[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-119573-en":3,"doc-seo-119573-105":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":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":27,"seo_description":14,"update_tm":28,"read_time":29},119573,1374391974585,"Genevieve","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","Parametric Nonlinear Volterra Series via Machine Learning - Transonic Aerodynamics","An approach models unsteady transonic aerodynamics within a parametric space by combining Volterra series and machine learning to enable interpolation of aerodynamic responses. First- and second-order Volterra kernels are identified from indicial aerodynamic responses produced by computational fluid dynamics, with the second-order kernel acting as a correction to the dominant linear behavior. Neural networks and Gaussian process regression interpolate kernel coefficients versus Mach number and angle of attack, validated on two- and three-dimensional transonic test cases.","This is the author accepted manuscript version of an article published by AIAA. The Version of Record is  \navailable at [https://doi.org/10.2514/1.c038288](https://doi.org/10.2514/1.c038288)  \nParametric Nonlinear Volterra Series via Machine Learning:  \nTransonic Aerodynamics  \nGabriele Immordino∗ , Andrea Da Ronch†  \nUniversity of Southampton, Southampton, United Kingdom, SO17 1BJ  \nMarcello Righi‡  \nZurich University of Applied Sciences ZHAW, Winterthur, Switzerland, 8401  \nThis study introduces an approach for modeling unsteady transonic aerodynamics within a parametric space, using Volterra series to capture aerodynamic responses and machine learning to enable interpolation. The first-and second-order Volterra kernels are derived from indicial aerodynamic responses obtained through computational fluid dynamics, with the second-order kernel calculated as a correction to the dominant linear response. Machine learning algorithms, specifically artificial neural network and Gaussian process regression, are used to interpolate kernel coefficients within a parameter space defined by Mach number and angle of attack. The methodology is applied to two and three dimensional test cases in the transonic regime. Results underscore the benefit of including the second-order kernel to address strong nonlinearity and demonstrate the effectiveness of neural networks. The approach achieves a level of accuracy that appears sufficient for use in conceptual design.  \nNomenclature  \nAcronyms  \n􀀗􀀨􀀘􀀬 = Benchmark Super Critical Wing 􀀘􀀛􀀙 = Computational Fluid Dynamics 􀀛􀀘􀀣􀀣 = fully–connected neural network 􀀜􀀥􀀧 = Gaussian process regression 􀀡􀀝􀀨 = Latin Hypercube Sampling 􀀢􀀡 = machine learning  \n􀀢􀀨􀀚 = mean squared error 􀀧􀀤􀀢 = reduced–order model  \n􀀨􀀫􀀙 = Singular Value Decomposition  \n∗ PhD Student, Faculty of Engineering and Physical Sciences, [Corresponding Author. Email: G.Immordino@soton.ac.uk](Corresponding Author. Email: G.Immordino@soton.ac.uk)  \n†Professor, Faculty of Engineering and Physical Sciences, AIAA Senior Member.  \n‡Professor, School of Engineering, AIAA Member, Lecturer at Federal Institute of Technology Zurich ETHZ  \nSymbols  \n􀀖 + = pseudo-inverse of the matrix 􀀖  \n􀁕0 = freestream angle of attack, deg  \n􀁕 = pitch angle, deg  \n¯􀁕 = mean pitch angle, deg  \n􀁕 􀀖 = angular amplitude, deg  \n􀀲 = chord  \n􀀘 􀀡 = lift coefficient  \n􀀘 􀀢 = pitching moment coefficient  \n􀀘 􀀥/􀀳􀀴􀀶 = pressure coefficient normalized by oscillation amplitude  \n(􀀘 􀀥)􀀞􀀼/􀀳􀀴􀀶 = imaginary component of pressure coefficient normalized by oscillation amplitude (􀀘 􀀥)􀀼􀀴􀀰􀀽 = mean pressure coefficient  \n(􀀘 􀀥)􀀧􀀴/􀀳􀀴􀀶 = real component of pressure coefficient normalized by oscillation amplitude ℎ() = convolution kernels  \nℎ0 ... ℎ 􀀽 = Volterra kernels  \n􀀝􀀽 = Volterra operator of order n  \n􀀺 = reduced frequency  \n􀀡 􀁁() = Laguerre polynomials of order r  \n􀀼 1...􀀼 􀀢 = memory lag terms  \n􀀢 = Mach number 􀀧􀀴 = Reynolds number  \n􀁜 􀁁 = weights of the Laguerre polynomials  \n􀁧 = reduced time  \nI. Introduction  \nIn aerospace and mechanical engineering, the design process for new products relies on hierarchies of mathematical models, the physical complexity of which may be imposed by computational costs or dictated by regulations. These models typically incorporate parameters to account for various operating conditions and configurations. In the framework of optimization, for example, hundreds of parameters (design variables) may be required to define the configuration of a system. Similarly, uncertainty propagation may necessitate defining a complex parameter space to account for variations in geometrical imperfections, material properties, or flow conditions.  \nThe design process, especially during optimization and uncertainty quantification, often involves numerous evaluations of the system’s mathematical models across a wide range of points in the parameter space. Computational costs vary with the level of model fidelity: lower fidelity models are traditionally used for computationally intensive","cbCaivpFjJQuIQE2","https://ap.wps.com/l/cbCaivpFjJQuIQE2","pdf",2561893,1,36,"English","en",105,"# Introduction\n## Modeling unsteady transonic aerodynamics in a parametric space\n## Reduced-order models and nonlinear behavior\n## Parametric interpolation and smoothness requirements","[{\"question\":\"How are the Volterra kernels obtained in the proposed method?\",\"answer\":\"First- and second-order Volterra kernels are derived from indicial aerodynamic responses computed via computational fluid dynamics. The second-order kernel is computed as a correction to the dominant linear response.\"},{\"question\":\"Which machine learning methods are used for interpolation of kernel coefficients?\",\"answer\":\"The study uses artificial neural networks and Gaussian process regression to interpolate Volterra kernel coefficients over the parameter space.\"},{\"question\":\"What parameter space and test cases are used to validate the approach?\",\"answer\":\"Interpolation is performed within a parameter space defined by Mach number and angle of attack. The methodology is applied to two- and three-dimensional test cases in the transonic regime.\"}]","Parametric Nonlinear Volterra Series via Machine Learning - Transonic Aerodynamics | PDF",1785725034,91,{"code":4,"msg":31,"data":32},"ok",{"site_id":24,"language":23,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"parametric-nonlinear-volterra-series-via-machine-learning-transonic-aerodynamics","",{"@graph":36,"@context":85},[37,54,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":20},"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":53},"https://docshare.wps.com/document/parametric-nonlinear-volterra-series-via-machine-learning-transonic-aerodynamics/119573/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":23,"description":14,"dateModified":62,"datePublished":62,"encodingFormat":61,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-08-03",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},"How are the Volterra kernels obtained in the proposed method?","Question",{"text":75,"@type":76},"First- and second-order Volterra kernels are derived from indicial aerodynamic responses computed via computational fluid dynamics. The second-order kernel is computed as a correction to the dominant linear response.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Which machine learning methods are used for interpolation of kernel coefficients?",{"text":80,"@type":76},"The study uses artificial neural networks and Gaussian process regression to interpolate Volterra kernel coefficients over the parameter space.",{"name":82,"@type":73,"acceptedAnswer":83},"What parameter space and test cases are used to validate the approach?",{"text":84,"@type":76},"Interpolation is performed within a parameter space defined by Mach number and angle of attack. The methodology is applied to two- and three-dimensional test cases in the transonic regime.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":89,"canonical":52},"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":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":53,"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"]