{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,10]],"date-time":"2026-04-10T05:41:24Z","timestamp":1775799684917,"version":"3.50.1"},"reference-count":41,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2020,9,12]],"date-time":"2020-09-12T00:00:00Z","timestamp":1599868800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>To generate more high-quality aerodynamic data using the information provided by different fidelity data, where low-fidelity aerodynamic data provides the trend information and high-fidelity aerodynamic data provides value information, we applied a deep neural network (DNN) algorithm to fuse the information of multi-fidelity aerodynamic data. We discuss the relationships between the low-fidelity and high-fidelity data, and then we describe the proposed architecture for an aerodynamic data fusion model. The architecture consists of three fully-connected neural networks that are employed to approximate low-fidelity data, and the linear part and nonlinear part of correlation for the low- and high-fidelity data, respectively. To test the proposed multi-fidelity aerodynamic data fusion method, we calculated Euler and Navier\u2013Stokes simulations for a typical airfoil at various Mach numbers and angles of attack to obtain the aerodynamic coefficients as low- and high-fidelity data. A fusion model of the longitudinal coefficients of lift CL and drag CD was constructed with the proposed method. For comparisons, variable complexity modeling and cokriging models were also built. The accuracy spread between the predicted value and true value was discussed for both the training and test data of the three different methods. We calculated the root mean square error and average relative deviation to demonstrate the performance of the three different methods. The fusion result of the proposed method was satisfactory on the test case, and showed a better performance compared with the other two traditional methods presented. The results provide evidence that the method proposed in this paper can be useful in dealing with the multi-fidelity aerodynamic data fusion problem.<\/jats:p>","DOI":"10.3390\/e22091022","type":"journal-article","created":{"date-parts":[[2020,9,13]],"date-time":"2020-09-13T21:11:32Z","timestamp":1600031492000},"page":"1022","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":44,"title":["Multi-Fidelity Aerodynamic Data Fusion with a Deep Neural Network Modeling Method"],"prefix":"10.3390","volume":"22","author":[{"given":"Lei","family":"He","sequence":"first","affiliation":[{"name":"Computational Aerodynamics Research Institute, China Aerodynamics Research and Development Center, Mianyang 621000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Weiqi","family":"Qian","sequence":"additional","affiliation":[{"name":"Computational Aerodynamics Research Institute, China Aerodynamics Research and Development Center, Mianyang 621000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Tun","family":"Zhao","sequence":"additional","affiliation":[{"name":"Computational Aerodynamics Research Institute, China Aerodynamics Research and Development Center, Mianyang 621000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Qing","family":"Wang","sequence":"additional","affiliation":[{"name":"Computational Aerodynamics Research Institute, China Aerodynamics Research and Development Center, Mianyang 621000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,9,12]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"587","DOI":"10.3724\/SP.J.1004.2013.00587","article-title":"Review of correlation analysis of aerodynamic data between flight and ground prediction for hypersonic vehicle","volume":"32","author":"Chen","year":"2014","journal-title":"Acta Aerodyn. Sin."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Hall, R.M., Biedron, R.T., Ball, D.N., Bogue, D.R., Chung, J., Green, B.E., Grismer, M.J., Brooks, G.P., and Chambers, J.R. (2005). Computational Methods for Stability and Control (COMSAC): The Time Has Come. AIAA J.","DOI":"10.2514\/6.2005-6121"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Kuya, Y., Takeda, K., and Zhang, X. (2009, January 4\u20137). Optimal Surrogate Modelling Approaches for Combining Experimental and Computational Fluid Dynamics Datasets. Proceedings of the 50th AIAA\/ASME\/ASCE\/AHS\/ASC Structures, Structural Dynamics, and Materials Conference, Palm Springs, CA, USA.","DOI":"10.2514\/6.2009-2216"},{"key":"ref_4","first-page":"777","article-title":"Application of data fusion technique in aerodynamics studies","volume":"32","author":"Kaifeng","year":"2014","journal-title":"Acta Aerodyn. Sin."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"604","DOI":"10.1080\/09544828.2013.788135","article-title":"A hybrid variable-fidelity global approximation modelling method combining tuned radial basis function base and kriging correction","volume":"24","author":"Zheng","year":"2013","journal-title":"J. Eng. Des."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"852","DOI":"10.1002\/zamm.201100112","article-title":"Comparing sampling strategies for aerodynamic Kriging surrogate models","volume":"92","author":"Rosenbaum","year":"2012","journal-title":"ZAMM J. Appl. Math. Mech."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.paerosci.2005.02.001","article-title":"Surrogate-based analysis and optimization","volume":"41","author":"Queipo","year":"2005","journal-title":"Prog. Aerosp. Sci."},{"key":"ref_8","unstructured":"Ai, Y. (2012). Research on Response Surface Method Optimisation Based on Radial Basis Function. [Master\u2019s Thesis, Huazhong University of Science and Technology]."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Barton, R.R. (2009, January 13\u201316). Simulation optimization using metamodels. Proceedings of the 2009 Winter Simulation Conference (WSC), Austin, TX, USA.","DOI":"10.1109\/WSC.2009.5429328"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"691","DOI":"10.1080\/03052150903386674","article-title":"Trends, features, and tests of common and recently introduced global optimization methods","volume":"42","author":"Younis","year":"2010","journal-title":"Eng. Optim."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"498","DOI":"10.2514\/1.32308","article-title":"Building Ecien t Response Surfaces of Aerodynamic Functions with Kriging and Cokriging","volume":"46","author":"Laurenceau","year":"2008","journal-title":"AIAA J."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Forrester, A., Sobester, A., and Keane, A. (2008). Engineering Design via Surrogate Modelling: A Practical Guide, John Wiley & Sons.","DOI":"10.1002\/9780470770801"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"273","DOI":"10.1080\/07408170500232495","article-title":"A review on design, modeling and applications of computer experiments","volume":"38","author":"Chen","year":"2006","journal-title":"IIE Trans."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Santner, T.J., Williams, B.J., Notz, W., and Williams, B.J. (2003). The Design and Analysis of Computer Experiments, Springer.","DOI":"10.1007\/978-1-4757-3799-8"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1029","DOI":"10.1007\/s00158-016-1550-y","article-title":"Remarks on multi-fidelity surrogates","volume":"55","author":"Park","year":"2017","journal-title":"Struct. Multidiscip. Optim."},{"key":"ref_16","first-page":"20160751","article-title":"Nonlinear information fusion algorithms for data-efficient multi-fidelity modelling","volume":"473","author":"Perdikaris","year":"2017","journal-title":"Proc. Math. Phys. Eng. Sci."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"109020","DOI":"10.1016\/j.jcp.2019.109020","article-title":"A composite neural network that learns from multi-fidelity data: Application to function approximation and inverse PDE problems","volume":"401","author":"Meng","year":"2020","journal-title":"J. Comput. Phys."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Hutchison, M., Mason, W., Grossman, B., and Haftka, R. (1993, January 11). Aerodynamic Optimization of an HSCT Configuration Using Variable-Complexity Modeling. Proceedings of the 31st Aerospace Sciences Meeting, Reno, NV, USA.","DOI":"10.2514\/6.1993-101"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"110","DOI":"10.2514\/3.46462","article-title":"Variable-complexity aerodynamic optimization of a high-speed civil transport wing","volume":"31","author":"Hutchison","year":"1994","journal-title":"J. Aircr."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"741","DOI":"10.2514\/2.3153","article-title":"Wing Optimization Using Design of Experiment, Response Surface, and Data Fusion Methods","volume":"40","author":"Keane","year":"2003","journal-title":"J. Aircr."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"297","DOI":"10.1023\/A:1023283917997","article-title":"A Knowledge-Based Approach To Response Surface Modelling in Multifidelity Optimization","volume":"26","author":"Leary","year":"2003","journal-title":"J. Glob. Optim."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Tang, C., Gee, K., and Lawrence, S. (2005, January 10\u201313). Generation of aerodynamic data using a design of experiment and data fusion approach. Proceedings of the 43rd AIAA Aerospace Sciences Meeting and Exhibit, Reno, NV, USA.","DOI":"10.2514\/6.2005-1137"},{"key":"ref_23","doi-asserted-by":"crossref","unstructured":"Tyan, M., Kim, M., Pham, V., Choi, C.K., Nguyen, T.L., and Lee, J.W. (2018, January 25\u201329). Development of Advanced Aerodynamic Data Fusion Techniques for Flight Simulation Database Construction. Proceedings of the 2018 Modeling and Simulation Technologies Conference, Atlanta, GA, USA.","DOI":"10.2514\/6.2018-3581"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Renganathan, A., Harada, K., and Mavris, D.N. (2019, January 17\u201321). Multifidelity Data Fusion via Bayesian Inference. Proceedings of the AIAA Aviation 2019 Forum, Dallas, TX, USA.","DOI":"10.2514\/6.2019-3556"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"972","DOI":"10.2514\/1.39626","article-title":"Accelerating the Numerical Generation of Aerodynamic Models for Flight Simulation","volume":"46","author":"Ghoreyshi","year":"2009","journal-title":"J. Aircr."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"289","DOI":"10.2514\/1.J050384","article-title":"Multifidelity Surrogate Modeling of Experimental and Computational Aerodynamic Data Sets","volume":"49","author":"Kuya","year":"2011","journal-title":"AIAA J."},{"key":"ref_27","unstructured":"Zhang, Q. (2017). Development of a Data Fusion Framework for the Aerodynamic Analysis of Launchers. [Master\u2019s Thesis, Delft University of Technology]."},{"key":"ref_28","first-page":"87","article-title":"Variable-Complexity Design of a Transport Wing","volume":"2","author":"Unger","year":"1992","journal-title":"Int. J. Syst. Autom. Res. Appl."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Knill, D., Giunta, A., Baker, C., Grossman, B., Mason, W., Haftka, R., and Watson, L. (1998, January 12\u201315). HSCT Configuration Design Using Response Surface Approximations of Supersonic Euler Aerodynamics. Proceedings of the 36th AIAA Aerospace Sciences Meeting and Exhibit, Reno, NV, USA.","DOI":"10.2514\/6.1998-905"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"215","DOI":"10.2514\/2.2941","article-title":"High-speed civil transport design space exploration using aerodynamic response surface approximations","volume":"39","author":"Baker","year":"2002","journal-title":"J. Aircr."},{"key":"ref_31","first-page":"3197","article-title":"Kriging surrogate model and its application to design optimization: A review of recent progress","volume":"37","author":"Han","year":"2016","journal-title":"Acta Aeronaut. Astronaut. Sin."},{"key":"ref_32","first-page":"3251","article-title":"Multi-fidelity optimization via surrogate modelling","volume":"463","author":"Forrester","year":"2007","journal-title":"Proc. R. Soc. Math. Phys. Eng. Sci."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Santos, M., Mattos, B., and Girardi, R. (2008, January 7\u201310). Aerodynamic Coefficient Prediction of Airfoils Using Neural Networks. Proceedings of the AIAA Aerospace Sciences Meeting & Exhibit, Reno, NV, USA.","DOI":"10.2514\/6.2008-887"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"993","DOI":"10.2514\/1.J057894","article-title":"Inverse Design of Airfoil Using a Deep Convolutional Neural Network","volume":"57","author":"Sekar","year":"2019","journal-title":"AIAA J."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"533","DOI":"10.1038\/323533a0","article-title":"Learning Representations by Back Propagating Errors","volume":"323","author":"Rumelhart","year":"1986","journal-title":"Nature"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"895","DOI":"10.1017\/jfm.2016.718","article-title":"Multi-fidelity modelling of mixed convection based on experimental correlations and numerical simulations","volume":"809","author":"Babaee","year":"2016","journal-title":"J. Fluid Mech."},{"key":"ref_37","unstructured":"Kingma, D., and Ba, J. (2014). Adam: A Method for Stochastic Optimization. arXiv."},{"key":"ref_38","unstructured":"Ruder, S. (2016). An overview of gradient descent optimization algorithms. arXiv."},{"key":"ref_39","first-page":"47","article-title":"Analysis and Improvement of HicksHenne Airfoil Parameterization Method","volume":"40","author":"Wang","year":"2010","journal-title":"Aeronaut. Comput. Tech."},{"key":"ref_40","first-page":"430","article-title":"Numerical simulation of ice shedding from ARJ21-700","volume":"31","author":"Wang","year":"2013","journal-title":"Acta Aerodyn. Sin."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"174","DOI":"10.1007\/s11804-008-7065-1","article-title":"Grid convergence study in the resistance calculation of a trimaran","volume":"7","author":"Chen","year":"2008","journal-title":"J. Mar. Sci. Appl."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/9\/1022\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T10:09:31Z","timestamp":1760177371000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/22\/9\/1022"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,9,12]]},"references-count":41,"journal-issue":{"issue":"9","published-online":{"date-parts":[[2020,9]]}},"alternative-id":["e22091022"],"URL":"https:\/\/doi.org\/10.3390\/e22091022","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,9,12]]}}}