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

NON-LINEAR ORTHOGONAL MODAL DECOMPOSITIONS IN
TURBULENT FLOWS VIA AUTOENCODERS

Hamidreza Eivazi
FLOW, Engineering Mechanics
KTH Royal Institute of Technology
Stockholm, Sweden
hamidre@kth.se

Soledad Le Clainche
School of Aerospace Engineering
Universidad Politécnica de Madrid
Madrid, Spain
soledad.leclainche@upm.es

Sergio Hoyas
Ricardo Vinuesa
Inst. Univ. Matemática Pura y Aplicada
FLOW, Engineering Mechanics
Universitat Politècnica de València
KTH Royal Institute of Technology
Valencia, Spain
Stockholm, Sweden
sergio.hoyas@mot.upv.es
rvinuesa@mech.kth.se

ABSTRACT

composition of flow-field data, deep neural networks (DNNs)
have shown promising performance in learning a compact latent representation of high-dimensional data by accounting forthe non-linearity in the low-dimensional mapping using nonlinear activation functions (Hinton & Salakhutdinov, 2006). In
particular, unsupervised learning based on autoencoders (AEs)has been shown suitable for efficient mode decomposition andreduced-order modeling with superior performance in flow reconstruction over the linear POD (Milano & Koumoutsakos,
2002; Eivazi et al., 2020).
Moreover, convolutional neural networks (CNNs) (Lecunet al., 1998) and their ability in pattern recognition have received increasing attention by the fluid-mechanics community (Lee & You, 2019; Fukami et al., 2019; Kim & Lee,
2020; Kim et al., 2021; Guastoni et al., 2021). Murata et al.
(2020) proposed a CNN-based autoencoder architecture fordecomposition of flow-fields into non-linear low-dimensionalmodes and to visualize each mode. They applied this socalled mode-decomposing convolutional-neural-network autoencoder (MD-CNN-AE) to a relatively simple laminar flow
around a circular cylinder at ReD = 100 (where ReD is the
Reynolds number based on freestream velocity and cylinder
diameter). Their results showed the superior performance of
the CNN-based autoencoder over POD where the reconstruction of the flow from only two MD-CNN-AE modes contains
also the higher-order POD modes. The architecture of AEbased methods allows a non-linear low-dimensional mapping
leading to a superior performance against linear-theory-based
methods. However, the AE-based methods do not benefit
from the useful properties of the eigenvalue or singular-valuedecomposition techniques, e.g., optimality and orthogonality.
In contrast to the POD modes, which are an orthogonal setof basis vectors arranged in the order of their energy content,
the AE-based modes are neither orthogonal nor ranked. This
may lead to the lack of interpretability and robustness of theAE-based modes (Vinuesa & Sirmacek, 2021). In order to obtain ranked modes, Fukami et al. (2020) proposed a hierarchical CNN-AE architecture inspired by the concept of hierarchical autoencoder (AE) (Saegusa et al., 2004). The proposed
method was first applied to a laminar cylinder wake and its

We propose a deep probabilistic-neural-network architecture for learning a minimal and near-orthogonal set of nonlinear modes from high-fidelity turbulent-flow data. Our approach is based on β -variational autoencoders (β -VAEs) andconvolutional neural networks (CNNs), which enable extracting non-linear modes from multi-scale turbulent flows while
encouraging the learning of independent latent variables andpenalizing the size of the latent vector. Moreover, we introduce an algorithm for ordering VAE-based modes with respect to their contribution to the reconstruction. We apply thismethod for non-linear mode decomposition of the turbulent
flow through a simplified urban environment. We demonstratethat by constraining the shape of the latent space, it is possible to motivate the orthogonality and extract a set of parsimonious modes sufficient for high-quality reconstruction. Our
results show the excellent performance of the method in the reconstruction against linear-theory-based decompositions. We
show the ability of our approach in the extraction of nearorthogonal modes with the determinant of the correlation matrix equal to 0.99, which may lead to interpretability.

INTRODUCTION
Modal-decomposition techniques offer methods to identify a low-dimensional coordinate system for capturing dominant flow features (Taira et al., 2017, 2020) useful for developing reduced-order models, analyzing non-linear and chaotic
dynamics, and designing efficient flow-control schemes.
Proper-orthogonal decomposition (POD) (Lumley, 1967) anddynamic-mode decomposition (DMD) (Rowley et al., 2009;
Schmid, 2010) are two mode-decomposition methods based onlinear algebra that have been widely used to extract the dominant spatio-temporal features in fluid flows. Balanced POD(BPOD) (Rowley, 2005), spectral POD (SPOD) (Towne et al.,2018), higher-order DMD (HODMD) (Le Clainche & Vega,
2017) and spatio-temporal Koopman decomposition (STKD)(Clainche & Vega, 2018) are several successful variants of
POD and DMD for analysis of turbulent flows.
Besides the aforementioned linear methods for modal de-1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
transient process and further to an in-plane cross-sectional velocity field of turbulent channel flow at Reτ = 180 (note thatReτ is the friction Reynolds number, based on channel halfheight and friction velocity). They showed that the hierarchicalautoencoder (AE) can rank the AE modes following their contributions to the reconstructed field while achieving efficientorder reduction. However, issues related to interpretability andnon-uniqueness remained unanswered.
In this paper, we propose a probabilistic method basedon β -variational autoencoders (β -VAEs) (Higgins et al., 2017)
and CNNs in order to extract a minimal (parsimonious) set ofnear-orthogonal non-linear modes from turbulent flows. We
applied the proposed machine-learning method for the modal
decomposition of high-fidelity turbulent-flow simulation dataof a simplified urban environment. The flow simulation iscarried through well-resolved large-eddy simulation (LES) bymeans of the spectral-element method. The results from the
proposed method are compared with the results from a conventional CNN-AE model, a hierarchical CNN-AE model, and
POD. Through the training process of the available AE-based
modal-decomposition methods, the objective is to only minimize the reconstruction loss. Therefore, the focus of the available AE-based methods for non-linear modal decompositionis only on the performance of the model in the reconstruction and not on the properties of the learned modes, such as
orthogonality and minimality. In contrast, here we minimize
the correlation between the latent variables and penalize thesize of the latent vector in addition to the minimization of
the reconstruction loss. By solving this multi-objective optimization problem using our CNN-β VAE approach, we seeka minimal set of uncorrelated non-linear modes that are able
to accurately describe the turbulent flow-field data. The obtained modes are extremely useful for the development of compact reduced-order surrogate models or designing flow-control
strategies. Moreover, understanding the physics of the turbulent flow through the extraction of uncorrelated non-linear
mechanisms is another incentive for applying CNN-β VAEs
for modal decomposition. In particular, the development of accurate predictive models and understanding flow structures inurban environments are of significant importance due to theirimpact on urban planning, air quality management, and pollutant dispersion (Vinuesa et al., 2015).
We applied POD using the singular-value decomposition(SVD) method on the urban-flow database discussed above.
Figure 1 shows the eigenvalues λi (left) and the cumulativej=ieigenvalue spectrum ∑ j=1 λ j (right) normalized with the cu-

Figure 1. Eigenvalues λi (left) and the cumulative eigenvalue
j=ispectrum ∑ j=1 λ j (right) normalized with the cumulative sum
j=mof the eigenvalues ∑ j=1 λ j , where i indicates the number ofmodes. The solid red line shows the number of modes required
to capture 99% of the energy.

required for reconstruction by optimizing the model parameters w to minimize the reconstruction loss Lrec . The autoencoder architecture is attractive for modal decomposition as itprovides a framework that can incorporate non-linearity in the
mappings through the use of non-linear activation functions.
Another challenge in the modal decomposition of turbulent flows is the process of information from input fields that
contain multiscale coherent features. The presence of coherent
features motivates the use of convolutional layers in autoencoder models to process the input information.
We consider three different types of AEs: CNN-based(CNN-AE), CNN-based hierarchical (CNN-HAE) and CNNbased β -variational (CNN-β VAE). For simplicity, we consider the fluctuating component of the streamwise velocity
u as the input/output of the model, but it is also possible
to consider all the velocity components (u, v, w) as the input/output. The first convolution layer contains 16 filters witha size of (3 × 3), and it is followed by a max-pooling layer
with P = 2. At each convolution step, we double the numberof feature maps to extract more information from the turbulentflow data while at each downsampling step we reduce the dimension. This allows the next layer to combine the featuresindividually identified in each feature map, enabling the extraction of larger and more complex features for progressively
deeper convolutional networks from simple non-linear combinations of the previous ones. Therefore, convolutional layers
can learn to recognize turbulent-flow patterns of various complexity and scales (Guastoni et al., 2021). After five steps ofconvolution and max pooling, the extracted features are flattened and fed to fully connected layers to reduce the dimension to the latent vector r with a size of d. The latent vector r is mapped back to the reference space through the consecutive upsampling and convolution operations using nearestneighbor interpolation. Throughout the model, we use a filter
size of (3 × 3) with the stride of one for convolution layersand (2 × 2) max pooling and upsampling operations with thesame stride. We employ the hyperbolic-tangent (tanh) function ϕ(z) = (ez − e−z )/(ez + e−z ) as the non-linear activation
function, since it led to the best performance in our study. For
all the models, we use mean-squared error as the loss function for reconstruction Lrec and the Adam algorithm (Kingma& Ba, 2017) to optimize the model parameters w . We employ
the early-stopping criterion and obtain the best model based on
the validation loss to avoid overfitting, where 20% of the data
is randomly selected for validation.
The CNN-HAE architecture was proposed by Fukami
et al. (2020), it is based on a hierarchy of CNN-AEs, and
it is aimed at extracting the modes ranked in terms of theircontribution to the reconstruction while achieving more efficient data compression. To this end, the first subnetwork F1
is trained to map the high-dimensional data to a latent vector

j=mmulative sum of the eigenvalues ∑ j=1 λ j , where i indicates thenumber of modes. We observed that 247 modes are required
to capture 99% of the energy as it is depicted by the vertical
red line in figure 1 (right). This result implies that it is impractical to represent turbulent flows as a linear superposition of afew modal functions, and thus, more sophisticated algorithms
enabling a non-linear modal decomposition are required.

AUTOENCODERS FOR MODAL DECOMPOSITION
An autoencoder is a deep neural network (DNN) with anarchitecture suitable for unsupervised feature extraction. The
network comprises two parts: an encoder that maps the input
data to a low-dimensional latent space x 7→ r , and a decoderthat projects the latent vector r back to the reference space
r 7→ x̃x. We refer to the encoder and decoder parts as E andD, respectively. Through the model training, the autoencoder
learns to extract the most important features in the data that are2

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022with size d = 1. The latent vector can be obtained using theencoder part of the first subnetwork as r 1 = E1 (xx). The second subnetwork F2 is then trained to reconstruct the input dataat the output from a two-dimensional latent vector comprising
the first latent vector r 1 , which has already been obtained, andthe second latent vector r 2 , being updated through the trainingof F2 , as [rr 1 r 2 ]. The subsequent networks are trained in asimilar way. In the present study, we employ the same hyperparameters as those of the CNN-AE model for the CNN-HAE.
Finally, we also use a modified version of the so-called
variational autoencoder (VAE) (Kingma & Welling, 2014),
which is a probabilistic generative neural architecture emerging from the combination of statistics and information theory. The goal is to map the data into a latent distribution, from which new meaningful samples can be generated.
VAEs have gained increasing attention in the scientific community (Maulik et al., 2020), both due to their strong probabilistic foundation and their valuable application in the fieldof representation learning. The CNN-β VAE employed here,
which is represented in Figure 2, has the goal of minimizing
the correlation between the latent variables, motivating the network to extract a set of orthogonal modes, also penalizing thesize of the latent vector d. This leads to an efficient representation of the high-dimensional data useful for flow analysis,
reduced-order modeling, and flow control. Note that the decoder part is the same as that of the CNN-AE model.

This term is usually called evidence lower bound (ELBO).
Since the operation that samples a latent vector from qφ (rr |xx) is
not differentiable, we need to perform a change of variable, theso-called reparameterization trick (Kingma & Welling, 2014),
to differentiate ELBO with respect to both θ and φ . We assume qφ (rr |xx) to be a Gaussian distribution,
log qφ (rr |xx) = log N (rr ; µ , σ 2 I),

where the mean µ and the standard deviation σ are outputsof the encoder, and I is the identity matrix. We sample fromqφ (rr |xx) using r = µ + σ ε where ε ∼ N (0, I) is an auxiliary normally-distributed random number, andindicates
an element-wise product. Note that for the testing steps themean part of the encoder output µ of the CNN-β VAE istaken as the vector of latent variables r . Moreover, the term
Eqφ (rr |xx) [log pθ (xx|rr )], which is the so-called log-likelihood,encourages accurate reconstruction of the data and can be estimated as a negative reconstruction error in an autoencoder
setting (Kingma & Welling, 2014). This leads to the VAE costfunction C, and we can take the negative of it as a loss function
L for training the NNs:

L(θθ , φ ; x ) = Lrec −Decoder4608(3 × 6 × 256)

6 × 12 × 256
12 × 24 × 128

24 × 48 × 64

48 × 96 × 32

96×192×16

96 × 192 × 1µ
d

96×192×16
96 × 192 × 1

48 × 96 × 32

24 × 48 × 64

12 × 24 × 128

6 × 12 × 256

128
4608
(3 × 6 × 256)

σ

µ+ε
d

J

σ

r

d
ε

EncoderdFigure 2. Schematic view of the CNN-β VAE. The color coding for each layer is: 2D-convolution ( ), tanh activation( ), max pooling ( ), reshape ( ), fully-connected layer( ), upsampling ( ).

L(θθ , φ ; x ) = Lrec −Let us consider a data sample x in some high-dimensional
space X with the distribution p(xx). VAEs define two probability distributions: qφ (rr |xx), a so-called probabilistic encoder(or recognition model) and in a similar vein pθ (xx|rr ), whichis referred to as a probabilistic decoder (or generative model).
θ and φ denote the unknown parameters. The objective is
optimize the unknown parameters to maximize the so-called
marginal likelihood, which can be defined for each x as:
log pθ (xx) = DKL (qφ (rr |xx)||pθ (rr |xx)) + C(θθ , φ ; x ),−DKL (qφ (rr |xx)||pθ (rr )) + Eqφ (rr |xx) [log pθ (xx|rr )] .

(4)

β d
∑ (1 + log(σi2 ) − µi2 − σi2 )
2 i=1

(5)

for β -VAEs. A detailed discussion on disentangling in β VAEs can be found in the works by Higgins et al. (2017);
Burgess et al. (2018); Achille & Soatto (2018); Locatello et al.
(2019).

RESULTS AND DISCUSSION(1)

The key insight of the present study is to encourage independence of the latent variables r 1 , . . . , r d to extract nearorthogonal modes from turbulent flows using CNN-β VAEs.
This is to motivate disentangled representations in the language of representation learning. We impose a limit on thecapacity of the latent information and motivate learning statistically independent latent variables using the penalization
factor β . The objective is to motivate orthogonality in the latent space to obtain modes that are useful for flow analysis,
reduced-order modeling, and flow control.

where the first right-hand-side (RHS) term is the Kullback–Leibler (KL) divergence DKL between qφ (rr |xx) and pθ (rr |xx).
The KL-divergence is non-negative, which indicates that thesecond RHS term C(θθ , φ ; x ) is a lower bound on the marginal
likelihood, and can be written as:log pθ (xx) ≥ C(θθ ,φφ ; x ) =1 d∑ (1 + log(σi2 ) − µi2 − σi2 ).
2 i=1

In the field of representation learning (Bengio et al.,2013), it is of interest to find a latent representation of thehigh-dimensional data as an uncorrelated representation witha minimal number of parameters (factors), the so-called disentangled representation, which can be useful for a large variety of tasks and domains. Higgins et al. (2017) proposed
to augment the original VAE loss function with a single hyperparameter β ≥ 0 that controls the extent of the learning
constraints. The goal is to encourage learning of statisticallyindependent latent variables r i and penalize the size of the latent vector r . This can be obtained by minimizing the distance DKL [p(rr )|| ∏i p(rr i )] between p(rr ) and the product of its
marginals. In practice, this is performed by upweighting the
KL term in the ELBO, see equation (2), with a penalization
factor β leading to the following loss function:

r
128

(3)

(2)

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
We define two evaluation metrics to measure the qualityof the reconstructions and the orthogonality (disentanglement)of the latent variables. For reconstruction quality, we evaluate
the energy percentage Eu that is captured by the model reconstructions as:n* ∑ (u − ũ)2 +

i=1
 × 100,
Eu = 
n
1 −

∑ u2

(6)
Figure 3. Reconstruction of the fluctuating component of thestreamwise velocity obtained from different methods, as indicated in each panel, in comparison with the reference data.
The value in brackets on each panel indicates the obtained Eu .

i=1

where h·i indicates ensemble averaging in time, u and ũ denote
the reference value of the fluctuating component of streamwise velocity and its reconstruction, respectively, and n is thenumber of grid points. To measure the independency of thelatent variables, we compute the determinant of the correlation matrix multiplied by 100 and refer to it as detR , whereR = (Ri j )d×d is the correlation matrix defined by:
Ci j
Rii = 1 and Ri j = p,
CiiC j j(7)

for all 1 ≤ i 6= j ≤ d, and Ci j denotes the components i, j of
the covariance matrix C. Note that detR is 100 when all thevariables are completely uncorrelated (Ri j = 0) and zero when
they are completely correlated (Ri j = 1). We report the valueof detR as a metric for independency of the latent variables.
We compare the performance of CNN-β VAEs with thatof CNN-HAEs, CNN-AEs and POD in terms of reconstruction accuracy and orthogonality of the latent variables. We
select d = 5 as the size of the latent vector for all models. For
the CNN-β VAE we use β = 10−3 . Figure 3 shows the reconstruction of the first time step in the dataset obtained fromdifferent methods in comparison with the reference data. It
can be observed that a more accurate reconstruction can beobtained using NN-based methods in comparison with 5 POD
modes due to the introduction of non-linearity in the algorithm. The CNN-AE model leads to the best reconstructions
with Eu = 94.22% while 32.41% of the energy is captured by
the 5 POD modes. Both CNN-HAE and CNN-β VAE models
also lead to excellent reconstructions with Eu of 91.84% and87.36%, respectively, which are slightly lower than that of the
CNN-AE. For the CNN-β VAE, it is due to the fact that theregularization with the KL term in the β -VAE loss function,
equation (5), induces a trade-off between the reconstruction
quality and learning independent representations. However,
our results show that excellent reconstructions can be obtainedusing all the three CNN-AE-based models leading to Eu of
about 90% using only 5 modes.
The reconstruction results are also reported in a detailed
area between the two obstacles in figure 4 to provide a clear insight into the fidelity of the reconstructions. It can be seen that
although some small-scale features are lost, all three CNN-AE
models are able to preserve the dominant structures of the turbulent flow. However, POD can not reconstruct the turbulent
flow properly from 5 modes.
Next, we compare the independence of the latent variables
obtained from different methods. Orthogonality of the modes
is a useful property for flow analysis, reduced-order modelingand flow control. Moreover, motivating the orthogonality ofthe modes may lead to interpretability. Results are depicted infigure 5 as the absolute value of the correlation matrix R corresponding to the latent variables from the CNN-HAE, CNNAE, CNN-β VAE with d = 5, and also POD with 5 modes. It

Figure 4. Reconstruction of the fluctuating component of thestreamwise velocity obtained from different methods, as indicated on each panel, in comparison with the reference data forthe zoomed-in area between the two obstacles marked by the
red rectangle.

can be seen that although the CNN-HAE model extract modesin the order of their contribution in the reconstruction, the latent variables are correlated leading to the lowest value for
detR among all methods. As mentioned above, it is possible
to motivate the disentanglement or independence of the latent variables using CNN-β VAEs and obtain near-orthogonalmodes. It can be observed that the correlation between thelatent variables is reduced for the CNN-β VAE in comparison
to that of the CNN-AE, where detR is equal to 99.20 for the
CNN-β VAE method and 87.59 for the CNN-AE technique.

Figure 5. Correlation matrix R for the latent variables obtained from different models as indicated on the panels. The
value in brackets indicates the corresponding detR for each
case.

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
RANKING THE CNN-β VAE MODES
mode 5 mode 4 mode 3 mode 2 mode 1

POD modes are sorted in terms of their energy content.
This property is extremely useful for understanding and analyzing the dominant patterns in complex flows. Fukami et al.
(2020) implemented hierarchical autoencoders to extract AEbased modes in the order of their contribution in the reconstruction, which requires training multiple NNs and might
be cumbersome especially for the extraction of higher-ordermodes. Here, we propose a strategy for ranking the CNNβ VAE modes. We showed that CNN-β VAEs are able to extract near-orthogonal and parsimonious modes from turbulent
flows. These properties allow us to rank these modes after the
training process and based on their contribution to the reconstruction. In particular, we rank CNN-β VAE modes based on
the maximum Eu that can be obtained from q modes, whereq represents the rank. To this end, after the training process,we first use the encoder to map high-dimensional data to thelatent vector E : x 7→ r . We zero out all the latent variables
except the ith variable, which leads to a latent vector r̂r i . Then,we employ the decoder part of the CNN-β VAE to send thislatent vector to the original space D : r̂r i 7→ x̃xi . This procedure
is performed for all the time steps. The energy percentage thatis captured by only considering the ith mode is evaluated as
Eki . The first mode is selected as the mode leading to the maximum value of Eki . For the second mode, we perform the sameprocedure while we preserve the first mode and look for themode, which in combination with the first mode, leads to the
maximum value of Eui . In a similar way, the third mode isselected as the mode which gives the maximum Eui in combination with the first and second modes. We continue thisprocedure to rank all the modes. Figure 6 illustrates the ranked
modes obtained from a CNN-β VAE model with d = 5 andβ = 10−3 , together with the modes obtained from the POD,
CNN-AE, and CNN-HAE methods with d = 5. The POD and
the CNN-HAE modes are already ranked and we also performthe ordering procedure for the CNN-AE modes. A clear resemblance can be observed between the first two modes of the
CNN-β VAE, see figure 6(a), and those from POD, as shown infigure 6(b), indicating the ability of CNN-β VAEs in the extraction of interpretable modes from turbulent flows. These modes
correspond to the large-scale vortex shedding from around the
obstacles into the wake region. Moreover, these results showthat using the ranking procedure it is possible to sort the CNNβ VAE modes based on their importance for reconstruction. It
also can be seen in figure 6(c) that it is extremely difficult torelate CNN-AE modes to physical processes, a fact that is referred to as the lack of interpretability. We observe that although the CNN-HAE model is able to extract large-scale features first, the obtained modes may not be physically interpretable, as shown in figure 6(d).

Figure 6. The ranked spatial modes obtained from the CNNβ VAE (a), POD (b), CNN-AE (c), and the CNN-HAE (d). The
size of the latent vector d is equal to 5.

bution to the reconstruction. We applied the proposed CNNβ VAE architecture for modal decomposition of the turbulent
flow through a simplified urban environment. Furthermore,
we compared the performance of the CNN-β VEs in terms of
the quality of the reconstructions and orthogonality of the extracted modes with that of the CNN-AEs and CNN-HAEs. Our
results from modal decomposition using POD indicates that
247 modes are required to obtain 99% of the energy from thereconstruction. This indicates that it is challenging to represent turbulent flows as a linear superposition of a few POD
modes. Our proposed CNN-β VAE model with a latent vector
size d = 5 and a penalization factor of β = 10−3 leads to Eu of
87.36% against 32.41% obtained from POD, which shows theexcellent performance of the CNN-β VAE in the reconstruction of the turbulent flow from only five modes. This model
also leads to near-orthogonal modes, where detR is equal to99.20. We showed that by constraining the shape of the latent space and motivating orthogonality of the modes, we canextract meaningful non-linear features where the first mode ofthis CNN-β VAE model represents the large-scale vortex shedding from around the obstacles into the wake. Our comparison
between the CNN-β VAE, CNN-HAE, and CNN-AE models
indicates that although motivating orthogonality of the modes
decreases the reconstruction accuracy, very good reconstructions can be obtained from five modes using the CNN-β VAEleading to Eu of 87.36% against 93.93% and 91.84% of the
CNN-AE and the CNN-HAE, respectively. The CNN-β VAE
model leads to a set of near-orthogonal modes with the highest
detR among the AE-based models.
The proposed CNN-β VAE architecture can be extended
in future works for the development of reduced-order surrogate models or advanced flow-control methods, among others.
In particular, the proposed method can be employed for nonlinear modal decomposition of the data obtained from thermalimaging cameras in complex urban environments, with extensive application in the context of urban air-quality forecast andcontrol.

SUMMARY AND CONCLUSIONSIn this study, we propose a probabilistic deep-neuralnetwork architecture based on β -VAEs and CNNs for nonlinear mode decomposition of turbulent flows. The objective
is to learn a compact, near-orthogonal and parsimonious latent
representation of high-dimensional data by introducing nonlinearity in the process of dimension reduction and also minimizing the correlation between the latent variables, as wellas penalizing the size of the latent vector. This may lead toa set of interpretable modes useful for flow analysis, reducedorder modeling and flow control. Since the correlations among
the learned latent variables are minimized, we proposed an algorithm to rank the VAE-based modes based on their contri-

ACKNOWLEDGMENTS
We acknowledge Álvaro Martı́nez for his contributions tothis work. RV acknowledges the Göran Gustafsson foundation
for the financial support of this research. SH has been supported by project RTI2018-102256-B-I00 of Mineco/FEDER.

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Osaka, Japan, July 19–22, 2022
based learning applied to document recognition. Proc. IEEE
86 (11), 2278–2324.
Lee, Sangseung & You, Donghyun 2019 Data-driven prediction of unsteady flow over a circular cylinder using deeplearning. J. Fluid Mech. 879, 217–254.
Locatello, Francesco, Bauer, Stefan, Lucic, Mario, Rätsch,
Gunnar, Gelly, Sylvain, Schölkopf, Bernhard & Bachem,
Olivier 2019 Challenging common assumptions in the unsupervised learning of disentangled representations.
Lumley, J L 1967 The structure of inhomogeneous turbulent
flows. Atmospheric Turbulence and Radio Wave Propogation. A. M. Yaglom and V. I. Tatarski (eds). Nauka, Moscow
pp. 166–78.
Maulik, Romit, Fukami, Kai, Ramachandra, Nesar, Fukagata,
Koji & Taira, Kunihiko 2020 Probabilistic neural networks
for fluid flow surrogate modeling and data recovery. Phys.
Rev. Fluids 5, 104401.
Milano, Michele & Koumoutsakos, Petros 2002 Neural network modeling for near wall turbulent flow. J. Comput.
Phys. 182 (1), 1–26.
Murata, Takaaki, Fukami, Kai & Fukagata, Koji 2020 Nonlinear mode decomposition with convolutional neural networks for fluid dynamics. J. Fluid Mech. 882, A13.
Rowley, C. W. 2005 Model reduction for fluids using balanced
proper orthogonal decomposition. Int. J. Bifurcation Chaos
15 (03), 997–1013.
Rowley, Clarence W., Mezić, Igor, Bagheri, Shervin, Schlatter,
Philipp & Henningson, Dan S. 2009 Spectral analysis ofnonlinear flows. J. Fluid Mech. 641, 115–127.
Saegusa, Ryo, Sakano, Hitoshi & Hashimoto, Shuji 2004 Nonlinear principal component analysis to preserve the order ofprincipal components. Neurocomputing 61, 57–70.
Schmid, Peter J. 2010 Dynamic mode decomposition of numerical and experimental data. J. Fluid Mech. 656 (November), 5–28.
Taira, Kunihiko, Brunton, Steven L., Dawson, Scott T. M.,
Rowley, Clarence W., Colonius, Tim, McKeon, Beverley J.,
Schmidt, Oliver T., Gordeyev, Stanislav, Theofilis, Vassilios& Ukeiley, Lawrence S. 2017 Modal analysis of fluid flows:
An overview. AIAA J. 55 (12), 4013–4041.
Taira, Kunihiko, Hemati, Maziar S., Brunton, Steven L., Sun,
Yiyang, Duraisamy, Karthik, Bagheri, Shervin, Dawson,
Scott T. M. & Yeh, Chi-An 2020 Modal analysis of fluid
flows: Applications and outlook. AIAA J. 58 (3), 998–1022.
Towne, Aaron, Schmidt, Oliver T. & Colonius, Tim 2018
Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis. J. Fluid Mech. 847, 821–867.
Vinuesa, Ricardo, Schlatter, Philipp, Malm, Johan, Mavriplis,
Catherine & Henningson, Dan S. 2015 Direct numericalsimulation of the flow around a wall-mounted square cylinder under various inflow conditions. J. Turbul. 16 (6), 555–
587.
Vinuesa, Ricardo & Sirmacek, Beril 2021 Interpretable deeplearning models to help achieve the sustainable development goals. Nat. Mach. Intell. 3 (11), 926–926.

SLC acknowledges the support of the Spanish Ministry of Science and Innovation under the grant PID2020-114173RB-100.

REFERENCES
Achille, Alessandro & Soatto, Stefano 2018 Information
dropout: Learning optimal representations through noisy
computation. IEEE Trans. Pattern Anal. Mach. Intell.
40 (12), 2897–2905.
Bengio, Yoshua, Courville, Aaron & Vincent, Pascal 2013
Representation learning: A review and new perspectives.
IEEE Trans. Pattern Anal. Mach. Intell. 35 (8), 1798–1828.
Burgess, Christopher P., Higgins, Irina, Pal, Arka, Matthey,
Loic, Watters, Nick, Desjardins, Guillaume & Lerchner,
Alexander 2018 Understanding disentangling in β -VAE.
Clainche, Soledad Le & Vega, José M. 2018 Spatio-temporal
Koopman decomposition. J. Nonlinear Sci. 28 (5), 1793–
1842.
Eivazi, Hamidreza, Veisi, Hadi, Naderi, Mohammad Hossein& Esfahanian, Vahid 2020 Deep neural networks for nonlinear model order reduction of unsteady flows. Phys. Fluids
32 (10), 105104.
Fukami, Kai, Fukagata, Koji & Taira, Kunihiko 2019 Superresolution reconstruction of turbulent flows with machinelearning. J. Fluid Mech. 870, 106–120.
Fukami, Kai, Nakamura, Taichi & Fukagata, Koji 2020 Convolutional neural network based hierarchical autoencoder
for nonlinear mode decomposition of fluid field data. Phys.
Fluids 32 (9), 095110.
Guastoni, Luca, Güemes, Alejandro, Ianiro, Andrea, Discetti, Stefano, Schlatter, Philipp, Azizpour, Hossein & Vinuesa, Ricardo 2021 Convolutional-network models to predict wall-bounded turbulence from wall quantities. J. Fluid
Mech. 928, A27.
Higgins, I., Matthey, Loı̈c, Pal, A., Burgess, Christopher P.,
Glorot, Xavier, Botvinick, M., Mohamed, S. & Lerchner,
Alexander 2017 beta-VAE: Learning basic visual concepts
with a constrained variational framework. In International
Conference on Learning Representations (ICLR).
Hinton, G. E. & Salakhutdinov, R. R. 2006 Reducing
the dimensionality of data with neural networks. Science
313 (5786), 504–507.
Kim, Hyojin, Kim, Junhyuk, Won, Sungjin & Lee, Changhoon
2021 Unsupervised deep learning for super-resolution reconstruction of turbulence. J. Fluid Mech. 910, A29.
Kim, Junhyuk & Lee, Changhoon 2020 Prediction of turbulent
heat transfer using convolutional neural networks. J. Fluid
Mech. 882, A18.
Kingma, Diederik P. & Ba, Jimmy 2017 Adam: A method forstochastic optimization.
Kingma, Diederik P & Welling, Max 2014 Auto-encoding
variational Bayes.
Le Clainche, Soledad & Vega, José M. 2017 Higher order
dynamic mode decomposition. SIAM J. Appl. Dyn. Syst.
16 (2), 882–925.
Lecun, Y., Bottou, L., Bengio, Y. & Haffner, P. 1998 Gradient-6