A multiscale and multicriteria Generative Adversarial Network to synthesize 1-dimensional turbulent fields

TL;DR

The paper addresses generating 1D turbulent velocity fields with correct energy distribution, energy cascade, and intermittency. It introduces a multiscale, multicriteria GAN where a fully convolutional generator $\mathcal{G}$ creates $u(x)$ from Gaussian noise, guided by four discriminators that enforce $S_2(l)$, $\mathcal{S}(l)$, $\mathcal{F}(l)$, and scale-invariance across segment sizes; training leverages Modane turbulence data. The approach yields fields whose structure functions satisfy inertial-range scaling $S_p(l) \propto l^{\zeta_p}$ with $\zeta_3=1$, produces scale-dependent PDFs resembling experimental turbulence, and captures intermittency through non-linear $\zeta_p$ and rising flatness at small scales, outperforming classical GANs and WGANs. The work enables realistic, scalable turbulence synthesis and sets the stage for extensions to 2D fields and Reynolds-number conditioning, with public code available for broader use.

Abstract

This article introduces a new Neural Network stochastic model to generate a 1-dimensional stochastic field with turbulent velocity statistics. Both the model architecture and training procedure ground on the Kolmogorov and Obukhov statistical theories of fully developed turbulence, so guaranteeing descriptions of 1) energy distribution, 2) energy cascade and 3) intermittency across scales in agreement with experimental observations. The model is a Generative Adversarial Network with multiple multiscale optimization criteria. First, we use three physics-based criteria: the variance, skewness and flatness of the increments of the generated field that retrieve respectively the turbulent energy distribution, energy cascade and intermittency across scales. Second, the Generative Adversarial Network criterion, based on reproducing statistical distributions, is used on segments of different length of the generated field. Furthermore, to mimic multiscale decompositions frequently used in turbulence's studies, the model architecture is fully convolutional with kernel sizes varying along the multiple layers of the model. To train our model we use turbulent velocity signals from grid turbulence at Modane wind tunnel.

Figures (14)

• Figure 1: Multiscale and multicriteria physics based GAN. In red the fully convolutional generator model $\mathcal{G}$, which produces realizations of a 1d stochastic field, $u(x)$, from realizations of a Gaussian noise, $w(x)$. In blue the physics-based discriminators used to train the model. The $\mathcal{D}_{\textrm{scale-invariance}}$ discriminator is fed with Modane realizations $v(x)$ and generated ones $u(x)$, while the other three discriminators are directly fed with the corresponding statistics across scales of $v(x)$ and $u(x)$. Each discriminator has its own loss function in green. The total loss function of the GAN, also in green, is a linear combination of the four loss functions of the discriminators.
• Figure 2: U-Net architecture of $\mathcal{G}$ taking as input a Gaussian white noise, $w(x)$, of size $N$ and providing as output a stochastic field with turbulent statistics, $u(x)$, of the same size. Convolutional and transpose convolutional blocks, made up of a convolutional layer (respectively transpose convolutional layer), batch normalization and ReLU activation function, are represented by the blue and purple rectangles respectively. The number at the top of each rectangle indicates the number of channels of the output of the block. The number at the left of each rectangle indicates the kernel size of the filter. The used padding is indicated with same when the size of the input and output are equal and $0$ when there is no padding. The stride used on each convolution and transpose convolution is always $1$. Green and red arrows mean respectively average pooling and upsampling. The concatenated long-skip connections are represented by the yellow rectangles.
• Figure 3: Architecture of the discriminator models $\mathcal{D}_{S_2}$, $\mathcal{D}_{\mathcal{S}}$ and $\mathcal{D}_{\mathcal{F}}$, which take as input respectively the $\log(S_2(l))$, $\mathcal{S}(l)$ and $\log(\mathcal{F}/3)$ curves and provide a scalar value in $\left(0,1\right)$. Each yellow rectangle represents a dense layer followed by Leaky ReLU activation. The number at the top of each rectangle indicates the output size of the dense layer. All Leaky ReLu use a slope of $0.2$ for negative values. The yellow circle represents the sigmoid activation function.
• Figure 4: Architecture of the four neural networks defining $\mathcal{D}_{\textrm{scale-invariance}}$. From top to bottom, each row of the discriminator takes as input a signal of size $N/2$, $N/4$, $N/8$ and $N/16$ respectively and provides a scalar value in $\left(0,1\right)$. Blue rectangles represent convolutional blocks made up of a convolutional layer, batch normalization and Leaky ReLU activation function. Yellow rectangles represent dense blocks made up of a dense layer followed by a Leaky Relu activation. The Leaky ReLu activations use a slope $0.2$ for negative values. The output size of each block is indicated by the number at the top of each rectangle. For convolutional blocks, the kernel size and stride are indicated by the numbers at the left of the blue rectangles and there is no padding. The yellow circle represents the sigmoid activation function.
• Figure 5: Illustration of three realizations of the process $u(x)$ generated with our GAN approach a), b) and c) and three realizations of Modane velocity measures $v(x)$ d), e) and f), in function of the spatial variable $x/L$. The red boxes correspond to the length of a Modane integral scale $L$.