[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83084-en":3,"doc-seo-83084-105":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},83084,1099514067415,"Rowan","https://ap-avatar.wpscdn.com/avatar/100002539d78ffe74a7?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779092875211072502",8,"Research & Report","Physics-Informed Neural Embeddings of PDE Solution Families","A physics-informed framework learns finite-dimensional embeddings of partial differential equation (PDE) solution families using a multihead Physics-Informed Neural Network. A shared network body captures a latent manifold of the solution space, while linear heads reconstruct solutions tied to specific initial conditions. An ahead-orthogonalization penalty removes latent degeneracies and stabilizes the principal-component spectrum across training runs. Applied to the 1D viscous Burgers equation with heat and wave equations as robustness checks, the method yields strong effective dimensional reduction and reproducible, band-resolved frequency profiles.","arXiv :2607 .06348v 1 [ cs .LG] 7 Jul 2026  \nPhysics-Informed Neural Embeddings of PDE Solution Families  \nRaul Jimenez 1,2 , Svitlana Mayboroda3 , Pavlos Protopapas4 ,  \nLeonid Sarieddine 1,5*, David N. Spergel6 , Pedro Taranc´on- ´Alvarez 1,5*  \n1* Institute of Cosmos Sciences (ICC), University of Barcelona, Mart´ı i  \nFranqu`es 1, ES-08028, Barcelona, Spain.  \n2 ICREA, Pg. Llu´ıs Companys 23, 08010, Barcelona, Spain.  \n3 Department of Mathematics, ETH Zurich, R¨amistrasse 101, 8092, Z¨urich, Switzerland.  \n4 Institute for Applied Computational Science, Harvard University, Cambridge, MA, USA.  \n5 Department of F´ısica Qu`antica i Astrof´ısica, Universitat de Barcelona, Mart´ı i Franqu`es 1, ES-08028, Barcelona, Spain.  \n6 Flatiron Institute, 162 Fifth Avenue, New York, NY, 10011, USA.  \n*Corresponding author(s). E-mail(s): [leonid.sarieddine@icc.ub.edu](leonid.sarieddine@icc.ub.edu) ; [pedro.tarancon@icc.ub.edu](pedro.tarancon@icc.ub.edu) ;  \nAbstract  \nWe introduce a physics-informed framework for learning finite-dimensional embeddings of solution families of partial differential equations. The method uses a multihead Physics-Informed Neural Network in which a shared body learns a latent manifold representing the solution space, while linear heads reconstruct individual solutions associated with different initial conditions. Ahead-orthogonalization penalty removes degeneracies in the latent representation and stabilizes the principal-component spectrum across training realizations. Because the initial condition is built into the network output by construction, these principal components measure the additional variability the network learnson top of the initial profile, not the full solution itself. We apply the method to the one-dimensional viscous Burgers equation, with the heat and wave equations as robustness checks. For a latent dimension nb = 20, the learned manifolds exhibit pronounced effective dimensional reduction: for Burgers dynamics, only 2–4 principal components capture about 95% of the latent-space variance, while  \n1  \n4–7 capture about 99%, depending on the initial-condition family; the same qualitative compression holds for the heat and wave equations. We also split the wavenumber axis into bands (“Fourier shells”) and measure how much each band contributes to every principal component. The resulting frequency profile is invariant under the change-of-basis freedom that the orthogonalization penalty leaves in the latent space, and is therefore reproducible across independent training runs. More broadly, this establishes the learned spectral profiles and principal components as robust observables of solution-manifold geometry.  \nKeywords: Physics-Informed Neural Networks, partial differential equations, latent representations, multihead neural networks, reduced-order modelling, Burgers  \nequation, solution manifolds  \n1 Introduction  \nA central question in physics-informed machine learning is whether the space of solutions of a nonlinear partial differential equation (PDE) can be understood as a low-dimensional geometric object, and whether neural networks (NNs) can be used to uncover the geometry directly from the governing equations. This question is physically motivated: nonlinear PDEs generate solution families with intricate structure arising from interactions across scales, from shock formation in viscous flows to multiscale transport in turbulence (Frisch (1995); Doering and Gibbon (1995); Alexakisand Biferale (2018)) . The Navier–Stokes equations stand as a paradigmatic example, but the question applies broadly to any system in which a compact description of the solution manifold would provide interpretable insight beyond individual numerical realizations. The mathematical complexity of these equations—global regularity in 3D remains an open Millennium Problem (Fefferman (2000)), and non-uniqueness of weak solutions has only recently been established (Buckmaster and Vicol (2019))  \n—underscores that s","cbCaigDeSZGB34c8","https://ap.wps.com/l/cbCaigDeSZGB34c8","pdf",830362,2,1,36,"English","en",105,"# Abstract\n# Keywords\n# 1 Introduction","[{\"question\":\"What problem does the framework address for PDEs?\",\"answer\":\"It studies whether solution families of nonlinear PDEs can be represented as low-dimensional geometric objects and learned directly from governing equations via embeddings.\"},{\"question\":\"How does the multihead Physics-Informed Neural Network structure work?\",\"answer\":\"A shared body learns a latent manifold, while separate linear heads reconstruct individual solutions corresponding to different initial conditions.\"},{\"question\":\"Why is ahead-orthogonalization penalty important?\",\"answer\":\"It removes degeneracies in the latent representation and stabilizes the principal-component spectrum across independent training 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problem does the framework address for PDEs?","Question",{"text":75,"@type":76},"It studies whether solution families of nonlinear PDEs can be represented as low-dimensional geometric objects and learned directly from governing equations via embeddings.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the multihead Physics-Informed Neural Network structure work?",{"text":80,"@type":76},"A shared body learns a latent manifold, while separate linear heads reconstruct individual solutions corresponding to different initial conditions.",{"name":82,"@type":73,"acceptedAnswer":83},"Why is ahead-orthogonalization penalty important?",{"text":84,"@type":76},"It removes degeneracies in the latent representation and stabilizes the principal-component spectrum across independent training 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