{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,15]],"date-time":"2025-10-15T18:13:43Z","timestamp":1760552023328,"version":"build-2065373602"},"reference-count":41,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2023,9,4]],"date-time":"2023-09-04T00:00:00Z","timestamp":1693785600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Advanced Aviation Power Innovation Workstation Project","award":["HKCX2022-01-022"],"award-info":[{"award-number":["HKCX2022-01-022"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Multi-output Gaussian process regression measures the similarity between samples based on Euclidean distance and assigns the same weight to each feature. However, there are significant differences in the aerodynamic performance of plane cascades composed of symmetric and asymmetric blade shapes, and there are also significant differences between the geometry of the plane cascades formed by different blade shapes and the experimental working conditions. There are large differences in geometric and working condition parameters in the features, which makes it difficult to accurately measure the similarity between different samples when there are fewer samples. For this problem, a metric learning for the multi-output Gaussian process regression method (ML_MOGPR) for aerodynamic performance prediction of the plane cascade is proposed. It shares parameters between multiple output Gaussian distributions during training and measures the similarity between input samples in a new embedding space to reduce bias and improve overall prediction accuracy. For the analysis of ML_MOGPR prediction results, the overall prediction accuracy is significantly improved compared with multi-output Gaussian process regression (MOGPR), backpropagation neural network (BPNN), and multi-task learning neural network (MTLNN). The experimental results show that ML_MOGPR is effective in predicting the performance of the plane cascade, and it can quickly and accurately make a preliminary estimate of the aerodynamic performance and meet the performance parameter estimation accuracy requirements in the early stage.<\/jats:p>","DOI":"10.3390\/sym15091692","type":"journal-article","created":{"date-parts":[[2023,9,4]],"date-time":"2023-09-04T10:14:11Z","timestamp":1693822451000},"page":"1692","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Plane Cascade Aerodynamic Performance Prediction Based on Metric Learning for Multi-Output Gaussian Process Regression"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0009-0003-6107-4070","authenticated-orcid":false,"given":"Lin","family":"Liu","sequence":"first","affiliation":[{"name":"School of Computer Science and Technology, Southwest University of Science and Technology, Mianyang 621010, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3035-4980","authenticated-orcid":false,"given":"Chunming","family":"Yang","sequence":"additional","affiliation":[{"name":"School of Computer Science and Technology, Southwest University of Science and Technology, Mianyang 621010, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Honghui","family":"Xiang","sequence":"additional","affiliation":[{"name":"AECC Sichuan Gas Turbine Establishment, Mianyang 621000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jiazhe","family":"Lin","sequence":"additional","affiliation":[{"name":"China Aerodynamics Research and Development Center, Computational Aerodynamic Research Institute, Mianyang 621000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,9,4]]},"reference":[{"key":"ref_1","first-page":"1755","article-title":"Progress and prospect of aerodynamic experimental research on linear cascade","volume":"54","author":"Qingdian","year":"2022","journal-title":"Chin. J. Theor. Appl. Mech."},{"key":"ref_2","first-page":"30","article-title":"Review of the cascade experimental technology","volume":"35","author":"Daijun","year":"2021","journal-title":"J. Exp. Fluid Mech."},{"key":"ref_3","first-page":"524689","article-title":"Prospect of artificial intelligence empowered fluid mechanics","volume":"42","author":"Weiwei","year":"2021","journal-title":"Acta Aeronaut. Astronaut. Sin."},{"key":"ref_4","first-page":"444","article-title":"Progresses in the application of machine learning in turbulence modeling","volume":"37","author":"Weiwei","year":"2019","journal-title":"Acta Aerodyn. Sin."},{"key":"ref_5","first-page":"185","article-title":"Progress in deep convolutional neural network based flow field recognition and its applications","volume":"42","author":"Shuran","year":"2021","journal-title":"Acta Aeronaut. Astronaut. Sin."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"588","DOI":"10.1002\/nme.6235","article-title":"A finite element reduced-order model based on adaptive mesh refinement and artificial neural networks","volume":"121","author":"Baiges","year":"2020","journal-title":"Int. J. Numer. Methods Eng."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"105949","DOI":"10.1016\/j.ast.2020.105949","article-title":"Fast pressure distribution prediction of airfoils using deep learning","volume":"105","author":"Hui","year":"2020","journal-title":"Aerosp. Sci. Technol."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"106","DOI":"10.1017\/jfm.2019.238","article-title":"Super-resolution reconstruction of turbulent flows with machine learning","volume":"870","author":"Fukami","year":"2019","journal-title":"J. Fluid Mech."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"477","DOI":"10.1146\/annurev-fluid-010719-060214","article-title":"Machine learning for fluid mechanics","volume":"52","author":"Brunton","year":"2020","journal-title":"Annu. Rev. Fluid Mech."},{"key":"ref_10","first-page":"936","article-title":"Data-driven Dynamic-classifiers-based Seismic Failure Mode Detection of Deep Steel W-shape Columns","volume":"67","author":"Barkhordari","year":"2023","journal-title":"Period. Polytech. Civ. Eng."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Andr\u00e9s-P\u00e9rez, E. (2020). Data mining and machine learning techniques for aerodynamic databases: Introduction, methodology and potential benefits. Energies, 13.","DOI":"10.3390\/en13215807"},{"key":"ref_12","first-page":"33","article-title":"Surrogate Model Construction for Rocket Aerodynamic Discipline Based on Support Vector Machine","volume":"4","author":"Bo","year":"2013","journal-title":"Missiles Space Veh."},{"key":"ref_13","first-page":"524093","article-title":"Missile aerodynamic performance prediction of Gaussian process through automatic kernel construction","volume":"42","author":"Weijie","year":"2021","journal-title":"Acta Aeronaut. Astronaut. Sin."},{"key":"ref_14","first-page":"138","article-title":"Aerodynamic inverse design method based on gradient-enhanced kriging model","volume":"38","author":"Han","year":"2017","journal-title":"Acta Aeronaut. Astronaut. Sin."},{"key":"ref_15","first-page":"2616","article-title":"Aerodynamic modeling method incorporating pressure distribution information","volume":"54","author":"Xuan","year":"2022","journal-title":"Chin. J. Theor. Appl. Mech."},{"key":"ref_16","first-page":"2251","article-title":"Prediction of Aerodynamic characteristics of compressor blades based on deep learning","volume":"38","author":"Du","year":"2023","journal-title":"J. Aerosp. Power"},{"key":"ref_17","first-page":"28","article-title":"Prediction of missile\u2019s aerodynamic parameters based on neural network","volume":"27","author":"Zhang","year":"2020","journal-title":"Aero Weapon."},{"key":"ref_18","first-page":"674","article-title":"Prediction of wing aerodynamic coefficient based on CNN","volume":"49","author":"Zhaoyang","year":"2021","journal-title":"J. Beiging Univ. Aeronaut. Astronaut."},{"key":"ref_19","unstructured":"Lin, J., Zhou, L., Wu, P., Yuan, W., and Zhou, Z. (2021). Research on rapid prediction technology of missile aerodynamic characteristics based on PIMTLNN. J. Beiging Univ. Aeronaut. Astronaut., 1\u201315."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Moin, H., Khan, H.Z.I., Mobeen, S., and Riaz, J. (2022, January 22\u201325). Airfoil\u2019s Aerodynamic Coefficients Prediction using Artificial Neural Network. Proceedings of the 2022 19th International Bhurban Conference on Applied Sciences and Technology (IBCAST), Bhurban, Pakistan.","DOI":"10.1109\/IBCAST54850.2022.9990112"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Zhang, Y., Sung, W.J., and Mavris, D.N. (2018, January 23\u201326). Application of convolutional neural network to predict airfoil lift coefficient. Proceedings of the 2018 AIAA\/ASCE\/AHS\/ASC Structures, Structural Dynamics, and Materials Conference, Honolulu, HI, USA.","DOI":"10.2514\/6.2018-1903"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Peng, W., Zhang, Y., and Desmarais, M. (2021, January 8\u201312). Spatial convolution neural network for efficient prediction of aerodynamic coefficients. Proceedings of the AIAA Scitech 2021 Forum, Orlando, FL, USA.","DOI":"10.2514\/6.2021-0277"},{"key":"ref_23","first-page":"299","article-title":"Identification of aerodynamic parameters of flapping-wing micro aerial vehicle based on double BP neural network","volume":"39","author":"Han","year":"2019","journal-title":"J. Comput. Appl."},{"key":"ref_24","first-page":"13","article-title":"Aerodynamic parameters prediction of airfoil ice accretion based on convolutional neural network","volume":"39","author":"Wang","year":"2021","journal-title":"Flight Dyn."},{"key":"ref_25","first-page":"294","article-title":"Aerodynamic coefficient prediction of airfoils based on deep learning","volume":"36","author":"Hai","year":"2018","journal-title":"Acta Aerodyn. Sin."},{"key":"ref_26","first-page":"835","article-title":"Structural damage identification using ensemble deep convolutional neural network models","volume":"134","author":"Barkhordari","year":"2022","journal-title":"Comput. Model. Eng. Sci."},{"key":"ref_27","first-page":"64","article-title":"A multi-task learning method for large discrepant aerodynamic data","volume":"40","author":"Jun","year":"2022","journal-title":"Acta Aerodyn. Sin."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"69","DOI":"10.1142\/S0129065704001899","article-title":"Gaussian processes for machine learning","volume":"14","author":"Seeger","year":"2004","journal-title":"Int. J. Neural Syst."},{"key":"ref_29","first-page":"153","article-title":"Multi-task Gaussian process prediction","volume":"Volume 20","author":"Bonilla","year":"2007","journal-title":"Advances in Neural Information Processing Systems"},{"key":"ref_30","first-page":"57","article-title":"Sparse convolved Gaussian processes for multi-output regression","volume":"Volume 21","author":"Alvarez","year":"2008","journal-title":"Advances in Neural Information Processing Systems"},{"key":"ref_31","unstructured":"Wilson, A., and Adams, R. (2013, January 16\u201321). Gaussian process kernels for pattern discovery and extrapolation. Proceedings of the 30th International Conference on International Conference on Machine Learning, Atlanta, GA, USA."},{"key":"ref_32","first-page":"1999","article-title":"GP kernels for cross-spectrum analysis","volume":"Volume 28","author":"Ulrich","year":"2015","journal-title":"Advances in Neural Information Processing Systems"},{"key":"ref_33","first-page":"6684","article-title":"Spectral Mixture Kernels for Multi-Output Gaussian Processes","volume":"30","author":"Parra","year":"2017","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"ref_34","unstructured":"Altamirano, M., and Tobar, F. (April, January 30). Nonstationary multi-output Gaussian processes via harmonizable spectral mixtures. Proceedings of the 25th International Conference on Artificial Intelligence and Statistics, PMLR, Valencia, Spain."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Zheng, W., Wang, C., Lu, J., and Zhou, J. (2021, January 19\u201325). Deep compositional metric learning. Proceedings of the 2021 IEEE\/CVF Conference on Computer Vision and Pattern Recognition (CVPR), Nashville, TN, USA.","DOI":"10.1109\/CVPR46437.2021.00920"},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Elezi, I., Vascon, S., Torcinovich, A., Pelillo, M., and Leal-Taix\u00e9, L. (2020, January 23\u201325). The group loss for deep metric learning. Proceedings of the Computer Vision\u2013ECCV 2020, Glasgow, UK.","DOI":"10.1007\/978-3-030-58571-6_17"},{"key":"ref_37","first-page":"207","article-title":"Distance Metric Learning for Large Margin Nearest Neighbor Classification","volume":"10","author":"Weinberger","year":"2009","journal-title":"J. Mach. Learn. Res."},{"key":"ref_38","unstructured":"Weinberger, K.Q., and Tesauro, G. (2007, January 25\u201327). Metric learning for kernel regression. Proceedings of the Eleventh International Conference on Artificial Intelligence and Statistics, PMLR, Valencia, Spain."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"408","DOI":"10.1109\/TPAMI.2018.2794976","article-title":"Metric learning for multi-output tasks","volume":"41","author":"Liu","year":"2018","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"107086","DOI":"10.1016\/j.patcog.2019.107086","article-title":"Metric learning-based kernel transformer with triplets and label constraints for feature fusion","volume":"99","author":"Kan","year":"2020","journal-title":"Pattern Recognit."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1016\/j.neucom.2020.09.085","article-title":"MOGPTK: The multi-output Gaussian process toolkit","volume":"424","author":"Cuevas","year":"2021","journal-title":"Neurocomputing"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/9\/1692\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T20:46:09Z","timestamp":1760129169000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/9\/1692"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,9,4]]},"references-count":41,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2023,9]]}},"alternative-id":["sym15091692"],"URL":"https:\/\/doi.org\/10.3390\/sym15091692","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2023,9,4]]}}}