Nonlinear sparse data regression and generation have been a long-term challenge in the field of aerodynamics, flow field reconstruction is an area of specific interest in this article. The high computational costs of computational fluid dynamics (CFD) make large scale CFD data production expensive, which is the reason why cheaper methods are needed. Traditional reduced-order models were promising but they cannot generate a large amount of full domain flow field data (FFD) to execute high-precision flow field reconstructions. Motivated by the problem of existing approaches, and inspired by the success of generative adversarial networks (GANs) in the field of computer vision, we prove a theorem that shows the optimal approximation to a GAN discriminator is a radial basis function neural network when engaged in with nonlinear sparse FFD regression and generation. Based on this theorem, a radial basis function-based GAN (RBF-GAN) and a RBF cluster-based GAN (RBFC-GAN) are proposed for regression and generation purposes. Three different datasets are applied to verify the feasibility of our models. The results show that the performance of RBF-GAN and RBFC-GAN are better than that of GANs and conditional GANs (cGANs) by both the mean square error and the MSPE measurements. In addition, compared with GANs/cGANs, the stability of the RBF-GAN and the RBFC-GAN are improved 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 requirements of high-precision flow field reconstructions.

中文翻译：

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基于径向基函数的 GAN 流场重建

非线性稀疏数据回归和生成一直是空气动力学领域的长期挑战，流场重建是本文特别感兴趣的一个领域。计算流体动力学 (CFD) 的高计算成本使大规模 CFD 数据的生产成本高昂，这就是需要更便宜的方法的原因。传统的降阶模型很有前景，但它们无法生成大量的全域流场数据（FFD）来执行高精度流场重建。受到现有方法问题的启发，并受到计算机视觉领域生成对抗网络 (GAN) 成功的启发，我们证明了一个定理，该定理显示 GAN 鉴别器的最佳近似是径向基函数神经网络，当从事非线性稀疏 FFD 回归和生成时。基于该定理，提出了基于径向基函数的 GAN（RBF-GAN）和基于 RBF 集群的 GAN（RBFC-GAN），用于回归和生成目的。应用三个不同的数据集来验证我们模型的可行性。结果表明，无论是均方误差还是 MSPE 测量，RBF-GAN 和 RBFC-GAN 的性能都优于 GAN 和条件 GAN (cGAN)。此外，与 GANs/cGANs 相比，RBF-GAN 和 RBFC-GAN 的稳定性分别提高了 34.62% 和 72.31%。最后，