[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83751-en":3,"doc-seo-83751-105":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},83751,137441390410,"Hazel","https://ap-avatar.wpscdn.com/avatar/2000252f4ab5702993?_k=1776741390130283984",8,"Research & Report","Approximate Dynamic Optimization via Deep Neural Operators","Nonlinear dynamic optimization problems determine optimal, time-varying manipulated input profiles to achieve desired output trajectories. Such trajectory optimization arises in process operations, notably chemical batch systems, where temperature and feeding strategies enforce time-dependent product quality by controlling reaction rates or heat-generation rates. The paper introduces deep neural operators as surrogates for function-to-function solution mappings, using DeepONets and Fourier-enhanced DeepONets. A batch polymerization reactor study enforces molecular-weight profiles and conversion through an optimal temperature program, yielding lower prediction error than standard DeepONets and feedforward networks.","Approximate Dynamic Optimization via Deep Neural Operators  \nAmin Nassaji 1 , Ilias Mitrai2 , Prodromos Daoutidis 1  \narXiv :2607 .03861v1 [ ee ss . SY] 4 Jul 2026  \nAbstract—This paper addresses the solution of nonlinear dynamic optimization problems that compute optimal manipulated input profiles to enforce desired output profiles. Such trajectory optimization problems commonly arise in chemical process applications, for example, batch processes where optimal temperature or feeding profiles (in case of fed-batch processes) are calculated to enforce time-varying product quality profiles, tightly controlling the reaction rate or rate of heat generation. We propose deep neural operators that approximate function to function mappings as surrogates for the solution of such dynamic optimization problems. We specifically employ deep operator networks (DeepONets) and Fourier-enhanced DeepONets in a batch polymerization reactor case study for which number-average and weight-average molecular weight profiles, together with a final conversion target, are enforced through an optimal temperature program. Our results show that the Fourier-enhanced DeepONet architecture performs very wellin approximating the solution of the dynamic optimization problem for different instances, achieving a lower prediction error compared to the standard DeepONet architecture and standard feedforward neural networks.  \nI. INTRODUCTION  \nDynamic optimization (DO) refers to the class of optimization problems that involve time-varying decisions and dynamical systems [1] . It is at the heart of numerous process operation and control problems, for example, in batch processes that are inherently dynamic, or in systems subject to frequent transitions in operating conditions [2] .  \nIn DO problems, one typically determines an optimal time-varying trajectory of the manipulated inputs (flow rates, heating/cooling loads, pressure) to optimize a criterion that can be economic (maximizing profit) or performance related (minimizing batch time, maximizing conversion, yield or selectivity, etc.) [3],[1] . A particularly important and challenging class of DO problems involves trajectory optimization. In such problems, the target is a trajectory rather than a single value of the output variable. Examples include batch polymerization reactors that operate at a time-varying temperature to enforce a desired product quality profile linked to the desired properties (molecular weight distribution, conversion, etc.), fed-batch bioreactors, control problems in batch processes that aim to enforce a precomputed optimal trajectory of manipulated inputs (for example, an optimal feeding or heating/cooling strategy), and process startup  \nThis work was supported by the McKetta Department of Chemical Engineering (IM) and NSF-CBET Award No. 2313289 (PD)  \n1Amin Nassaji and Prodromos Daoutidis are with the Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, MN, [USA](USA {nassa033@umn.edu)[ {](USA {nassa033@umn.edu)[nassa033@umn.edu](USA {nassa033@umn.edu) , [daout001@umn.edu](daout001@umn.edu})[}](daout001@umn.edu})  \n2Ilias Mitrai is with the McKetta Department of Chemical Engineering, The University of Texas at Austin, 78712, Austin, TX, USA[imitrai@che.utexas.edu](imitrai@che.utexas.edu)  \nor shutdown where the process needs to follow a specific operating path for safety reasons [4] . Another major class involves transient operations and load changes, including grade transitions in polymerization and specialty chemicals plants, where constraints must be continuously satisfied asthe process moves from one product grade to another (often specified by allowable property ranges) [2], and the goal is to find feasible trajectories that avoid unsafe regions while minimizing transition time and off-spec production [5], [6] .  \nIn all of these applications, the key point is that performance and feasibility depend on the entire time evolution of the syste","cbCais6UgjmmYnkf","https://ap.wps.com/l/cbCais6UgjmmYnkf","pdf",549949,4,1,6,"English","en",105,"# Introduction\n## Dynamic Optimization and Trajectory Optimization\n## Operator Learning Framework for Function-to-Function Mappings","[{\"question\":\"What problem does the paper address?\",\"answer\":\"The paper targets nonlinear dynamic optimization problems that compute optimal time-varying manipulated inputs to enforce desired output trajectories.\"},{\"question\":\"How do the proposed deep neural operators help?\",\"answer\":\"They approximate the function-to-function mapping required by dynamic trajectory optimization, acting as surrogates to avoid solving the full optimization repeatedly for each instance.\"},{\"question\":\"What model variants are evaluated and what is the key result?\",\"answer\":\"The study uses DeepONets and Fourier-enhanced DeepONets, and the Fourier-enhanced architecture achieves lower prediction error than standard DeepONet and standard feedforward neural networks in a batch polymerization reactor case.\"}]",1784190209,15,{"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},"approximate-dynamic-optimization-via-deep-neural-operators","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"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":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":20},"https://docshare.wps.com/document/approximate-dynamic-optimization-via-deep-neural-operators/83751/",{"url":52,"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","2026-07-27","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 paper address?","Question",{"text":75,"@type":76},"The paper targets nonlinear dynamic optimization problems that compute optimal time-varying manipulated inputs to enforce desired output trajectories.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How do the proposed deep neural operators help?",{"text":80,"@type":76},"They approximate the function-to-function mapping required by dynamic trajectory optimization, acting as surrogates to avoid solving the full optimization repeatedly for each instance.",{"name":82,"@type":73,"acceptedAnswer":83},"What model variants are evaluated and what is the key result?",{"text":84,"@type":76},"The study uses DeepONets and Fourier-enhanced DeepONets, and the Fourier-enhanced architecture achieves lower prediction error than standard DeepONet and standard feedforward neural networks in a batch polymerization reactor 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