Learning the nonlinear manifold ofextreme aerodynamics

Kai Fukami and Kunihiko Taira∗
Department of Mechanical and Aerospace Engineering,
University of California, Los Angeles
Corresponding author: ktaira@seas.ucla.edu

CL(e): |uθ max|/u∞= 0.5: |uθ max|/u∞= 0.7: |uθ max|/u∞= 0.9

α = 60°α = 20°α = 30°α = 40°α = 50°α = 50°CL

α = 60°Disturbed airfoil wakes
(d)

(a)

{uθ max/u∞, y0/c}={-0.5,-0.3} {uθ max/u∞, y0/c}={0.9,-0.3}

α = 40°(f){uθ max/u∞, y0/c}={0.9,0.1} (c)t(e){uθ max/u∞, y0/c}={-0.9,-0.3} {uθ max/u∞, y0/c}={0.7,-0.3} {uθ max/u∞, y0/c}={-0.9,0.1}

CL(c)α = 30°(b)(f)

Identified
manifold

t

γ1…

…

…

…

γ6

t

…

4…
…

α = 20°
2…

1.6
1.4
1.2
1.0
0.8
0.6
0.4
0(d)

(b)

(a)

CL
γ3

γ2

Figure 1: Physics-embedded autoencoder-based manifold identification of extreme aerodynamics.

Abstract
With the increased occurrence of extreme events and miniaturization of aircraft,it has become an urgent task to understand aerodynamics in highly turbulent
flight environments. We propose a physics-embedded autoencoder to discover alow-dimensional compact manifold representation of extreme aerodynamics. The
present method is demonstrated with the highly nonlinear dynamics of vortex gustairfoil wake interaction around a NACA0012 airfoil over a range of configurations.
The present model extracts key features of the high-dimensional airfoil wake
dynamics on a physically interpretable and compact manifold, covering a massivenumber of wake scenarios across a huge parameter space that determines thecharacteristics of complex gusty flow conditions. Our data-driven approach offers
a new avenue for expressing the seemingly high-dimensional fluid flow systems
by identifying the low-dimensional data coordinates that can also be leveraged fordata compression and flow control.

36th Conference on Neural Information Processing Systems (NeurIPS 2022).

1Introduction

Global warming fueled by climate change is contributing to extreme weather conditions, which makes
flight difficult for drones, helicopters, and airplanes. It is now critically important to understandand control such highly unsteady and chaotic conditions to fly safely and efficiently. Traditionally,
these flight conditions were rarely studied. However, enormously large combinations of parameters
govern the extreme weather conditions associated with flow disturbances and gusts. Moreover, theseparameters nonlinearly affect the overall aerodynamics [1]. These factors make analyzing unsteadyflows incredibly challenging using existing theoretical techniques, calling for data-driven approaches.
This study aims to capture the key dynamics of complex airfoil wake-vortex gust interaction. Transitioning to examine fluid flow data in high-dimensional space-time coordinates to low-dimensional
phase-based space enables us to achieve this objective. We leverage a nonlinear autoencoder toidentify a compact manifold of high-dimensional airfoil wakes. We demonstrate that a nonlinear
autoencoder becomes a powerful tool to obtain a physically-interpretable manifold for rich dynamics
of transient airfoil wakes by incorporating the lift response into the low-dimensional latent spacedesign. The present manifold offers a diverse possibility of extensions including reduced-ordermodeling, data compression, and flow control in complex aerodynamics.

2Methods

2.1

Example: a family of wakes around a NACA0012 airfoil

This study finds a universal manifold that describes a broad range of complex wake behaviors ina low-dimensional space. We consider a variety of wake patterns generated around a NACA0012
airfoil at a chord-based Reynolds number Rec = 100 and a Mach number M∞ = 0.1, as presented in figure 1. In addition to the steady and periodic shedding cases at the angles of attackof α ∈ [20◦ , 30◦ , 40◦ , 50◦ , 60◦ ], the present data set from the numerical simulation also includes
complex transient wakes associated with strong vortex disturbance interacting with the airfoil atα ∈ [20◦ , 30◦ , 40◦ ]. These complex wakes also provide a diversity of lift responses.
For the cases with vortical gust, single disturbance modeled by a compressible Taylor vortex isinitially added to the flows, enabling us to emulate the practical aerodynamics scenario when a gusttraverses through an airfoil during flight. The vortex disturbance is parameterized with its radius R
and max vortex velocity uθmax . We set the radius R/c to be 0.25. At the initial condition, the vortex
is placed at (x0 , y0 )/c while x0 being set at −2.
For the cases without any disturbances, the wake at α = 20◦ is steady with no vortex shedding,while the wakes at α ≥ 30◦ exhibit unsteady periodic shedding. In contrast, the cases with thedisturbance show the different types of nonlinear and intrinsic responses depending on the max vortex
velocity uθmax and the vertical position y0 of the vortex [2]. We consider disturbed flows with aparameter combination of uθmax /u∞ ∈ [−0.9, −0.7, −0.5, 0.5, 0.7, 0.9] and y0 /c ∈ [−0.3, 0.1] atα ∈ [20◦ , 30◦ , 40◦ ]. Although the flows considered in the present study are laminar, these wakes
involve strong nonlinearities associated with the interaction among a disturbance, an airfoil, and thewake. As a result, a variety of wake patterns are generated, as presented in figure 1.
The present wake data set includes an extremely large spatiotemporal degree of freedom corresponding
to Nx × Ny × Nt × Nc , where Nx and Ny are respectively the number of the grid points in thex and y directions, Nt is the number of the snapshots for each case, and Nc is the number of thecases considered in the present data sets. In the present study, we consider Nt = 1200 snapshots ofvorticity field ω on Nx × Ny = 240 × 120 grids for each of Nc = 50 cases, resulting in O(109 ). We
demonstrate that the present autoencoder successfully expresses such enormously high-dimensionaland rich nonlinear wake dynamics in a low-dimensional latent space while conventional linear modal
reduction techniques cannot distinguish these features in their traditionally identified low-dimensional
coordinates.
2.2

Physics-embedded autoencoder

Principal component analysis (PCA) has often been used to study wake patterns [3]. However, thestrong complexity of the present transient wakes associated with disturbance causes difficulty in2

PCA

Regular autoencoder

α = 60°α = 60°α = 50°a3

α = 20°Physics-embedded autoencoder

α = 40°α = 50°γ3

α = 20°γ1

α = 30°a2

α = 30°a1

γ2

α = 40°γ4γ2γ3Figure 2: Comparison of low-dimensional latent spaces with various compression techniques.
acquiring a low-dimensional manifold with such conventional linear techniques (we will also demonstrate it later). Hence, we consider nonlinear dimensionality reduction [4, 5] to low-dimensionalize
the present wake family. This study uses a physics-embedded autoencoder [6, 7] composed ofconvolutional neural network [8] and multi-layer perceptron [9] to optimally find a low-order andnonlinear representation of the high-dimensional wakes q. An autoencoder outputs the same dataas that given as the input through the bottleneck structure using nonlinear activation functions. The
lowest-dimensional vector γ (pink circles in figure 1) can be regarded as a low-dimensional representation of the original data q if the model successfully outputs the same data as the input. Following
our preliminary test, we set the number of latent variables to 4 and use the hyperbolic tangent function(tanh) as the activation function.
While efficient data compression can be achieved with a nonlinear autoencoder, we also aim tofind coordinates that explicitly express the disturbed dynamics about the baseline states (the wakeswithout disturbance). To find such an interpretable manifold, we enforce the low-dimensional latentvector γ to have a coherent relationship with the lift coefficient CL , as illustrated in figure 1. The lift
coefficient is an observable characteristic from sensors, having a physical relationship with angles ofattack. This physics embedding not only promotes the interpretability of the latent vectors but also
prevents a scenario in which the latent variables cannot distinguish the wakes with different angles ofattack. Our optimization technique also allows for a general representation of the manifold shapeoriented towards visualization or the design of flow control.
The optimization process of the present physics-embedded autoencoder is expressed asw∗ = argminw ||q − q̂||2 + β1 ||CL − CˆL ||2 + β2 ||f (γ)||2 ,(1)where w denotes the weights inside the autoencoder model, and β1 and β2 decide the weighting(balance) in the loss function among the reconstruction loss and the lift-based regularization, and themanifold identification. This study uses a paraboloid-based manifold, such that f (γ) = (γ1 + δγ1 ) −(γ22 + γ32 ), inspired by the traditional reduced-order modeling for transient wake dynamics [3]. Theoffset parameter δγ1 is needed to successfully represent the disturbing influence at α = 20◦ , this casewithout the disturbance is steady. We set {β1 , β2 , δγ1 } = {0.05, 1, 0.025}. The Adam optimizer [10]is used for updating the weights in training. The training of the autoencoder is performed with theNVIDIA Tesla V100 GPU.

3Results and discussion

Let us first compare a three-dimensional space composed of latent variables derived from PCA, aregular autoencoder (without the lift and manifold shape loss functions), and the present physicsembedded autoencoder, in figure 2. PCA models the flow fields without disturbances at α = 20◦ and30◦ into the different portions, suggesting that these two cases can be distinguished with the lineartechnique. However, the low-dimensional variables for the cases at α = 40◦ , 50◦ , and , 60◦ overlap
around the region of a1 > 0. In addition, it is also difficult to identify the influence of disturbancein the low-dimensional space, although we observe some non-interpretable fluctuations in the PCA3γ1Reference

Decodedγ2γ3Figure 3: Wake fields projected on the present manifold. Decoded flow fields are also shown.

space that may correspond to the disturbed wake snapshots. Hence, PCA cannot find a universal
mode to successfully classify these cases, especially in the presence of disturbed transient flows withstrong nonlinearities.
In contrast, a regular autoencoder with tanh nonlinear activations clearly distinguishes the cases in
the present wake data sets. While the cases at α = 40◦ , 50◦ , and, 60◦ are mapped into the different
portions in the latent space, the influence of the disturbance is expressed in the radius direction ofeach periodic orbit. The reconstruction L2 error of the regular autoencoder for all cases consideredherein is 0.0858, while that of the PCA is 0.456, clearly indicating that the latent variables from thenonlinear autoencoder are more informative compared to that from the linear PCA. However, since
there is no constraint for the data distribution in the latent space, the low-dimensional variables residein the latent space in an unorganized manner. For instance, there are no clear axes to express thedifference in angles of attack. This is because the latent distribution with this no-constraint setup
is just needed to be distinguishable among the cases, implying that the autoencoder attempts toefficiently and widely use area in the latent space rather than possessing its interpretability. Towards
a deeper understanding of the high-dimensional and complex wake dynamics and controlling thembased on the manifold, it is desirable to obtain a more compact manifold, which clearly represent thedisturbed dynamics against the baseline limit cycles, through the nonlinear autoencoder.
From this aspect, the present physics-embedded autoencoder provides a compact and understandable
low-dimensional representation of high-dimensional wake dynamics. The axis of γ1 corresponds
to the angle of attack. Moreover, the influence of the disturbance is represented on the manifold
surface, providing a similarity between disturbed wake fields and the undisturbed baseline flow fields,
as depicted in figure 3. This indicates that the present autoencoder-based manifold captures effective
angle of attacks of the disturbed wake cases from the vorticity snapshots and the lift coefficients.
The reconstruction L2 error of the present physics-embedded autoencoder is 0.0901, which is in aqualitative agreement of the decoded field with the reference simulation data as presented in figure 3.
In other words, the present manifold offers a low-dimensional expression (only 3 variables) of theoriginal key dynamics (O(109 )) while reducing its complexity.

4Concluding remarks

We proposed a physics-embedded autoencoder to optimally derive a compact representation ofhigh-dimensional and complex extreme aerodynamic wakes. The current method successfully offered4

a simple and informative representation of the original wake dynamics associated with vortex gustairfoil wake interaction. As demonstrated, the physics-embedded autoencoder can be used forcompressing high-dimensional fluid flow data sets into a few latent variables while keeping theiressential and key features of transient dynamics. Although not shown, we have also confirmed that thepresent autoencoder is robust against not only noisy field measurements but also untrained parameterconditions. We expect that the present parabolic-type manifold can be leveraged for flow control inextreme aerodynamic conditions by combining with energy-based control [11] with the modificationof angle of attack to mitigate the impact of the gusty disturbance on the airfoil.

Acknowledgements
This work was supported by the US Department of Defense Vannevar Bush Faculty Fellowship(N00014-22-1-2798) and the US Air Force Office of Scientific Research (FA9550-21-1-0178).

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