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Geometrical shape of airfoils, together with the corresponding flight conditions, are crucial factors for aerodynamic performances prediction. The obtained airfoils geometrical features in most existing approaches (e.g., geometrical parameters extraction, polynomial description and deep learning) are in Euclidean space. State-of-the-art studies showed that curves or surfaces of an airfoil formed a manifold in Riemannian space. Therefore, the features extracted by existing methods are not sufficient to reflect the geometric-features of airfoils. Meanwhile, flight conditions and geometric features are greatly discrepant with different types, the relevant knowledge of the influence of these two factors that on final aerodynamic performances predictions must be evaluated and learned to improve prediction accuracy. Motivated by the advantages of manifold theory and multi-task learning, we propose a manifold-based airfoil geometric-feature extraction and discrepant data fusion learning method (MDF) to extract geometric-features of airfoils in Riemannian space (we call them manifold-features) and further fuse the manifold-features with flight conditions to predict aerodynamic performances. Experimental results show that our method could extract geometric-features of airfoils more accurately compared with existing methods, that the average MSE of re-built airfoils is reduced by 56.33%, and while keeping the same predicted accuracy level of CL, the MSE of CD predicted by MDF is further reduced by 35.37%.

The quality of datasets is one of the key factors that affect the accuracy of aerodynamic data models. For example, in the uniformly sampled Burgers' dataset, the insufficient high-speed data is overwhelmed by massive low-speed data. Predicting high-speed data is more difficult than predicting low-speed data, owing to that the number of high-speed data is limited, i.e. the quality of the Burgers' dataset is not satisfactory. To improve the quality of datasets, traditional methods usually employ the data resampling technology to produce enough data for the insufficient parts in the original datasets before modeling, which increases computational costs. Recently, the mixtures of experts have been used in natural language processing to deal with different parts of sentences, which provides a solution for eliminating the need for data resampling in aerodynamic data modeling. Motivated by this, we propose the multi-task learning (MTL), a datasets quality-adaptive learning scheme, which combines task allocation and aerodynamic characteristics learning together to disperse the pressure of the entire learning task. The task allocation divides a whole learning task into several independent subtasks, while the aerodynamic characteristics learning learns these subtasks simultaneously to achieve better precision. Two experiments with poor quality datasets are conducted to verify the data quality-adaptivity of the MTL to datasets. The results show than the MTL is more accurate than FCNs and GANs in poor quality datasets.

Nonlinear sparse data regression and generation have been a long-term challenge, to cite the flow field reconstruction as a typical example. The huge computational cost of computational fluid dynamics (CFD) makes it much expensive for large scale CFD data producing, which is the reason why we need some cheaper ways to do this, of which the traditional reduced order models (ROMs) were promising but they couldn't generate a large number of full domain flow field data (FFD) to realize high-precision flow field reconstructions. Motivated by the problems of existing approaches and inspired by the success of the generative adversarial networks (GANs) in the field of computer vision, we prove an optimal discriminator theorem that the optimal discriminator of a GAN is a radial basis function neural network (RBFNN) while dealing with nonlinear sparse FFD regression and generation. Based on this theorem, two radial basis function-based GANs (RBF-GAN and RBFC-GAN), for regression and generation purposes, are proposed. Three different datasets are applied to verify the feasibility of our models. The results show that the performance of the RBF-GAN and the RBFC-GAN are better than that of GANs/cGANs by means of both the mean square error (MSE) and the mean square percentage error (MSPE). Besides, compared with GANs/cGANs, the stability of the RBF-GAN and the RBFC-GAN improve by 34.62% and 72.31%, respectively. Consequently, our proposed models can be used to generate full domain FFD from limited and sparse datasets, to meet the requirement of high-precision flow field reconstructions.