12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

PREDICTION AND CONTROL OF TURBULENT FLOWSUSING DEEP LEARNING

Jiyeon Kim
School of Mathematics and Computing(Computational Science and Engineering)
Yonsei University
Seoul 03722, Korea
jykim3994@yonsei.ac.kr

Junhyuk Kim
Department of Mechanical Engineering
Yonsei University
Seoul 03722, Korea
junhyuk6@yonsei.ac.kr

Changhoon Lee
School of Mathematics and Computing (Computational Science and Engineering)
Department of Mechanical Engineering
Yonsei University
Seoul 03722, Korea
clee@yonsei.ac.krABSTRACT

diction and flow control that can be applied to practical applications and more complex turbulence.

In this study, prediction and control of 2D decaying homogeneous isotropic turbulence (DHIT) using deep learning
have been performed as an example of a fundamental study toimprove understanding of the dynamic behavior of turbulence.
PredictionNet, a neural network used for prediction, showed
high accuracy in the prediction of flow up to one integral timescale, 𝑇𝐿 , with a correlation coefficient of 0.855. Our model
based on generative adversarial network (GAN) also showedmuch higher accuracy in the enstrophy spectrum than convolutional neural network (CNN) model. Predicability dependingon the scale is also analyzed using scale decomposition. Another neural network used for control, ControlNet, was able togenerate disturbances that allow the time-evolution of the flow
field to be in a direction that fits the objective function. In addition, it was possible to bring some physical understanding of
the input vorticity fields through the analysis of the disturbancefields.

MACHINE LEARNING METHODOLOGY
In this section, target, loss function, applied model andnetwork architecture of PredictionNet and ControlNet will beexplained.

PredictionNet
We applied two models of CNN and conditional GAN(cGAN, Mirza & Osindero, 2014) for PredictionNet. CNN
was used as a baseline model because it is well known for
its capability of learning spatial information of variables embedded in data. However, CNN has a problem of producing blurry output sometimes due to the objective function
which is simply minimizing the mean squared difference ofthe ground truth and the prediction (Kim & Lee, 2020; Kimet al., 2021). On the other hand, GAN returns a well-trainedgenerator (𝐺) containing statistical aspects of the data throughadversarial training to a discriminator (𝐷). In the basic GAN,
the parameters of 𝐺 and 𝐷 are updated in directions to minimize log(1 − 𝐷 (𝐺 (𝒛))) and log(𝐷 (𝒙)), respectively, which
stands for the two-player min-max game with value function,
𝑉 (𝐺, 𝐷):INTRODUCTION
Recently, a lot of studies applying data-driven learning tovarious applications in turbulence such as flow control (Yehet al., 2021), closure modeling (Maulik et al., 2018), andflow reconstruction (Deng et al., 2019) are actively conducted.
However, there are not many examples yet of turbulence prediction and control, especially for the time-varying turbulence,
due to complex characteristics such as high-dimensionality,
multi-scale, and nonlinearity. Thus, we tried to develop a prediction model using GAN (Goodfellow et al., 2014) that predicts the evolution of turbulence with high accuracy by reflecting not only spatial information but also statistical characteristics of turbulence and to present a control model which enables
turbulence control for some specific purposes using this highaccuracy prediction model as a surrogate model. Although it isa prediction and control of a relatively simple 2D turbulence,
the results of the current study provide a new approach to pre-min max𝑉 (𝐺, 𝐷) = E 𝒙∼ 𝑝 ( 𝒙) [log 𝐷 (𝒙)]𝐺𝐷(1)+ E𝒛∼ 𝑝 (𝒛) [log(1 − 𝐷 (𝐺 (𝒛)))]First, for the cGAN model used in this study, eq.(1) is modified using Earth-Mover (Wasserstein-1) distance (WGAN, Arjovsky et al., 2017) to stabilize the training procedure. Then,some extra information, 𝒚, the input of the generator in ourcase, is given as a condition to the input of the discriminator to1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

Figure 1.

Schematic configurations of (a) PredictionNet and (b) ControlNet.

improve the output quality of the generator like the following:based on eq.(2) as follows:

𝑁1 ∑︁𝐿 𝑖 − 𝐿 𝑎𝑑𝑣
𝑁𝑖=1
DE
𝐿 𝑖 = (𝜔∗𝑖 − 𝜔𝑡𝑖 +Δ𝑡 ) 2𝐿 𝑎𝑑𝑣 = 𝐿 𝑓 𝑎𝑙𝑠𝑒

min max 𝑉 (𝐺, 𝐷) = E 𝒙∼ 𝑝 ( 𝒙) [𝐷 (𝒙 | 𝒚)]
𝐺𝐷𝐿 𝑐𝐺 𝐴𝑁 = 𝛾(2)− E𝒛∼ 𝑝 (𝒛) [𝐷 (𝐺 (𝒛 | 𝒚))] .

PredictionNet is a network that predicts the vorticity fieldof a later time, Δ𝑡, from an input field (𝑃𝑟𝑒𝑑 (𝑋 (𝑡 𝑖 )) = 𝑋𝑖∗ ≈𝑋 (𝑡𝑖 + Δ𝑡)). It is trained using direct numerical simulations(DNS) data to play a functional role in predicting the later timefrom each time point of our dataset. Therefore, the following
objective function is the optimization target regardless of theapplied model:

argmin ∥ 𝑋𝑖∗ − 𝑋 (𝑡 𝑖 + Δ𝑡)∥ 22𝐿 𝐷 = −𝐿 𝑡𝑟𝑢𝑒 + 𝐿 𝑓 𝑎𝑙𝑠𝑒 + 𝛼𝐿 𝑔 𝑝 + 𝛽𝐿 𝑑𝑟𝑖 𝑓 𝑡𝑁𝐿 𝑡𝑟𝑢𝑒 =1 ∑︁𝐷 (𝜔𝑡𝑖 +Δ𝑡 , 𝜔𝑡𝑖 )𝑁𝑁𝐿 𝑓 𝑎𝑙𝑠𝑒 =𝑖=1

1 ∑︁𝐷 (𝜔∗𝑖 , 𝜔𝑡𝑖 )
𝑁𝑖=1

(6)

where 𝛼, 𝛽, and 𝛾 determine the portion of each loss term in
the total loss, and fixed to 100, 10, and 0.001. The schematicconfiguration of PredictionNet can be found from figure 1(a).
One thing to note that is if there’s no discriminator then it isthe configuration of the baseline CNN model.

(3)

𝑤𝑝

where 𝑋 and 𝑋 ∗ represent the real (DNS) data and prediction
result, respectively, and 𝑤 𝑝 means weight parameters of PredictionNet. Based on the objective function eq.(3), additionalterms such as regularization loss or adversarial loss are added
to form loss functions depending on the model applied. ForCNN, an L2 regularization loss is used to prevent overfitting
and increase stability as follows:

ControlNet
For ControlNet, the same generative CNN as baseline
CNN or cGAN generator of PredictionNet was used. ControlNet is trained to generate disturbance fields that fit the objective function combined with PredictionNet (𝐶𝑜𝑛𝑡𝑟𝑜𝑙 (𝑋 (𝑡𝑖 )) =Δ𝑋𝑖 , 𝑃𝑟𝑒𝑑 (𝑋 (𝑡𝑖 ) + Δ𝑋𝑖 ) = 𝑋˜ 𝑖 ). Examples of the objectivefunction used in this study include maximizing the change ofvorticity at a later time (propagation of control effect) and minimizing the enstrophy at a later time as follows:

𝑁1 ∑︁
𝐿 𝑖 + 𝜎𝑅(𝑤 𝑝 )𝑁𝑖=1E
D1 ∑︁ 2𝑅(𝑤 𝑝 ) =𝐿 𝑖 = (𝜔∗𝑖 − 𝜔𝑡𝑖 +Δ𝑡 ) 2
𝑤 𝑝,𝑘2(5)

𝐿𝐶 𝑁 𝑁 =(4)argmax ∥ 𝑋𝑖∗ − 𝑋˜ 𝑖 ∥ 22(7)argmin ∥ 𝑋˜ 𝑖 ∥ 22(8)𝑤𝑐𝑘𝑤𝑐

where 𝑁 and 𝜎 are the batch size and the strength of regularization, each of which is set to 32 and 0.0001 by finetuning. The loss function of cGAN was set by adding a gradient penalty (GP) loss term (WGAN-GP, Gulrajani et al., 2017)

where 𝑋˜ and 𝑤 𝑐 are disturbed prediction at time 𝑡 + Δ𝑡 andweight parameters of ControlNet, respectively. A spatial gradient loss term is additionally used to remove non-physical2

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022Table 1. Quntified predictive accuracy of CNN and cGAN bytest MSE & C.C. depending on the prediction time distance.

Test MSE

C.C.

Δ𝑡CNN

cGAN

CNN

cGAN

0.25𝑇𝐿

0.01440.0086

0.98980.99400.5𝑇𝐿

0.04040.03900.96990.9708𝑇𝐿

0.12980.17240.88700.8547

2𝑇𝐿

0.38630.5056

0.52270.4530Figure 2. The decaying level of vorticity RMS and dissipation rate within the dataset time interval, 100𝛿𝑡.
noise for the ControlNet loss function as follows:
𝑁𝐿 𝑐𝑜𝑛𝑡𝑟 𝑜𝑙 =1 ∑︁
𝐿 𝑖 + 𝜃 𝐿 𝑔𝑟 𝑎𝑑𝑁𝑖=1* 𝑝+1,𝑞𝑝,𝑞 2 + *𝑝,𝑞+1𝑝,𝑞 2 + 𝑁 − 𝜔𝑡𝑖 ª− 𝜔𝑡𝑖 ª 1 ∑︁  © 𝜔𝑡𝑡© 𝜔𝑡𝑖𝐿 𝑔𝑟 𝑎𝑑 =® + ­®  ­𝑁𝑑𝑥𝑑𝑦
𝑖=1  «¬«¬ (9)The schematic configuration of ControlNet can be found fromfigure 1(b).

DATA COLLECTION
Because the flow is 2-dimensional, DNS were performedon the incompressible Navier-Stokes equations in the form of
the vorticity transport equation and the stream function formulation to basically satisfy the continuity equation, as follows:𝜕𝜔𝜕2𝜔𝜕𝜔
= −𝑢 𝑗+𝜈𝜕𝑡
𝜕𝑥 𝑗
𝜕𝑥 𝑗 𝜕𝑥 𝑗&𝜕2𝜓= −𝜔
𝜕𝑥 𝑗 𝜕𝑥 𝑗(10)where 𝜔, 𝜓, and 𝜈 are vorticity, stream function, and kinematic
viscosity, respectively. They are solved by the pseudo-spectral
method with 3/2 zero-padding for de-aliasing. The simulation
time step, 𝑑𝑡, is 0.0025, and the Crank-Nicolson method forthe viscous term and the 2nd order Adams-Bashforth method
for the convective term are used for the time integration. Computational domain size is 2𝜋 × 2𝜋, the number of spatial grids
is 128 × 128, and the bi-periodic boundary condition is applied
spatially. 500, 100, and 50 independent simulations with random phase and fixed amplitude initialization are performed fortraining, validation and test, respectively. The time interval ofthe dataset was determined after statistically sufficiently developed (after satisfying the power-law spectrum), and the datatime step, 𝛿𝑡, equals 20 simulation time steps (𝛿𝑡 = 20𝑑𝑡). Although we are reflecting the decaying nature of the flow in
our models, it is undesirable if those flow fields almost converged to the steady solution were contained in the dataset. So,
only 100𝛿𝑡 interval showing a relatively high rate of change instatistics is chosen. Among the representative statistics, theroot mean square (RMS) of vorticity decreased by about 25%and dissipation rate (𝜖) decreased by about 44% in this interval(see figure 2).

Figure 3. Examples of PredictionNet result using one of test
simulations. (a) Input field. And results of prediction time
distances of (b) Δ𝑡 = 0.25𝑇𝐿 , (c) Δ𝑡 = 0.5𝑇𝐿 , (d) Δ𝑡 = 𝑇𝐿 , and(e) Δ𝑡 = 2𝑇𝐿 .

RESULTS
PredictionNetThe prediction results of PredictionNet were comparedaccording to the prediction time distance, Δ𝑡. It is set to 0.25,
0.5, 1, and 2 times the integral time scale, 𝑇𝐿 . It corresponds
to around 40𝛿𝑡 or 800𝑑𝑡 where the autocorrelation drops toaround 0.25. As a result, it was confirmed that both CNN andcGAN show high accuracy in both qualitative and quantitative
measures up to one 𝑇𝐿 (see Table 1, figure 3). However, as
shown in enstrophy spectra of figure 4, it was found that CNN
failed to predict the high-wavenumber region even from a veryclose Δ𝑡. On the other hand, in the case of cGAN, high-order3

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

Figure 4. Comparison of enstrophy spectra for the input time
point of 𝑡 0 . (a) Δ𝑡 = 0.25𝑇𝐿 , (b) Δ𝑡 = 0.5𝑇𝐿 , (c) Δ𝑡 = 𝑇𝐿 , and(d) Δ𝑡 = 2𝑇𝐿 .

statistics are predicted well only with a small amount of error
even at 𝑇𝐿 .
It is conjectured that cGAN reflects the statistical characteristics of the target flow field well while the generator
is trained adversarially to the discriminator and contains thespatial correlation between the input and the target by conditioning the discriminator to take the input field of the generator together with its own inputs (i.e., the prediction target
and the generated flow field). In other words, cGAN seems
to contain small-scale characteristics better by learning thenon-linear features of the flow field based on complex dynamics compared to CNN that simply minimizes MSE based on
point-wise values. This can be verified using scale decomposition as shown in figure 5. While CNN fails to generatesmall-scale details and underpredicts even intermediate-scalefeatures, cGAN shows very high accuracy for both large-scale
and intermediate-scale. Small-scale details are also generated
well, although the structures are slightly shifted and distorted.

Figure 5. Results of scale decomposition for Δ𝑡 = 𝑇𝐿 . (a)Original fields, (b) large-scales (𝑘 ≤ 4), (c) intermediate-scales(5 ≤ 𝑘 ≤ 20), and (d) small-scales (𝑘 ≥ 21).

ControlNet
The first target of ControlNet is to maximize the propagation of the control effect at a later time, as in eq.(7). Here,
we set the target time distance as 0.5𝑇𝐿 , and the only constraint applied to ControlNet is the RMS of the disturbancefield (Δ𝑋𝑟 𝑚𝑠 = 0.05𝑋𝑟 𝑚𝑠 ). Looking at the disturbance fieldand enstrophy spectra in figures 6(c) and 6(e), a disturbancewith large-scale structures in which most of the energy isconcentrated in the low-wavenumber region is generated. In
addition, this energy-concentrated disturbance structures aremostly acting on the vortex structures of the input field. Thus,
we can think of it as the control effect of ControlNet output
is propagated and amplified well to the later time by transposing the dynamic path of the flow evolution. As a result, it
is shown that the vortex structures of original prediction anddisturbance added prediction have slightly different positions
and directions (see figures 6(b) and 6(d)). To see whether our
disturbance is the optimum case for maximizing the changeof vorticity field under the given condition, we performed two
separate validation cases. The first one is shown in figure 7(a),showing the results of quantifying the degree of change at theFigure 6. Examples of ControlNet with the targetargmax 𝑤𝑐 ∥ 𝑋𝑖∗ − 𝑋˜ 𝑖 ∥ 22 . (a) Input field, (b) original prediction, (c) disturbance field, (d) disturbance added prediction,and (e) enstrophy spectra.

later time from the original prediction by MSE. The ControlNet output, red dotted line, is the largest control effect compared to the subsequent 1000 nodes, MSE using other disturbance fields modified by random phase shift or spatial shift.
The second one is shown in figure 7(b). It presents normalized
RMS difference of simulations by adding various disturbancefields to the input, including disturbances suggested by previous researches, from the original simulation. Obviously, thebest case was our ControlNet output (red line).
The second target is to minimize the enstrophy at a later
time, as in eq.(8). Looking at the disturbance field and enstrophy spectra in figures 8(c) and 8(e), we can see that disturbancewith an almost similar structure to the main vortex structures

412th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
bulence is temporally decorrelated were developed. Amongthem, the cGAN model was able to predict even small-scalefeatures well by reflecting the complex nonlinear dynamics
and interactions of turbulence, and scale decomposition thatcan more precisely analyze the accuracy of turbulence prediction is presented. In the control part, we developed a modelthat can produce a control effect suitable for the desired target.
Although this study is a fundamental example of a relatively
simple flow field, it has the potential to be applied to practicalapplications or more complex turbulence.
Figure 7. (a) Quantifed results about the degree of the propagation of the control effect by MSE, and (b) normalized RMS
difference of various simulations from the original simulation.

ACKNOWLEDGMENTS
This work was supported by National Research Foundation of Korea (NRF) grants funded by the Korean government(MSIP) (2017R1E1A1A03070282, 2022R1A2C2005538).

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Figure 8. Examples of ControlNet with the target
argmin 𝑤𝑐 ∥ 𝑋˜ 𝑖 ∥ 22 . (a) Input field, (b) original prediction,(c) disturbance field, (d) disturbance added prediction, and (e)enstrophy spectra.

of the input field but in opposite sign is generated. The spectrum also showed a very similar result to the input. Checking
the disturbed prediction fields, there was only minor change inthe behavior and evolution of the flow field, and only the rateof decay and min/max values are affected to minimize enstrophy at the later time.

CONCLUSION
In this study, prediction and control of 2D decaying turbulence were performed as a fundamental example of prediction and control of turbulence. In the prediction part, models
that can predict with high accuracy up to a time where the tur-5