[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-128444-en":3,"doc-seo-128444-105":31,"detail-sidebar-cat-0-en-105":92},{"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":28,"seo_description":14,"update_tm":29,"read_time":30},128444,962085662650,"Jiven","https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0",8,"Research & Report","Efficient hybrid modeling and sorption model discovery for non-linear advection-diffusion-sorption systems - A systematic scientific machine learning approach","This study proposes a systematic scientific machine learning framework to build efficient hybrid models and to discover sorption uptake kinetics in non-linear advection-diffusion-sorption systems. The method trains models with gradient-based optimizers, adjoint sensitivity analysis, and JIT-compiled vector Jacobian products, paired with spatial discretization and adaptive integration. Sparse and symbolic regression are used to identify missing terms inside an artificial neural network, while robustness is evaluated on noisy in-silico breakthrough curve data for fixed-bed adsorption.","Efficient hybrid modeling and sorption model discovery for non-linear  \nadvection-diffusion-sorption systems: A systematic  \nscientific machine learning approach  \narXiv :2303 . 13555v1 [ cs .CE] 22 Mar 2023  \nVinicius Viena Santana*  \nLSRE–LCM †, ALiCE‡  \nFaculty of Engineering University of Porto [up201700649@edu.fe.up.pt](up201700649@edu.fe.up.pt)  \nErbet Costa  \nDepartment of Chemical Engineering Norwegian University of Science and Technology  \nCarine de Menezes Rebello  \nDepartment of Chemical Engineering Norwegian University of Science and Technology  \nAna Mafalda Ribeiro  \nLSRE–LCM †, ALiCE‡  \nFaculty of Engineering University of Porto  \nChristopher Rackauckas  \nJuliaLab  \nComputer Science and Artificial Intelligence Laboratory  \nMassachusetts Institute of Technology  \nIdelfonso B.R. Nogueira*  \nDepartment of Chemical Engineering  \nNorwegian University of Science and Technology  \nidelfonso .b.d .r.nogueira@ntnu .no  \nMarch 27, 2023  \nAbstract  \nThis study presents a systematic machine learning approach for creating efficient hybrid models and discovering sorption uptake models in non-linear advection-diffusion-sorption systems. It demonstrates an effective method to train these complex systems using gradientbased optimizers, adjoint sensitivity analysis, and JIT-compiled vector Jacobian products, combined with spatial discretization and adaptive integrators. Sparse and symbolic regression were employed to identify missing functions in the artificial neural network. The robustness of the proposed method was tested on an in-silico data set of noisy breakthrough curve observations of fixed-bed adsorption, resulting in a well-fitted hybrid model. The study successfully reconstructed sorption uptake kinetics using sparse and symbolic regression, and accurately predicted breakthrough curves using identified polynomials, highlighting the potential of the proposed framework for discovering sorption kinetic law structures.  \n∗ To whom correspondence should be addressed  \n†LSRE-LCM (Laboratory of Separation and Reaction Engineering – Laboratory of Catalysis and Materials, Faculty of Engineering, University of Porto, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal)  \n‡ALiCE (Associate Laboratory in Chemical Engineering,, Faculty of Engineering, University of Porto, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal)  \nA preprint March 27, 2023  \nKeywords Hybrid Modeling 􀀁 Scientific Machine Learning 􀀁 Sparse Regression 􀀁 Advection-DiffusionSorption 􀀁 Partial Differential Equations  \n1 Introduction  \nMathematical modeling has been an essential part of science and engineering since its early stages. The use of mathematical models for problem-solving in chemical engineering has become increasingly important due to the exponential growth of computer power. Two main approaches have been used to model systems in chemical engineering: The mechanistic/classical and the empirical approaches [1] . The classical approach encompasses conservation equations, transport, and thermodynamic expressions and can accurately describe the problem. However, simplifications are often employed to make the problem computationally tractable. The empirical/data-driven models do not make any assumptions about the physics behind the observations, making them flexible. Nevertheless, they are suitable for use in a limited range of process conditions due to their limited extrapolation power [2] .  \nOn the other hand, empirical models have gained substantial attention in the last few years with machine learning and deep learning. At the same pace, it is becoming more accepted that mechanistic model errors caused by simplifications can be compensated with other forms of knowledge, e.g., empirical models [3] . In this sense, hybrid models have received considerable attention in engineering [4, 5] . The \"hybrid\" term refers to a structure where empirical equations and physics knowledge (conservation laws, operation invariance, etc.) are merged. The merging process for engineering appl","cbCaiqi4mGblCh75","https://ap.wps.com/l/cbCaiqi4mGblCh75","pdf",1364504,2,1,25,"English","en",105,"# Introduction\n## Hybrid modeling in chemical engineering\n## Universal Differential Equations and universal approximators","[{\"question\":\"What is the main goal of the proposed method?\",\"answer\":\"The approach builds efficient hybrid models and discovers sorption uptake kinetics for non-linear advection-diffusion-sorption systems.\"},{\"question\":\"How are the hybrid models trained and optimized?\",\"answer\":\"Training uses gradient-based optimizers, adjoint sensitivity analysis, and JIT-compiled vector Jacobian products, together with spatial discretization and adaptive integrators.\"},{\"question\":\"How are missing functions or kinetics terms identified?\",\"answer\":\"Sparse and symbolic regression are employed to reconstruct missing functions inside the artificial neural network and recover sorption uptake kinetics structures.\"}]","Efficient hybrid modeling and sorption model discovery for non-linear advection-diffusion-sorption systems - A systematic scientific machine learning approach | PDF",1786001099,63,{"code":4,"msg":32,"data":33},"ok",{"site_id":25,"language":24,"slug":34,"title":13,"keywords":35,"description":14,"schema_data":36,"social_meta":87,"head_meta":89,"extra_data":91,"updated_unix":29},"efficient-hybrid-modeling-and-sorption-model-discovery-for-non-linear-advection-diffusion-sorption-systems-a-systematic-scientific-machine-learning-approach","",{"@graph":37,"@context":86},[38,54,69],{"@type":39,"itemListElement":40},"BreadcrumbList",[41,45,48,51],{"item":42,"name":43,"@type":44,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":46,"name":47,"@type":44,"position":20},"https://docshare.wps.com/document/","Document",{"item":49,"name":12,"@type":44,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":44,"position":53},"https://docshare.wps.com/document/efficient-hybrid-modeling-and-sorption-model-discovery-for-non-linear-advection-diffusion-sorption-systems-a-systematic-scientific-machine-learning-approach/128444/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":24,"description":14,"dateModified":62,"datePublished":63,"encodingFormat":61,"isAccessibleForFree":64,"interactionStatistic":65},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":42,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-08-23","2026-08-06",true,{"@type":66,"interactionType":67,"userInteractionCount":20},"InteractionCounter",{"@type":68},"ViewAction",{"@type":70,"mainEntity":71},"FAQPage",[72,78,82],{"name":73,"@type":74,"acceptedAnswer":75},"What is the main goal of the proposed method?","Question",{"text":76,"@type":77},"The approach builds efficient hybrid models and discovers sorption uptake kinetics for non-linear advection-diffusion-sorption systems.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"How are the hybrid models trained and optimized?",{"text":81,"@type":77},"Training uses gradient-based optimizers, adjoint sensitivity analysis, and JIT-compiled vector Jacobian products, together with spatial discretization and adaptive integrators.",{"name":83,"@type":74,"acceptedAnswer":84},"How are missing functions or kinetics terms identified?",{"text":85,"@type":77},"Sparse and symbolic regression are employed to reconstruct missing functions inside the artificial neural network and recover sorption uptake kinetics structures.","https://schema.org",{"og:url":52,"og:type":88,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":90,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":93},[94,98,102,106,111,116,121,124,129,132,136],{"id":21,"doc_module":4,"doc_module_name":47,"category_name":95,"show_sort_weight":96,"slug":97},"Story & Novel",90,"story-novel",{"id":20,"doc_module":4,"doc_module_name":47,"category_name":99,"show_sort_weight":100,"slug":101},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":47,"category_name":103,"show_sort_weight":104,"slug":105},"Exam",70,"exam",{"id":107,"doc_module":4,"doc_module_name":47,"category_name":108,"show_sort_weight":109,"slug":110},5,"Comic",60,"comic",{"id":112,"doc_module":4,"doc_module_name":47,"category_name":113,"show_sort_weight":114,"slug":115},6,"Technology",50,"technology",{"id":117,"doc_module":4,"doc_module_name":47,"category_name":118,"show_sort_weight":119,"slug":120},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":47,"category_name":12,"show_sort_weight":122,"slug":123},30,"research-report",{"id":125,"doc_module":4,"doc_module_name":47,"category_name":126,"show_sort_weight":127,"slug":128},9,"Religion & Spirituality",20,"religion-spirituality",{"id":127,"doc_module":4,"doc_module_name":47,"category_name":130,"show_sort_weight":127,"slug":131},"World Cup","world-cup",{"id":133,"doc_module":4,"doc_module_name":47,"category_name":134,"show_sort_weight":133,"slug":135},10,"Lifestyle","lifestyle",{"id":137,"doc_module":4,"doc_module_name":47,"category_name":138,"show_sort_weight":107,"slug":139},19,"General","general"]