[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83800-en":3,"doc-seo-83800-105":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},83800,5909877438554,"Maeve","https://ap-avatar.wpscdn.com/avatar/5600025385ad2bf12a7?_k=1778553567797529272",8,"Research & Report","Approximate Feedback Linearization for a Nonlinear Hyperbolic PDE Class—Part II: Neural Operator","Volterra series feedback linearizes a nonlinear hyperbolic PDE class but yields an implementational burden: a truncated controller still requires solving a tower of plant-specific kernel PDEs and computing nested integrals. This paper proves the truncated controller is jointly Lipschitz in plant and state, then learns it as a single neural operator mapping plant nonlinearity and state to boundary control. After training, kernel PDEs are never solved again for any admissible plant in the class, and closed-loop performance remains practically stable with a residual ball scaling linearly with training accuracy.","Approximate Feedback Linearization fora Nonlinear Hyperbolic PDE Class—Part II: Neural Operator  \nMiroslav Krstic  \narXiv :2607 .04362v1 [ ee ss . SY] 5 Jul 2026  \nAbstract  \nVolterra series feedback linearizes a class of nonlinear hyperbolic PDEs but produces a controller that, even after truncation, demands solving a tower of plant-specific kernel PDEs and evaluating nested integrals. We prove the truncated controller is jointly Lipschitz in plant and state, and learn it as a single neural operator from plant nonlinearity and state to boundary control. Once trained, no kernel is ever solved again, for any plant in the trained class. The closed loop is practically stable in class-KL form, with a residual ball scaling linearly with training accuracy.  \n1 Introduction  \nThis is the second of two companion papers on approximate feedback linearization of a class of nonlinear hyperbolic PDEs. Part I [6] replaces the exact infinite Volterra linearizer of [7] with a finite truncation and establishes its closed-loop stability in L∞ . Part II, the present paper, replaces the truncated controller with a learned neural-operator surrogate that eliminates the online kernel solve and the nested Volterra integration, and re-establishes the same closed-loop guarantees under two simultaneous approximations.  \nVolterra series feedback turns a nonlinear hyperbolic PDE into a transport equation, exactly. The price is a controller defined by an infinite series of nested integrals over kernels living on simplices of growing dimension—a feedback no implementation can evaluate and no offline solver can precompute. Truncation at some finite order N is forced upon us, and the truncated controller is what gets implemented. The question is what truncation does to the closed loop.  \nPart I [6] answers it. With the sup-norm machinery developed there, the truncated feedback delivers, on a sup-norm ball whose radius is set by the contraction radius of the inverse backstepping transformation, four properties: forward invariance, a practical-stability bound holding for all time, finite-time attractivity to a residual ball determined by the truncation tail εN (r0 ), and a class-KL asymptotic estimate. Stabilization of [7] is recovered in the high-order limit.  \nBut the truncated controller, while finite, is  \nEmail address: [mkrstic@ucsd.edu](mkrstic@ucsd.edu) (Miroslav Krstic).  \n1 Department of Mechanical and Aerospace Engineering, University of California San Diego, La Jolla, CA 92093-0411 .  \n2 The principal AI aid in developing the paper was Claude.  \nstill computationally expensive: each evaluation of UN (fN , u (·, t)) requires an N-fold nested integration against N − 1 separately-precomputed kernelsk2 ,..., kN , with each kn obtained by solving a linear PDE on an n-simplex. Worse, this kernel computation is plant-specific: change the nonlinearity F, and every kn must be solved for again from scratch. The kernels are not the controller; they are an intermediate object the controller is forced to traverse.  \nThe proper object of study is the operator UN (fN , u) that maps the plant nonlinearity, with Volterra coefficients fN = (f2 ,..., fN ), together with the state, to the boundary input. This is what gets implemented in the loop, and it is what we propose to learn. A neural operator trained on a class of admissible plants emulates UN on that whole class at once. The kernel PDEs are solved offline only as part of generating training data; once the network is trained, no kernel is ever solved again—not when the plant nonlinearity is changed within the trained class, not when the state is updated in real time, not at all. A single forward pass of fixed cost replaces the nested-integral computation, for any plant in the class.  \nThis note is the analysis of the closed loop under such a neural surrogate. Once UN is replaced by aoningsupththe-njeootriamntrg-aec(ftur,rauan)tsesppoaarpctpeero, tqxhuimeatiabotonionunarerwesitsiahduneaewrlrodcrriovnε--tpεcutefri","cbCaiu4HecqeZiYL","https://ap.wps.com/l/cbCaiu4HecqeZiYL","pdf",961586,1,11,"English","en",105,"# Introduction\n# Operator Definition\n## Neural-operator surrogate and stability framework","[{\"question\":\"What problem does the neural-operator surrogate solve in the truncated feedback linearization method?\",\"answer\":\"It removes the need to solve plant-specific kernel PDEs and to compute nested Volterra integrals during online control. The trained neural operator replaces the operator mapping from plant nonlinearity and state to boundary input.\"},{\"question\":\"How does the paper characterize robustness when the controller is truncated and replaced by a neural operator?\",\"answer\":\"The truncated controller is proven jointly Lipschitz in plant and state, and the closed loop is practically stable in a class-KL form. A residual ball remains, scaling linearly with training accuracy, and the conventional class-KL behavior is recovered as the approximation error ε approaches zero.\"},{\"question\":\"Where are the kernel PDEs solved after training?\",\"answer\":\"Kernel PDEs are solved offline only to generate training data. Once the network is trained, no kernel is ever solved again for any plant within the trained admissible class.\"}]",1784190495,28,{"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},"approximate-feedback-linearization-for-a-nonlinear-hyperbolic-pde-classpart-ii-neural-operator","",{"@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/approximate-feedback-linearization-for-a-nonlinear-hyperbolic-pde-classpart-ii-neural-operator/83800/",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 neural-operator surrogate solve in the truncated feedback linearization method?","Question",{"text":75,"@type":76},"It removes the need to solve plant-specific kernel PDEs and to compute nested Volterra integrals during online control. The trained neural operator replaces the operator mapping from plant nonlinearity and state to boundary input.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the paper characterize robustness when the controller is truncated and replaced by a neural operator?",{"text":80,"@type":76},"The truncated controller is proven jointly Lipschitz in plant and state, and the closed loop is practically stable in a class-KL form. A residual ball remains, scaling linearly with training accuracy, and the conventional class-KL behavior is recovered as the approximation error ε approaches zero.",{"name":82,"@type":73,"acceptedAnswer":83},"Where are the kernel PDEs solved after training?",{"text":84,"@type":76},"Kernel PDEs are solved offline only to generate training data. 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