[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83384-en":3,"doc-seo-83384-105":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},83384,13056703020460,"Valentina","https://ap-avatar.wpscdn.com/avatar/be000253dac470eee5d?_k=1778207105932848923",8,"Research & Report","Neural and Spectral Operator Surrogates on Gaussian Spaces","The work establishes expression rate bounds for finite-parametric spectral and neural surrogate operators approximating holomorphic maps between separable Hilbert spaces. Surrogates use an encoder–approximator–decoder (Karhunen–Loève) architecture and cover two approximation families: spectral surrogates from N-term truncations of Wiener polynomial chaos expansions and neural surrogates based on deep feedforward ReLU/RePU networks with uniformly bounded weights. Under algebraic eigenvalue decay of the Gaussian measure covariance, mean-square errors and first-order Gaussian Sobolev errors for gradient approximation are quantified.","NEURAL AND SPECTRAL OPERATOR SURROGATES ON GAUSSIAN SPACES  \nCARLO MARCATI, MARIO MARI, CHRISTOPH SCHWAB, AND JAKOB ZECH  \narXiv :2607 .08492v1 [math .NA] 9 Jul 2026  \nAbstract. We prove expression rate bounds of finite-parametric, spectral and neural surrogates for holomorphic maps between separable Hilbert spaces. The surrogateshave an encoder-approximatordecoder architecture, with Karhunen-Lo . We prove expression rate bounds for two classes of finite-parametric surrogates: i) spectral surrogates obtained by Nterm truncations of Wiener polynomial chaos expansions and ii) neural surrogates obtained by approximation of parametric maps with deep feedforward neural networks, ReLU and RePU activation functions and uniformly bounded weights. We work under an algebraic decay assumption on the eigenvalues of the covariance of the Gaussian measure on the input space. We obtain convergence rates for mean-square errors, and additionally in first-order Gaussian Sobolev spaces, to account for errors in the approximation of gradients.  \nContents  \n1. Introduction 2  \n1.1. Existing Results 2  \n1.2. Contributions 3  \n1.3. Layout 3  \n1.4. Notation and Terminology 4  \n2. Preliminaries 5  \n2.1. Gaussian Measures on Hilbert Spaces 5  \n2.2. Hermite Polynomials 6  \n2.3. Frames 7  \n2.4. Scales of Hilbert Spaces 8  \n2.5. Operator Derivatives 9  \n3. FrameNet Architecture 10  \n3.1. Encoder 10  \n3.2. Decoder 11  \n3.3. Neural Networks 11  \n3.4. FrameNet 11  \n4. Main Results 12  \n4.1. Mean-Squared Approximation Rates 12  \n4.2. Gauss-Sobolev Approximation Rates 13  \n5. Proofs of Expression Rates 13  \n5.1. Domains of Parametric Holomorphy 13  \n5.2. Weighted Summability Results 14  \n5.3. Spectral Operator Surrogates: Expression Rates 17  \n5.4. DNN Expression Rates 21  \nDate: July 10, 2026 .  \nCM and JZ acknowledge the support of the National Group for Scientific Computing (GNCS-INDAM) through the Visiting Professors program. MM and JZ acknowledge support from the German Research Foundation (DFG) within the Priority Programme SPP 2298, Theoretical Foundations of Deep Learning (project number 543965776) .  \n1  \nNEURAL AND SPECTRAL OPERATOR SURROGATES ON GAUSSIAN SPACES 2  \n6. Coefficient-to-solution map for a linear elliptic PDE 27  \n7. Conclusion 29  \nReferences 30  \nAppendix A. ReLU/RePU Neural Network Theory 32  \nAppendix B. DNN Emulation of Hermite Polynomials with Weight Bounds 36  \nB.1 . Univariate Hermite Polynomials 36  \nB.2 . Multivariate Hermite Polynomials 42  \nAppendix C. Auxiliary Results 47  \n1. Introduction  \nNeural operators are finite-parametric approximations GN of in general nonlinear maps G : X → Y between (subsets of) infinite-dimensional, separable function spaces X and Y. In recent years, several classes of such surrogate maps have been proposed and developed. See, e.g., the surveys [16, 17] and references there. Starting with the foundational work [4], to date a large range of finite-parametric maps have been indentified with mathematically ensured universal approximation properties.  \nSurrogate families with expression rate bounds are less well investigated: due to the infinitedimensionality of the input-and output-spaces, viability of finite-parametric surrogate maps will incur (and have to overcome, under conditions) the so-called “curse of dimension”(CoD) .  \nOne possible architecture of neural surrogate operators (considered, e.g., in [19, 15]) takes the generic form GN = D ◦ gN ◦ E. Here, E and D are en-and decoders on the input and output spaces X and Y respectively. The maps gN depend on N parameters, and leverage approximation properties of suitable approximators. The proof of approximation rate bounds, i.e. estimates on the approximation error G − GN in terms of N, on suitable sets of inputs in X, involves first selecting en-and decoders, and then constructing parametric approximators gN . Due to the infinite dimension of X and Y, establishing approximation rate bounds for the approximators gN requires overcoming the CoD.  ","cbCaifTZ0Mb6Hhbt","https://ap.wps.com/l/cbCaifTZ0Mb6Hhbt","pdf",708323,3,1,51,"English","en",105,"# Introduction\n## Existing Results\n## Contributions\n## Layout\n# Preliminaries\n## Gaussian Measures on Hilbert Spaces\n## Hermite Polynomials\n## Frames\n## Scales of Hilbert Spaces\n## Operator Derivatives\n# FrameNet Architecture\n## Encoder\n## Decoder\n## Neural Networks\n## FrameNet\n# Main Results\n## Mean-Squared Approximation Rates\n## Gauss-Sobolev Approximation Rates\n# Proofs of Expression Rates\n## Domains of Parametric Holomorphy\n## Weighted Summability Results\n## Spectral Operator Surrogates: Expression Rates\n## DNN Expression Rates","[{\"question\":\"What types of surrogate operators are studied, and what do they approximate?\",\"answer\":\"The paper analyzes finite-parametric spectral and neural surrogate operators approximating holomorphic maps between separable Hilbert spaces, using an encoder–approximator–decoder architecture.\"},{\"question\":\"How are the spectral and neural surrogates constructed?\",\"answer\":\"Spectral surrogates are obtained via N-term truncations of Wiener polynomial chaos expansions, while neural surrogates approximate parametric maps with deep feedforward ReLU/RePU networks with uniformly bounded weights.\"},{\"question\":\"What error metrics and assumptions are used to derive convergence rates?\",\"answer\":\"Convergence is quantified using mean-square errors and also errors in first-order Gaussian Sobolev spaces to capture gradient-approximation effects. Rates rely on an algebraic decay assumption on the covariance eigenvalues of the Gaussian measure.\"}]",1784187135,129,{"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},"neural-and-spectral-operator-surrogates-on-gaussian-spaces","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,50],{"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":20},"https://docshare.wps.com/document/research-report/",{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/neural-and-spectral-operator-surrogates-on-gaussian-spaces/83384/",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","2026-07-23","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 types of surrogate operators are studied, and what do they approximate?","Question",{"text":75,"@type":76},"The paper analyzes finite-parametric spectral and neural surrogate operators approximating holomorphic maps between separable Hilbert spaces, using an encoder–approximator–decoder architecture.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How are the spectral and neural surrogates constructed?",{"text":80,"@type":76},"Spectral surrogates are obtained via N-term truncations of Wiener polynomial chaos expansions, while neural surrogates approximate parametric maps with deep feedforward ReLU/RePU networks with uniformly bounded weights.",{"name":82,"@type":73,"acceptedAnswer":83},"What error metrics and assumptions are used to derive convergence rates?",{"text":84,"@type":76},"Convergence is quantified using mean-square errors and also errors in first-order Gaussian Sobolev spaces to capture gradient-approximation effects. Rates rely on an algebraic decay assumption on the covariance eigenvalues of the Gaussian measure.","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":47,"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"]