.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the method represent each density?",{"text":80,"@type":76},"Each density is represented using the Kantorovich potential that transports a fixed reference density to the target density, and then that potential is mapped to a transport signature.",{"name":82,"@type":73,"acceptedAnswer":83},"How is the online evaluation made efficient?",{"text":84,"@type":76},"The transport-signature mapping is treated as a parameter-indexed continuous matrix, decomposed via a low-rank skeleton (maximal volume) approach, and the parameter-to-coefficient map is learned with a neural network for non-intrusive online 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