Grasping extreme aerodynamics on a low-dimensional manifold

Domain: earth_science
Published: 2023-10-14
Editorial outcome: minor_revision

Abstract
Modern air vehicles perform a wide range of operations, including transportation, defense, surveillance, and rescue. These aircraft can fly in calm conditions but avoid operations in gusty environments, encountered in urban canyons, over mountainous terrains, and in ship wakes. With extreme weather becoming ever more frequent due to global warming, it is anticipated that aircraft, especially those that are smaller in size, will encounter sizeable atmospheric disturbances and still be expected to achieve stable flight. However, there exists virtually no theoretical fluid-dynamic foundation to describe the influence of extreme vortical gusts on wings. To compound this difficulty, there is a large parameter space for gust-wing interactions. While such interactions are seemingly complex and different for each combination of gust parameters, we show that the fundamental physics behind extreme aerodynamics is far simpler and lower-rank than traditionally expected. We reveal that the nonlinear vortical flow field over time and parameter space can be compressed to only three variables with a lift-augmented autoencoder while holding the essence of the original high-dimensional physics. Extreme aerodynamic flows can be compressed through machine learning into a low-dimensional manifold, which can enable real-time sparse reconstruction, dynamical modeling, and control of extremely unsteady gusty flows. The present findings offer support for the stable flight of next-generation small air vehicles in atmosphere conditions traditionally considered unflyable.

Introduction
Since the Wright brothers accomplished the first human-powered and controlled flight in 1903, a wide variety of aircraft has been developed for transportation, defense, observation, search, and rescue missions. What makes these aircraft uniquely different from other modes of transportation is their ability to stay aloft by taking advantage of aerodynamics. To support the development of these aircraft, the field of aerodynamics has undergone tremendous growth over the past century. Despite its expansive theory, the current aerodynamics is generally based on steady (cruise) or quasi-steady flight conditions with linear analysis or its nonlinear extensions for small perturbations1–3.

We are now at an important transition point for aerodynamics. With novel materials and enhanced power plants/batteries becoming available over the past couple of decades, there have been tremendous efforts in developing and operating smaller size personalized air vehicles and unmanned air systems4–8. They can traverse unconventional terrain, including mountainous and urban environments9–11 that were traditionally avoided by conventional aircraft. As a matter of fact, flight demonstrations of such air vehicles in calm weather have taken place in recent years12. These new and amazing flying vehicle concepts will likely revolutionize air-based transportation5,13,14 and have already been realized for some cases15,16.

However, there are major challenges in operating small-scale aircraft under these complex airspace when adverse weather generates highly turbulent environments17–21. Moreover, the increased occurrence of extreme weather caused by global warming limits the operations of aircraft. Flying small-scale aircraft near large natural or manmade structures in adverse weather face additional complications as these vehicles need to navigate through severe turbulence comprised of gusts and vortical disturbances, as illustrated in Fig. 1. These disturbances come in a variety of forms22 and are far more disruptive than what present-day commercial aircraft experience in inclement weather23. These extreme aerodynamic environments can be characterized by a manifestation of a large number of strong vortices with different strengths, sizes, and orientations generated by the surrounding structures24. Such a flight environment has been off-limits due to the fact that there is virtually no available theory for extreme aerodynamic problems and to avoid possible loss of aircraft.

With infinite scenarios of large and strong atmospheric disturbances hitting a flying vehicle, we cannot only focus on a single cruise condition but must also consider a whole array of cases in which wings experience extreme aerodynamic disturbances. These disturbances are characterized by a variety of parameters, including the size, strength, orientation, position, and geometry of the disturbances, necessitating massive experimental and computational campaigns if approached naively. With a single extreme aerodynamic simulation already producing a very large amount of flow field data, extensive parametric sweeps lead to an enormous collection of aerodynamic flow data and calls for significant computational and experimental resources. These extreme aerodynamic flows exhibit rich nonlinear behavior over a range of spatiotemporal scales that cannot be easily analyzed and modeled with existing theories.

One of the important parameters under such extreme aerodynamic situations is the gust ratio G=u gust/u ∞, which is ratio between the characteristic gust velocity u gust and the translational velocity of the wing u ∞. For conditions of G≳1, sustaining stable flight becomes challenging18,25. In the present work, we consider high levels of aerodynamic disturbance with 0≤G≤10, and refer to cases of G>1 as extreme aerodynamics. Strong gusts with G>1 can be encountered in urban canyons, mountainous environments, and severe atmospheric turbulence. The goal of this study is to identify the unifying dynamics that a wing experiences from extreme gust disturbances. At the most fundamental level, the present problem requires the identification of the underlying nonlinear dynamics of the complex separated flows from an enormous amount of data in an efficient manner while gaining physical insights into extreme aerodynamics.

Although the aerodynamic influence of large-scale disturbances on lifting bodies take various forms, the underlying dynamics are generally shared. In this study, we seek these dominant dynamics embedded in complex extreme aerodynamic flows. To achieve this objective, we examine the reduction of massive fluid flow data into a low-dimensional space in which the right set of variables describe the underlying extreme aerodynamic physics. This process is enabled by incorporating physical observables and ensuring that the gained insights are interpretable and beneficial for future aircraft operations and designs. In fact, we find that extreme aerodynamic flows can be compressed by a carefully designed nonlinear machine-learning technique to only three variables for a model problem of a strong vortex hitting a canonical airfoil. The present findings further suggest that the discovered manifold hold potential to support downstream tasks such as real-time flow estimation, dynamical modeling, flow control, and vehicle design.

Results
Extreme vortex–airfoil Interactions
We consider a strong vortex gust impacting an airfoil as a representative model problem for wings experiencing extreme atmospheric disturbances. The present model problem involves wake vortices shedding from a high-rise building, ships in rough seas, and mountain ridges18. The size of such strong vortices can be comparable to the size of the wing, exerting tremendously large lift and drag forces. In this study, we simulate two-dimensional incompressible flows with a vortex placed upstream of the wing with varied vortex size, strength, and initial position, as shown in Fig. 1. Because the airfoil wake responds differently for each combination of these disturbance settings, the resulting flow fields exhibit vastly different wake patterns and aerodynamic forces from case to case due to the nonlinear vortex dynamics, as presented in Fig. 2. As the vortex passes around an airfoil, the wing experiences massive flow separation (stall), which also causes the emergence of additional vortical structures. All of these flow structures interact nonlinearly, making the dynamics complex and difficult to predict and control.

This study considers two-dimensional extreme aerodynamic flows around a NACA 0012 airfoil at a chord-based Reynolds number Re=u ∞c/ν=100. Here, u ∞ is the free-stream velocity, c is the chord length, and ν is the kinematic viscosity. The flow field is obtained with direct numerical simulation using an incompressible flow solver26,27. The airfoil is positioned in the free stream with its leading edge at the origin with angles of attack of α∈[20°,60°], enabling us to cover cases of steady and unsteady wakes in undisturbed (baseline) cases. These wakes are disturbed with a gust vortex having an angular velocity profile28 of u θ=u θ,max (r/R)exp[1/2(1-r2/(2R2))], where the radius of the vortex is R.

The present disturbance vortex is characterized by the gust ratio G=u θ,max/u ∞ ∈[-10,10] with its size relative to the wing chord L≡2R/c∈[0.5,2] and is introduced upstream of the wing at x 0/c=-2 and y 0/c∈[-0.5,0.5]. The combinations of these parameters provide a wide range of large and strong gust vortices hitting the wing at various locations. The flow field around the airfoil exhibits a rich dynamical response to the disturbance vortex characterized by a large parameter space comprised of (α,G,L,y 0/c). To fully resolve the dynamics over this parameter space, a substantial number of flow cases for different combinations of these parameters would be required. In general, Re is also another parameter but is fixed for this study at 100. While the gust vortices contained in actual atmospheric turbulence can be much more complex than what is considered here, the primary dynamics of large vortex core interacting with the wing is captured well at this Re at least in a two-dimensional manner. What is particularly important is to resolve the large vorticity source from the surface under local flow acceleration29. These phenomena have relatively short time-scales compared to the much longer viscous scales associated with Re=100, making the current problem setup an appropriate testbed. In this study, we set the convective time to be zero when the center of the vortex arrives at the leading edge of the wing x 0/c=0. The snapshots of the vorticity field and the lift history are examined in detail with data-driven analyses in what follows.

Without any external disturbances, the wing experiences steady aerodynamic lift forces at low angles of attack (α≲20°) and moderate unsteady aerodynamic force fluctuations at higher angles of attack (α≳30°). These lift values are shown by the dashed lines for the different angles of attack in Fig. 2. The unsteady lift fluctuations are exerted by the von Karman vortices shedding from the leading and trailing edges, as visualized in Fig. 2. At this Reynolds number, these cases are periodic in time and constitute limit cycles.

When the wing encounters a strong gust vortex, the flow around the wing is significantly modified. The approaching vortex strongly influences the vorticity field around the wing and triggers large-scale vortex formation. For example, let us consider the flow field for a case shown in Fig. 2 for which a strong positive vortex with G=3.8 hits a wing at α=20°. Due to this disturbance, two large vortices are formed shortly after impact, generating massive separation due to the interaction of the gust vortex and the wing wake. These dramatic transient wake dynamics exert sharp increase in aerodynamic forces on the wing. In fact, the lift force increases 714% and drops 656% within a short duration of 1.8 convective time. Such a significant variation in the lift force makes controlling air vehicles tremendously difficult. While not shown, the moment experienced by the wing also undergoes a tremendous change. We also display a total of 100 force histories for cases of flows disturbed by extreme level of gust vortices in Fig. 2. In all of these cases, the airfoil wake dynamics undergo large transient changes within a short amount of time with large aerodynamic force fluctuations with similar order of magnitudes. These violent disturbances not only destabilize flight but can also damage vehicle structures, making the analysis of these flows critically important.

Because of the nonlinear nature of the dynamics, flow around the airfoil responds differently to each different gust vortex with massive flow separation and large-scale formation of additional vortices, as visualized in Fig. 2. There is no aerodynamic model or theory that can easily describe the highly nonlinear nature of the extreme gust-airfoil interactions. What is especially challenging is that there is no simple scaling that collapses the collection of lift curves or the vortical flow fields due to the strong nonlinearity. This is an enormous burden for studying extreme aerodynamic flows since each and every case needs to be examined through painstaking computational or experimental campaigns that require significant resources. We re-emphasize that even for the present problem setup, there are a good number of parameters (vortex strength G, size L, position y 0/c, and airfoil angle of attack α) that necessitate a very large number of cases of extreme aerodynamic flows to map out the response dynamics. Practically speaking, such a campaign may not be possible for all gust encounters with limited computational resources. Therefore, it is desirable to capture the underlying dominant dynamics that form the basis of extreme gust response characteristics without having to rely on expensive simulations with a very large degree of freedom, which in this case is proportional to 2.88×104 grid (spatial) points to describe the instantaneous flow field and 1.26×105 temporal frames for sufficient spatiotemporal solutions for all cases. While we do have the Navier–Stokes equations as the governing partial differential equations to fully describe the dynamics, solving them in real time for practical air vehicle operations is out of the question.

For the aforementioned reasons, it is important to extract the dominant low-dimensional dynamics from the possible collection of extreme aerodynamic datasets. The rich responses to different gust vortices appear uniquely different from one snapshot to another but possess some common and identifiable features, including the disturbance vortex, flow separation, wake vortices, secondary vortices, shear layers, vortex roll-up, vortex pinch-off, and vortex deformation. The fact that these features are indeed identifiable by the trained eyes of fluid dynamicists suggests that there is likely some underlying low-dimensional representation of the high-dimensional complex dynamics. Thus, we aim to capture the key dynamics in a space comprised of a very small number of variables that can estimate the full state of the flow field and offer insights into the nonlinear dynamics of the vortex-gust interaction. We find that a nonlinear autoencoder with physical observables illustrated in Fig. 3 achieves the present objective.

Identification of extreme aerodynamic manifold
To find a low-dimensional space that captures the essential physics of extreme aerodynamic interactions between the gust vortex and the airfoil wake, we perform data-driven compression of the flow field. Herein, we consider cases with randomly-sampled parameters from G∈[-4,4], L∈[0.5,2], and y 0/c∈[-0.5,0.5].

First, let us consider the most commonly used linear dimensionality reduction technique, namely the principal component analysis (PCA), which is also known as the proper orthogonal decomposition (POD)30–32. With this method, the low-dimensional representation of a fluctuating variable is found by identifying the primary modes (or vectors) that best capture the variance about the mean. In the present study, we first analyze the vorticity fields over (x,y)/c∈[-1.4,4]×[-1.2,1.2], shown in Fig. 2 by applying PCA to determine the most vortically energetic components of the flow field.

Shown in Fig. 4 are the temporal variations of the first three PCA components (ξ1(t), ξ2(t), and ξ3(t)) of the vorticity field data. The gray curves represent the whole collection of extreme aerodynamic data plotted in this coordinate space. Also highlighted are undisturbed baseline cases for α=20° to 60°. Here, we observe that the gray curves span a range of values over ξ1, ξ2, and ξ3 in a seemingly incoherent manner. What is further problematic with this compression is the overlap of the baseline cases. These observations reveal that PCA struggles to compress the extreme aerodynamic data in a meaningful manner while keeping different angles of attack cases distinct. Because different flows over different angles of attack are collapsed as the same, the overlapping low-dimensional representations produced by PCA cannot distinguish important flow characteristics and yield grossly inaccurate flow reconstructions, as shown in Fig. S.1.

The challenges of reducing degrees of freedom (dimension) of extreme vorticity dynamics by PCA can be mitigated by utilizing a nonlinear compression technique. For this purpose, we utilize a nonlinear convolutional autoencoder, presented in Fig. 3 (excluding the green-shaded side network), to reduce a large number of vorticity field data to very few variables. An autoencoder is a neural network composed of an encoder and a decoder with a bottleneck in the middle33–35. Generally, this neural network framework is used to take an input data and replicate the same data at the output. The variables that lie in the middle are referred to as latent variables (red circles in Fig. 3), which hold the compressed information about the input data. When the nonlinear autoencoder can replicate the same input data at the output, this means that both the encoder and the decoder function effectively to nonlinearly transform the full dataset to a low-dimensional latent variable ξ and vice versa effectively. The present autoencoder first compresses a flow field using a convolutional neural network (CNN)36 to capture global features of the vortical flow field. The compressed vector (extracted feature) through the CNN is then flattened at the reshape layer in Fig. 3 to pass into a multi-layer perceptron (MLP)37 towards the latent space. A similar operation is performed for the decoder side to expand the dimension of the latent variable back to the size of the original flow field. The present autoencoder is trained with the same datasets as that used for PCA. The details of the autoencoder setup used in the present study are provided in Supplemental Material.

The nonlinear autoencoder is able to compress the flow field data and reproduce the flow field accurately, as shown in Fig. S.1. We also present the latent space comprised of only three latent variables (ξ1, ξ2, and ξ3) in Fig. 4 (middle). The ability of the nonlinear autoencoder to compress the vorticity field to mere three variables is not only surprising but also reaffirms that the flow field is indeed comprised of common flow features. The compression capability offered by a nonlinear autoencoder is promising to capture violent flow physics that appear tremendously rich. Nonetheless, we should note that the full dataset shown in Fig. 4 is distributed over the latent space without a meaningful collapse of the latent variables. This is sufficient if the objective is to ensure that data in latent space are distinct, thus covering as much space as necessary to ensure uniqueness of the information. However, with regard to this study, we are aiming not only to compress the extreme aerodynamic flow data but also to identify universal features among the large number of flow field data holding dynamical information.

The above autoencoder analysis was performed purely from a data-centric perspective. Instead, let us consider incorporating a physical measurement (observable) into the autoencoder to facilitate the identification of a low-dimensional subspace defined by the appropriate latent variable coordinates. Capturing the low-dimensional nature of extreme aerodynamic flows can support the flight stabilization of air vehicles in extreme levels of turbulence. For this reason, we weight the latent space variables ξ with lift force acting on the wing. This is achieved by supplementing the autoencoder with a multi-layer perceptron that outputs the lift force for each vorticity field over time, as shown by the green-shaded network in Fig. 3. In this case, training is performed to compress the vorticity field to the latent variables and to estimate the lift force accurately from the latent variables.

Incorporating lift into the learning process of the autoencoder assists in effectively extracting the essence of extreme aerodynamics due to three main reasons. First, using the lift force with the vorticity input comes as a natural choice since vorticity is theoretically known as the main mechanism for generating lift on a body. We take advantage of this intimate relationship between the vorticity field and lift. Second, the latent variables are guided to retain important information held by the vorticity field correlated with the lift force. This means that vortical structures that apply large vortical forces are well-captured by the lift-augmented autoencoder. Because extreme aerodynamic disturbances exert enormous amounts of transient forces as presented in Fig. 2, the autoencoder weights these extreme event appropriately and is able to accurately identify the responsible vortical structures. Third, the latent variables are encouraged to distinguish cases that yield different lift responses to impinging gust vortices. This is crucial to avoid latent variables from overlapping unnecessarily as observed in the struggling cases of PCA.

Now, let us examine the compression results from the lift-augmented autoencoder. The latent space comprised of three variables ξ1, ξ2, and ξ3 is presented in Fig. 4. We observe that the entire collection of extreme aerodynamic cases collapses well into these latent space coordinates, confirming that the extreme aerodynamic responses to gust vortices possess a fundamentally low-dimensional behavior if captured appropriately with a nonlinear compression method. Here, the asymptotic periodic shedding states of the airfoil wakes provide shows a “cone” or a “chocolate cornet” like structure with the extreme aerodynamic trajectories lying in its vicinity in the three-dimensional latent space. As the dynamical trajectories converge to the cone-shaped structure, this structure serves as the inertial manifold38–40. This manifold geometry can be considered as an hour-glass shape since there is a mirrored manifold for negative angle of attack cases. It should be noted that the geometry of this structure is not specified a priori and is discovered in an unsupervised manner. Here, this surface constitutes a manifold on which the key dynamics of extreme gust response reside. That is, the trajectories of the presently considered extreme aerodynamic flows are mapped onto the discovered geometry or to its vicinity.

Given the lift-augmented autoencoder collapsing all extreme aerodynamic response data onto this cone-shaped manifold, let us examine the accuracy of state reconstruction for the disturbed flow and lift based on the three latent variables (ξ1, ξ2, ξ3). As representative examples, we present the performance of the autoencoder for the cases of (α,G,L,y 0/c)=(40°, -2.2, 0.5, 0.3) and (60°, -2.8, 1.5, 0), which are unused in training but chosen from the training parameter range. Here, the gust ratios G for these two cases are much higher than what are traditionally considered in gust response studies, but are within the training data range of |G|≤4. The latent variable trajectories for these cases and the reconstruction of lift and vortical flows are also shown in Fig. 5. To assess the reconstruction performance, we evaluate the structural similarity index (SSIM)41 between the reference and the decoded flow fields. The SSIM value for each decoded flow field is listed under the visualized flow reconstruction. Even the flow states exhibiting nonlinear interactions between the wing and the extreme vortex gust can be reconstructed well by the present autoencoder, as presented in Figs. 5 and S.1. Note that the structural similarity index for a regular autoencoder without lift being higher compared to that for the lift-augmented autoencoder is expected. This is because a regular autoencoder is able to tune its weights solely to obtain accurate reconstruction of the flow field from the latent variables. It is also possible to reconstruct lift solely from the vorticity field. However, the lift-augmented autoencoder is critical for revealing the manifold for extreme aerodynamic response dynamics. These successful reconstructions indicate that high-dimensional extreme aerodynamic flows can be compressed into only three variables without significant loss of key physics.

The trajectory in the present latent space for the disturbed cases reflects key features of nonlinear vortex-gust interaction appearing in the high-dimensional space. For the extreme aerodynamic case of α=40°, the latent vector first drops towards the direction of the undisturbed periodic orbit of α=30° then comes back to the original undisturbed orbit of α=40°. This is due to the approach of negative vortex disturbance to the wing, decreasing the effective angle of attack. This indicates that the lift-augmented autoencoder captures the relationship between high-dimensional extreme aerodynamic flows and lift force in the low-order space. In fact, the reduction in ξ3 toward the direction of α=30° in the latent space coincides with the temporal evolution of lift responses, as shown in Fig. 4. A similar trend is also observed in the case of α=60° in which the latent vector first heads to the direction of the periodic orbit of α=50° corresponding to the decrease of the lift response. Being able to capture the extreme aerodynamic response of the wing on this manifold enables us to relate the instantaneous dynamics to the effective angle of attack, which is critically important for the flight stability of air vehicles. It is particularly encouraging that only three values in latent space are required to accurately estimate the state of the violent flow around the wing and transient lift force, which is promising for future development of sensors.

The discovered manifold captures dynamics beyond the trained gust strength G. Let us demonstrate how the present autoencoder approach is able to capture even more severe vortex gust conditions with |G|≥4 by presenting two cases (α,G,L,y 0/c)=(40°, 6.4, 1.0, 0.1) and =(60°, -6.0, 2.0, -0.3), as shown in Fig. 5. The trajectory of these seemingly extrapolative cases exhibits a larger radius on the ξ1–ξ2 plane compared to the variables of the interpolation cases while also presenting the effective angle of attack as the latent vector moves in the ξ3 direction. The wider radial trajectory is due to the stronger disturbance, which produces a higher level of fluctuations in the vortical flow response likely away from the body without affecting lift. The present decoder can also recover the high-dimensional flow states while estimating the very large transient lift dynamics as presented in Fig. 5. This indicates that the present autoencoder can be robustly applied for such extreme gust vortex-airfoil interactions while achieving a nearly lossless compression of high-dimensional data. These findings also suggest that even under the extrapolating condition of |G|>4, the underlying vortex dynamics shares common physics and can be nonlinearly compressed to a low-dimensional and universal manifold. This provides hope in estimating the flow states for cases that were not part of the extreme aerodynamic training data.

To further examine the robustness of the identified manifold and the autoencoder, let us consider cases that are different from the training cases, namely flows with noise and two extreme vortices, as shown in Fig. 6. Here, the noisy flow field is generated by adding Gaussian noise that is 30% of the original extreme aerodynamic flow field (same as the cases shown in Fig. 5(b)). The present lift-augmented autoencoder not only reconstructs a vortical flow but also estimates the lift response well from a noisy flow. The autoencoder noise rejection characteristics is beneficial in obtaining real-time situational awareness and working with turbulent flows in which less influential smaller-scale structures may also be present around the extreme gust vortices.

We also consider cases of vortex-dominated gust flows that are challenging for most reconstruction techniques trained only with single-gust disturbances. Here, we take two vortices that are introduced vertically and horizontally upstream of the airfoil, as depicted in Fig. 6. In both cases, the dynamical lift responses can be accurately estimated while the decoder reproduces the presence of two vortex disturbances very well, as shown in Figs. 6 and S.2. We can also notice from the latent space that the trajectory of the two-horizontal-vortex case presents two inflections at ξ2 ≈ 0. This coincides with the observation in the lift dynamics which possesses two valleys due to the impingement of two negative vortices. We note that these particular examples are difficult for linear techniques, including PCA which completely fails to reconstruct the flow field as shown in Fig. S.2.

Finally, let us demonstrate the potential of the present lift-augmented autoencoder for handling a more challenging and realistic extreme flight condition. In this last example, we introduce randomly generated five strong vortices upstream of the wing to simulate severe wake turbulence striking the airfoil, as shown in Fig. 7 (top left). The decoded lift, reconstructed flow fields, and latent trajectory are presented in Fig. 7. Even under this extreme operating condition, the present autoencoder robustly provides accurate reconstruction of the flow variables despite the model being trained only with single-gust disturbances. This success in flow compression/reconstruction and lift estimation corroborates that the discovered low-dimensional manifold universally capture the extreme gust vortex-airfoil interaction dynamics. This discovery provides great hope in establishing flight in extreme gusts, which was traditionally considered impossible.

Methods
Simulations of extreme vortex–airfoil interactions
The present study considers the unsteady flow field generated by the extreme gust vortex-airfoil interactions. The flow fields examined in this study are obtained from direct numerical simulations of flows over a NACA 0012 airfoil at a chord-based Reynolds number Re≡u ∞c/ν=100. Here, u ∞ is the free-stream velocity, c is the chord length, and ν is the kinematic viscosity. The simulations are performed with an incompressible flow solver26,27 for an airfoil at six different angles of attack of α=20°, 30°, 40°, 50°, and 60°. For the undisturbed cases, the flow at α=20° is steady while that at α≥30° exhibits unsteady periodic wake shedding. The computational domain extends over (x,y)/c∈[-15,30]×[-20,20] with the leading edge of the wing positioned at the origin.

To study the gust-airfoil interactions, a very strong vortex with an angular velocity profile prescribed by equation28 is introduced upstream of the airfoil at x 0/c=-2 and y 0/c∈[-0.5,0.5]. The present disturbance vortex is parameterized by the gust ratio G≡u θ,max/u ∞ ∈[-10,10], its size L≡2R/c∈[0.5,2], and the vertical position of the disturbance y 0/c. From the parameter space composed of these three variables, 40 randomly-sampled cases of the disturbed flows are simulated for each angle of attack. For the purpose of learning the extreme aerodynamics with the autoencoder, 20 cases are used for training and the remaining 20 cases are used for testing. The simulated flows were validated with previous studies45,51–53, in particular with a study that considered a vortex-airfoil interaction problem45.

For each of the cases considered in the present study, we prepare 1200 snapshots of vorticity field over 10.2 non-dimensional convective time t*≡u ∞ t/c. We refer to this convective time as simply ‘time’ in the main text. Of the entire flow field, a subdomain (x,y)/c∈[-1.4,4]×[-1.2,1.2] with spatial grid points (N x,N y)=(240,120) is considered for the data-driven analysis since vortex-airfoil interactions primarily occur in this region. Moreover, the history of the lift fluctuations is provided by the numerical simulations. The non-dimensional lift coefficient C L ≡F lift/(1/2ρu ∞2c), where F lift is the lift force on the wing body and ρ is the density. In the main text, C L is referred to as ‘lift.’ Overall, the training data used for the present models amounts to 1.26×105 frames comprised of 100 extreme aerodynamic gust response cases and 5 undisturbed wake cases with 1200 snapshots for each case.

Autoencoder setup
To discover the universal nonlinear manifold that represents the high-dimensional extreme aerodynamic flows in a low-dimensional latent space, we use an autoencoder33 (see Fig. 3). Here, we consider a convolutional neural-network-based autoencoder F, which is trained to output q^ to be the same data as the input q ∈ R n such that q^ ≈ F(q;w), where w denotes the weights inside the autoencoder. This autoencoder is comprised of an encoder F e and a decoder F d connected through a low-dimensional variable ξ ∈ R m in the middle, where m≪n. Here, the high-dimensional input q can be compressed into the latent vector ξ if the autoencoder F successfully recovers the data accurately. That is, we seek to have an autoencoder that achieves
q≈q^=F(q;w)=F d(ξ)=F d(F e(q)). (1)
The autoencoder F is found based on data such that its weights w are optimized to minimize a desired cost (loss) function E, yielding the following optimization problem
w=argminw [E(q,F(q;w))]=argminw ||q-q^||2. (2)
The weights w are determined with the Adam optimizer54.

We use an autoencoder composed of convolutional neural networks (CNN)36 and multi-layer perceptrons (MLP)37, as illustrated in Fig. 3. In the encoder, the CNN captures global features of the extreme aerodynamic flow field and the MLP is used to extract features from the CNN while further reducing the size of the data. By leveraging nonlinear activation functions, an autoencoder can achieve better compression than linear compression techniques such as principal component analysis (PCA)33. Note that using autoencoder with linear activation functions is mathematically equivalent to performing principal component analysis (PCA)33,34. As for the nonlinear activation function, we use the hyperbolic tangent function φ(s)=(es-e-s)/(es+e-s), enabling us to consider the positive and the negative gust influence in latent space. The hyperparameters used in the MLP and CNN follow previous work with similar settings34,55. Full details on the parameters of the present convolutional nonlinear autoencoder are shown in table S.1.

In addition to PCA and a regular autoencoder, we develop a lift-augmented convolutional autoencoder in this study. The present lift-augmented autoencoder trains the model with a lift coefficient C L(t) in addition to a vorticity field q(t) such that [q^(t), C^ L(t)]=F(q(t)). The additional side network based on an MLP is illustrated in the green-shaded portion of Fig. 3. This additional network ensures that the latent vector ξ(t) holds relevant information related to the lift coefficient C L(t) to support the manifold identification. The cost function in this case becomes
w*=argminw ||q-q^||2 + β||C L-C^ L||2, (3)
where β balances the vorticity field and lift reconstruction losses. In this study, we choose β=0.05 based on the L-curve analysis56. With the lift decoder F L, the reconstructed lift coefficient C^ L(t) is given by
C^ L(t)=F L(ξ(t))=F L(F e(q(t))). (4)

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "Main comments:\nThis work focuses on modeling extreme unsteady gusty flows, specifically addressing the\nchallenges associated with modeling extreme disturbance vortexes due to their strong nonlinearity.\nDifferent from traditional modeling methods, the paper proposes a reduction modeling approach\nusing a nonlinear autoencoder. It reduces the temporal frames into a latent space with 3\ndimensions. Surprisingly, despite the low dimensionality, this latent space is capable of effectively\nreconstructing data under various conditions such as different airfoil angles and measurement\nnoise. Moreover, it demonstrates generalizability under complex scenarios involving multiple\nvortices. In general, this paper provides a new perspective for complex system modeling, and\ncould contribute to the control of air vehicles under this type of extreme environment.\nThis work is expected to have a certain level of significance to the field. Autoencoders is designed\nfor reduction modeling for various complex (especially nonlinear) systems. The reviewer believes\nthe methodology provided in this paper is sound. Autoencoders are generally possible to perform\nreduced-order modeling (proven in many tasks), as demonstrated in [1]. By tackling the modeling\nof extreme unsteady gusty flows with autoencoders, this paper offers valuable insights that could\nbenefit researchers and practitioners working in the field of fluid dynamics and aerodynamics. This\nwork is likely to be of interest to a wide range of researchers.\nIn terms of the modeling/control of UAVs under extreme gusty flows, this work has the potential to\nopen up the field of extreme gusty flow control. One of the baseline deep learning model is similar\nto prior works like [2] (cited in draft). Part of this work further extends the study of [2] to extreme\ngust vortices.\nAll the experiments are well demonstrated with details to support the claims. The analysis and\ninterpretation is clean. The experiments are thoughtfully designed, covering various interesting\naspects of the problem at hand.\n------------------------------------\nIn the following, the reviewer aims to point out certain concerns and minor flaws identified in the\npaper. The reviewer intends to address these questions to the authors in order to seek further\nclarification.\ni) [Question] The experimental study is only demonstrated in 2D cases. Would the authors plan to\nbuild more experiments in 3D for future work? How would this expect to cost computationally?\nii) [Suggestion] This paper is mostly using simulation data. What would be the difficulty/gap when\none tries to apply this method to real data? Would it cost the required latent dimension to be more\nthan three? Moreover, this paper uniformly suggests a model with latent dimension=3. It might be\nbetter to write a few lines to discuss the reason.\niii) [Discussion] AE with lift is acceptable. Strictly speaking, AE with lift should be compared to AE\nwithout lift using latent dimension=4 (3 original latent dimensions + 1 lift). This would create a\nmore cohesive modeling approach. However, the reviewer acknowledges that such a modeling\nmight pose challenges for interpretation. The reviewer would appreciate some further discussion\nfrom the authors regarding this particular point.\nMinor points:\ni) [Question] For the reconstruction percentage in Fig. 5, 6, 7, do they represent the\nreconstruction rate of this specific snapshot, or is it averaged over all the snapshots?\nii) [Discussion] In Fig. 8, the performance of AE with lift slightly underperforms AE without lift.\nMaybe some very 'advanced' hyper-parameter tuning could help to fix the problem?\niii) In abstract, there should be a 'will/would' in front of the first 'encounter'.\n[1] Chen, Boyuan, et al. \"Automated discovery of fundamental variables hidden in experimental\ndata.\" Nature Computational Science 2.7 (2022): 433-442.\n[2] Omata, Noriyasu, and Susumu Shirayama. \"A novel method of low-dimensional representation\nfor temporal behavior of flow fields using deep autoencoder.\" Aip Advances 9.1 (2019): 015006.", "reviewer_2": "The manuscript entitled “Grasping Extreme Aerodynamics on a Low-Dimensional Manifold” by Kai\nFukami and Kunihiko Taira presents a data-driven approach to identify a low-dimensional manifold\non which the key dynamics between strong gust vortices and airfoil wakes can be collapsed. As\nalso stated by the authors, the interaction between the vortical gusts and wings is seemingly\ncomplex and different for each combination of gust parameters, however they achieved to discover\na manifold with an autoencoder designed to retain the knowledge of aerodynamic lift as part of the\nlatent variables and showed that the fundamental physics behind extreme aerodynamics is far\nsimpler and low-rank than traditionally expected. It is a very interesting, significant work, a\nvaluable research presented in a clear and complete form. Therefore, I would recommend that the\npaper be published as is. However, I would like to point out a couple of concerns for the\nperfectness of the manuscript.\nIf I understand correctly, ξ3 is related to the effective angle of attack and the width\nof the radial trajectory on the ξ1 and ξ2 plane (combination of\nξ1 and ξ2) shows the strength of the disturbance. I assume the\neffective angle of attack will depend on all of the parameters of the study, namely α, G, L, and\ny0/c. On the other hand the strength of the disturbance or its consequence in terms\nof lift force depends on all of the parameters of the study as well. Therefore, I expect the latent\nvariables are independent and represents a physical and probably a non-dimensional value made\nof a combination of those parameters. It would be very insightful if the authors could also\ncomment on how physically the latent variables are related to the parameters of the study.\nAccording to the results presented in Figure S1, structural similarity indices for AE with lift is a bit\nsmaller than those for AE w/o lift. A brief expanation for the reason behind will be helpful to\nunderstand the approach and the flow physics. Having noticed this slight difference, I wonder if\nthe authors obtained the lift from the reconstructed flow field data with the regular autoencoder\nand compared with that obtained with lift-augmented autoencoder. And finally, I believe a\nmeasure for the accuracy of the lift may support the success of the discovered manifold.\nIn terms of the structure of the manuscript, I would like to point out only that in the Suplemental\nMaterial, the references go up to #35 while in the reference list of the main text, the final\nreferefence is numbered as #30.", "reviewer_3": "The paper presents findings that offer a new perspective on modeling gusts using machine\nlearning. It has interesting results and utilizes multi-disciplinary research to achieve them. I\nbelieve the scoping of the paper needs further work and the big claims made need to be toned\ndown a little. The paper will benefit focusing on the ability to reduce the problem to 3 parameters.\nThe abstract and introduction don’t introduce the content of the paper properly or clearly. For\nexample, the application of the work is explained at end of paper whereby the low dimensional\nrepresentation can help the use of sensors on an aircraft. I think this needs to be highlighted at\nthe start of the paper for context. But also note that there has already been work on the use of ML\nto predict stall over a 3D wing by Shane Windsor in University of Bristol and others. Discussing this\nearly will set the scene and enable the reader to follow the vision of why this work is important\nand how it can be used to improve safety. There are some bold claims which I don’t agree with\nentirely such as “However, there exists virtually no foundation to describe the influence of extreme\nvortical gusts on flying bodies”. I think the foundation through CFD simulation and experimental\nwork does exist and the review papers by Anya Jones list them. There has even been attempts to\nexplore worst case gusts in urban environments by simulating flight through shear layers which\nare far more “extreme” than the case being considered in this paper:\n• Mohamed, A.; Marino, M.; Watkins, S.; Jaworski, J.; Jones, A. Gusts Encountered by Flying\nVehicles in Proximity to Buildings. Drones 2023, 7, 22. https://doi.org/10.3390/drones7010022\nThis gust has even been characterized as extreme in this publication:\n• Colin M. Stutz, John T. Hrynuk, Douglas G. Bohl, Dimensional analysis of a transverse gust\nencounter, Aerospace Science and Technology, Volume 137, 2023, 108285, ISSN 1270-9638,\nhttps://doi.org/10.1016/j.ast.2023.108285.\nI suggest the authors somehow explain the scope of the gusts (G>1) they will explore earlier in\nthe manuscript in a clearer way. I also suggest avoiding the term “extreme aerodynamics” which\nseems subjective and is not representative since it doesn’t cite prior work (example Colin et al).\nI think just calling it a strong gust is sufficient just like the authors already did in Paragraph1 of\nExtreme Vortex Airfoil Interactions section whereby it was called strong.\nMore specific comments below:\nIntroduction Para2: It would be good to acknowledge that UAVs already operate to deliver parcels\nfor example in Australia by SwoopAero, and Google X since 2019. Therefore statements like this\nneeds to be updated since the technology has already started operation, “These new and amazing\nflying vehicle concepts will likely revolutionize air-based transportation5, 10, 11”.\nIntroduction Para3: Suggest to not list gusts and vortical disturbances as two separate things. It\nmay be good to reference the gust taxonomy and use the terminology from it:\n• Mohamed, A.; Marino, M.; Watkins, S.; Jaworski, J.; Jones, A. Gusts Encountered by Flying\nVehicles in Proximity to Buildings. Drones 2023, 7, 22. https://doi.org/10.3390/drones7010022\nAlso the statement of “Such a flight environment has been off-limits due to the fact that there is\nvirtually no available theory for extreme aerodynamic problems and to avoid possible loss of\naircraft”, is not accurate since drones have operated in these environments (see references\nbelow).\n• Mohamed, A., Abdulrahim, M., Watkins, S., & Clothier, R. (2016). Development and flight testing\nof a turbulence mitigation system for micro air vehicles. Journal of Field Robotics, 33(5), 639-660.\n• Prudden, S., Fisher, A., Marino, M., Mohamed, A., Watkins, S., & Wild, G. (2018). Measuring\nwind with small unmanned aircraft systems. Journal of Wind Engineering and Industrial\nAerodynamics, 176, 197-210.\n• Gavrilovic, N., Mohamed, A., Marino, M., Watkins, S., Moschetta, J. M., & Bénard, E. (2018).\nAvian-inspired energy-harvesting from atmospheric phenomena for small UAVs. Bioinspiration &\nbiomimetics, 14(1), 016006.\nAlso you don’t need to understand the theory for a controller to mitigate the disturbances which\nhas already been demonstrated for fixed and rotary wings… As mentioned before researchers are\neven exploring the use of ML to create controllers that use an array of sensors during stall (see\nbelow)… The narrative needs to therefore be scoped down and more focused.\n• Araujo-Estrada, S. A., & Windsor, S. P. (2021). Aerodynamic state and loads estimation using\nbioinspired distributed sensing. Journal of Aircraft, 58(4), 704-716.\nIntroduction Para6 / last sentence: Based on the fact that the focus of the paper is on use of ML to\nonly model a vortex hitting a canonical airfoil as stated in this sentence; its best to rescope the\nabstract and introduction accordingly since you only considered 1 orientation of flight through a\nvortex in a 2D sense. There are other flow disturbances (such as shear layers) and even flying\nthrough them at different orientations/angles which is important from a 3D perspective that you\nhave not yet explored. There is also the fact that experiencing a single discrete vortex can be\ndifferent to experiencing a series of vortices of different scales (more realistic) which is probably\nworth mentioning as a condition to consider in the future.\nFigure1 (Right) a scale is missing.\nExtreme Vortex Airfoil interaction Para1: Can you add any validation of the simulations undertaken\nto increase confidence in the simulations. I couldn’t find it in the Supp material too. Comparing the\nsimulated flow behavior to published experiments is important here.\nFigure2: What is unit of Lift on y-axis? Can it be non-dimensionalised? Also Im not sure what all\nthe light colored curves are on the plot? Finally I assume that the contour represents pressure?\nIts stated in figure.\nExtreme Vortex Airfoil interaction Para3: It would be good to explicitly state the RE here. Also\nworth noting that atmospheric flow is rarely laminar which affects flow behavior significantly.\nAcknowledging this is important as part of stating the underlying assumptions and explaining it’s a\nstep towards more complex and realistic simulations.\nThe descriptions of the reconstructed lift isn't very clear. Does this mean that the state variables\nare used to produce the lift value given the latent state values? If so, why isn't lift just another\npart of the auto-encoder input/output? Why is the augmentation required?\n\"autoencoder to compress the vorticity field to mere three variables is not only surprising but also\nreaffirms that the flow field is indeed comprised of common flow features\" → not really surprising\ngiven autoencoder literature - the whole point is to use few features! the PCA analysis' highlight\nfew field being used.\nImages start switching \"input\" and \"reference\" image in the figures, but doesn't clearly describe\nthis distinction in the text. Is the reference input? or is this what should be being produced by the\nnetwork? if so, where does this reference image come from?\nI note these can mean different things depending on the context\nIs this just showing the input and output can be replicated? Why is this significant for nature? How\nthis this actually applied?", "reviewer_4": "This paper reveals that extreme aerodynamic fields can be beautifully embedded in low-\ndimensional space by data-driven methods.\nThe idea of introducing a lift-augmented structure to the traditional autoencoder is simple and\nclear: the features are left with enough information to estimate the lift. However, the difference\nemerged by this modification is significant, and the feature space that the autoencoder composes\nbecomes completely different. In particular, it is surprising that it can be generalized to\nextrapolative situations such as those shown in the lower part of Figure 5, Figure 6, and Figure 7.\nThe description of the paper is basically clear and well-organized.\nIt would be desirable to write more clearly what is to be done with the obtained feature space and\nwhy we are happy to have this embedding throughout the text.\nSeveral interesting expected specific applications are given in the discussion section, e.g:\n“low-dimensional representation of the extreme aerodynamic flows suggests that only a small\nnumber of sensors on the airfoil may be able to accurately reconstruct the surrounding flow field in\nreal time”,\n“it is possible to develop a reduced-order model that can capture the dynamics in the latent space\nto desired level of accuracy and complexity”,\n“With the discovered manifold, active flow control and vehicle stability control strategies can be\ndeveloped for mitigating the effects of extreme aerodynamic disturbances”.\nIt would be helpful for readers to additionally explain the connection between the applications and\nthe proposed method. It would be even better if this explanation could be related to existing works\nincluding the authors' ones dealing with flow field data based on autoencoders.\nOther more specific comments are as follows:\nWhile the title and other text uses word \"manifold\", can the destination of this embedding be\nconsidered as a manifold? It seems to me that it is just a \"space\" of three dimensions. Are you\nsaying that 3-dimensional $\\xi$ space is a manifold in some sense? Or are you referring to the\ncone-shaped structure? If so, is the structure really within 2 dimensions (even approximately)?\nFrom the right side of Fig. 4, it seems to be widely distributed in the three-dimensional space. The\nclaim in the discussion section that the embedding can be reduced to two variables is also\nquestionable.\nIn the abstract, it is stated that “there is an enormous parameter space for gusty conditions wings\nencounter”. What does this mean? I could not find a detailed description in the main text. Do you\nmean the space of four parameters (or, plus Re) in the experiment? Or do you mean a much\nlarger space of parameters? If the latter, is 4 or 5 dimensions in the authors' experiment\nsufficient?\nThe abstract states that “the fundamental physics behind extreme aerodynamics is far simpler and\nlow-rank than traditionally expected”. How much dimensionality has conventionally expected by\nwhom? It should be stated in the main text.\nThe Reynolds number used in this study is fixed at 100. What specific application is this Reynolds\nnumber set with in mind? Is it not too low?\nThe following sentences in page 3:\n“The current choice of Re captures flows that are representative of a wide range of realistic flow\nscenarios since the extreme aerodynamic disturbance creates high local flow acceleration injecting\na large amount of vorticity from the surface. This generally leads to laminarizing the flow near the\nbody.”\nmay be in response to this question, but if so, could you provide more detailed explanation?\nIn page 5, the authors stated that “While powerful at compressing Gaussian-distributed data, PCA\nis known to have difficulty in compressing transient physics and those with strong nonlinearities\nthat produce non-Gaussian distributions, which is certainly the case with extreme aerodynamic\nflows”, which seems unclear to me.\nIt is true that PCA can be regarded as a probabilistic modeling with multidimensional normal\ndistribution, but, in my understanding, Gaussianity and the compression performance is almost\nunrelated. What determines the compression rate by PCA is how much data lies in a limited linear\nsubspace. The reason AE works is that this subspace can be bent by nonlinear transformations.\nIn addition, I don't understand well the term “nonlinearity”. The meaning of the term nonlinear\nseems unclear throughout the main text. What is nonlinear object in this sentence, for example?\nDo you mean that the Navier-Stokes equation is a nonlinear differential equation? Is it OK to\ndiscuss the linearity/nonlinearity of PCA/AE transformations and the nonlinearity of differential\nequations as the same thing?\nFigures 5, 6 and 7 show the structural similarity index. Since it is not a general index value, it\nshould be mentioned briefly in the text.\nIn page 7, the authors stated that “it can also be considered as an hour-glass shape since there is\na mirrored manifold for negative angle of attack cases.”\nSince AE performs nonlinear transformations, I don't think that simply mirroring the geometry\nworks in general. Have you tried embedding the negative angle of attack cases? If it works, it is\nrather interesting result and should be further discussed."}, "author_response": "Response to Reviewer 1\nJournal: Nature Communications (Manuscript ID: NCOMMS-23-21038)\nTitle: Grasping Extreme Aerodynamics on a Low-Dimensional Manifold\nAuthors: Kai Fukami, Kunihiko Taira\nDate sent: August 12, 2023\nThe authors would like to thank the referees for taking his/her time to review our manuscript. Based on the\nreviewers’ insightful comments, revisions have been made to the manuscript, which are highlighted in red.\nWe hope our responses fully address the comments from the referee. In what follows, we list the referee’s\ncomments with our responses shown in blue.\n...........................................................................................................\nThisworkfocusesonmodelingextremeunsteadygustyflows,specificallyaddressingthechallengesassociated\nwithmodelingextremedisturbancevortexesduetotheirstrongnonlinearity. Differentfromtraditionalmod-\neling methods, the paper proposes a reduction modeling approach using a nonlinear autoencoder. It reduces\nthe temporal frames into a latent space with 3 dimensions. Surprisingly, despite the low dimensionality, this\nlatent space is capable of effectively reconstructing data under various conditions such as different airfoil\nanglesandmeasurementnoise. Moreover,itdemonstratesgeneralizabilityundercomplexscenariosinvolving\nmultiple vortices. In general, this paper provides a new perspective for complex system modeling, and could\ncontribute to the control of air vehicles under this type of extreme environment.\nThis work is expected to have a certain level of significance to the field. Autoencoders is designed for\nreductionmodelingforvariouscomplex(especiallynonlinear)systems. Thereviewerbelievesthemethodology\nprovided in this paper is sound. Autoencoders are generally possible to perform reduced-order modeling\n(proven in many tasks), as demonstrated in [1]. By tackling the modeling of extreme unsteady gusty flows\nwithautoencoders,thispaperoffersvaluableinsightsthatcouldbenefitresearchersandpractitionersworking\nin the field of fluid dynamics and aerodynamics. This work is likely to be of interest to a wide range of\nresearchers.\nInterms ofthemodeling/controlof UAVs underextremegustyflows, thisworkhas thepotentialtoopen\nup the field of extreme gusty flow control. One of the baseline deep learning model is similar to prior works\nlike [2] (cited in draft). Part of this work further extends the study of [2] to extreme gust vortices.\nAlltheexperimentsarewelldemonstratedwithdetailstosupporttheclaims. Theanalysisandinterpreta-\ntion is clean. The experiments are thoughtfully designed, covering various interesting aspects of the problem\nat hand. In the following, the reviewer aims to point out certain concerns and minor flaws identified in the\npaper. The reviewer intends to address these questions to the authors in order to seek further clarification.\nResponse: Thank you for the encouraging remarks. We have revised our manuscript based on your\ncomments and hope that the updated manuscript addresses your comments.\ni) [Question] The experimental study is only demonstrated in 2D cases. Would the authors plan to build\nmore experiments in 3D for future work? How would this expect to cost computationally?\nResponse: Yes, the extension to three-dimensional flow fields is definitely one of our upcoming tasks.\nIn fact, we are preparing for the public an open-access data set of three-dimensional vortical flows around a\nwingwithagoodnumberofconfigurationscoveringavarietyofwakepatterns(Ribeiroetal.,arXivpreprint,\n2023). We are very excited about extending the proposed method for such three-dimensional flows.\nFrom the aspect of model formulation, applications to three-dimensional flows can be achieved by solely\nreplacing the two-dimensional convolutional operations with three-dimensional operations. Regarding the\ncomputational cost, it would be more expensive compared to the present two-dimensional cases. However, it\nwillstillbemanageablebecausetheconvolutionaloperationsinthenetworkgreatlyreducethecomputational\ncost compared to the fully-connected models. We now mention extensions for three-dimensional flows in the\nthird paragraph of Discussions on page 10. We also note that our group has extensive experience with two-\n1\nand three-dimensional machine-learning-based super-resolution analysis that will help with extending our\nanalysis to three-dimensional flows (Fukami et al., Theor. Comput. Fluid Dyn., 2023).\n• Ribeiro, J. H. M., Neal, J., Burtsev, A., Amitay, M., Theofilis, V., & Taira, K. (2023). Laminar\npost-stall wakes of tapered swept wings. arXiv:2304.07587.\n• Fukami,K.,Fukagata,K.,&Taira,K.(2023). Super-resolutionanalysisviamachinelearning: asurvey\nfor fluid flows. Theoretical and Computational Fluid Dynamics, 1-24.\nii) [Suggestion] This paper is mostly using simulation data. What would be the difficulty/gap when one\ntries to apply this method to real data? Would it cost the required latent dimension to be more than three?\nMoreover, this paper uniformly suggests a model with latent dimension = 3. It might be better to write a\nfew lines to discuss the reason.\nResponse: This is a great point. When we apply the proposed approach to real data, for example,\nmeasuredfromexperiments,onechallengeishowtodealwithnoiseinthemeasuredvorticalflowdata. How-\never, it is generally known that nonlinear autoencoders are able to extract key features of high-dimensional\ndata in a low-order manner while performing denoising. In fact, we have recently demonstrated for noisy\nexperimental data that a nonlinear autoencoder with the assistance of a topology-based concept can reveal\nthe low-dimensional manifold of high-dimensional aerodynamic flows with transient gust encounter (which is\nadifferenttypeofgust;Smithetal.,2023). Thelow-dimensionallatentspaceforsuchaflowiscoincidentally\nrank 3.\nThe number of the latent vector required to capture the dynamics is generally dependent on the variety\nof flow features held in the data set, more than the level of noise contained therein. Hence, the handling\nof experimental or computational data does not seem to alter our findings in any substantial manner. For\nthe present study, three latent variables are needed to express the amplitude, timing, and effective angle of\nattack, in high-dimensional extreme aerodynamic flows. These parameters lead to the emergence of certain\nflow features. We have now added this description to the second paragraph of the Discussions.\n• Smith, L., Fukami, K., Sedky, G., Jones, A., & Taira, K. (2023). A cyclic perspective on transient gust\nencounters through the lens of persistent homology. arXiv:2306.15829.\niii)[Discussion]AEwithliftisacceptable. Strictlyspeaking,AEwithliftshouldbecomparedtoAEwithout\nlift using latent dimension= 4 (3 original latent dimensions + 1 lift). This would create a more cohesive\nmodeling approach. However, the reviewer acknowledges that such a modeling might pose challenges for\ninterpretation. The reviewer would appreciate some further discussion from the authors regarding this par-\nticular point.\nResponse: The comparison between the present lift-augmented autoencoder with 3 latent variables\nand a regular autoencoder with 4 latent variables is actually not appropriate because the lift augmentation\ndoes not correspond to the addition of the latent dimension. To discuss how manifold identification can be\npromoted with the same latent dimension, the comparison should be performed with a regular autoencoder\nwith 3 latent variables. In fact, a regular autoencoder slightly outperforms the lift-augmented autoencoder\nin terms of vortical flow reconstruction, as shown in figure S1, although their latent variable distribution is\nnot physically interpretable.\nMinor points:\ni) [Question] For the reconstruction percentage in Fig. 5, 6, 7, do they represent the reconstruction rate of\nthis specific snapshot, or is it averaged over all the snapshots?\nResponse: They are time-averaged values over all snapshots. We have revised the text to clarify this\npoint. Thank you.\n2\nii) [Discussion] In Fig. 8, the performance of AE with lift slightly underperforms AE without lift. Maybe\nsome very ’advanced’ hyper-parameter tuning could help to fix the problem?\nResponse: Related to comment iii) above, this is expected because the decoded vorticity field is only\na function of latent variable ξ ∈ R3 (and not a function of C ). Again, the present augmentation with C\nL L\ndoes not correspond to the addition of the latent dimension. Since the weighting for the reconstruction loss\nof the lift-augmented autoencoder becomes smaller compared to a regular autoencoder (see equation 3 in\nsupplemental material), the reconstructed vorticity field from a regular autoencoder is generally better than\nthat from the lift-augmented autoencoder.\niii) In abstract, there should be a ’will/would’ in front of the first ’encounter’.\nResponse: Thank you. We have added “would” in the appropriate location.\nJournal: Nature Communications (Manuscript ID: NCOMMS-23-21038)\nTitle: Grasping Extreme Aerodynamics on a Low-Dimensional Manifold\nAuthors: Kai Fukami, Kunihiko Taira\nDate sent: August 12, 2023\nThe authors would like to thank the referees for taking his/her time to review our manuscript. Based on the\nreviewers’ insightful comments, revisions have been made to the manuscript, which are highlighted in red.\nWe hope our responses fully address the comments from the referee. In what follows, we list the referee’s\ncomments with our responses shown in blue.\n...........................................................................................................\nThe manuscript entitled “Grasping Extreme Aerodynamics on a Low-Dimensional Manifold” by Kai Fukami\nandKunihikoTairapresentsadata-drivenapproachtoidentifyalow-dimensionalmanifoldonwhichthekey\ndynamics between strong gust vortices and airfoil wakes can be collapsed. As also stated by the authors,\ntheinteractionbetweenthevorticalgustsandwingsisseeminglycomplexanddifferentforeachcombination\nof gust parameters, however they achieved to discover a manifold with an autoencoder designed to retain\nthe knowledge of aerodynamic lift as part of the latent variables and showed that the fundamental physics\nbehindextremeaerodynamicsisfarsimplerandlow-rankthantraditionallyexpected. Itisaveryinteresting,\nsignificant work, a valuable research presented in a clear and complete form. Therefore, I would recommend\nthat the paper be published as is. However, I would like to point out a couple of concerns for the perfectness\nof the manuscript.\nResponse: Thank you for the encouraging remarks. We have revised our manuscript based on your\ncomments and hope that the updated manuscript addresses your comments.\n1. IfIunderstandcorrectly,ξ isrelatedtotheeffectiveangleofattackandthewidthoftheradialtrajectory\n3\non the ξ and ξ plane (combination of ξ and ξ ) shows the strength of the disturbance. I assume the\n1 2 1 2\neffective angle of attack will depend on all of the parameters of the study, namely α, G, L, and y /c. On\n0\nthe other hand the strength of the disturbance or its consequence in terms of lift force depends on all of\nthe parameters of the study as well. Therefore, I expect the latent variables are independent and represents\na physical and probably a non-dimensional value made of a combination of those parameters. It would be\nvery insightful if the authors could also comment on how physically the latent variables are related to the\nparameters of the study.\nResponse: This is a great comment. The parameters mentioned above have an implicit influence on the\nlatent variables. The influence of the parameters on the flow fields is not as clear as what may be expected,\nas commented by the referee. We have actually considered this point for a separate study of turbulent flows\n(Fukami, Goto, and Taira, 2023). Within the context of the present paper, we have not commented on this\npoint in detail since the compression of the flow field data examined here is based on the flow features and\nnot directly on the parameters. We hope to make a stronger case for examining the connection between the\nproblem parameters and the flow field as we extend the separate study in preparation. Please stay tuned.\n• Fukami, K., Goto, S., & Taira, K. (2023). Assessing inter- and extrapolation in learning turbulence by\ndata-driven Buckingham Pi scaling. to be submitted.\n2. According to the results presented in Figure S1, structural similarity indices for AE with lift is a bit\nsmaller than those for AE w/o lift. A brief explanation for the reason behind will be helpful to understand\ntheapproachandtheflowphysics. Havingnoticedthisslightdifference,Iwonderiftheauthorsobtainedthe\nlift from the reconstructed flow field data with the regular autoencoder and compared with that obtained\nwith lift-augmented autoencoder. And finally, I believe a measure for the accuracy of the lift may support\n1\nthe success of the discovered manifold.\nResponse: The structural similarity indices for AE without lift being higher for the results from an AE\nwithliftisexpected. ThisisbecausetheAEwithoutliftisabletosolelytuneitsweightstooptimizeaccurate\nreconstruction of the flow field from the latent variables.\nIt is also possible to reconstruct lift directly from the vorticity field. However, let us emphasize that\nthe lift-augmented AE is utilized to reveal the manifold for extreme aerodynamic responses, more so than\nestimating the lift (although that aspect is important as well).\nWe appreciate the comment on the connection between the accuracy of lift estimation and the success of\ndiscovering the manifold. This is certainly true and is incorporated into the formulation by how we select\nthe parameter β in the loss function. We have added these descriptions to the fourth paragraph on page 7.\n3. Intermsofthestructureofthemanuscript,IwouldliketopointoutonlythatintheSuplementalMaterial,\nthereferencesgoupto#35whileinthereferencelistofthemaintext,thefinalreferenceisnumberedas#30.\nResponse: We have made appropriate corrections. Thank you.\nJournal: Nature Communications (Manuscript ID: NCOMMS-23-21038)\nTitle: Grasping Extreme Aerodynamics on a Low-Dimensional Manifold\nAuthors: Kai Fukami, Kunihiko Taira\nDate sent: August 12, 2023\nThe authors would like to thank the referees for taking his/her time to review our manuscript. Based on the\nreviewers’ insightful comments, revisions have been made to the manuscript, which are highlighted in red.\nWe hope our responses fully address the comments from the referee. In what follows, we list the referee’s\ncomments with our responses shown in blue.\n...........................................................................................................\nThe paper presents findings that offer a new perspective on modeling gusts using machine learning. It has\ninteresting results and utilizes multi-disciplinary research to achieve them. I believe the scoping of the paper\nneeds further work and the big claims made need to be toned down a little. The paper will benefit focusing\non the ability to reduce the problem to 3 parameters.\nResponse: Thank you for very careful review and valuable comments. We have revised our manuscript\nbased on your comments and hope that the updated manuscript addresses your comments.\n1. The abstract and introduction don’t introduce the content of the paper properly or clearly. For example,\ntheapplicationoftheworkisexplainedatendofpaperwherebythelowdimensionalrepresentationcanhelp\nthe use of sensors on an aircraft. I think this needs to be highlighted at the start of the paper for context.\nBut also note that there has already been work on the use of ML to predict stall over a 3D wing by Shane\nWindsor in University of Bristol and others. Discussing this early will set the scene and enable the reader to\nfollow the vision of why this work is important and how it can be used to improve safety.\nResponse: Thank you for the comment. We have now highlighted how the discovered manifold can be\nleveraged for not only understanding extreme aerodynamic flows but also aiding downstream tasks such as\nreal-time flow estimation, dynamical modeling, flow control, and vehicle design, throughout the manuscript.\nIn addition to the abstract and the introduction, we also have additional discussions in the last section of\nthe paper to present possible future directions based on our findings. We already started a few efforts on the\naforementioned items and have achieved promising results. These ongoing works will appear soon — please\nstay tuned.\nWe also thank the referee for mentioning a study by Araujo-Estrada & Windsor (2021). We feel that it\nis important to cite their work and have now included it in the text.\n2. There are some bold claims which I don’t agree with entirely such as “However, there exists virtually\nno foundation to describe the influence of extreme vortical gusts on flying bodies”. I think the foundation\nthrough CFD simulation and experimental work does exist and the review papers by Anya Jones list them.\nTherehasevenbeenattemptstoexploreworstcasegustsinurbanenvironmentsbysimulatingflightthrough\nshear layers which are far more “extreme” than the case being considered in this paper:\n• Mohamed, A.; Marino, M.; Watkins, S.; Jaworski, J.; Jones, A. Gusts Encountered by Flying Vehicles\nin Proximity to Buildings. Drones 2023, 7, 22. https://doi.org/10.3390/drones7010022\nThis gust has even been characterized as extreme in this publication:\n• ColinM.Stutz,JohnT.Hrynuk,DouglasG.Bohl,Dimensionalanalysisofatransversegustencounter,\nAerospaceScienceandTechnology,Volume137,2023,108285,ISSN1270-9638"}]

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