[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81618-en":3,"doc-seo-81618-105":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},81618,34359740700684,"Finn","https://ap-avatar.wpscdn.com/avatar/1f400023980c374ae676?_k=1777273430885731487",8,"Research & Report","Tucker Tensor Train Taylor Series","Learning derivative-accurate surrogate models for implicit simulators is a central scientific machine learning challenge. High-order Taylor surrogates are traditionally viewed as intractable in large dimensions since derivative tensors are huge and only accessible through probing. The method Tucker tensor train Taylor series (T4S) makes such surrogates practical by expressing each derivative tensor from a truncated Taylor expansion as a Tucker tensor train. It learns locally from randomized symmetric derivative probes at a single expansion point, enabling efficient linearized solves, dimension reduction, Riemannian Gauss-Newton/Cauchy SGD fitting, and provable guarantees under spectral decay. Experiments validate accuracy and recovery of high-order Taylor structure for Poisson PDEs.","arXiv :2603 .21141v2 [math .NA] 9 Jul 2026  \nTucker Tensor Train Taylor Series  \nTucker Tensor Train Taylor Series  \nNick Alger [nalger@oden.utexas.edu](nalger@oden.utexas.edu)  \nOden Institute for Computational Engineering and Sciences The University of Texas at Austin  \nAustin, TX 78712, USA  \nBlake Christierson [bechristierson@utexas.edu](bechristierson@utexas.edu)  \nOden Institute for Computational Engineering and Sciences The University of Texas at Austin  \nAustin, TX 78712, USA  \nPeng Chen [pchen402@gatech.edu](pchen402@gatech.edu)  \nSchool of Computational Science and Engineering Georgia Institute of Technology  \nAtlanta, GA 30308, USA  \nOmar Ghattas [omar@oden.utexas.edu](omar@oden.utexas.edu)  \nOden Institute for Computational Engineering and Sciences and Walker Department of Mechanical Engineering  \nThe University of Texas at Austin Austin, TX 78712, USA  \nEditor:  \nAbstract  \nLearning derivative-accurate surrogates for implicit simulators is a key challenge in scientific machine learning. High-order Taylor surrogates have long been considered intractable in high dimensions, because the derivative tensors are enormous and accessible only through probes. We make such surrogates tractable with the Tucker tensor train Taylor series (T4S), a local surrogate that represents each derivative tensor of a truncated Taylor expansion as a Tucker tensor train. T4S targets a different learning problem than global operator learning: rather than training from input-output pairs at many parameter values, it is trained from random directionally symmetric derivative probes at a single expansion point. Computing m probes of the kth derivative requires only O (mk) linearized solves sharing one operator, cheaper than the O (m) nonlinear solves for function evaluations or O (m2k ) linearized solves for asymmetric probes. We develop derivativeinformed dimension reduction, Riemannian Gauss-Newton and Cauchy SGD fitting algorithms with rank continuation, requiring little hyperparameter tuning, and fast sweeping routines for the Riemannian Jacobian. We prove representational guarantees under spectral decay of the input covariance. Experiments show that our methods match quasi-optimal T3-SVD accuracy on random tensors from probes alone, up to data-limited ranks, and recover high-order Taylor structure in Poisson PDE examples.  \nKeywords: scientific machine learning; surrogate modeling; derivative-informed learning; tensor networks; uncertainty quantification  \nAlger, Christierson, Chen and Ghattas  \nx θ u q  \n−−−−−→  \nEvaluate θ0+Cx  \n−−−−−−→ Solve  \n0=R(θ,u)  \n−−−−−→  \nEvaluate Q (θ,u)  \nFigure 1: Illustration of the components of the covariance-whitened mapping f : x 7→ q for the example in Section 8.2.1 . The state equation is a Poisson PDE modeling steady state heat conduction from a source at the center of the domain Ω = [0 , 1]2 . The parameter θ is a random spatially varying log conductivity coefficient, and the output q is the trace of the temperature, u, along the top boundary of the domain. The noise function x is the covariance-whitened version of θ . We approximate f using a surrogate model based on a Taylor series.  \n1 Introduction  \nLearning surrogate models for expensive simulators is a central challenge in scientific machine learning. In many scientific and engineering applications, a model evaluation requires solving an implicit system, such as a parameterized partial differential equation (PDE), rather than applying an explicit formula. These model evaluations are often embedded in outer-loop tasks, such as inverse problems, prediction and control under uncertainty, optimal experimental design, and digital-twin updating. Such tasks may require many model evaluations, and many of them also need gradients, Hessians, or higher derivatives. Even when derivatives are not used directly, derivative accuracy can determine the quality of local optimization steps, posterior approximations, proposal distributions, preconditioners, and u","cbCaikhkgEJomqc5","https://ap.wps.com/l/cbCaikhkgEJomqc5","pdf",2389781,1,81,"English","en",105,"# Abstract\n# Introduction","[{\"question\":\"What is the Tucker tensor train Taylor series (T4S) and what problem does it address?\",\"answer\":\"T4S is a local surrogate modeling approach that represents truncated Taylor derivative tensors in a Tucker tensor train format. It targets the difficulty of learning derivative-accurate surrogates for implicit simulators in high dimensions where conventional high-order Taylor surrogates are intractable.\"},{\"question\":\"How does the training data differ from standard operator-learning approaches?\",\"answer\":\"Instead of training from input-output pairs at many parameter values, T4S is trained from randomized derivative tensor probes at a single expansion point. Increasing data corresponds to probing more random directions and potentially higher derivative orders.\"},{\"question\":\"Why are derivative probes computationally cheaper for implicit simulators?\",\"answer\":\"Computing multiple directional probes of a kth derivative can be done with only O(mk) linearized solves that share one operator, rather than O(m) nonlinear solves for function evaluations or O(m^2k) linearized solves for asymmetric probes.\"}]",1784174847,204,{"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},"tucker-tensor-train-taylor-series","",{"@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/tucker-tensor-train-taylor-series/81618/",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","2026-07-25","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 is the Tucker tensor train Taylor series (T4S) and what problem does it address?","Question",{"text":75,"@type":76},"T4S is a local surrogate modeling approach that represents truncated Taylor derivative tensors in a Tucker tensor train format. It targets the difficulty of learning derivative-accurate surrogates for implicit simulators in high dimensions where conventional high-order Taylor surrogates are intractable.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the training data differ from standard operator-learning approaches?",{"text":80,"@type":76},"Instead of training from input-output pairs at many parameter values, T4S is trained from randomized derivative tensor probes at a single expansion point. Increasing data corresponds to probing more random directions and potentially higher derivative orders.",{"name":82,"@type":73,"acceptedAnswer":83},"Why are derivative probes computationally cheaper for implicit simulators?",{"text":84,"@type":76},"Computing multiple directional probes of a kth derivative can be done with only O(mk) linearized solves that share one operator, rather than O(m) nonlinear solves for function evaluations or O(m^2k) linearized solves for asymmetric probes.","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":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":45,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":45,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":45,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":45,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":45,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":45,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":45,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]