1. Introduction
Transonic buffet phenomena determine the high-speed limit of a flight envelope. To extend the flight envelope towards the high-speed side with a better and safer design of modern commercial aircraft that includes asymmetric supercritical wings, the transonic buffet, specifically referred to as Type II buffet (Giannelis, Vio & Levinski Reference Giannelis, Vio and Levinski2017), needs to be tamed, rooted in profound understandings of its physics and practically useful models to describe the buffet. While extensive analyses using simulations and experiments have been performed providing a variety of posits to describe complex behaviours of transonic airfoil buffet, what is commonly believed is that there exists a self-sustained shock buffet cycle (Giannelis et al. Reference Giannelis, Vio and Levinski2017). We pose a question of whether such a seemingly complex, but cyclic dynamics of transonic buffet phenomena can be described in a low-order manner with nonlinear machine learning.
The aerodynamic instability known as transonic buffet, characterised by self-sustained shock-wave oscillations on aircraft wings, needs to be taken into account during transonic or high-subsonic flight. This phenomenon arises because shock waves can form when the wing geometry accelerates the flow along the leading edge of the suction side, generating a localised supersonic region (Tijdeman & Seebass Reference Tijdeman and Seebass1980). The occurrence of transonic buffet depends on a flow condition characterised by a combination of parameters such as Mach number, Reynolds number and angle of attack.
To facilitate characterisation of transonic buffet phenomena, a range of numerical and experimental endeavours have been carried out. Such studies on the transonic buffet are classified based on their focus on dimension in phenomena, namely two-dimensional and three-dimensional. In the two-dimensional airfoil buffet, chordwise large-scale shock oscillations occur, which are numerically and experimentally reproduced by confining a flow field in a narrow spanwise domain (Lusher, Sansica & Hashimoto Reference Lusher, Sansica and Hashimoto2024). The chordwise shock oscillations result in a distinct spectral peak at a low frequency generally smaller than
$0.1$
, for example, a Strouhal number
$St\approx 0.06$
for the OAT15A supercritical airfoil (Deck Reference Deck2005; Jacquin et al. Reference Jacquin, Molton, Deck, Maury and Soulevant2009; Fukushima & Kawai Reference Fukushima and Kawai2018; Cuong Nguyen, Terrana & Peraire Reference Cuong Nguyen, Terrana and Peraire2022).
On the other hand, the three-dimensional buffet is caused due to characteristics associated with the three-dimensionality of the wing, such as swept and taper effects. One notable feature of the three-dimensional buffet, absent in the two-dimensional buffet, is the occurrence of buffet cells (Iovnovich & Raveh Reference Iovnovich and Raveh2015). Buffet cells refer to a cellular flow structure propagating outboard. A range of numerical (Ohmichi, Ishida & Hashimoto Reference Ohmichi, Ishida and Hashimoto2018; Tamaki & Kawai Reference Tamaki and Kawai2024) and experimental (Meneveau & Katz Reference Meneveau and Katz2000; Dandois Reference Dandois2016; Sugioka et al. Reference Sugioka, Koike, Nakakita, Numata, Nonomura and Asai2018, Reference Sugioka, Nakakita, Koike, Nakajima, Nonomura and Asai2021; Masini, Timme & Peace Reference Masini, Timme and Peace2020) studies have reported the occurrence of buffet cells.
It has widely been observed that the power spectrum density of relevant quantities, such as the pressure coefficient fluctuation, typically presents a broadband spectrum peak with a Strouhal number ranging from 0.2 to 0.6 (Dandois Reference Dandois2016; Koike et al. Reference Koike, Ueno, Nakakita and Hashimoto2016), 10 times higher in frequency than that of the two-dimensional buffet counterpart, depending on the sweep angle (Plante, Dandois & Laurendeau Reference Plante, Dandois and Laurendeau2020; Sugioka, Kouchi & Koike Reference Sugioka, Kouchi and Koike2022; Lusher, Sansica & Hashimoto Reference Lusher, Sansica and Hashimoto2025). Particularly considering a full-aircraft configuration of the NASA Common Research Model, understanding of the buffet cell structure has been deepened with modal analysis, including tri-global stability analysis (Timme Reference Timme2020; Sansica & Hashimoto Reference Sansica and Hashimoto2023), tri-resolvent analysis (Houtman, Timme & Sharma Reference Houtman, Timme and Sharma2023), dynamic mode decomposition (Ohmichi et al. Reference Ohmichi, Ishida and Hashimoto2018) and its Hankel variant (Asada & Kawai Reference Asada and Kawai2025). Based on them, the buffet cell has been recognised as a key player in the self-sustaining instability mechanism of a three-dimensional buffet. However, there is still no widely accepted physical model that explains the self-sustaining mechanism of a three-dimensional buffet.
While acknowledging the significance of buffet cells, this study focuses on the two-dimensional airfoil buffet mechanism, which remains active and critical even under three-dimensional buffet conditions. Sugioka et al. (Reference Sugioka, Koike, Nakakita, Numata, Nonomura and Asai2018) experimentally demonstrated that shock-wave oscillations over the NASA Common Research Model at high angles of attack exhibit behaviour similar to that of a two-dimensional buffet. Paladini et al. (Reference Paladini, Beneddine, Dandois, Sipp and Robinet2019) showed that a two-dimensional global instability mode, akin to that observed in airfoil buffet (Crouch et al. Reference Crouch, Garbaruk, Magidov and Travin2009), can coexist with a spanwise-varying three-dimensional mode associated with buffet cells. Similar modal structures have been reported by Crouch et al. (Reference Crouch, Garbaruk and Strelets2018, Reference Crouch, Garbaruk and Strelets2019). Paladini et al. (Reference Paladini, Beneddine, Dandois, Sipp and Robinet2019) performed a wavemaker analysis to reveal that the two-dimensional mode is primarily linked to the shock-wave dynamics, whereas the spanwise-varying mode originates from the separated shear layer. These findings highlight the importance of considering not only the three-dimensional buffet cells but also the underlying two-dimensional instability mechanisms that remain fundamental to understanding buffet phenomena.
For these reasons, the mechanism of self-sustained large-scale shock oscillations is of particular interest in the community (Lee Reference Lee2001; Iwatani et al. Reference Iwatani, Asada, Yeh, Taira and Kawai2023). While a Reynolds-averaged formulation had been considered for numerical investigations (Crouch et al. Reference Crouch, Garbaruk, Magidov and Travin2009; Iovnovich & Raveh Reference Iovnovich and Raveh2012; Sartor, Mettot & Sipp Reference Sartor, Mettot and Sipp2015), recent advancements in computational resources along with wall-modelling approaches enable performing large-eddy simulations (LES) (Fukushima & Kawai Reference Fukushima and Kawai2018; Tamaki & Kawai Reference Tamaki and Kawai2024; Goc et al. Reference Goc, Agrawal, Bose and Moin2025). This offers further reliable assessments of transonic buffet flows by accurately capturing the interaction between the shock wave and the turbulent boundary layer. Along with spatiotemporal high-resolution measuring techniques such as laser Doppler velocimetry (Jacquin et al. Reference Jacquin, Molton, Deck, Maury and Soulevant2009), particle image velocimetry (D’Aguanno et al. Reference D’Aguanno, Schrijer and van Oudheusden2021) and schlieren visualisation (Schauerte & Schreyer Reference Schauerte and Schreyer2023), experimental studies have not only provided a simplified model of transonic buffet supporting the understanding of buffet phenomena (Lee Reference Lee1990; Crouch, Garbaruk & Magidov Reference Crouch, Garbaruk and Magidov2007) but also suggested passive control devices to suppress buffet-associated instabilities (Lagemann et al. Reference Lagemann, Brunton, Schröder and Lagemann2024). However, the self-sustaining mechanisms of the transonic airfoil buffet still require further clarification.
In analysing transonic buffet flows with a large degree of freedom in the direction of space, time and flow parameters, one can consider applying data-driven order-reduction techniques to flow-field snapshots made available through simulations and experiments. For example, proper orthogonal decomposition (POD) (Lumley Reference Lumley1967) has been considered to obtain a low-order representation of transonic buffet phenomena (Ohmichi et al. Reference Ohmichi, Ishida and Hashimoto2018; Poplingher, Raveh & Dowell Reference Poplingher, Raveh and Dowell2019; Iwatani, Asada & Kawai Reference Iwatani, Asada and Kawai2022; Sansica et al. Reference Sansica, Loiseau, Kanamori, Hashimoto and Robinet2022). However, seeking a minimal representation of unsteady flows with such a linear technique is generally challenging because given data are linearly projected onto a flat manifold (Graham & Floryan Reference Graham and Floryan2021).
To extract a low-order representation that best captures the underlying characteristics of transonic buffet flows from data, this study considers a nonlinear autoencoder-based compression (Hinton & Salakhutdinov Reference Hinton and Salakhutdinov2006). Nonlinear activation functions inside an autoencoder enable better compression of unsteady flow data compared with linear techniques, which has been discussed with wake shedding (Omata & Shirayama Reference Omata and Shirayama2019; Murata, Fukami & Fukagata Reference Murata, Fukami and Fukagata2020), channel flow (Fukami et al. Reference Fukami, Nabae, Kawai and Fukagata2019; Yousif, Yu & Lim Reference Yousif, Yu and Lim2022), Kolmogorov turbulence (Page et al. Reference Page, Holey, Brenner and Kerswell2024) and aerodynamic flows under gusty environments (Mousavi & Eldredge Reference Mousavi and Eldredge2025). Compressed representations obtained from the autoencoder can be used for a range of analyses including mode decomposition (Fukami, Nakamura & Fukagata Reference Fukami, Nakamura and Fukagata2020; Mo, Traverso & Magri Reference Mo, Traverso and Magri2024), dynamical modelling (Fukami et al. Reference Fukami, Murata, Zhang and Fukagata2021b ; Constante-Amores & Graham Reference Constante-Amores and Graham2024; Solera-Rico et al. Reference Solera-Rico, Sanmiguel Vila, Gómez-López, Wang, Almashjary, Dawson and Vinuesa2024), shape optimisation (Tran et al. Reference Tran, Fukami, Inada, Umehara, Ono, Ogawa and Taira2024) and flow control (Linot, Zeng & Graham Reference Linot, Zeng and Graham2023; Liu, Beckers & Eldredge Reference Liu, Beckers and Eldredge2025).
Although a nonlinear autoencoder can be employed as a powerful data compressor of unsteady flows, it is important to note that careful use of an autoencoder by incorporating prior knowledge of physics is essential to promote understanding of flows in a low-order latent space (Fukami & Taira Reference Fukami and Taira2023). It is challenging to use compressed variables obtained through a naive application of a standard autoencoder for characterising and controlling unsteady flows (Fukami, Nakao & Taira Reference Fukami, Nakao and Taira2024; Smith et al. Reference Smith, Fukami, Sedky, Jones and Taira2024). In response, we incorporate aerodynamic coefficients into the nonlinear autoencoder formulation in identifying a low-order subspace. Equipped with this observable-augmented autoencoder, this study reveals the existence of a three-dimensional representation of transonic airfoil buffet flows, which describes the complex phenomena over the buffet cycle dynamics in a compact manner. Furthermore, the current model trained at a wind-tunnel-scale Reynolds number based on a chord length
${Re}\sim 10^6$
can be used for sparse-sensor reconstruction of aerodynamic responses at the level of a real-aircraft-operation high Reynolds number
${Re}\sim 10^7$
. The present approach may facilitate data-driven analysis of transonic buffet flows across a range of Reynolds numbers.
This paper is organised as follows. The simulation set-up used for data generation and flow physics are expressed in § 2. The present autoencoder technique is described in § 3. Results and discussion are presented in § 4. Conclusions are offered in § 5.
2. Transonic airfoil buffet flows at high Reynolds numbers
This study seeks a low-dimensional representation of two-dimensional transonic airfoil buffet flows, capturing time-varying characteristics over the buffet cycle using nonlinear machine learning. We consider datasets of Fukushima & Kawai (Reference Fukushima and Kawai2018) generated by wall-modelled LES of the transonic buffet over the OAT15A supercritical airfoil at a high Reynolds number of
${Re} = u_\infty c/\nu _\infty = 3\times 10^6$
for nonlinear machine-learning compression. Here,
$u_\infty$
,
$c$
and
$\nu _\infty$
describe the free-stream velocity, the chord length and the kinematic viscosity, respectively. Following the observation in our previous study (Fukushima & Kawai Reference Fukushima and Kawai2018), we consider two different Mach numbers of
$M_\infty = u_\infty /a_\infty = (0.715, 0.730)$
, where
$a_\infty$
is the free-stream sonic speed. While a steady shock wave is observed at
$M_\infty = 0.715$
, the unsteady shock oscillating buffet phenomena emerge on increasing the Mach number to 0.730. Involving both non-buffet and buffet conditions in the present datasets for the nonlinear machine-learning analysis enables extracting the difference between them in a low-order manner. All the physical variables throughout the paper are normalised using combinations of
$c$
,
$a_\infty$
and the density
$\rho _\infty$
. We further consider a higher-Reynolds-number case of
${Re} = 3\times 10^7$
with
$M_\infty = 0.730$
, exhibiting unsteady buffet phenomena, to evaluate the applicability of the current technique trained at a wind-tunnel-scale Reynolds number
${Re} \sim 10^6$
to a scenario at a real aircraft-scale Reynolds number
${Re} \sim 10^7$
.
The computational mesh used in the present study is shown in figure 1. The spatially filtered compressible Navier–Stokes equations are numerically solved, where the LES with modelled wall shear stresses and wall heat fluxes resolves the outer-layer turbulence (Fukushima & Kawai Reference Fukushima and Kawai2018). We follow our previous studies (Kawai & Larsson Reference Kawai and Larsson2012, Reference Kawai and Larsson2013; Fukushima & Kawai Reference Fukushima and Kawai2018) for the numerical schemes as well as the treatment of the boundary conditions.
Figure 1. The computational grid used in the present wall-modelled LES of two-dimensional transonic airfoil buffet at a high Reynolds number (Fukushima & Kawai Reference Fukushima and Kawai2018). An instantaneous streamwise velocity field
$u$
near the wall and the density gradient magnitude
$|\boldsymbol{\nabla }\rho |$
are superposed. The grey grid lines are displayed every fifth point in the
$g_1$
and
$g_2$
(wall-normal) directions. The inset is focused on the region of the shock wave–turbulent boundary layer interactions with the grey grid lines plotted every fifteenth point in the
$g_1$
direction and every fifth point in the
$g_2$
direction.
The spatial derivatives at interior grid points are evaluated using the sixth-order compact differencing scheme (Lele Reference Lele1992). Time integration is performed with the third-order total variation diminishing Runge–Kutta scheme (Gottlieb & Shu Reference Gottlieb and Shu1998). To accurately resolve the shock wave, the localised artificial diffusivity method is employed with the sixth-order compact scheme (Kawai, Shankar & Lele Reference Kawai, Shankar and Lele2010). While we compute the subgrid-scale turbulent eddy viscosity with a selective mixed-scale model (Lenormand, Sagaut & Ta Phuoc Reference Lenormand, Sagaut and Ta Phuoc2000), the equilibrium wall model (Kawai & Larsson Reference Kawai and Larsson2012) is considered.
The computational mesh for the present wall-modelled LES is designed based on the grid resolution requirements (Kawai & Larsson Reference Kawai and Larsson2012; Larsson et al. Reference Larsson, Kawai, Bodart and Bermejo-Moreno2016). Although we use the same mesh at both Reynolds numbers,
${Re} = 3\times 10^6$
and
$3\times 10^7$
, the employed mesh satisfies the resolution requirements across the streamwise domain of the attached fully turbulent boundary layer upstream of the shock wave (
$0.2\lesssim x/c \lesssim 0.35$
), providing more than 23–25 grid points in each direction per boundary-layer thickness. Specifically, the mesh resolves the boundary layer with at least 29, 34 and 38 points in the wall-normal direction at
$x/c \approx 0.2$
, 0.25 and 0.3, respectively. In the wall-parallel directions, the resolution corresponds to at least 23, 28 and 33 grid points per local boundary-layer thickness at the same stream locations. These values meet the standards for wall-modelled LES resolution (Kawai & Larsson Reference Kawai and Larsson2012).
Furthermore, previous studies have reported that wall-modelled LES with the equilibrium wall model can reasonably produce the flow states associated with the interaction between the shock waves and turbulent boundary layer even with the simplification of the equilibrium wall model (Bermejo-Moreno et al. Reference Bermejo-Moreno, Campo, Larsson, Bodart, Helmer and Eaton2014; Fukushima & Kawai Reference Fukushima and Kawai2018; De Vanna et al. Reference De Vanna, Bernardini, Picano and Benini2022; Sashida et al. Reference Sashida, Aoyama, Kawai and Kawai2024). Therefore, the present wall-modelled LES provides a high-fidelity dataset for the present nonlinear machine-learning analysis. Further details on the simulation set-up are provided in Fukushima & Kawai (Reference Fukushima and Kawai2018).
The temporal evolution of lift coefficient
$C_L$
and a sectional pressure field
$p$
extracted from the wing centre in the spanwise direction at
${Re}=3\times 10^6$
obtained through the present simulation is presented in figure 2. The case for
$M_\infty = 0.715$
shows statistically steady states, producing small fluctuations of lift over time. The shock mostly appears at
$x/c\approx 0.55$
while slightly oscillating in the streamwise direction on the wing.
Figure 2. Lift coefficient and pressure fields at
$M_\infty =0.715$
(a–d) and 0.730 (e–h). A note concerning the shock location is provided underneath each contour of
$M_\infty = 0.730$
. The arrow in each subcontour represents the direction of shock movement.
In contrast, the case for
$M_\infty = 0.730$
clearly presents its time-varying feature associated with self-sustained large-scale shock oscillation. The shock wave periodically moves in large amplitude over the wing while the separation near the trailing edge is triggered depending on the shock location, which coincides with observations in wind-tunnel experiments (Jacquin et al. Reference Jacquin, Molton, Deck, Maury and Soulevant2009). Correspondingly, the lift response also exhibits a periodic signal over the buffet cycle. Hence, the phase of shock location over the buffet cycle is almost identical to that of lift. The separation height is particularly increased when the shock wave moves upstream, which is shown later. The interaction between the wake and separation at this stage causes the upstream-travelling wave (Lee Reference Lee2001; D’Aguanno et al. Reference D’Aguanno, Schrijer and van Oudheusden2021; Iwatani et al. Reference Iwatani, Asada, Yeh, Taira and Kawai2023). The lift response is greatly affected by the time-varying area size of supersonic flow along with the aforementioned processes. Note that these buffet dynamics are further discussed and quantified later with observation in the machine-learning-based low-dimensional subspace.
3. Nonlinear machine-learning-based compression of transonic airfoil buffet flows
To seek a low-dimensional representation of transonic airfoil buffet flows from data, we consider a nonlinear autoencoder-based data compression (Hinton & Salakhutdinov Reference Hinton and Salakhutdinov2006). An autoencoder
${\mathcal F}_{{AE}}$
aims to reconstruct (or output) the same data as the input data
$\boldsymbol{q}\in \mathbb{R}^n$
. The autoencoder is designed to possess a bottleneck, referred to as a latent space
$\boldsymbol{\xi } \in \mathbb{R}^{m}$
, as illustrated in figure 3. The latent dimension
$m$
is generally set to be much smaller than the original data dimension
$n$
such that
$m\ll n$
. Hence, the latent vector
$\boldsymbol{\xi }$
can be considered as a compressed representation of the given data
$\boldsymbol{q}$
if the autoencoder
${\mathcal F}_{{AE}}$
accurately reconstructs the data. The aforementioned process is described as
\begin{align} \boldsymbol{q}\approx {\mathcal F}_{{AE}}(\boldsymbol{q}) = {\mathcal F}_d({\mathcal F}_e(\boldsymbol{q})), \quad \boldsymbol{\xi } = {\mathcal F}_e(\boldsymbol{q}),\quad \boldsymbol{q} \approx \widehat {\boldsymbol{q}} = {\mathcal F}_d(\boldsymbol{\xi }), \end{align}
where
$\widehat {(\boldsymbol{\cdot })}$
denotes a reconstructed variable and
${\mathcal F}_e$
and
${\mathcal F}_d$
correspond to an encoder and a decoder, respectively. A range of neural-network models with nonlinear activation functions can be considered for the construction of autoencoder
${\mathcal F}_{{AE}}$
. The use of nonlinear activation functions promotes network capabilities, providing better compression than linear techniques, which is mathematically proven through the relationship between a linear activation autoencoder and other linear compression approaches (Oja Reference Oja1982; Bourlard & Kamp Reference Bourlard and Kamp1988; Fukami et al. Reference Fukami, Hasegawa, Nakamura, Morimoto and Fukagata2021a
).
Figure 3. Lift-augmented nonlinear autoencoder (Fukami & Taira Reference Fukami and Taira2023).
We consider a sectional pressure field sampled from the wing centre in the spanwise direction as the input and output
$\boldsymbol{q}$
of a nonlinear autoencoder to extract the underlying characteristics of transonic airfoil buffet flows. While a standard autoencoder achieves significant data compression of fluid flows, it is often challenging to interpret the identified subspace in a physically understandable manner. To facilitate the present latent identification from the viewpoint of aerodynamics, this study uses a lift-augmented nonlinear autoencoder (Fukami & Taira Reference Fukami and Taira2023) producing a lift response from the latent vector through a branch network, as illustrated in figure 3. The optimisation for the parameters (or weights)
$\boldsymbol{w}$
inside the lift-augmented autoencoder is performed with
\begin{align} \boldsymbol{w}^* = \textrm {argmin}_{\boldsymbol{w}}\biggl [||\boldsymbol{q}-\hat {\boldsymbol{q}}||^2_2 + \beta ||C_L - \hat {C_L}||^2_2 \biggr], \end{align}
where
$\beta$
balances the pressure field and lift reconstruction loss terms. This weighting parameter
$\beta$
is set to 0.03 and 0.05 based on the L-curve analysis (Hansen & O’Leary Reference Hansen and O’Leary1993) for the observable-augmented autoencoder, while a regular autoencoder, i.e.
$\beta = 0$
, is also considered for comparison. To minimise the above cost function, the model needs to accurately estimate
$C_L(t)$
while performing data compression of the pressure field data
$\boldsymbol{q}(t)$
. In other words, the current formulation enables
$\boldsymbol{w}$
to be tuned to capture structures appearing over the buffet cycle that are associated with the lift response. As the periodic shock movement over an OAT15A airfoil, clearly observed in the pressure field, is highly correlated with the lift coefficient
$C_L(t)$
, the resulting low-dimensional representation is expected to emphasise aerodynamically important events during the buffet cycle.
The current dataset for the nonlinear autoencoder analysis is composed of 6800 snapshots with
$M_\infty = 0.715$
(non-buffet condition) over 30.8 non-dimensional time,
$t/(c/u_\infty)$
, and 17 300 snapshots with
$M_\infty = 0.730$
(buffet condition) over 26.4 non-dimensional time. We consider a subdomain of
$(x, y)/c \in [-0.6, 1.5] \times [-0.5, 1.3]$
with spatially uniform grid points
$(N_x, N_y) = (480, 200)$
extracted from the entire computational domain for the data-driven analysis, where the leading edge of the wing is positioned at the origin. The interior of the wing is set to be zero. As a fixed angle of attack is considered for all the data in this study, the model is not affected by this operation. The present autoencoder is composed of convolutional neural networks (LeCun et al. Reference LeCun, Bottou, Bengio and Haffner1998) and multi-layer perceptrons (Rumelhart, Hinton & Williams Reference Rumelhart, Hinton and Williams1986) following the original study of the lift-augmented autoencoder, as summarised in table 1. While the convolutional network learns large-scale structures in a flow field through filter-based operations, the multi-layer perceptrons are used for the bottleneck part of the autoencoder, where the data dimension is very low and the spatial coherence is less important than the complex relationship among the latent variables (Fukagata & Fukami Reference Fukagata and Fukami2025). This combination enables data-driven compression of fluid flow data with reasonable computational costs compared with a model based solely on a multi-layer perceptron that often encounters the curse of dimensionality (Fukami et al. Reference Fukami, Hasegawa, Nakamura, Morimoto and Fukagata2021a
; Morimoto et al. Reference Morimoto, Fukami, Zhang, Nair and Fukagata2021). Further details on machine-learning set-ups with the present L-curve analysis for the decision of
$\beta$
are given in Appendix A and a sample code at https://github.com/kfukami/Observable-AE.
Table 1. The architecture of observable-augmented nonlinear autoencoder. The convolutional layers are denoted as ‘Conv.’. The size of the convolutional filter
$F$
and the number of the filter
$K$
are shown for each convolutional layer as
$(F, F, K)$
. The maxpooling/upsampling ratio
$R$
is shown for each layer as
$(R, R)$
.
4. Results and discussion
4.1. Latent space identification of transonic airfoil buffet flows
This section discusses data-driven compression and the resulting subspace identification of transonic airfoil buffet flows. Let us first examine the latent dimension that accurately reproduces the original flow state. The relationship between the latent dimension
$n_{\boldsymbol{\xi }}$
and the
$L_2$
reconstruction error norm
${\varepsilon }_{\boldsymbol{q}}$
is shown in figure 4. Here, the
$L_2$
reconstruction error norm between a variable
$\boldsymbol{f}$
and its reconstruction
${\boldsymbol{\hat{f}}}$
is defined as
$\varepsilon _{\boldsymbol{f}} = ||\boldsymbol{f}-{\boldsymbol{\hat{f}}}||^2_2/||{\boldsymbol{f}}^\prime ||^2_2$
, where
$\boldsymbol{f}^\prime$
represents the fluctuation of
$\boldsymbol{f}$
from the time-averaged value. While a standard nonlinear autoencoder without lift incorporation, i.e.
$\beta =0$
, is considered for this analysis, linear POD is also used for comparison.
Figure 4. Comparison of compression performance for transonic airfoil buffet flow data between linear POD and a standard nonlinear autoencoder (AE,
$\beta =0$
).
$(a)$
The relationship between the latent dimension
$n_{\boldsymbol{\xi }}$
and the
$L_2$
reconstruction error
$\varepsilon$
.
$(b)$
Representative reconstructed pressure snapshots with
$n_{\boldsymbol{\xi }} = (1,3,5)$
for
$M_\infty = 0.730$
with
$(c)$
the reference field.
$(d)$
The absolute error field
$e_{L_1} = |\boldsymbol{q}-\hat {\boldsymbol{q}}|$
corresponding to panels in
$(b)$
.
The nonlinear autoencoder is superior to POD across the latent dimension, suggesting that the use of nonlinear activation functions inside the model facilitates compression performance. Compared with the POD-based reconstruction exhibiting high error near the shock, the autoencoder accurately reproduces a flow state, as presented in figure 4. We also find that the error curve of the autoencoder plateaus once the latent dimension reaches three. This reveals that the primary large-scale feature of the pressure fields for the present transonic airfoil buffet flows at
${Re}=3\times 10^6$
can be represented with solely three-dimensional latent variables with nonlinear machine learning. To achieve a similar reconstruction level of
$\varepsilon _{\boldsymbol{q}} \approx 0.1$
to a nonlinear autoencoder with
$n_{\boldsymbol{\xi }}=3$
, 85 linear POD modes are needed.
The plateau behaviour for the autoencoder is in part due to the present network architecture shown in table 1, which compresses data with 480 dimensions given by the portion of the convolutional network to be
${\mathcal O}(10^0)$
using multi-layer perceptrons. A similar observation of producing plateau behaviour in capturing dominant large-scale features has recently been found (Fukami, Smith & Taira Reference Fukami, Smith and Taira2025) for extremely strong vortex–airfoil interactions with turbulent vortical structures. It is anticipated that the error would be further reduced once fine-scale structures begin to be captured in the latent space with much larger latent dimensions. Since large-scale motions have already been extracted with
$n_{\boldsymbol{\xi }} = 3$
, the resulting curve for the autoencoder likely exhibits a step-type behaviour in which the plateaued error reduces again once the latent dimension becomes sufficiently large. Hereafter, we choose a latent dimension of 3 for the discussions.
Next, we examine the behaviour of low-dimensionalised transonic airfoil buffet flows in the latent space. The three-dimensional subspace identified by a standard autoencoder (
$\beta =0$
) and the lift-augmented autoencoder (
$\beta =0.03$
and 0.05) is exhibited in figure 5. For all cases, the trajectory for the non-buffet and buffet cases appears in different regions of the latent space. The non-buffet case for
$M_\infty = 0.715$
across the autoencoders is described in a similar way, that is, a small-sized circle-like orbit. This representation likely corresponds to the statistically steady dynamics with small oscillations of aerodynamic responses for the present non-buffet flows, which is evident from the reconstruction of lift response and pressure fields for the non-buffet case presented in Appendix B.
Figure 5. Latent subspace identified by a standard autoencoder (
$\beta =0$
) and the lift-augmented autoencoder (
$\beta =0.03$
and 0.05) coloured by the cases of different Mach numbers
$M_\infty = (0.715, 0.730)$
(a) and the time-varying lift coefficient
$C_L(t)$
(b). The pressure fields over time corresponding to the points
$(i){-}(\textit{iv})$
in the latent space are also shown. The arrow in each subcontour represents the direction of shock movement. The zoomed-in view of wake and the downstream region visualised with a different colour scheme are also depicted to emphasise the interaction between the wake, shock and turbulent boundary layer.
While all the present subspaces capture the relationship between the non-buffet and buffet cases and the characteristics of the non-buffet flow in a low-order manner, the latent expression for the buffet case of
$M_\infty = 0.730$
shows a clear difference by introducing the lift augmentation. This can be observed with the difference in the relative location of the low-dimensionalised flow states
$(i)$
,
$(\textit{iv})$
and
$(v)$
. Here, the shock in the flow field
$(i)$
moves downstream while that in
$(\textit{iv})$
and
$(v)$
moves upstream. The standard model encodes them into nearby regions in the latent space. In contrast, their locations begin to differ due to the lift augmentation. Consequently, the low-order trajectory with
$\beta = 0.05$
presents a geometric structure possessing two wings, while that with
$\beta =0$
and 0.03 rather shows a regular cyclic orbit.
To discuss what physics are captured in the present low-order representation, the temporal behaviour of latent vectors
$\boldsymbol{\xi }(t)$
is compared with the shock location
$x_s(t)$
, the lift coefficient
$C_L(t)$
and the separation height
$h(t)$
, as shown in figure 6. Here, the shock location
$x_s$
is defined as a streamwise position at which the density gradient magnitude
$|\boldsymbol{\nabla }\rho |$
takes the maximum value. The separation height
$h$
is set to be a distance from the wall in which the streamwise momentum
$\rho u$
becomes 0 at
$x/c = 0.6$
in measuring across the wall-normal direction.
Figure 6. Time trace of latent vectors
$\boldsymbol{\xi }$
obtained from nonlinear autoencoders, shock location
$x_s$
, lift coefficient
$C_L$
and separation height
$h$
for the buffet case.
The latent expression from the standard autoencoder emphasises the cyclic behaviour of shock location as the notable peak of latent vectors at
$t\approx 13$
. With the lift incorporation of
$\beta = 0.05$
, the latent vectors possess an additional dominant peak around
$t=20$
, corresponding to the emergence of the wing-type geometric structure in the low-order subspace. While this moment is under-evaluated with
$\beta = 0$
and 0.03, we find that the peak appearing at
$t\approx 20$
coincides with the timing when the separation height
$h$
is increased, as shown in figure 6. This increase in the separation height
$h$
is attributed to the upstream-moving shock wave, not only producing a strong shock due to the increase of relative shock Mack number but also inducing a large separation due to a strong shock adverse pressure gradient. In this manner, the separation height varies depending on the direction of shock movement across the streamwise direction, i.e. relative shock Mach number, in addition to the shock location. Hence, it can be argued that the current lift augmentation well captures the relationship between the shock motion and the aerodynamic responses in its latent representation. Although the flow field data themselves given as the input may also include phase information of the buffet cycle as the phase of shock location matches that of lift response as presented in figure 6, the present observation suggests that providing an aerodynamic variable as an observable output through the subnetwork is essential to identify the physically interpretable subspace. The dependence of the latent representation geometry on the number of training samples and the initial random seed assigned to the weights in the observable-augmented autoencoder is examined in Appendices C and D, respectively.
Note that all the latent spaces across
$\beta$
represent the cyclic transonic buffet dynamics while achieving the same level of reconstruction through the decoder. The latent expression hence becomes stretched by highlighting the events associated with a given observable. In other words, all the latent subspaces are regarded as the compact representation of transonic airfoil buffet flows, although their ways of presentation are different from each other. The present lift augmentation can highlight aerodynamically important events as a manifold geometry while a regular model does not capture them in an interpretable manner, e.g. points (i), (v) and (iv) in figure 5.
4.2. Sparse-sensor reconstruction of transonic airfoil buffet flows via low-order subspace
The current findings through autoencoder compression imply that the right set of variables may capture the essence of transonic airfoil buffet flows. This also makes us anticipate that sparse sensors could also be such a set of low-order variables, thereby achieving sparse-sensor-based reconstruction. Furthermore, of interest here is whether it is possible to gain situational awareness from sparse sensors towards guiding flight operations based on insights into the physically interpretable latent subspace. Based on this viewpoint, we further consider leveraging the discovered low-order subspace for the data-driven global flow field reconstruction.
Since the decoder
${\mathcal F}_d$
provides the pressure field from the latent vector, we aim to estimate the latent vector
$\boldsymbol{\xi }(t)$
from sparse sensors
$\boldsymbol{s}(t)$
by preparing an independent machine-learning model
${\mathcal F}_s$
. By feeding the estimated latent vector
$\hat {\boldsymbol{\xi }}={\mathcal F}_s(\boldsymbol{s}(t))$
into the pretrained decoder
${\mathcal F}_d$
, a pressure field
$\boldsymbol{q}(t)$
is reconstructed, as illustrated in figure 7. The above-mentioned procedure is expressed as
\begin{align} \boldsymbol{q}(t) \approx \hat {\boldsymbol{q}}(t) = {\mathcal F}_d(\hat {\boldsymbol{\xi }}(t)) = {\mathcal F}_d({\mathcal F}_s({\boldsymbol{s}}(t))), \end{align}
with an optimisation for the weights
$\boldsymbol{w}_s$
of the latent vector estimator
${\mathcal F}_s$
:
\begin{align} \boldsymbol{w}_s^* = \textrm {argmin}_{\boldsymbol{w}_s}||\boldsymbol{\xi } - {\mathcal F}_s(\boldsymbol{s;w}_s)||_2^2. \end{align}
We use multi-layer perceptrons (Rumelhart et al. Reference Rumelhart, Hinton and Williams1986) with the units of 14–32–64–128–32–3 across the layers for constructing the latent vector estimator
${\mathcal F}_s$
that maps sensor measurements
$\boldsymbol{s}\in \mathbb{R}^{n_s}$
to
$\boldsymbol{\xi } \in \mathbb{R}^3$
, where
$n_s$
represents the number of sensors. This low-order mapping between sparse sensors and the latent vector enables avoiding a naive learning for the relationship between the sensor inputs and the global field output (Fukami, Fukagata & Taira Reference Fukami, Fukagata and Taira2023; Eldredge & Mousavi Reference Eldredge and Mousavi2025). While such a field reconstruction problem often becomes computationally expensive due to a significant difference in data dimension between the input and output, this approach can save costs by leveraging the pretrained decoder.
Figure 7. Sparse-sensor-based reconstruction via the low-order subspace.
$(a)$
Pressure sensor placements on the wall and responses in time.
$(b)$
The present full state reconstruction combined with a latent vector estimator
${\mathcal F}_s$
and the pretrained decoder
${\mathcal F}_d$
. An example of the reconstructed field with the
$L_2$
error norm
$\varepsilon _{\boldsymbol{q}}$
and reproduced lift coefficient from 14 sensors is shown.
An example of the reconstructed pressure field and estimated lift coefficient from 14 sensors is shown in figure 7. Here, these sensors are placed along the airfoil surface in an equispaced manner, enabling a comprehensive analysis of data-driven sensor reduction performed later. We use the latent vector
$\boldsymbol{\xi }$
extracted from the lift-augmented autoencoder with
$\beta = 0.05$
. In addition to the flow state including wake shedding and shock location, the lift response is accurately reproduced from the sensor readings. As implied through the discovery of a low-dimensional subspace, sparse-sensor-based reconstruction is indeed possible for the present transonic airfoil buffet flow.
Furthermore, the minimal number and appropriate placements of sensors can be quantified with the latent vector estimator trained with 16 sensors above and the lift subnetwork prepared for subspace identification. This is achieved by performing a sensitivity analysis between a machine-learning estimate and a given input (Morimoto et al. Reference Morimoto, Fukami, Zhang and Fukagata2022; Chen et al. Reference Chen, Kaiser, Hu, Rival, Fukami and Taira2024). Considering the gradient between the sensor input
$\boldsymbol{s}$
and the output of machine-learning model
$\hat {\boldsymbol{z}}$
,
$\boldsymbol{\gamma }(t) = \partial {\hat {\boldsymbol{z}}}(t)/\partial \boldsymbol{s}(t)$
, the importance of each sensor for estimation, i.e. sensitivity
$S(t)$
, is quantified as a weighted input:
\begin{align} S_j(t) = \gamma _j(t)s_j(t), \end{align}
where
$j$
is an index of pressure sensor
$s_j$
. As an output variable
$\hat {\boldsymbol{z}}$
, the estimated lift coefficient
$\hat {C_L}$
and latent vector
$\hat {\boldsymbol{\xi }}$
are considered.
The sensitivity
$S$
with respect to the lift and latent vectors is shown in figure 8. In addition to the time trace, the absolute time-averaged values are also presented to further gain insights into the general trend of sensitivities over the buffet cycle. As the present autoencoder is trained such that the latent vector extracts the flow features associated with the lift coefficient, both sensitivity maps present a consistent trend in the direction of time and sensor index. Note that high-frequency fluctuations of the sensitivity are caused because the present sensitivity is calculated using the estimate by the machine-learned model, which includes the estimation error varying in time. We have confirmed that the rank of sensor importance is not affected by such high-frequency fluctuations through a preliminary analysis by taking moving averages.
Focusing on the lift estimation, the sign of sensor sensitivity seems to be opposite between the suction (index 2–7) and pressure (index 9–14) sides due to their different role in contributing to lift. The responsible sensors are clearly shown where
$\overline {|S|}\gt 0.02$
: sensor 1 at the leading edge, sensor 8 at the trailing edge, sensors 5, 6 and 7 placed on the suction side and sensors 10 and 13 placed on the pressure side. In turn, less sensitive sensors are also identified. Sensors 2 and 3, placed in the supersonic region, particularly show very small
$\overline {|S|}$
, likely because their sensor signals are less affected by the shock movement compared with others according to figure 7
$(a)$
.
The present sensitivity information is further leveraged to reduce the number of sensors for subspace estimation. Let us consider removing the sensors following the rank of absolute time-averaged sensitivity
$\overline {|S|}$
so that sensors with small contribution to estimation are eliminated while keeping the highly contributing sensors. The relationship between the number of sensors
$n_s$
and the estimation errors is shown in figure 9. The error for the latent vector and lift response is depicted on a single plot. The error curves are flat between
$n_s=7$
and 14, exhibiting that accurate estimation of lift and flow fields is achieved up to
$n_s=7$
. This is also evident from the reconstructed flow field shown in figure 9, and the estimated latent subspace and lift response presented in figure 10.
Figure 9. Sensitivity-based sensor reduction. The relationship between the number of sensors
$n_s$
and the estimation errors of latent vectors
$\varepsilon _{\boldsymbol{ \xi }}$
and lift
${\varepsilon }_{C_L}$
is shown. The reconstructed fields are presented with the
$L_2$
error norm
$\varepsilon _{\boldsymbol{q}}$
underneath each contour.
Figure 10. The estimated latent subspace and estimated lift coefficient across
$n_s$
reduced via the sensitivity analysis.
All seven sensor readings here report the absolute time-averaged value of
$\overline {|S|}\gt 0.02$
, exhibiting a relatively larger value compared with other less-contributing sensors observed in figure 8. Once the sensors are further removed, the error of the latent vector starts to increase. However, the error curve for the lift coefficient presents a slower slope at
$n_{s}\leqslant 6$
compared with that for the latent space. In fact, the lift response at
$n_s = 3$
still exhibits reasonable agreement with the reference data. This is likely because a global quantity of lift coefficient aggregating the flow information over the entire body is easier to estimate than the latent subspace, a representation of the whole flow field itself.
To examine the dependence of reconstruction performance on the choice of sensor-selection technique and compression approach, we further consider the QR pivot-based sensor placement optimisation (Manohar et al. Reference Manohar, Brunton, Kutz and Brunton2018) with
$n_s = 7$
. Their approach finds the optimal sensor locations through QR factorisation with column pivoting applied to the POD bases. Further details on this linear technique are provided in Manohar et al. (Reference Manohar, Brunton, Kutz and Brunton2018). The original placements of sensors before performing the QR pivot-based reduction are constrained on the wing surface and the same as those used in the autoencoder-based analysis shown in figure 7
$(a)$
. Here, three approaches are considered:
-
(i) Estimate the three-dimensional latent vectors
$\boldsymbol{\xi }$
based on the sensors reduced via the gradient sensitivity and decode a flow using the nonlinear decoder
${\mathcal F}_d$
(the original formulation). -
(ii) Estimate the dominant three POD coefficients
$\boldsymbol{a}$
based on the sensors reduced via the QR pivot and decode a flow with POD modes. -
(iii) Estimate the three-dimensional latent vectors
$\boldsymbol{\xi }$
based on the sensors reduced via the QR pivot and decode a flow using the nonlinear decoder
${\mathcal F}_d$
.
For fair comparison, we use the same multi-layer perceptron architecture for all three cases in estimating the latent vectors and the three dominant POD coefficients. The flow fields are then decoded using the nonlinear decoder
${\mathcal F}_d$
or POD modes
$\boldsymbol{\varPhi }$
. While (4.1) is applied for cases (i) and (iii), case (ii) with the POD multi-layer perceptron model with QR pivot-based sensor reduction is expressed as
\begin{align} \boldsymbol{q}(t) \approx \boldsymbol{\varPhi }\hat {\boldsymbol{a}}(t) = \boldsymbol{\varPhi }({\mathcal F}_{s}(\boldsymbol{ s}(t))). \end{align}
Let us compare the reduced sensor placements in figure 11. Four sensors (index 5, 7, 8 and 13), reporting high
$\overline {|S|}$
with the gradient-based approach, are commonly kept with both sensor-reduction methods through the reduction process. However, the remaining three sensors are placed in a different way. While sensors chosen by the QR pivot are grouped with neighbours (
$s_3$
–
$s_5$
,
$s_7$
–
$s_8$
and
$s_{13}$
–
$s_{14}$
), the gradient-based method seems to attempt to cover the entire wing surface. This result suggests that the dominant features captured by both POD and the autoencoder make the reduction approach keep the common four sensors, while the subdominant characteristics that are better compressed with the nonlinear autoencoder cause the difference in the location of the remaining three sensors.
Figure 11. Dependence of sparse-sensor reconstruction performance on the choice of sensor-reduction technique and compression approach with
$n_{\boldsymbol{\xi }} = 7$
.
The reconstruction fields with cases (i)–(iii) are also shown in figure 11. When using the nonlinear decoder, the reconstruction with the gradient-based approach is slightly better than that with the QR pivot. These accurate reconstructions suggest that the high error for case (ii) is primarily due to the use of linear POD modes as a decoder rather than the sensor placements determined by the QR pivot. We note that the error for case (iii) of the QR pivot and the autoencoder latent variables starts to increase with
$n_s \leqslant 6$
, similarly to case (i) using the gradient-based method, although not shown. While both the gradient-based method and the QR pivot currently provide a similar level of sensor reduction performance, they could be further improved by accounting for redundancy between sensor readings, which can be quantified with inter-correlations and mutual information.
The present analysis is focused on transonic airfoil buffet flow at
${Re} = 3\times 10^6$
. While the current Reynolds number may be higher than those often considered for numerical and data-driven analyses in the community, this still resides in the range of wind-tunnel-scale conditions. Of particular interest here is whether the current model trained at a wind-tunnel-scale Reynolds number can be applied to a scenario under a real aircraft operation level of Reynolds number. In response, this study lastly evaluates the applicability of the present method to a transonic airfoil buffet flow at
${Re}=3\times 10^7$
with
$M_\infty = 0.730$
.
The wall-modelled LES is performed for the case with
$({Re},M_\infty) = (3\times 10^7, 0.730)$
at
$\alpha = 3.5^\circ$
, as presented in figure 12
$(a)$
. There is a self-sustained shock buffet cycle that produces almost the same frequency and oscillation amplitude of aerodynamic coefficients as those for
${Re}=3\times 10^6$
, as seen in figure 12
$(b)$
. The difference in the flow between the two Reynolds numbers is examined with the instantaneous streamwise velocity
$u$
sampled at the same phase
$t/T=0.70$
, where
$T$
denotes the time window across the buffet cycle, as depicted in figure 12
$(c)$
. The shock location moves downward and the separation height becomes greater on increasing the Reynolds number, strengthening the shock wave accompanied by a large adverse pressure gradient and triggering a larger separation, which is also evident from the time-averaged flow fields shown in figure 12
$(d)$
. Due to the trade-off relationship between the suppression effect of separation due to the increment of Reynolds number and the separation induced by the strong shock wave, the resulting shock-wave oscillation is sustained.
Figure 12.
$(a)$
An instantaneous snapshot of transonic airfoil buffet flows at
${Re} = 3\times 10^7$
visualised by the isocontours of the
$Q$
-criterion. Comparison of
$(b)$
lift coefficient and
$(c)$
instantaneous streamwise velocity fields sampled at
$t/T = 0.70$
with
${Re}=3\times 10^6$
and
${Re}=3\times 10^7$
.
$(d)$
Time- and spanwise-averaged streamwise velocity fields at
${Re}=3\times 10^6$
and
${Re}=3\times 10^7$
.
Let us finally apply the present sensor-based reconstruction model trained at
${Re} = 3\times 10^6$
to the level of real aircraft operation at
${Re} = 3 \times 10^7$
, as shown in figure 13. Here, we use the latent vector estimator
${\mathcal F}_p$
trained with seven sensors following the observation in figures 9 and 10. The reconstructed fields exhibit a smaller height of shock compared with the reference snapshots as such a shock with a greater height does not appear in the training data at
${Re}=3\times 10^6$
. However, it is worth noting that the shock locations of the machine-learning reconstruction are constantly evaluated forward compared with that of the reference at
${Re} = 3\times 10^7$
. Since the shock moves downward on increasing the Reynolds number while keeping its phase as presented in figure 12, this constant shift indicates that the present model may correctly capture the phase information across the buffet cycle even at the current real-aircraft-level Reynolds number. This is further evident from the reproduced lift response. While the magnitude of lift is underestimated due to the difference in Reynolds number between the training and testing data, the temporal trend of the lift signal accurately matches the reference. This observation suggests that nonlinear machine learning can be transferred to scenarios where the characteristics of variables of interest remain relatively consistent across different Reynolds numbers.
Figure 13. Application of the sparse-sensor reconstruction model trained at
${Re}=3\times 10^6$
to a flow at the level of a real aircraft operation of
${Re}=3\times 10^7$
. The reconstructed pressure field and lift response are shown.
5. Concluding remarks
This study examined a low-dimensional representation of transonic airfoil buffet flows at a high Reynolds number with nonlinear machine learning. Wall-modelled LES of flow over the OAT15A supercritical airfoil at Mach numbers
$M_\infty = 0.715$
and 0.730, corresponding to non-buffet and buffet conditions, were performed at a chord-based Reynolds number of
${Re} = 3\times 10^6$
to generate the datasets used in the present data-driven analysis. To derive a low-order expression from the data, we considered nonlinear lift-augmented autoencoder-based compression. We found that there exists a compact three-dimensional latent subspace reflecting the characteristics of transonic airfoil buffet flow. The discovered representation captures key flow features, including shock movement and shock-induced separation, in a reduced-order manner.
Based on these physical implications, sparse-sensor-based reconstruction via the learned representation was further performed. Equipped with the sensitivity analysis, the sensor configuration required for accurately reproducing aerodynamic responses can be determined. Finally, the model trained at a wind-tunnel-scale Reynolds number of
${Re} = 3\times 10^6$
was assessed at a real aircraft operational level of
${Re} = 3\times 10^7$
, revealing its ability to reasonably predict phase dynamics of aerodynamic loads from sparse sensors.
While we considered two configurations of buffet/non-buffet conditions at a fixed angle of attack, additional cases with a range of different parameters, including angle of attack, Reynolds number and Mach number, would be needed to fully characterise the whole picture of buffet onset. Although it is anticipated that a low-order subspace capturing the difference in such parameters and the occurrence of transonic buffet could be identified, a major challenge arises from a collection of datasets through large-scale simulations. From this aspect, one can consider data fusion between LES, unsteady Reynolds-averaged Navier–Stokes and experimental measurements to supplement the pros and cons across different datasets with each other in extracting a low-order submanifold with observable-augmented learning (Fukami & Taira Reference Fukami and Taira2025).
The present analysis reveals that three latent variables are needed to represent transonic airfoil buffet flows. Although buffet dynamics is often modelled as a self-sustained oscillator subjected to stochastic forcing (Feldhusen-Hoffmann et al. Reference Feldhusen-Hoffmann, Lagemann, Loosen, Meysonnat, Klaas and Schröder2021; Sansica et al. Reference Sansica, Loiseau, Kanamori, Hashimoto and Robinet2022; Crouch, Ahrabi & Kamenetskiy Reference Crouch, Ahrabi and Kamenetskiy2024), our findings suggest the necessity of a third dimension. This additional latent dimension likely corresponds to aerodynamic phenomena related to the separation height, according to the observation in figure 6. To characterise this dynamics more precisely, it is essential to investigate the nonlinear modal structures associated with each latent variable. This can be achieved by integrating mode-decomposing autoencoders (Fukami et al. Reference Fukami, Nakamura and Fukagata2020; Murata et al. Reference Murata, Fukami and Fukagata2020), which we plan to pursue in future work.
With the present formulation of observable-augmented learning, users have to choose an appropriate observable from the candidates, and it currently takes some level of computational effort to find a physically relevant subspace. Note, however, that the former point of the non-automatic process enables us to have the opportunity to incorporate physical or mathematical knowledge based on what we would like to associate with, while the computational cost for the latter point is still manageable as the degree of freedom of observables is much less than that of the original simulations. A series of recent studies on observable-augmented manifold learning have revealed that an appropriate choice of observable assists in compactly extracting physics for a range of unsteady flow scenarios including vortex–airfoil interactions (Fukami & Taira Reference Fukami and Taira2023; Fukami et al. Reference Fukami, Nakao and Taira2024; Liu et al. Reference Liu, Beckers and Eldredge2025; Mousavi & Eldredge Reference Mousavi and Eldredge2025), vehicle aerodynamics (Tran et al. Reference Tran, Fukami, Inada, Umehara, Ono, Ogawa and Taira2024), turbulent boundary layers (Fukami & Taira Reference Fukami and Taira2025) and roughness turbulence (Nair et al. Reference Nair, Kunz, Zhang and Yang2025), enabling the enjoyment to learn physics from data for fluid mechanicians. More broadly, an ‘observable’ here does not need to be a variable. Some applied mathematical techniques, such as persistent homology (Smith et al. Reference Smith, Fukami, Sedky, Jones and Taira2024) and information theory (Fukami & Araki Reference Fukami and Araki2025), can also be considered as observables depending on the physics of interest. Hence, adding an observable may be regarded as one approach to support data-driven analysis for unsteady flows.
Based on the current findings considering flows around a wing, the applicability of the present data-driven subspace identification to transonic buffet conditions around a full-aircraft configuration would also be of interest (Asada et al. Reference Asada, Tamaki, Takaki, Yumitori, Tamura, Hatanaka, Imai, Maeyama and Kawai2023; Tamaki & Kawai Reference Tamaki and Kawai2024). For such cases, a combination of linear, scalable compression techniques such as POD and the present observable augmentation would be helpful to reduce the computational burden (Linot & Graham Reference Linot and Graham2023; Tran et al. Reference Tran, Fukami, Inada, Umehara, Ono, Ogawa and Taira2024; Asada & Kawai Reference Asada and Kawai2025). The current study may offer a new perspective on the analysis and determination of flight envelopes towards next-generation air vehicle operations.
Acknowledgements
We thank H. Itsui and T. Hattori for fruitful discussions.
Funding
We acknowledge support from MEXT as a programme for promoting research on the Supercomputer Fugaku (research toward DX in aircraft development led by digital flight, JPMXP1020230320). K.F. acknowledges support from the JSPS KAKENHI (grant no. JP25K23418), the JST PRESTO (grant no. JPMJPR25KA), and the MEXT Coordination Funds for Promoting Aerospace Utilization (grant no. JPJ000959). Y.I. acknowledges support from JSPS Grant-in-Aid for JSPS Fellows (JP23KJ0167). S.M. acknowledges support from the Japan Science and Technology Agency (JST) SPRING (JPMJSP2114).
Declaration of interests
The authors report no conflict of interest.
Appendix A. Training procedures and L-curve analysis
Here, we provide details on training procedures and the choice of weighting parameter
$\beta$
in (3.2) for the present observable-augmented nonlinear autoencoder. The Adam optimiser (Kingma & Ba Reference Kingma and Ba2014) with the default parameter sets in Keras is used to update the weights through machine-learning training. The maximum number of training iterations is set to be 50 000, while early stopping (Prechelt Reference Prechelt1998) with the criterion of a series of 100 continuous epochs is employed to avoid overfitting. We use 70 % of the datasets for training and the remaining 30 % are prepared for validation. The number of grid points
$(N_x,N_y) = (480,200)$
for the current data-driven analysis is determined such that the shock can be represented without exhibiting any discontinuous artefacts, which is evident from a comparison with other resolutions
$(N_x,N_y) = (240,100)$
and
$(960,400)$
shown in figure 14. In using the entire data set of 24 100 snapshots, the training process takes approximately two hours in an NVIDIA A100 GPU environment, and the inference time for each snapshot is 0.003 seconds.
Figure 14. Pressure field interpolated onto a spatially uniform grid with a resolution of
$(n_x,n_y) = (240,100)$
,
$(480,200)$
and
$(960,400)$
.
Figure 15. L-curve analysis for the present observable-augmented autoencoder.
The weighting parameter
$\beta$
in (3.2) is determined based on the L-curve analysis (Hansen & O’Leary Reference Hansen and O’Leary1993) that finds an appropriate regularisation parameter of the cost function, as shown in figure 15. We consider nine different values of
$\beta$
(0.005, 0.01, 0.03, 0.05, 0.1, 0.5, 1, 5 and 10). The cases with
$\beta =0.03$
and 0.05, providing low reconstruction errors for the lift response and the pressure field in a balanced manner, are chosen for the present analysis.
Appendix B. Reconstructed variables for the non-buffet case
Figure 16. Decoded lift coefficient and pressure fields via a lift-augmented autoencoder with
$\beta = 0.05$
for the non-buffet case with
$M_\infty = 0.715$
. The flow fields (a–d) correspond to those shown in figure 2. The whole (bottom left) and zoom-in (bottom right) views of the pressure coefficient
$C_p$
on the wing surface for the snapshots (a–d) are also presented.
We exhibit in figure 16 the decoded lift coefficient and pressure fields obtained from the present lift-augmented autoencoder with
$\beta = 0.05$
. While achieving accurate estimation of the lift coefficient, the reconstructed pressure fields are in agreement with the reference data, reporting less than 8 %
$L_2$
norm error over time. Along with the observation of a small-sized cyclic orbit in figure 5 and small oscillations of the pressure coefficient
$C_p$
in figure 16, it is argued that the present model well represents statistically steady dynamics of the non-buffet case in the identified low-order subspace.
Appendix C. Effect of the number of training samples
We examine the dependence of reconstruction performance and latent space geometry on the number of training snapshots by subsampling them to be 25 % and 50 % of the original amount, as presented in figure 17. We use the same autoencoder network with the same weighting parameter
$\beta$
of 0.05 as that used in the original case. The case with 50 % presents a similar result to the original model. However, the latent geometry with the 25 % case starts to deform from the original shape, although there still exists a two-wing-shaped submanifold. Since the dimensionality in the subspace is determined based on whether the given data cover the entire space of the attractor or not, rather than the number of snapshots, the latent dimension is not affected for this analysis, in which we subsample the snapshots while keeping the entire time window.
The deformation of latent space geometry is caused by several factors. There may exist an optimal weighting parameter
$\beta$
for the case with 25 % data. Furthermore, the primary reason is likely less temporal density of data compared to the original case, which may cause miscapturing of some events over the buffet cycle. A sufficient temporal resolution is needed to obtain an interpretable low-order subspace in a data-driven manner.
Appendix D. Uniqueness of latent representation
Figure 17. Dependence of field reconstruction performance and latent space geometry on the number of training snapshots.
Figure 18. Dependence of the latent geometry on the initial random seed assigned to the weights in the observable-augmented autoencoder.
To consider the uniqueness of latent representation, we examine the dependence of the latent geometry on the initial random seed assigned to the weights in the observable-augmented autoencoder, as shown in figure 18. A weighting parameter
$\beta$
of 0.05 is used for this analysis. The model exhibits reasonable robustness across the three runs, presenting a two-wing-shaped submanifold while distinguishing the non-buffet and buffet cases in a low-order manner. Although this paper only considers a single-network configuration of observable-augmented autoencoder, the results above indicate that a model may provide a similar wing-shaped geometry over a range of the network capacities by choosing the optimal value of
$\beta$
through the L-curve analysis.
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