High-fidelity reconstruction of large-area damaged turbulent fields with a physically constrained generative adversarial network

The reconstruction of incomplete information in flow fields is a pervasive challenge in numerous turbulence-related applications. This paper proposes a framework for the high-fidelity reconstruction of large-area damaged turbulent fields with high resolution based on a physically constrained generative adversarial network. The network incorporates some special designs, such as leveraging complete or sparse fields of velocity components as physical constraints, employing a ResNet-like network with a fast Fourier convolution module, and adopting a PatchGAN discriminator network. To train the network, we employ a combination of loss functions, which comprise mean absolute error, network-level loss, and feature matching loss. We validate our method on a dataset of compressible isotropic turbulent flow and investigate three distinct damaged distributions with gap ratios of $39.06\%$ and $56.25\%$. The proposed reconstruction framework has been shown to achieve excellent reconstruction performance. The reconstructed flow fields are consistent with the raw flow fields in terms of magnitude, power spectrum, and two-point correlation function. We also provide extensive ablation studies to validate our approach. The results indicate that the utilization of physical constraints significantly improves the reconstruction performance in damaged regions, particularly in cases that involve a large damaged area. Furthermore, in the proposed reconstruction framework, the size of patch in the PatchGAN discriminator network can be flexibly adjusted according to the scale of turbulence structures, and the perceptual loss function plays a crucial role in evaluating differences between feature maps of flow fields at network levels based on a pretrained network. These special designs ensure consistency across diverse scales of turbulence structures and improve the accuracy and efficiency of network training.

Figure 1

(a) The instantaneous isosurface of velocity divergence, and (b) the velocity power spectrum of compressible isotropic turbulent flow; (c) three different masks with a gap ratio of $56.25\%$: a single-damaged mask, a multiple-damaged mask, and a random-damaged mask.

Figure 2

(a) Framework of generator network for flow field reconstruction. The framework is based on ResNet-like feedforward networks with a fast Fourier convolution module; (b) specific implementation of the fast Fourier convolution; (c) specific implementation of the PatchGAN discriminator network.

Figure 3

(a) Three different masks with a gap ratio of 39.06%; (b) reconstructed density fields corresponding to the three different masks; (c) original density field; (d) gradient fields corresponding to the reconstructed density fields; (e) original density gradient field.

Figure 4

(a) Three different masks with a gap ratio of 56.25%; (b) reconstructed density fields corresponding to the three different masks; (c) original density field; (d) gradient fields corresponding to the reconstructed density fields; (e) original density gradient field.

Figure 5

${\mathrm{MAE}}_{\text{elem},d}^{*}$ contour for the single-damaged form with gap ratios of (a) $39.06\%$ and (b) $56.25\%$.

Figure 6

${\mathrm{MAE}}_{\text{elem},g}^{*}$ contour for the single-damaged form with gap ratios of (a) $39.06\%$ and (b) $56.25\%$.

Figure 7

(a), (b) Power spectra of the reconstructed fields with various damaged forms, as well as that of the original and the random-damaged fields; (c), (d) power spectrum ratio of the reconstructed to the original fields $[E_R\left(k\right)/E_{GT}\left(k\right)]$.

Figure 8

Two-point correlation functions of density in the reconstructed and the original fields with different damaged forms. (a) Gap Ratio = $39.06\%$; (b) Gap Ratio = $56.25\%$.

Figure 9

(a) Framework of generator network for flow reconstruction, which is based on encoder and decoder networks; (b), (c) specific implementation of the encoder and decoder networks; (d) specific implementation of the global GAN discriminator network.

Figure 10

(a) Random-damaged mask with a gap ratio of 56.25%; (b) reconstructed density field with the random-damaged form by DCGAN; (c) original density field.

Figure 11

(a) Random-damaged mask with a gap ratio of 56.25%; (b) gradient field based on the reconstructed density field by DCGAN with the random-damaged form; (c) original density gradient field.

Figure 12

Comparison of reconstruction performance of DCGAN and PCGAN for the random-damaged form with a gap ratio of $56.25\%$, based on (a) the power spectrum function and (b) the two-point correlation function.

Figure 13

Comparison of reconstruction performance of PCGAN and its reduced frameworks (${\mathrm{PCGAN}}_1\text{–}{\mathrm{PCGAN}}_3$) with the random-damaged form and a gap ratio of $56.25\%$, based on (a) the power spectrum function and (b) the two-point correlation function.

Figure 14

(a) Random-damaged mask with a gap ratio of 56.25%; (b) original density field (ground truth); (c) reconstructed density field based on the proposed reconstruction framework ($\mathrm{PCGAN}$) and its reduced versions (${\mathrm{PCGAN}}_1\text{–}{\mathrm{PCGAN}}_3$) with the random-damaged form.

Figure 15

Comparison of reconstruction performance of $\mathrm{PCGAN}$ for different damaged forms with a gap ratio of $56.25\%$ and additional physical constraints that contain flow information extracted randomly from the complete reference field at different proportions. (a), (c), (e) Power spectrum function; (b), (d), (f) two-point correlation function.

Figure 16

(a) Original density field (ground truth); (b) additional physical constraints that contain flow information extracted randomly from the complete reference field at different proportions; (c) single-damaged, multiple-damaged and random-damaged masks with a gap ratio of 56.25%; (d)–(f) reconstructed density fields with corresponding masks and additional physical constraints.

Figure 17

Schematic diagrams of (a) the global GAN discriminator network and (b) the PatchGAN discriminator network.

Figure 18

(a) Schematic diagram of the pretrained network and the calculation of perceptual loss; (b) instantaneous contour of the input field and (c) its representative feature maps generated in stage 1 of the pretrained network.

Figure 19

Effects of the amount of training data on the reconstruction performance with the random-damaged form and a gap ratio of 56.25%; (a) the power spectrum function and (b) the two-point correlation function.

Figure 20

(a) Original density field (ground truth); (b) original density field combined with Gaussian noise of different variances; (c) random-damaged masks with a gap ratio of $56.25\%$; (d), (e) reconstructed density fields with corresponding masks and additional physical constraints.

Figure 21

Power spectrum of the reconstructed fields corresponding to input containing Gaussian noise, and that of the original field and the noisy input; (a) ${\mathrm{PCGAN}}_{\mathrm{R}}^{100\%}$ results and (b) ${\mathrm{PCGAN}}_{\mathrm{R}}^{0\%}$ results.

Figure 22

(a) Original density field (ground truth); (b) additional physical constraints from reference fields, where figure $b_2$ as an example denotes that reference fields have the same damaged form as the density input; (c) single-damaged, multiple-damaged and random-damaged masks with a gap ratio of 56.25%; (d)–(f) reconstructed density fields with corresponding masks and additional physical constraints.

Figure 23

(a) Power spectra of the reconstructed fields with various damaged forms, as well as that of the original and the random-damaged fields; (b) two-point correlation functions of density in the reconstructed and the original fields.