††thanks:

# High-fidelity reconstruction of turbulent flow from spatially limited data
using enhanced super-resolution generative adversarial network

Mustafa Z. Yousif  Linqi Yu  Hee-Chang Lim  Corresponding author,
<EMAIL_ADDRESS>School of Mechanical Engineering, Pusan National University,
2, Busandaehak-ro 63beon-gil, Geumjeong-gu, Busan, 46241, Rep. of KOREA

###### Abstract

In this study, a deep learning-based approach is applied with the aim of
reconstructing high-resolution turbulent flow fields using minimal flow fields
data. A multi-scale enhanced super-resolution generative adversarial network
with a physics-based loss function is introduced as a model to reconstruct the
high-resolution flow fields. The model capability to reconstruct high-
resolution laminar flows is examined using data of laminar flow around a
square cylinder. The results reveal that the model can accurately reproduce
the high-resolution flow fields even when limited spatial information is
provided. The case of turbulent channel flow is used to assess the ability of
the model to reconstruct the high-resolution wall-bounded turbulent flow
fields. The instantaneous and statistical results obtained from the model
agree well with the ground truth data, indicating that the model can
successfully learn to map the coarse flow fields to the high-resolution ones.
Furthermore, the computational cost of the proposed model, which is examined
carefully, is found to be effectively low. This demonstrates that using high-
fidelity training data with physics-guided generative adversarial network-
based models can be practically efficient in reconstructing high-resolution
turbulent flow fields from extremely coarse data.

##  I Introduction

The reconstruction of flow fields using spatially limited data has long been a
topic of interest in the fluid dynamics community. With the rapid development
of particle image velocimetry (PIV) and computational power for performing
direct numerical simulations (DNS), high-fidelity turbulence data can be
generated. However, as turbulence exhibits a chaotic behaviour with a wide
range of spatio-temporal scales, an expensive experimental setup and a
substantial computational cost are required to obtain high-resolution
turbulent flow fields. With the availability of enormous amounts of data that
can be obtained from experimental and DNS studies, deep learning techniques
have a great potential to serve as an alternative data-driven methods to
tackle turbulent flow problems. Deep learning is a subset of machine learning,
where deep neural networks are used for classification, prediction, and
feature extraction [1]. In this study, we demonstrate a deep learning-based
approach to reconstruct high-resolution laminar and turbulent flow fields
using coarse data, which are represented by a limited number of distributed
points.

Recently, there have been considerable developments in deep learning
algorithms that can be practically utilised in the field of fluid dynamics [2,
3]. Various deep learning-based approaches have been applied for different
applications in turbulence, such as turbulence modelling [4, 5, 6, 7], flow
prediction [8, 9] and flow control [10, 11]. On the other hand, high-
resolution turbulent flow reconstruction has recently become an active
research topic after the introduction of various supervised and unsupervised
deep learning-based algorithms designed to deal with high-resolution image
reconstruction [12]. These developments in deep learning techniques
accompanied with the rapid growth in graphics processing unit (GPU) power have
opened the doors to explore novel methods that could help reconstruct super-
resolution flow fields using extremely low-resolution experimental
measurements or simulation results rather than applying traditional
handcrafted super-resolution methods, such as bicubic interpolation [13].

Fukami et al. [14, 15] proposed models based on convolutional neural networks
(CNNs) for spatial and spatio-temporal super-resolution reconstruction of
turbulent flows. They reported good reconstruction of velocity and vorticity
fields using extremely low-resolution data along with good prediction of the
temporal evolution for the intervals for which the model was trained.

Onishi et al. [16] presented a super-resolution method based on CNN for
reconstructing high-resolution data from low-resolution urban meteorology
simulation data. Their results revealed that the CNN-based model highly
outperformed the conventional interpulation methods. Liu et al. [17]
considered the temporal behaviour of the flow in the reconstruction of the
high-resolution turbulent flow fields using CNN-based model. They showed that
by applying this approach, the reconstruction accuracy could be remarkably
improved compared with the static CNN-based model.

Recently, Kim et al. [18] showed that unsupervised deep learning has great
potential for reconstructing high-resolution turbulence using unpaired
training data. They used a cycle-consistent generative adversarial network
(CycleGAN) [19] to reconstruct high-resolution velocity fields from low-
resolution DNS and large eddy simulation (LES) data. Their results showed
better reconstruction accuracy compared with that of bicubic interpolation and
CNN-based model.

In terms of experimental studies, Deng et al. [20] applied a super-resolution
GAN (SRGAN) [21] and enhanced SRGAN (ESRGAN) [22] to reconstruct high-
resolution flow fields using PIV measurements of flow around a cylinder. They
reported an accurate reconstruction of the mean and fluctuation flow fields.
They observed that the reconstruction capability of ESRGAN was better than
that of SRGAN. Cai et al. [23] proposed a CNN-based model for estimating high-
resolution velocity fields from PIV measurments. Their model showed a better
performance compared with the traditional cross-correlation algorithms.
Morimoto et al. [24] applied a CNN-based model to estimate the velocity fields
using PIV measurements with missing regions. They reported a relatively good
estimation of the missing regions in the velocity.

In this study, we focus on the reconstruction of high-resolution flow fields
using extremely spatially limited data that are represented by distributed
points in the flow. This approach can mimic high-resolution flow fields
reconstruction from spatially limited PIV or hot wire measurements. We apply
ESRGAN-based model, i.e. multi-scale ESRGAN (MS-ESRGAN), to reconstruct high-
resolution flow fields using data with various coarseness levels. As
representative examples, we consider the DNS of laminar flow around a square
cylinder and turbulent channel flow.

The remainder of this paper is organised as follows. In Section 2, the
methodology of reconstruction high-resolution flow fields using the proposed
deep learning model is explained. The generation of the training data using
DNS is described in Section 3. In Section 4, the results of testing the
proposed model are discussed. Finally, the conclusions of this study are
presented in Section 5.

##  II Methodology

Since the first version of GAN was introduced by Goodfellow et al. [25],
variants of GAN have been proposed to tackle different types of image
transformation and super-resolution problems [21, 26, 22, 19]. The
architecture of GAN is designed to be different from the traditional
architecture of multilayer perceptron (MLP) or CNN-based models. In GAN, two
adversarial networks, i.e. the generator ($G$) and the discriminator ($D$),
compete with each other. Here, $G$ generates fake images similar to the real
ones, whereas $D$ distinguishes the fake images from the real ones. $G$ and
$D$ are usually MLPs or CNNs that are trained simultaneously. The goal of the
training process is to make $G$ generate fake images that are difficult to
distinguish using $D$. This process can be expressed as a min-max two-player
game with a value function $V(D,G)$ such that:

$\begin{split}\begin{subarray}{c}min\\\
G\end{subarray}~{}\begin{subarray}{c}max\\\
D\end{subarray}~{}V(D,G)=\mathbb{E}_{x_{r}\sim P_{data(x_{r})}}[{\rm
log}D(x_{r})]+\mathbb{E}_{z\sim P_{z}(z)}[{\rm log}(1-D(G(z)))],\end{split}$
(1)

where $x_{r}$ is the image from the ground truth data, whereas
$P_{data(x_{r})}$ is the real image distribution. $\mathbb{E}$ represents the
operation of calculating the average of all the data in the training mini-
batch. In the second right term of Eq. 1, $z$ is a random vector used as an
input to $G$, whereas $D(x_{r})$ represents the probability that the image is
real and not generated by $G$. $G(z)$ is the output from $G$, which is
expected to generate an image that is similar to the real image, such that the
value of $D(G(z))$ is close to 1. On the other hand, in $D$, $D(x_{r})$
returns a value close to 1, whereas $D(G(z))$ returns a value close to 0.
Thus, in the training process, $G$ is trained in a direction that minimises
$V(D,G)$, and $D$ is trained in a direction that maximises $V(D,G)$. After
successful training, $G$ is expected to produce an image with a distribution
similar to the real image that $D$ cannot judge whether it is real or fake.

This study applies a newly developed high-fidelity deep learning framework
based on ESRGAN [22] to reconstruct a high-resolution flow fields using
extremely coarse data as input to $G$. We adopted the generator network to
obtain MS-ESRGAN. The architecture of $G$ in MS-ESRGAN is shown in Fig. 1(a).
Here, $G$ consists of a deep convolution neural network represented by
residuals in residual dense blocks (RRDBs) and multi-scale branches.

The coarse input data are first passed through a convolution layer and then
through a series of RRDBs. The multi-scale part, which consists of three
parallel convolutional sub-models with different kernel sizes, is applied to
the data features that are extracted by the RRDBs. Finally, the outputs of the
three branches are simply summed and passed through a final convolutional
layer to generate a high-resolution fake image ($x_{f}$). Fig. 1(b) shows the
architecture of $D$. As mentioned earlier, $D$ is designed to distinguish
between the fake high-resolution image and the real image. The fake and real
images are fed to $D$ and passed through a series of convolutional, batch
normalisation, and leaky ReLU layers. Then, the data are passed through a
final convolutional layer. The non-transformed discriminator outputs using the
real and fake images, i.e. $C(x_{r})$ and $C(x_{f})$, are used to calculate
the relativistic average discriminator value $D_{Ra}$ [27]:

$D_{Ra}(x_{r},x_{f})=\sigma(C\left(x_{r})\right)-\mathbb{E}_{x_{f}}\left[C(x_{f})\right],$
(2)

$D_{Ra}(x_{f},x_{r})=\sigma(C\left(x_{f})\right)-\mathbb{E}_{x_{r}}\left[C(x_{r})\right],$
(3)

where $\sigma$ is the sigmoid function. In Eqs. 2 and 3, $D_{Ra}$ predicts the
probability that the output from $D$ using the real image is relatively more
realistic than the output using the fake image. The discriminator loss is then
defined as:

$L_{D}^{Ra}=-\mathbb{E}_{x_{r}}\left[{\rm
log}(D_{Ra}(x_{r},x_{f}))\right]-\mathbb{E}_{x_{f}}\left[{\rm
log}(1-D_{Ra}(x_{f},x_{r}))\right].$ (4)

The adversarial loss of the generator can be expressed in a symmetrical form
as:

$L_{G}^{Ra}=-\mathbb{E}_{x_{r}}\left[{\rm
log}(1-D_{Ra}(x_{r},x_{f}))\right]-\mathbb{E}_{x_{f}}\left[{\rm
log}(D_{Ra}(x_{f},x_{r}))\right].$ (5)

In addition to the adversarial loss, four additional loss terms are used to
form the combined loss function of the generator: pixel loss $(L_{pixel})$,
perceptual loss $(L_{perceptual})$, gradient loss $(L_{gradient})$, and
Reynolds stress loss $(L_{Reynolds~{}stress})$. $L_{pixel}$ is the pixel-based
error between the generated data and the ground truth data. $L_{perceptual}$
represents the difference in the extracted features of the real and fake data.
The pre-trained CNN VGG-19 [28] is used to extract the features. While one
layer of VGG-19 was applied in the model of Wang et al. [22], we apply three
different layers to extract the features. This strategy has showed a
remarkable improvement in the training stability. $L_{gradient}$ represents
the difference in the gradient of the generated fake data and real data,
whereas $L_{Reynolds~{}stress}$ is the error that represents the difference
between the Reynolds stress tensor of the velocity data that are obtained from
the generator and the Reynolds stress tensor of the ground truth velocity
data. The mean squared error (MSE) is used to calculate all the loss terms
except $L_{G}^{Ra}$.

The combined loss function of the generator is expressed as:

$\mathcal{L}_{G}=L_{G}^{Ra}+\lambda_{1}L_{pixel}+L_{perceptual}+\lambda_{2}L_{gradient}+\lambda_{3}L_{Reynolds~{}stress},$
(6)

where $\lambda_{1},\lambda_{2}$, and $\lambda_{3}$ represent the coefficients
that are used to balance the different loss terms whose values are set to be
5000, 10, and 100, respectively. $L_{gradient}$, is used to consider the non-
uniform distribution of the grid points in the training process, whereas
$L_{perceptual}$ can help in overcoming the training instability. The
turbulence statistics can be improved by applying $L_{Reynolds~{}stress}$,
which forces the model to consider the components of the Reynolds stress
tensor in the training process.

In this study, the adaptive moment estimation (ADAM) optimisation algorithm
[29] is applied to update the weights of the model. The training data are
divided into mini-batches, and the size of each mini-batch is set to be 16.

Figure 1: MS-ESRGAN architecture: (a) the generator ($\beta$ is the residual
scaling parameter = 0.2), and (b) the discriminator.

##  III Generation of training data

Two examples are used in this study, two-dimensional laminar flow around a
square cylinder at $Re_{d}=100$ is used as a demonstration, and turbulent
channel flow at $Re_{\tau}=180$ is used as a test case for reconstructing
high-resolution wall-bounded turbulent flow. The training data of each example
are generated by performing DNS.

The momentum equation for an incompressible viscous fluid is:

$\frac{\partial{\bf u}}{\partial t}+{\bf u}\cdot\nabla{\bf
u}=-\frac{1}{\rho}\nabla p+{\it\nu}\nabla^{2}{\bf u},$ (7)

where ${\bf u}$ is the velocity of the fluid, $\rho$ is the density, $p$ is
the pressure, and $\nu$ is the kinematic viscosity. The continuity equation
that expresses the incompressibility of the fluid is defined as:

$\nabla\cdot{\bf u}=0.$ (8)

The open-source computational fluid dynamics (CFD) finite-volume code
OpenFOAM-5.0x is used to perform the DNS. In the case of two-dimensional
laminar flow around a square cylinder, the Reynolds number is based on the
free-stream velocity and cylinder width, i.e. $Re_{d}=U_{\infty}d/{\it\nu}$,
where $U_{\infty}$ is the free-stream velocity and $d$ is the cylinder width.
The domain size is set to be $(xd{\times}yd)=(15{\times}20)$ with the
corresponding grid size of $(381{\times}221)$. The time step of the simulation
is set to be $\Delta t=10^{-2}$. The statistics obtained from the simulation
have been validated agnainst DNS results obtained by Anzai et al.[30]

In the case of turbulent channel flow, the friction Reynolds number, i.e.
$Re_{\tau}=u_{\tau}\delta/{\it\nu}$ is set to be 180, where $u_{\tau}$ is the
friction velocity and $\delta$ is half of the channel height. The dimensions
of the computational domain are set to be $4\pi\delta$, $2\delta$, and
$2\pi\delta$ in the streamwise ($x$), wall-normal ($y$), and spanwise ($z$)
directions, respectively. The corresponding grid points are 256, 128, and 256,
respectively. Uniform grid points distributions with spaceing $\Delta x^{+}$
$\approx$ 6.3 and $\Delta z^{+}$ $\approx$ 2.8 are used in the streamwise and
spanwise directions. Note that the superscript $+$ indicates that the quantity
is made dimensionless using the wall variables, i.e. $u_{\tau}$ and $\it\nu$.
A non-uniform grid points distribution is used in the wall-normal direction.
The first grid piont away from the wall is located at $y^{+}$ $\approx$ 0.63
and the maximum spacing (at the centreline of the channel), i.e. $\Delta
y^{+}_{max}$ $\approx$ 6.4. The periodic boundary condition is assigned to the
streamwise and spanwise directions, whereas the no-slip condition is applied
to the upper and lower walls of the channel. The time step of the simulation
is set to be $\Delta t=10^{-2}$ corresponding to $\Delta t^{+}=0.1134$. The
turbulent statistics obtained from the simulation have been validated using
DNS data obtained by Moser et al. [31].

For both simulations, the pressure implicit split operator algorithm is
employed to solve the coupled pressure momentum system. The convective fluxes
are discretised with a second-order accurate linear upwind scheme and all
other discretisation schemes that are used in each simulation have second-
order accuracy. The maximum Courant–Friedrichs–Lewy (CFL) number is maintained
to be less than 1 to ensure simulation stability.

The training data are obtained with 6,000 snapshots from the simulation of
two-dimensional laminar flow around a square cylinder and 10,000 snapshots
from a single ($y-z$) plane of the turbulent channel simulation. The domain
size used for training in the case of two-dimensional laminar flow around
square cylinder is fixed to be $(xd\times yd)=(16.82\times 8)$, which is
equivalent to a grid size of ($320\times 160$). The same grid size obtained
from the simulation is used for training in the turbulent channel flow case.
In both cases, the interval between the collected snapshots of the flow fields
is 10 times the time step used in the simulation.

As mentioned earlier, the low-resolution data are obtained by selecting a
distributed points in the flow, i.e. no filtering operation is used. The
distribution of the selected points at different coarseness levels for the two
cases is shown in Fig. 2. The distribution of the points
($n_{x}{\times}n_{y}$) in the case of two-dimensional laminar flow around a
square cylinder, as shown in Fig. 2(a), has three levels of coarseness: case 1
(40 $\times$ 20), case 2 (20 $\times$ 10), and case 3 (10 $\times$ 5). Figure
2(b) shows the three coarseness levels of the points distribution
($n_{y}\times n_{z}$) in the case of turbulent channel flow: case 1 (16
$\times$ 32), case 2 (8 $\times$ 16), and case 3 (4 $\times$ 8). To prepare
the data for the training process, the data are normalised using the min-max
normalisation function to produce values between 0 and 1. The shape of the
input data to $G$ is fixed to be (40 $\times$ 20) in the case of laminar flow
around a square cylinder and (16$\times$32) in the case of turbulent channel
flow. To achieve these shapes, upsampling is performed on cases 1 and 2 for
each of the two cases used in this study.

Figure 2: The distribution of the selected points at three different
coarseness levels for the case of (a) two-dimensional laminar flow around a
square cylinder, and (b) turbulent channel flow.

##  IV Results and discussion

###  IV.1 Flow around a square cylinder

In this section, we examine the ability of MS-ESRGAN to reconstruct high-
resolution flow fields using coarse data of two-dimensional laminar flow
around a square cylinder. Note that all the results are obtained using the
test data that are not included in the training process. The reconstructed
instantaneous velocity fields ($u$ and $\upsilon$), pressure field, and root-
mean-square error (RMSE) of the reconstruction are shown in Fig. 3. Here, the
velocity components are normalised by $U_{\infty}$, and the dimensionless
pressure is given as $C_{p}=(p-p_{\infty})/0.5\rho U_{\infty}$, where
$p_{\infty}$ is the free-stream pressure. The reconstructed fields show a
commendable agreement with the DNS results even when a high coarseness level
is used (i.e. case 3). As shown in the figure, the RMSE is proportional to the
coarseness level with a well-accepted maximum value.

Figure 3: Reconstructed instantaneous flow fields and RMSE for the case of
two-dimensional laminar flow around a square cylinder: (a) streamwise
velocity, (b) spanwise velocity, and (c) pressure.

To further validate the model, the capability of the model to reconstruct
high-resolution flow fields is examined statistically. Figure 4 shows the
probability density function (PDF) of the reconstructed velocity components
and the pressure. All the reconstruction results show excellent agreement with
the results obtained from the DNS, thus indicating the ability of the model to
reconstruct instantaneous high-resolution flow fields.

Figure 4: Probability density functions of the reconstructed velocity
components and pressure for the case of two-dimensional laminar flow around a
square cylinder.

To observe the flow characteristics, the profiles of the mean streamwise
velocity and mean pressure are calculated using 5000 reconstructed snapshots.
As shown in Fig. 5(a) and (b), the mean streamwise velocity and pressure are
in commendable agreement with the results obtained from the DNS. However, a
scattering in the pressure values near the rear of the cylinder, i.e. $x/d$
$\approx$ $0.5$ can be seen in Fig. 5(b) while it is not shown in the velocity
profile in Fig. 5(a). This behaviour can be attributed to the rapid change of
the pressure values in this region and the fewer physics constraints in the
combined loss function that are dedicated to the pressure compared with those
that are dedicated to the velocity components. For the pressure, there is only
the gradient term, while for the velocity, there are the gradient and the
Reynolds stress tenser terms. We believe that to achieve more accurate
reconstruction for the pressure field, more studies that focus on the
pressure-based physics constrains are required.

Figure 5: Profiles of the reconstructed mean streamwise velocity (a) and mean
pressure (b) for the case of two-dimensional laminar flow around a square
cylinder.

Figure 6 shows the power spectrum density (PSD) of the streamwise velocity
fluctuations at two different streamwise locations plotted against Strouhal
number ($St$=$fd$/$U_{\infty}$), where $f$ is the frequency. The results
obtained from the reconstructed velocity data are in excellent agreement with
the results obtained from the DNS for all the three levels of coarseness,
indicating that the reconstructed data match the temporal behaviour of the
ground truth data accurately.

The aforementioned results suggest that MS-ESRGAN can reconstruct laminar flow
fields with high spatial resolution and reproduce the same dynamics as the
ground truth data.

Figure 6: Power spectrum density of the reconstructed streamwise velocity
fluctuations at two different streamwise locations for the case of two-
dimensional laminar flow around a square cylinder.

###  IV.2 Turbulent channel flow

The ability of MS-ESRGAN to reconstruct high-resolution wall-bounded turbulent
flow fields is validated in this section using a plane normal to the
streamwise direction in the turbulent channel flow case, i.e. ($y-z$) plane.
The reconstructed instantaneous velocity fields (${u^{+}}$, ${\upsilon^{+}}$,
and ${w^{+}}$), and RMSE are shown in Fig. 7. The results of all the velocity
components are in agreement with the DNS data, including that of case 3, where
minimal information about the flow field is available. While the RMSE of the
streamwise velocity component is noticeably affected by the level of
coarseness, the wall-normal and spanwise velocity components show less
sensitivity to the coarseness level.

As shown in Fig. 8, the PDF plots of the reconstructed velocity components
reveal a commendable agreement with the results of the streamwise and wall-
normal velocities that obtained from the DNS. However, a slight deviation can
be observed for the spanwise velocity, which increases with the increase in
coarseness level. This can be attributed to the limited information about the
spanwise velocity component that is available considering the more random
behaviour of the spanwise velocity component compared with that of the other
two velocity components.

Figure 7: Reconstructed instantaneous velocity fields and RMSE for the case of
turbulent channel flow: (a) streamwise velocity, (b) wall-normal velocity, and
(c) spanwise velocity. Figure 8: Probability density functions of the
reconstructed velocity components for the case of turbulent channel flow.

To further examine the capability of the model to reproduce the velocity
fields with accurate spatial resolution, two-dimensional cross-correlation
$(R_{ii}(\Delta y,\Delta z))$ plots of the velocity fluctuations are examined,
as shown in Fig. 9. The correlations of all the three levels of coarseness are
generally in good agreement with the correlations obtained from the DNS
results, indicating the excellent ability of MS-ESRGAN to reproduce the high-
resolution velocity fields with an accurate spatial distribution.

Figure 9: Two-dimensional cross-correlations of the reconstructed velocity
components for the case of turbulent channel flow.

Furthermore, the statistics of 20,000 generated velocity fields corresponding
to $t^{+}=22,680$ are compared with the statistics obtained from the DNS
results, as shown in Fig. 10. As can be seen in Fig. 10(a), the mean
streamwise velocity profile for all the three cases of coarseness levels shows
excellent agreement with the DNS data obtained within the wall distance range,
i.e. the linear viscous sublayer, buffer layer, and logarithmic region.
Similarly, the root-mean-square (RMS) profiles of the streamwise and wall-
normal velocity components ($u_{rms}^{+}$ and $\upsilon_{rms}^{+}$) are also
in good agreement with the DNS results for all the coarseness levels, as shown
in Fig. 10(b) and (c). Although the RMS profile of the spanwise velocity
component ($w_{rms}^{+}$) for cases 1 and 2 is in good agreement with the
profile obtained using the DNS results, it shows an offset in case 3, as shown
in Fig. 10(d). As mentioned earlier, this can be regarded as the limited
information regarding spanwise velocity in case 3. Fig. 10(e) shows the mean
shear stress profile $-\overline{u^{\prime+}\upsilon^{\prime+}}$. The values
are generally in good agreement with the DNS results for all the coarseness
levels. Nevertheless, a noticeable scattering of the values can be seen near
$y^{+}$ $\approx$ 20 - 25 for cases 1 and 2. This can be attributed to the
maximum shear stress values that appear in this region which are harder to
capture by the model compared with the values that appear in the other regions
along the wall distance. Interestingly, the values of
$-\overline{u^{\prime+}\upsilon^{\prime+}}$ for case 3 show more smooth
profile compared with the values for cases 1 and 2 which is contrary to the
general expectations considering the previous results. This might be atributed
to the under and over-prediction of the streamwise and wall-normal velocity
components that can be compensated during the multiplication and averaging
processes.

Figure 10: Turbulence statistics of the reconstructed data for the case of
turbulent channel flow: (a) mean streamwise velocity profile, (b) RMS profile
of the streamwise velocity, (c) RMS profile of the wall-normal velocity, (d)
RMS profile of the spanwise velocity, (e) mean shear stress profile, and (f)
RMS profile of the streamwise vorticity

The RMS profiles of the streamwise vorticity ($\omega_{rms}^{+}$) are shown in
Fig. 10(f). The results for cases 1 and 2 show commendable match with the DNS
results. However, case 3 shows less values for most of regions along the wall
distance, indicating that the effect of the coarseness level in case 3 is more
noticeable compared with those of cases 1 and 2. This would be attributed to a
lack of information resulting in impaired reproduction of the streamwise
vorticity.

To further investigate the capability of MS-ESRGAN to reconstruct high-
resolution velocity fields with realistic behaviour, the one-dimensional
spanwise energy spectra for the three cases at different wall distances are
shown in Fig. 11. It can be observed from the figure that in all the three
cases, the spectral content is reproduced appropriately with a slight
deviation at high wavenumbers. These results suggest that the model could
successfully reproduce spectra similar to that obtained from the DNS.

Figure 11: One-dimensional spanwise energy spectra of the reconstructed
velocity components for the case of turbulent channel flow at different wall
distances.

The temporal evolution of the reconstructed snapshots is examined using the
time correlation $(R_{ii}(t))$ of each velocity component at $y^{+}$ $\approx$
177.6, as shown in Fig. 12. The results obtained from the reconstructed data
are in excellent agreement with the DNS results. Here, the model shows a
remarkable ability to reconstruct the velocity data with the same dynamics as
the ground truth data.

Figure 12: Time correlations of the reconstructed velocity components for the
case of turbulent channel flow.

In summary, the results obtained for the turbulent channel flow case indicate
that the proposed MS-ESRGAN, which can successfully reconstruct the velocity
fields with high spatial resolution, can reproduce turbulence statistics that
are similar to the turbulence statistics of the ground truth data.

Our primitive studies revealed that using only $L_{G}^{Ra}$, $L_{pixel}$, and
$L_{perceptual}$ in the generator loss function can result in distorted
fluctuations of the velocity components. Furthermore, the reconstructed images
showed less sharpness in terms of the flow field details compared with the
images reconstructed using the combined loss function. This suggests that
using physics-based constraints can remarkably improve the accuracy of the
model output.

###  IV.3 Computational cost

As a final remark, the computational cost of the proposed MS-ESRGAN is
presented in Table 1. The total number of trainable parameters for both
examples in this study is approximately 51 million (45.8 million for $G$ and
5.2 million for $D$). The training of the model on a single GPU machine
(Nvidia TITAN RTX) requires approximately 18.8 h for the case of two-
dimensional laminar flow around a square cylinder, whereas the training of the
turbulent channel flow case requires approximately 65.6 h. This computational
cost is required only once to learn how to map the low-resolution flow fields
to the high-resolution ones. The reconstruction process of the high-resolution
flow fields using the proposed MS-ESRGAN is considered to be computationally
inexpensive, as shown at the bottom of Table 1.

| Flow around a square cylinder |  Turbulent channel flow
---|---|---
|  $G$ $D$ |  $G$ $D$
No. of trainable parameters (million) | 45.8 5.2 | 45.8 5.2
Training time(h) | 18.8 | 65.6
Reconstruction time (s) | $1.22\times 10^{-2}$ | $7.81\times 10^{-3}$
Table 1: Number of trainable parameters and computational cost of MS-ESRGAN.

##  V Conclusions

In this study, a deep learning-based framework was proposed for the
reconstruction of high-resolution turbulent flow fields from spatially limited
flow data. We developed an improved version of ESRGAN, i.e. MS-ESRGAN, and
applied it to reconstruct the flow fields using distributed points at
different levels of coarseness. A combined loss function that includes
physics-based loss terms was utilised in $G$ to obtain more realistic results.
First, two-dimensional laminar flow around a square cylinder at $Re_{d}=100$
simulated using DNS was considered as an illustration of the careful
observation of high-resolution flow fields reconstruction using MS-ESRGAN. The
model showed a remarkable ability to reconstruct the laminar flow with precise
spatial and temporal details even when only minimal spatial flow information
was available. The ability of the model to reconstruct wall-bounded turbulence
was examined using data from DNS of turbulent channel flow at $Re_{\tau}=180$.
The model reproduced the instantaneous velocity fields successfully with
commendable accuracy for all the three coarseness levels used in the study.
Moreover, the turbulence statistics were reproduced appropriately with a
slight deviation, which was noticed when very limited spatial information of
the velocity fields was provided. The spectra, spatial correlations, and time
correlations were also in agreement with the data obtained from the DNS,
indicating that the developed model could accurately reconstruct the velocity
fields with similar spatial and temporal accuracy as that of the DNS data. In
this study, the developed MS-ESRGAN could effectively map flow fields with
minimal spatial distribution to a high-resolution ones by utilising the
principle of GAN combined with the physics-based loss function. This motivates
us to explore more physics-guided deep learning models that can serve as an
efficient and inexpensive data-driven methods for recovering high-resolution
turbulent flow fields from limited spatial information.

###### Acknowledgements.

This work was supported by ’Human Resources Program in Energy Technology’ of
the Korea Institute of Energy Technology Evaluation and Planning (KETEP),
granted financial resource from the Ministry of Trade, Industry & Energy,
Republic of Korea (no. 20214000000140). In addition, this work was supported
by the National Research Foundation of Korea (NRF) grant funded by the Korea
government (MSIP) (no. 2019R1I1A3A01058576).

## Data Availability

The data that supports the findings of this study are available within this
article.

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# This is the title

THE $(abc,pqr)$-PROBLEM FOR APPROXIMATE SCHAUDER FRAMES FOR BANACH SPACES
K. MAHESH KRISHNA

Post Doctoral Fellow

Statistics and Mathematics Unit

Indian Statistical Institute, Bangalore Centre

Karnataka 560 059 India

Email<EMAIL_ADDRESS>
Date:

Abstract: Motivated from the complete solution of important $abc$-problem for
Gabor system for the Hilbert space $\mathcal{L}^{2}(\mathbb{R})$ by Dai and
Sun [Memoirs of Amer. Math. Soc., 2016] and from the existential result of
approximate Schauder frames for $\mathcal{L}^{p}(\mathbb{R})$ using
translation operators on $\mathcal{L}^{p}(\mathbb{R})$ by Freeman, Odell,
Schlumprecht, and Zsak [Israel. J. Math, 2014], we formulate
$(abc,pqr)$-problem for approximate Schauder frames for Banach spaces
$\mathcal{L}^{p}(\mathbb{R})$, $1<p<\infty$.

Keywords: $abc$-problem for Gabor system, Approximate Schauder Frame.

Mathematics Subject Classification (2020): 42C15, 46E30.

###### Contents

  1. 1 Introduction
  2. 2 The $(abc,pqr)$-Problem for Approximate Schauder Frames

##  1\. Introduction

Let $\mathcal{H}$ be a Hilbert space over $\mathbb{C}$. Recall that [16, 7,
29] a sequence $\\{\tau_{n}\\}_{n}$ in $\mathcal{H}$ is said to be a frame for
$\mathcal{H}$ if there exist $a,b>0$ such that

$\displaystyle a\|h\|^{2}\leq\sum_{n=1}^{\infty}|\langle
h,\tau_{n}\rangle|^{2}\leq b\|h\|^{2},\quad\forall h\in\mathcal{H}.$

Among various examples of frames for Hilbert spaces, one of the most studied
classes of frames for the Hilbert space $\mathcal{L}^{2}(\mathbb{R})$ is the
frames arising from the Gabor systems

$\displaystyle\\{E_{mb}T_{na}g\\}_{m,n\in\mathbb{Z}},$

where $g\in\mathcal{L}^{2}(\mathbb{R})$ and for $c,d\in\mathbb{R}$,

$\displaystyle T_{c}:\mathcal{L}^{2}(\mathbb{R})\ni f\mapsto
T_{c}f\in\mathcal{L}^{2}(\mathbb{R});~{}T_{c}f:\mathbb{R}\ni
x\mapsto(T_{c}f)(x)\coloneqq f(x-c)\in\mathbb{C},$ $\displaystyle
E_{d}:\mathcal{L}^{2}(\mathbb{R})\ni f\mapsto
E_{d}f\in\mathcal{L}^{2}(\mathbb{R});~{}E_{d}f:\mathbb{R}\ni
x\mapsto(E_{d}f)(x)\coloneqq e^{2\pi idx}f(x)\in\mathbb{C}.$

A detailed analysis of Gabor frames for Hilbert spaces can be found in [21,
13, 19, 20, 23, 25, 55]. A fundamental problem in Gabor analysis is to
characterize parameters $a>0$, $b>0$ and the function $g$ such that
$\\{E_{mb}T_{na}g\\}_{m,n\in\mathbb{Z}}$ is a frame for
$\mathcal{L}^{2}(\mathbb{R})$. The problem has been solved fully or partially
for certain classes of functions in [32, 33, 18, 24, 38, 15, 27, 28, 56, 40,
37, 39, 53, 58, 57, 52, 3, 34, 14, 26, 51, 48, 49, 28, 30, 6, 31, 36, 35, 12,
47, 41]. Let $c>0$. When $g=\chi_{[0,c]}$, the characteristic function on
$[0,c]$, the problem of characterizing parameters $a>0$, $b>0$ and $c>0$ such
that

$\displaystyle\\{E_{mb}T_{na}\chi_{[0,c]}\\}_{m,n\in\mathbb{Z}}$

is a frame for $\mathcal{L}^{2}(\mathbb{R})$ is known as $abc$-problem for
Gabor system. A list of partial answers for this are obtained in [49, 25, 48,
51, 28, 30, 31, 6, 20]. In 2013, Dai and Sun fully resolved this problem in
their Memoirs [11, 10]. In this paper, we formulate a similar problem for
approximate Schauder frames for function spaces.

##  2\. The $(abc,pqr)$-Problem for Approximate Schauder Frames

We are motivated from the following two results.

  1. (i)

The complete solution of $abc$-problem for Gabor system by Dai and Sun [11].

  2. (ii)

For every $2<p<\infty$, there is an approximate Schauder frame for
$\mathcal{L}^{p}(\mathbb{R})$ obtained using translation operators on
$\mathcal{L}^{p}(\mathbb{R})$ [22].

Generalized Fourier expansion which resulted from the theory of frames for
Hilbert spaces led the notion of framing in [5] which is generalized to
Schauder frames in [4] which is further generalized in [22, 59] to approximate
Schauder frames defined as follows.

######  Definition 2.1.

[22, 59] Let $\mathcal{X}$ be a separable Banach space and $\mathcal{X}^{*}$
be its dual. Let $\\{\tau_{n}\\}_{n}$ be a sequence in $\mathcal{X}$ and
$\\{f_{n}\\}_{n}$ be a sequence in $\mathcal{X}^{*}.$ The pair
$(\\{f_{n}\\}_{n},\\{\tau_{n}\\}_{n})$ is said to be an approximate Schauder
frame (ASF) for $\mathcal{X}$ if the map

$\displaystyle S_{f,\tau}:\mathcal{X}\ni x\mapsto
S_{f,\tau}x\coloneqq\sum_{n=1}^{\infty}f_{n}(x)\tau_{n}\in\mathcal{X}$

is a well-defined bounded linear, invertible operator.

We now consider the Banach space $\mathcal{X}=\mathcal{L}^{p}(\mathbb{R})$,
for $1<p<\infty$. Let $q$ be the conjugate index of $p$. Since dual of
$\mathcal{L}^{p}(\mathbb{R})$ is isometrically isomorphic to
$\mathcal{L}^{q}(\mathbb{R})$, given $\phi\in\mathcal{X}^{*}$, there exists a
unique $\omega\in\mathcal{L}^{q}(\mathbb{R})$ such that

$\displaystyle\phi(u)=\int_{\mathbb{R}}u(\alpha)\omega(\alpha)\,d\alpha\eqqcolon[u,\omega],\quad\forall
u\in\mathcal{L}^{p}(\mathbb{R}).$

In particular, if $\\{f_{n}\\}_{n}$ is a sequence in $\mathcal{X}^{*}$, then
there exists a unique sequence $\\{\omega_{n}\\}_{n}$ in $\mathcal{X}$ such
that

$\displaystyle f_{n}(x)=[x,\omega_{n}],\quad\forall x\in\mathcal{X},\forall
n\in\mathbb{N}.$

Thus asking whether a pair $(\\{f_{n}\\}_{n},\\{\tau_{n}\\}_{n})$ is an ASF
for $\mathcal{X}$ is same as asking whether the map

$\displaystyle S_{\omega,\tau}:\mathcal{X}\ni x\mapsto
S_{\omega,\tau}x\coloneqq\sum_{n=1}^{\infty}[x,\omega_{n}]\tau_{n}\in\mathcal{X}$

is a well-defined bounded linear, invertible operator. We then have the
following problem.

######  Problem 2.2.

Let $1<p<\infty$ and $q$ be the conjugate index of $p$. Characterize
parameters $a>0$, $b>0$, $p>0$, $q>0$ and functions
$g\in\mathcal{L}^{p}(\mathbb{R})$, $h\in\mathcal{L}^{q}(\mathbb{R})$ such that
the pair

$\displaystyle(\\{\tilde{E}_{mq}\tilde{T}_{np}h\\}_{m,n\in\mathbb{Z}},\\{E_{mb}T_{na}g\\}_{m,n\in\mathbb{Z}}),$

is an ASF for $\mathcal{L}^{p}(\mathbb{R})$, where

$\displaystyle T_{c}:\mathcal{L}^{p}(\mathbb{R})\ni f\mapsto
T_{c}f\in\mathcal{L}^{p}(\mathbb{R});~{}T_{c}f:\mathbb{R}\ni
x\mapsto(T_{c}f)(x)\coloneqq f(x-c)\in\mathbb{C},~{}\text{ for
}c\in\mathbb{R},$ $\displaystyle E_{d}:\mathcal{L}^{p}(\mathbb{R})\ni f\mapsto
E_{d}f\in\mathcal{L}^{p}(\mathbb{R});~{}E_{d}f:\mathbb{R}\ni
x\mapsto(E_{d}f)(x)\coloneqq e^{2\pi idx}f(x)\in\mathbb{C},~{}\text{ for
}d\in\mathbb{R}$

and

$\displaystyle\tilde{T}_{s}:\mathcal{L}^{q}(\mathbb{R})\ni
f\mapsto\tilde{T}_{s}f\in\mathcal{L}^{q}(\mathbb{R});~{}\tilde{T}_{s}f:\mathbb{R}\ni
x\mapsto(\tilde{T}_{s}f)(x)\coloneqq f(x-s)\in\mathbb{C},~{}\text{ for
}s\in\mathbb{R},$ $\displaystyle\tilde{E}_{t}:\mathcal{L}^{q}(\mathbb{R})\ni
f\mapsto\tilde{E}_{t}f\in\mathcal{L}^{q}(\mathbb{R});~{}\tilde{E}_{t}f:\mathbb{R}\ni
x\mapsto(\tilde{E}_{t}f)(x)\coloneqq e^{2\pi itx}f(x)\in\mathbb{C},~{}\text{
for }t\in\mathbb{R}.$

Note that all operators $T_{c}$, $E_{c}$, $\tilde{T}_{s}$, $\tilde{E}_{t}$ are
invertible isometries. We also note that Problem 2.2 can be stated for space
$\mathcal{L}^{p}[a,b].$ A particular case of Problem 2.2 is the following
$(abc,pqr)$-problem.

######  Problem 2.3.

(The $(abc,pqr)$-problem for approximate Schauder frames) Let $1<p<\infty$ and
$q$ be the conjugate index of $p$. Characterize parameters $a>0$, $b>0$,
$c>0$, $p>0$, $q>0$, $r>0$ such that the pair

$\displaystyle(\\{\tilde{E}_{mq}\tilde{T}_{np}\chi_{[0,r]}\\}_{m,n\in\mathbb{Z}},\\{E_{mb}T_{na}\chi_{[0,c]}\\}_{m,n\in\mathbb{Z}}),$

is an ASF for $\mathcal{L}^{p}(\mathbb{R})$.

We remark that we can formulate problems similar to Problems 2.2, 2.3 for
p-approximate Schauder frames, p-approximate Bessel sequences, p-approximate
orthonormal bases and p-approximate Riesz bases [44, 43, 45, 46]. Following is
one such in the discrete case. Our problem is motivated from the discrete
Gabor analysis in $\ell^{2}(\mathbb{Z})$ [50, 9, 54, 2, 8, 1].

######  Problem 2.4.

(The $(MN,PQ)$-problem for p-approximate Schauder frames) Let $1<p<\infty$ and
$q$ be the conjugate index of $p$. Characterize natural numbers $N,M,P,Q$ and
sequences $\\{x_{j}\\}_{j\in\mathbb{Z}}\in\ell^{p}(\mathbb{Z})$,
$\\{y_{j}\\}_{j\in\mathbb{Z}}\in\ell^{q}(\mathbb{Z})$ such that the pair

$\displaystyle(\\{\tilde{E}_{m/P}\tilde{T}_{nQ}\\{y_{j}\\}_{j\in\mathbb{Z}}\\}_{0\leq
m\leq
P-1,n\in\mathbb{Z}},\\{E_{m/M}T_{nN}\\{x_{j}\\}_{j\in\mathbb{Z}}\\}_{0\leq
m\leq M-1,n\in\mathbb{Z}})$

is a

  1. (i)

p-orthonormal basis for $\ell^{p}(\mathbb{Z})$.

  2. (ii)

p-approximate Riesz basis for $\ell^{p}(\mathbb{Z})$.

  3. (iii)

p-approximate Schauder frame for $\ell^{p}(\mathbb{Z})$.

  4. (iv)

p-approximate Bessel sequence for $\ell^{p}(\mathbb{Z})$,

where

$\displaystyle\tilde{E}_{m/P}\tilde{T}_{nQ}:\ell^{q}(\mathbb{Z})\ni\\{a_{j}\\}_{j\in\mathbb{Z}}\mapsto\tilde{E}_{m/P}\tilde{T}_{nQ}\\{a_{j}\\}_{j\in\mathbb{Z}}\coloneqq\\{e^{2\pi
ijm/P}a_{j-nQ}\\}_{j\in\mathbb{Z}}\in\ell^{q}(\mathbb{Z}),$ $\displaystyle
E_{m/M}T_{nN}:\ell^{p}(\mathbb{Z})\ni\\{b_{j}\\}_{j\in\mathbb{Z}}\mapsto
E_{m/M}T_{nN}\\{a_{j}\\}_{j\in\mathbb{Z}}\coloneqq\\{e^{2\pi
ijm/M}b_{j-nN}\\}_{j\in\mathbb{Z}}\in\ell^{p}(\mathbb{Z}).$

Next we formulate the continuous version of Problem 2.2 (see [42] for the
definition of continuous approximate Schauder frame and [17] for the
definition of continuous Schauder frame).

######  Problem 2.5.

Let $1<p<\infty$ and $q$ be the conjugate index of $p$. Characterize functions
$g\in\mathcal{L}^{p}(\mathbb{R})$, $h\in\mathcal{L}^{q}(\mathbb{R})$ such that
the pair

$\displaystyle(\\{\tilde{E}_{b}\tilde{T}_{a}h\\}_{a,b\in\mathbb{R}},\\{E_{b}T_{a}g\\}_{a,b\in\mathbb{R}})$

is a continuous approximate Schauder frame for $\mathcal{L}^{p}(\mathbb{R})$.

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