[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-86245-en":3,"doc-seo-86245-105":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},86245,687197207919,"Theodora","https://ap-avatar.wpscdn.com/avatar/a000253d6f5f7c60be?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779446848396160552",8,"Research & Report","IG-GAN Intrinsic Geometry-Based Generative Adversarial Network for Aerodynamic Data Generation","Existing generative models assume data lie in flat Euclidean space, yet real aerodynamic datasets often form low-dimensional manifolds embedded in high-dimensional space. IG-GAN introduces an intrinsic-geometry-based generative adversarial network that models aerodynamic data as a piecewise smooth manifold built from Bézier surfaces, learning Bézier surface coefficients to automatically assemble a smooth manifold. A radial-basis-function discriminator improves realism learning. Experiments reduce predicted MSE versus baselines, including 97.41% on Burgers’ velocity u and 82.95% on ONERA M6 nine coefficients.","arXiv :2607 . 1 1497v 1 [ cs .LG] 13 Jul 2026  \nIG-GAN: A Generative Adversarial Network for Aerodynamic Data Generation Based on Intrinsic  \nGeometry  \nYing Yana , Liwei Hua,∗, Xiaoming Zhanga,∗  \na School of Information Science and Engineering, Hebei University of Science and  \nTechnology, Shijiazhuang, 050091, China  \nAbstract  \nExisting generative models learn data distributions in flat Euclidean space. However, most data in our real world are manifolds embedded in high dimensional Euclidean space. Therefore, we propose an intrinsic-geometry-based generative adversarial network (IG-GAN) for data generation in the field of aerodynamics. The generator of the IG-GAN represents aerodynamic data as a piecewise smooth manifold constructed by Bézier surfaces, and the generator tries to learn the coefficients of each Bézier surface to further combine multiple Bézier surfaces into a smooth manifold automatically. The discriminator in the IG-GAN is a radial-basis-function based discriminator (RBF-D) . Experimental results show that IG-GAN achieves lower predicted Mean Squared Errors (MSEs) than those of three baselines. Specifically, on the Burgers’ equation dataset, IG-GAN reduces the predicted MSE of velocity u by 97.41% compared with state of the art SSL-Transformer. Additionally, on the ONERA M6 aircraft dataset, IG-GAN reduces the overall MSE of nine aerodynamic coefficients by 82.95% compared with SSL-Transformer.  \nKeywords: Generative adversarial networks, Manifold learning, Bézier surfaces, Neural networks  \n⋆This work is Funded by Science Research Project of Hebei Education Department.(No. QN2026857)  \n∗ Corresponding authors.  \nEmail addresses: [yingyan@stu.hebust.edu.cn](yingyan@stu.hebust.edu.cn) (Ying Yan), [liweihu@hebust.edu.cn](liweihu@hebust.edu.cn) (Liwei Hu), [zhangxiaom@hebust.edu.cn](zhangxiaom@hebust.edu.cn) (Xiaoming Zhang)  \n1. INTRODUCTION  \nExisting generative methods, e.g. generative adversarial networks (GANs)  \n[1], diffusion models [2], and transformers [3], perform data generation under flat Euclidean spaces [4, 5] . Using these methods to generate data is essentially a way to learn the geometric structure of the data (or data distribution) . Data generation under flat Euclidean space contains a hybrid representation of data geometry, i.e. these methods learn not only the intrinsic geometry (e.g. , curvature and torsion [6]) of the data itself, but also the extrinsic geometry (e.g. the specific coordinate values [7], normal vector [8], etc. ) determined by the external Cartesian coordinate system. However, most real-world data are low-dimensional embeddings with complex and curved geometries in highdimensional space, which implies that those geometries of these data are governed by their intrinsic geometric properties, independent of their position in the external coordinate system [9] .  \nRecent studies that use intrinsic geometry-based methods to represent objects have demonstrated advantages in capturing the inherent structural properties of objects in high-dimensional space [10] . For example, a hyperbolic manifold can be used to represent data with tree-like geometry [11], and a sphere manifold can be used to approximate the hyper surface induced by deep learning loss functions [12] . In aerodynamics, there is a growing trend in intrinsic geometry representations for aircraft. Xiang et al. [13] used a set of Bézier curves to represent the geometry of UIUC airfoils, and further calculated the intrinsic geometric features to reduce the coefficient of drag ( CD ) prediction errors. Based on the study of Xiang, Hu et al. used a set of Bézier surfaces to represent the intrinsic geometry of DLR-F11 wing in 3-D space, and using connection and curvature to reduce the coefficient of pressure ( CP ) prediction error on the wing surface [6] . The above studies demonstrated that the intrinsic geometry methods can be used to accurately represent the aircraft geometry and further to reduce the prediction error","cbCaidZa2BlBmebc","https://ap.wps.com/l/cbCaidZa2BlBmebc","pdf",3501214,4,1,27,"English","en",105,"# Abstract\n# Introduction\n## Motivation and Problem Setting\n## Related Work in Intrinsic Geometry for Aerodynamics\n## Proposed Method and Contributions\n# Mathematical Formulation and Generator Design\n## Bézier-based Generator","[{\"question\":\"Why does IG-GAN move beyond flat Euclidean assumptions for generative modeling?\",\"answer\":\"Most real-world aerodynamic data are low-dimensional embeddings with curved, intrinsic geometries in high-dimensional space. IG-GAN targets intrinsic geometric structure rather than extrinsic coordinate-dependent geometry learned by flat-space models.\"},{\"question\":\"How does the IG-GAN generator represent aerodynamic data?\",\"answer\":\"The generator represents aerodynamic data as a piecewise smooth manifold constructed by Bézier surfaces, and it learns the coefficient matrices for each Bézier surface to combine them into a smooth manifold automatically.\"},{\"question\":\"What discriminator architecture does IG-GAN use, and why?\",\"answer\":\"IG-GAN uses a radial-basis-function based discriminator (RBF-D). As the generator improves its data generation ability, the discriminator must also strengthen its distinguishability.\"}]",1784209754,68,{"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},"ig-gan-intrinsic-geometry-based-generative-adversarial-network-for-aerodynamic-data-generation","",{"@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/ig-gan-intrinsic-geometry-based-generative-adversarial-network-for-aerodynamic-data-generation/86245/",{"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","2026-07-26","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},"Why does IG-GAN move beyond flat Euclidean assumptions for generative modeling?","Question",{"text":75,"@type":76},"Most real-world aerodynamic data are low-dimensional embeddings with curved, intrinsic geometries in high-dimensional space. IG-GAN targets intrinsic geometric structure rather than extrinsic coordinate-dependent geometry learned by flat-space models.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the IG-GAN generator represent aerodynamic data?",{"text":80,"@type":76},"The generator represents aerodynamic data as a piecewise smooth manifold constructed by Bézier surfaces, and it learns the coefficient matrices for each Bézier surface to combine them into a smooth manifold automatically.",{"name":82,"@type":73,"acceptedAnswer":83},"What discriminator architecture does IG-GAN use, and why?",{"text":84,"@type":76},"IG-GAN uses a radial-basis-function based discriminator (RBF-D). 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