2.1. Physical model and resolvent formulation

We consider the compressible flow over a flat plate governed by the linearised Navier–Stokes (LNS) equations in conservative form. The governing system is derived by linearising the full compressible equations about a two-dimensional steady base flow $\overline {\boldsymbol{q}}$ . The resulting conservative perturbation state vector, $\boldsymbol{q}'$ , comprises fluctuations of density, streamwise, wall-normal and spanwise momentum, and total energy. The LNS system can be expressed in compact form as

(2.1) \begin{align} \frac {\partial \boldsymbol{q}'}{\partial t}= \mathcal{L}(\overline {\boldsymbol{q}}) \boldsymbol{q}'+ \boldsymbol{f}', \end{align}

where $\mathcal{L}$ denotes the linearised Navier–Stokes operator, including both inviscid and viscous contributions evaluated about the base flow, and $\boldsymbol{f}'$ represents an external forcing term.

Assuming harmonic disturbances of the form $ (\boldsymbol{q}',\boldsymbol{f}')=(\hat {\boldsymbol{q}},\kern2.5pt \hat {\boldsymbol{\kern-2.5pt f}}) \,\text{e}^{{i}(\beta z+\omega t)},$ where $\omega$ and $\beta$ denote the angular frequency and spanwise wavenumber, respectively, substitution into (2.1) yields the harmonic LNS system

(2.2) \begin{align} \left ( \mathrm{i}\omega \boldsymbol{I}-\mathcal{L}_{\beta } \right ) \hat {\boldsymbol{q}}=\kern2.5pt \hat {\boldsymbol{\kern-2.5pt f}}, \end{align}

where $\mathcal{L}_{\beta }$ denotes the spanwise wavenumber-dependent harmonic LNS operator.

After spatial discretisation, equation (2.2) can be written as $\boldsymbol{A}_{\omega ,\beta }\hat {\boldsymbol{q}}=\kern2.5pt \hat {\boldsymbol{\kern-2.5pt f}}$ , where $\boldsymbol{A}_{\omega ,\beta }$ denotes the complete discrete harmonic LNS operator. The corresponding resolvent operator is defined as

(2.3) \begin{align} \hat {\boldsymbol{q}}=\mathcal{H}(\omega ,\beta ,\overline {\boldsymbol{q}}) \kern2.5pt \hat {\boldsymbol{\kern-2.5pt f}}=\boldsymbol{A}_{\omega ,\beta }^{-1} \kern2.5pt \hat {\boldsymbol{\kern-2.5pt f}}, \end{align}

where $\mathcal{H}$ represents the forcing-to-response mapping associated with the base flow $\overline {\boldsymbol{q}}$ .

In the classical resolvent framework, the optimal forcing and response are obtained from the leading singular vectors of the resolvent operator $\mathcal{H}$ through singular-value decomposition (SVD), while the corresponding singular value represents the maximum amplification gain. The resulting optimal forcing response pairs constitute the reference datasets used throughout this work. Throughout the paper, $\boldsymbol{A}_{\omega ,\beta }$ denotes the discrete harmonic LNS operator, whereas $\mathcal{H}=\boldsymbol{A}_{\omega ,\beta }^{-1}$ denotes the corresponding resolvent operator. The discretised harmonic LNS operator and the reference resolvent datasets are constructed using the BROADCAST-based resolvent analysis framework developed by Poulain et al. (Reference Poulain, Content, Sipp, Rigas and Garnier2023).

The numerical cases investigated in this study cover supersonic and hypersonic boundary layers, including laminar, turbulent and actively controlled configurations. Table 1 summarises the cases in the present study. The first six rows correspond to neural operator training or evaluation tasks, while the CFD simulations case is used as a realisable control validation to connect the inverse forcing prediction with boundary actuation.

Table 1.

Summary of numerical cases and their roles in this study.

Table 1. Long description

The table has seven rows and four columns. The columns are labeled Case, Flow condition, Main variation, and Task. The rows are labeled with different cases: F, S, M, T, BS, PI, and CFD simulations. Each row provides details about the flow condition, main variation, and task for each case. For example, Case F involves a laminar boundary layer with a Mach number of 4.5 and focuses on forward TS-wave prediction. Case S also involves a laminar boundary layer with a Mach number of 4.5 but focuses on second-mode prediction. Case M involves a laminar boundary layer with Mach numbers ranging from 3.0 to 4.5 and focuses on Mach-number interpolation. Case T involves a turbulent boundary layer with a Mach number of 2.25 and focuses on data-driven inverse forcing reconstruction. Case BS involves a blowing/suction boundary layer with a Mach number of 5.86 and focuses on data-driven inverse forcing reconstruction. Case PI involves spatially attenuated instability waves with a Mach number of 4.5 and focuses on physics-informed inverse forcing. The CFD simulations case involves upstream forced TS response and focuses on realisable wall control validation.

Detailed numerical parameters, data partitions, attenuation target sampling settings and implementation details are provided in Appendix A. A supplementary wall-clock efficiency comparison is given in Appendix B.

2.2. The ResFormer architecture

The overall ResFormer framework is illustrated in figure 1. Starting from a prescribed compressible boundary layer configuration, the base flow $\overline {\boldsymbol{q}}$ and modal parameters $(\omega ,\beta )$ are used to construct harmonic resolvent data. In this study, the reference resolvent fields are generated following the BROADCAST-based resolvent analysis framework (Poulain et al. Reference Poulain, Content, Sipp, Rigas and Garnier2023). The resulting data include forcing response pairs $(\kern2.5pt \hat {\boldsymbol{\kern-2.5pt f}},\hat {\boldsymbol{q}})$ and optimal mode targets $(\hat {\boldsymbol{q}}_{\textit{opt}},\kern2.5pt \hat {\boldsymbol{\kern-2.5pt f}}_{\textit{opt}})$ . For inverse control-oriented studies, prescribed attenuation targets $\hat {\boldsymbol{q}}_{\textit{target}}$ are subsequently constructed from these reference responses using spatial masks. The neural operator backbone is adapted from the physics attention Transformer architecture of Transolver (Wu et al. Reference Wu, Luo, Wang, Wang and Long2024).

Figure 1.

Overall ResFormer framework for forward and inverse resolvent modelling. The physical set-up and base flow are first used to generate harmonic resolvent datasets. ResFormer then learns three parameter-conditioned operator mappings: forcing-to-response approximation, rapid optimal mode prediction and response-to-forcing reconstruction for prescribed target responses.

ResFormer is designed to learn parameter-conditioned field-to-field mappings. Compared with conventional convolution-based networks, the physics attention mechanism first aggregates grid point features into physics-aware tokens, applies attention in a latent physical space and then projects the learned information back to the physical domain. This structure is suitable for resolvent mode prediction because instability waves involve long-range streamwise correlations, multi-scale spatial structures and strong dependence on modal parameters.

As shown in figure 1, the framework contains three operator learning pathways. The first pathway approximates the forward resolvent mapping from forcing to response,

(2.4) \begin{align} \mathcal{G}_{\textit{op}}: ( \kern2.5pt \hat {\boldsymbol{\kern-2.5pt f}}, \omega , \beta ) \rightarrow \hat {\boldsymbol{q}} . \end{align}

This pathway directly learns the forcing-to-response relation associated with the resolvent operator. It is used to evaluate whether the neural operator can reproduce the linear response of the flow to prescribed harmonic forcing.

The second pathway performs rapid optimal mode prediction,

(2.5) \begin{align} \mathcal{G}_{\textit{pred}}: ( \overline {\boldsymbol{q}}, \omega , \beta ) \rightarrow \big( \hat {\boldsymbol{q}}_{\textit{opt}}, \kern2.5pt \hat {\boldsymbol{\kern-2.5pt f}}_{\textit{opt}} \big). \end{align}

Instead of performing a new SVD-based resolvent computation for each parameter combination, this pathway maps the base-flow state and modal parameters directly to the corresponding optimal response and forcing. It therefore provides a fast surrogate for constructing high-resolution modal databases.

The third pathway is a response-to-forcing reconstruction model, denoted by

(2.6) \begin{align} \mathcal{G}_{\textit{in}v}: \big( \hat {\boldsymbol{q}}_{\textit{target}}, \omega , \beta \big) \rightarrow \kern2.5pt \hat {\boldsymbol{\kern-2.5pt f}}_{\textit{in}v} . \end{align}

This pathway learns a surrogate response-to-forcing mapping for control-oriented guidance. When the prescribed target response is the optimal resolvent response, $\kern2.5pt \hat {\boldsymbol{\kern-2.5pt f}}_{\textit{in}v}$ corresponds to the associated optimal forcing $\kern2.5pt \hat {\boldsymbol{\kern-2.5pt f}}_{\textit{opt}}$ . When the target is a spatially attenuated response, $\kern2.5pt \hat {\boldsymbol{\kern-2.5pt f}}_{\textit{in}v}$ represents the compatible volumetric forcing under the discrete harmonic LNS formulation.

All supervised models employ a relative $L_2$ objective. The detailed architecture and optimisation settings, the precise samplewise averaging convention and the random seed sensitivity analysis are provided in Appendix A.2.

3.1. Capturing instability modes

This section validates the capability of ResFormer to capture instability mechanisms across distinct spectral regimes, ranging from low-frequency Tollmien–Schlichting (TS) waves to high-frequency second-mode (Mack mode) instabilities. Case F evaluates low-frequency TS-wave responses by varying $\beta$ at fixed $\omega$ , whereas Case S evaluates high-frequency second-mode responses by varying $\omega$ at $\beta =0$ . The optimal forcing at $ \textit{Ma}=4.5$ and $ \textit{Re}_u=3.4\times 10^6$ serves as the model input, with the corresponding optimal response designated as the prediction target.

To quantify the prediction accuracy, we use a consistent global $L_2$ response norm for both the ground truth (GT) and the prediction:

(3.1) \begin{align} \mu _{L_2}= \left \| \hat {\boldsymbol{q}} \right \|_{\varOmega }= \left ( \int _{\varOmega } \sum _{i=1}^{5} | \hat {q}_i |^2 \,\mathrm{d}\varOmega \right )^{\!1/2} \!, \end{align}

where $\hat {q}_i$ denotes the conservative variable components of $\hat {\boldsymbol{q}}$ . This metric is used because the neural network predicts conservative variables directly. It provides a consistent amplitude measure for comparing predicted and reference response fields in the same variable representation.

Unless otherwise stated, the reported prediction error is the mean samplewise relative $L_2$ error defined in (A2). The modal field errors reported in figures 2(c) and 3(c) are evaluated using spatial quadrature, with the arbitrary global modal sign aligned where applicable. Random seed variability for representative forward and inverse tasks is reported in table 8.

As shown in figures 2 and 3, ResFormer accurately reconstructs the global response amplitude, mode profile, peak location and spatial structure. Figures 2(a) and 3(a) compare the reference and predicted global response norms, whereas figures 2(c) and 3(c) report the spatially weighted relative $L_2$ error of the complete five-component response field at the test points distributed throughout the training parameter range. For Case F, the mean relative error of the predicted response norm $\mu _{L_2}$ is $0.16\,\%$ , and the mean relative $L_2$ field error is $0.0096$ . The near-wall TS wavepacket and its streamwise evolution are accurately reproduced. For Case S, the corresponding mean errors are $0.034\,\%$ for $\mu _{L_2}$ and $0.0026$ for the complete response field. The predicted field also reproduces the acoustic mode structures trapped inside the boundary layer. The agreement at the distributed test points demonstrates that ResFormer maintains good interpolation accuracy throughout the considered training parameter ranges.

Figure 2.

Forward prediction for Case F (TS instability). (a) Global response norm $\mu _{L_2}$ as a function of spanwise wavenumber $\beta$ at $\omega =1.0\times 10^{-5}$ . The black dashed line denotes the reference result, and the red hollow circles denote the ResFormer predictions at the test points distributed throughout the training parameter range. (b) Reference and predicted density mode profiles at $ \textit{Re}_y=1.12\times 10^4$ for the representative interpolation sample $(\omega =1.0\times 10^{-5},\ \beta =2.2\times 10^{-4})$ . (c) Spatially weighted relative $L_2$ error of the complete five-component response field at the distributed test points; the black dashed line denotes the mean value. (d–f) Reference density fluctuation field, prediction and pointwise absolute error field for the same representative sample.

Figure 3.

Forward prediction for Case S (second-mode instability) at $ \textit{Ma}=4.5$ and $\beta =0$ . (a) Global response norm $\mu _{L_2}$ as a function of frequency $\omega$ . The black dashed line denotes the reference result, and the red hollow circles denote the ResFormer predictions at the test points distributed throughout the training parameter range. (b) Reference and predicted density mode profiles at $ \textit{Re}_y=8.78\times 10^3$ for the representative interpolation sample $\omega =3.1\times 10^{-4}$ . (c) Spatially weighted relative $L_2$ error of the complete five-component response field at the distributed test points; the black dashed line denotes the mean value. (d–f) Reference density fluctuation field, prediction and pointwise absolute error field for the same representative sample.

Figure 3. Long description

Panel A: A line graph shows the global response norm as a function of frequency. The x-axis is labeled with frequency (fω) in units of 10^-4, and the y-axis is labeled with the global response norm (μL2) in units of 10^6. The black dashed line represents the ground truth, and the red hollow circles denote the ResFormer predictions at test points. Panel B: A line graph displays the reference and predicted density mode profiles. The x-axis is labeled with Re_x in units of 10^5, and the y-axis is labeled with density (ρ). The blue line represents the reference, and the red line represents the prediction. Panel C: A scatter plot shows the spatially weighted relative error of the complete five-component response field at distributed test points. The x-axis is labeled with frequency (fω) in units of 10^-4, and the y-axis is labeled with the relative error (εrelL2). The red circles denote test samples, and the black dashed line denotes the mean value. Panel D: A heatmap shows the reference density fluctuation field. The x-axis is labeled with Re_x in units of 10^5, and the y-axis is labeled with Re_y. The color scale ranges from -600 to 600. Panel E: A heatmap shows the predicted density fluctuation field. The x-axis is labeled with Re_x in units of 10^5, and the y-axis is labeled with Re_y. The color scale ranges from -600 to 600. Panel F: A heatmap shows the pointwise absolute error field. The x-axis is labeled with Re_x in units of 10^5, and the y-axis is labeled with Re_y. The color scale ranges from 0 to 2.

As an additional architecture baseline, Case S was evaluated using the Fourier neural operator (FNO) of Li et al. (Reference Li, Kovachki, Azizzadenesheli, Liu, Bhattacharya, Stuart and Anandkumar2021), implemented with the official Neural Operator package (Kossaifi et al. Reference Kossaifi2025). In a separate fixed budget comparison, both FNO and ResFormer were trained for 5000 epochs using the same dataset, train test split, preprocessing, optimiser settings and unweighted samplewise relative $L_2$ loss. Under this common budget, ResFormer ( $0.274$ M parameters) achieved a relative $L_2$ error of $1.443\,\%\pm 0.069\,\%$ , whereas FNO ( $0.505$ M parameters) achieved $2.713\,\%\pm 0.128\,\%$ . Thus, ResFormer attained a lower fixed budget error despite using fewer trainable parameters. Further details are provided in table 9.

3.2. Mach-number interpolation

Following the validation of frequency-regime instabilities, we assess the model’s interpolation capability under Mach-number-induced base-flow variations. For Case M, the Mach number varies within $ \textit{Ma} \in [3.0, 4.5]$ while keeping $ \textit{Re}_u = 3.4 \times 10^6$ , $\omega = 1.0 \times 10^{-5}$ and $\beta = 1.0 \times 10^{-5}$ fixed.

As shown in figure 4, ResFormer smoothly interpolates the variation of the instability response with Mach number. For the displayed interpolation test points, the mean relative error of the global response norm $\mu _{L_2}$ is $0.50\,\%$ , while the mean spatially weighted relative $L_2$ error of the complete five-component response field is $0.0231$ . The predicted response amplitude curve agrees closely with the reference data. Despite the variations in boundary layer thickness and mode structure induced by compressibility effects, the model accurately preserves the spatial distribution and streamwise evolution of the momentum fluctuation $\widehat {\rho u}$ . These results demonstrate good interpolation accuracy within the considered Mach-number range and suggest that a relatively sparse sampling of the flow parameter space may be used to construct high-resolution stability databases at reduced computational cost.

Additional norm-sensitive effective gain and response alignment diagnostics for Case M are reported in Appendix C.2.

Figure 4.

Mach-number interpolation for Case M. (a) Global response norm $\mu _{L_2}$ as a function of Mach number over $ \textit{Ma}\in [3.0,4.5]$ . The black dashed line denotes the reference result, and the red hollow circles denote ResFormer predictions at the displayed interpolation test points. (b) Reference and predicted streamwise momentum fluctuation profiles $\widehat {\rho u}$ at $ \textit{Re}_y=7.52\times 10^3$ for the representative test sample $ \textit{Ma}=3.50$ . (c) Spatially weighted relative $L_2$ error of the complete five-component response field at the displayed test points; the black dashed line denotes the mean value. (d–f) Reference streamwise momentum fluctuation field, prediction and pointwise absolute error field for the same representative sample.

The present evaluation focuses on Mach-number interpolation over $ \textit{Ma}\in [3.0,4.5]$ at a fixed Reynolds number. Because these cases retain a consistent family of boundary layer states and instability structures, the results demonstrate the predictive capability of ResFormer for unseen Mach-number conditions within the represented flow family. A substantial Reynolds-number variation may additionally modify the viscous scaling and instability spectrum or activate a different instability branch. Extension to such topology-changing regimes can be achieved through targeted database enrichment, adaptive sampling or transfer learning using representative samples from the newly encountered regime.

Figure 5.

Inverse forcing reconstruction for the supersonic turbulent boundary layer at $ \textit{Ma}=2.25$ (Case T). (a) Global forcing norm $\|\kern2.5pt \hat {\boldsymbol{\kern-2.5pt f}}\|_{\varOmega }$ as a function of spanwise wavenumber $\beta$ at $\omega =1.0\times 10^{-5}$ . The black dashed line denotes the reference result, and the red hollow circles denote ResFormer predictions on the test set. (b) Reference and predicted profiles of the streamwise momentum forcing component $f_{\rho u}$ at $ \textit{Re}_y=5.29\times 10^3$ for the representative test sample $(\omega =1.0\times 10^{-5},\ \beta =1.92\times 10^{-4})$ . (c) Spatially weighted relative $L_2$ error of the complete five-component forcing field at the test points; the black dashed line denotes the mean value. (d–f) Reference $f_{\rho u}$ forcing field, prediction and pointwise absolute error field for the same representative sample.

Figure 5. Long description

Panel A: A line graph shows the global forcing norm as a function of spanwise wavenumber. The x-axis is labeled ‘Wavenumber β’ with units in 10^-4, and the y-axis is labeled ‘Forcing norm ||f||2’ with units in 10^-1. The black dashed line represents the reference result, and the red hollow circles denote ResFormer predictions on the test set. Panel B: A line graph displays the reference and predicted profiles of the streamwise momentum forcing component. The x-axis is labeled ‘Re_x’ with units in 10^6, and the y-axis is labeled ‘f_rho u’. The blue line represents the reference, and the red line represents the prediction. Panel C: A scatter plot shows the spatially weighted relative error of the complete five-component forcing field at the test points. The x-axis is labeled ‘Wavenumber β’ with units in 10^-4, and the y-axis is labeled ‘ε_rel L2’. The red circles denote test samples, and the black dashed line denotes the mean value. Panels D, E, and F: Heatmaps illustrate the reference forcing field, prediction, and pointwise absolute error field. The x-axis is labeled ‘Re_x’ with units in 10^6, and the y-axis is labeled ‘Re_y’ with units in 10^4. The color scale indicates the magnitude of the forcing field or error.

4.1. Inverse forcing reconstruction over a turbulent mean flow

Beyond forward prediction, ResFormer is used for data-driven response-to-forcing reconstruction. It learns the mapping from prescribed response fields to their associated forcing fields, and identifies the dominant forcing structures in the investigated configurations. Here, we apply ResFormer to a supersonic turbulent boundary layer at $ \textit{Ma}=2.25$ to reconstruct the forcing fields corresponding to resolvent response modes across a range of spanwise wavenumbers. Unlike Case M, the mean flow profile remains fixed; consequently, the network must uniquely distinguish variations in the forcing field from changes inherent to the modal structures.

The turbulent boundary layer data are generated from the fully developed turbulent flow reported by Feng et al. (Reference Feng, Wang, Chang and Zhou2025). The time- and spanwise-averaged mean flow is used as the base flow, and the inverse task is to reconstruct the forcing fields associated with resolvent response modes over a range of spanwise wavenumbers.

As illustrated in figure 5, ResFormer accurately reconstructs the forcing structures associated with coherent modes in the turbulent mean flow. The predicted global forcing norm agrees closely with the reference result, with a mean relative error of $0.11\,\%$ on the test set. The mean spatially weighted relative $L_2$ error of the complete five-component forcing field is $0.0053$ . The comparison of the forcing profiles and spatial fields further shows that the near-wall forcing structures are reconstructed with good spatial consistency. The global forcing norm is used to quantify the reconstructed forcing magnitude:

(4.1) \begin{align} \| \kern2.5pt \hat {\boldsymbol{\kern-2.5pt f}}\|_{\varOmega }= \left ( \int _{\varOmega } \sum _{i=1}^{5} \big| \hat {f}_i \big|^2 \,\mathrm{d}\varOmega \right )^{\!1/2}\!. \end{align}

These results indicate that ResFormer can reconstruct resolvent forcing structures associated with a turbulent mean profile, demonstrating its potential for rapid inverse diagnosis of coherent structures in wall-bounded turbulence.

4.2. Inverse forcing reconstruction with localised wall blowing/suction

In this section, we investigate a local blowing/suction control case in a hypersonic boundary layer at $ \textit{Ma}=5.86$ (Zhuang et al. Reference Zhuang, Wan, Liu, Sun and Lu2024). The dataset spans a two-parameter frequency–wavenumber space, allowing the inverse model to infer forcing structures associated with different controlled response modes.

As shown in figure 6, ResFormer accurately reconstructs the upstream forcing distribution associated with the target response. Despite the non-parallel effects induced by local blowing/suction, the predicted forcing location, wavepacket shape and phase distribution are consistent with the reference forcing for the representative test sample $(\omega =4.6\times 10^{-5},\ \beta =4.0\times 10^{-5})$ .

The predicted global forcing norms agree closely with the reference values across the test set, with a mean relative error of $0.077\,\%$ . The mean spatially weighted relative $L_2$ error of the complete five-component forcing field is $0.0149$ . These results indicate that the model can estimate both the magnitude and spatial structure of the forcing associated with a target response, supporting its potential use for rapid inverse diagnosis and control-oriented screening under the ideal forcing formulation.

Figure 6.

Inverse forcing reconstruction for the local blowing/suction control case (Case BS). (a) Global forcing norm $\|\kern2.5pt \hat {\boldsymbol{\kern-2.5pt f}}\|_{\varOmega }$ for all samples sorted according to the reference norm. The black dashed line denotes the reference result, and the red hollow circles denote ResFormer predictions on the test set. (b) Reference and predicted profiles of the streamwise momentum forcing component $f_{\rho u}$ at $ \textit{Re}_y=4.07\times 10^4$ for the representative test sample $(\omega =4.6\times 10^{-5},\ \beta =4.0\times 10^{-5})$ . (c) Spatially weighted relative $L_2$ error of the complete five-component forcing field at the test points; the black dashed line denotes the mean value. (d–f) Reference $f_{\rho u}$ forcing field, prediction and pointwise absolute error field for the same representative sample.

5.1. Adaptive attenuation formulation and physics-informed objective

For a given naturally evolving instability wave field $\hat {\boldsymbol{q}}_{{nat}}$ , the streamwise peak location varies with frequency, wavenumber and modal type. To avoid prescribing a fixed attenuation location for all samples, we introduce an adaptive peak tracking strategy. A smooth streamwise mask $W(x)$ is applied to construct the target attenuated response,

(5.1) \begin{align} \hat {\boldsymbol{q}}_{\textit{decay}}(x,y)= W(x) \hat {\boldsymbol{q}}_{{nat}}(x,y). \end{align}

The mask is defined as

(5.2) \begin{align} W(x)= \dfrac {1}{2} \left [1- \tanh \left ( \dfrac {x-x_{\textit{target}}}{\sigma } \right ) \right ]\!, \end{align}

where $x_{\textit{target}}$ is the half-amplitude location of the attenuation profile and $\sigma$ controls the transition width. This adaptive construction preserves the upstream natural growth region while generating diverse downstream attenuation targets.

The ideal forcing is computed by applying the complete discretised harmonic LNS operator to the attenuated target response,

(5.3) \begin{align} \kern2.5pt \hat {\boldsymbol{\kern-2.5pt f}}_{\textit{ideal}}= \boldsymbol{A}_{\omega ,\beta } \hat {\boldsymbol{q}}_{\textit{decay}}. \end{align}

Because both the complete target response $\hat {\boldsymbol{q}}_{\textit{decay}}$ and the discrete operator $\boldsymbol{A}_{\omega ,\beta }$ are known, (5.3) defines a direct and deterministic operator application. In Case PI, the resulting matrix-consistent forcing serves as the common reference for the normalised data loss and the decoded operator consistency loss defined in equations (5.5) and (5.6), respectively.

The inverse neural operator $F_{\theta }$ maps the target attenuated response to its compatible forcing field. To enforce both supervised accuracy and physical consistency, the training objective combines a data loss in the normalised training space and a physics loss in the decoded physical variable space:

(5.4) \begin{align} \mathcal{L} = \lambda _d \mathcal{E}^{d}_{2} + \lambda _p \mathcal{E}^{p}_{2}. \end{align}

Let $\widetilde {\boldsymbol{f}}_{\theta }$ and $\widetilde {\boldsymbol{f}}_{\textit{ideal}}$ denote the normalised predicted and reference forcing fields. The data loss is computed as

(5.5) \begin{align} \mathcal{E}^{d}_{2} = \dfrac { \big \| \widetilde {\boldsymbol{f}}_{\theta } - \widetilde {\boldsymbol{f}}_{\textit{ideal}} \big \|_2 }{ \big \| \widetilde {\boldsymbol{f}}_{\textit{ideal}} \big \|_2 }. \end{align}

For the physics loss, the network output and the input response are first decoded back to physical variables, denoted by $\kern2.5pt \hat {\boldsymbol{\kern-2.5pt f}}_{\theta }^{\textit{phys}}$ and $\hat {\boldsymbol{q}}_{\textit{decay}}^{\textit{phys}}$ , respectively. The physics loss is then evaluated as

(5.6) \begin{align} \mathcal{E}^{p}_{2} = \dfrac { \big \| \boldsymbol{P} \big( \kern2.5pt \hat {\boldsymbol{\kern-2.5pt f}}_{\theta }^{\textit{phys}} - \boldsymbol{A}_{\omega ,\beta } \hat {\boldsymbol{q}}_{\textit{decay}}^{\textit{phys}} \big) \big \|_2 }{ \big \| \boldsymbol{P} \boldsymbol{A}_{\omega ,\beta } \hat {\boldsymbol{q}}_{\textit{decay}}^{\textit{phys}} \big \|_2 }, \end{align}

where $\boldsymbol{P}$ retains the complex momentum forcing entries used in the physics loss evaluation.

Figure 7.

Streamwise amplitude envelopes and corresponding forcing distributions for a TS instability wave. The predicted forcing captures the localised peak required for spatial attenuation.

Figure 8.

Streamwise amplitude envelopes and corresponding forcing distributions for a second-mode instability wave. The result demonstrates the applicability of the inverse formulation to high-frequency modes.

The data term provides supervised learning in the normalised training space, whereas the physics term evaluates consistency with the discrete operator after decoding the fields to physical variables. The weighting coefficients $\lambda _d$ and $\lambda _p$ balance these two contributions. The physics term acts as an operator consistency regulariser within the directly constrained forcing subspace. For the principal physics-informed Case PI results reported in § 5.2, we set $(\lambda _d,\lambda _p)=(1,1)$ . Because both loss terms are formulated as dimensionless relative $L_2$ errors, equal nominal weights provide a natural reference setting; the ablation reported in table 2 evaluates this choice against weaker physics weights.

5.2. Case PI forcing prediction and ablation results

The construction in the previous section is applied to Case PI, which is derived from the Case S laminar flat-plate boundary layer resolvent database with $\beta =0$ and $\omega \in [0.0,40.0]\times 10^{-5}$ . This frequency range contains both low-frequency TS-wave responses and high-frequency second-mode responses, allowing the physics-informed inverse model to be evaluated on two representative instability mechanisms within the same base-flow configuration. For each naturally evolving response, a spatially attenuated target is generated by applying the smooth streamwise mask defined in equation (5.2), and the corresponding ideal forcing is obtained by applying the discrete harmonic LNS operator.

Figures 7 and 8 show representative TS-wave and second-mode cases. In the upstream natural growth regions, the predicted forcing remains close to zero. In the downstream attenuation regions, where the target response is actively reduced by the mask, the model captures the localised forcing required to produce the prescribed attenuation. The predicted forcing peaks remain close to those of the matrix-consistent reference. The forcing alignment, global phase error and peak localisation indicators used below are defined in Appendix C.3, and their test set statistics are summarised in table 12. Quantitatively, the physics-informed model achieves a mean peak location error of $1.534\pm 0.177\,\mathrm{mm}$ .

To further examine the spatial and phase structure of the prediction, figures 9 and 10 compare two-dimensional fields for representative TS-wave and second-mode cases. The model preserves the main wavepacket topology and reproduces the localised forcing structures associated with the prescribed attenuation. The one-dimensional streamwise slices in figure 11 show close phase and amplitude agreement between the prediction and the matrix-consistent reference. Over the complete held-out test set, the corresponding global phase error is $0.0708^{\circ }\pm 0.0078^{\circ }$ , and the phase-invariant momentum forcing alignment is $0.998215\pm 0.000340$ .

Figure 9.

Two-dimensional spatial comparison for a TS wave $(\omega =8.7\times 10^{-5})$ : (a) full domain and (b) locally magnified view.

Figure 9. Long description

Panel A: Two heatmaps compare ground truth and prediction data for a TS wave across the full domain. The horizontal axis is labeled Re_x with units in the range of 0 to 2.0 times 10 to the power of 6. The color bar ranges from -3.00 to 3.00 times 10 to the power of -3. Panel B: Two heatmaps compare ground truth and prediction data for a TS wave in a locally magnified view. The horizontal axis is labeled Re_x with units in the range of 1.45 to 1.80 times 10 to the power of 6. The color bar ranges from -3.00 to 3.00 times 10 to the power of -3.

Figure 10.

Two-dimensional spatial comparison for a second-mode case $(\omega =33\times 10^{-5})$ : (a) full domain and (b) locally magnified view.

Figure 11.

One-dimensional streamwise slice comparisons of the density-equation forcing component $f_\rho$ for representative (a) TS-wave and (b) second-mode cases. The predictions show close phase and amplitude agreement with the matrix-consistent reference; the corresponding test set phase statistics are reported in table 12.

The close agreement for the second-mode case in figure 11(b) indicates that ResFormer can resolve the shorter-wavelength instability structures represented in the present database. A frequency-resolved evaluation of the held-out cases shows no systematic deterioration in prediction accuracy towards higher frequencies. In the present implementation, the normalised spatial coordinates are incorporated into the input representation, while physics attention constructs adaptive physics-aware tokens and models their long-range correlations. Together with the adopted spatial resolution and training data coverage, these features support stable reconstruction over the investigated frequency range.

To isolate the contribution of the physics loss, we conducted a controlled ablation between data-only and physics-informed training. All branches used the same architecture, data partition and optimisation settings, and the same model checkpoint obtained after the first 500 data-only epochs. The data loss weight was fixed at $\lambda _d=1$ throughout all branches. Starting from the common checkpoint, the data-only branch retained $\lambda _p=0$ , whereas the physics-informed branches linearly increased $\lambda _p$ from zero to their prescribed values of $0.01$ , $0.1$ or $1$ over epochs 500–599, followed by joint optimisation until epoch 1000. The comparison was repeated using three training seeds. Prediction accuracy was evaluated in decoded physical variables using the samplewise relative forcing error

(5.7) \begin{align} E_{f} = \frac {1}{N_{\textit{test}}} \sum _{n=1}^{N_{\textit{test}}} \frac { \big \| \boldsymbol{P} \big( \kern2.5pt \hat {\boldsymbol{\kern-2.5pt f}}_{\theta }^{(n)} - \boldsymbol{f}_{\textit{ideal}}^{(n)} \big) \big \|_2 }{ \big \| \boldsymbol{P} \boldsymbol{f}_{\textit{ideal}}^{(n)} \big \|_2 }. \end{align}

Because $\boldsymbol{f}_{\textit{ideal}}^{(n)}=\boldsymbol{A}_{\omega ,\beta }\hat {\boldsymbol{q}}_{\textit{decay}}^{(n)}$ , each summand in (5.7) is identical to the projected operator residual in (5.6) for the corresponding decoded test sample.

As shown in table 2, the data-only model already achieves accurate response-to-forcing prediction under complete target response supervision. Increasing the operator consistency weight progressively reduces the interior momentum channel relative error. Among the tested settings, $\lambda _p=1$ yields the lowest mean error, decreasing it from $0.074468\pm 0.013534$ for data-only training to $0.069628\pm 0.012959$ , corresponding to a paired mean improvement of $6.55\,\%\pm 0.59\,\%$ over three training seeds and supporting its use for the principal physics-informed model. Because this error is equivalent to the projected operator residual for the corresponding decoded test samples, the result shows that the physics term provides a modest but consistent improvement, in agreement with the discrete LNS relation within the directly constrained forcing subspace.

Table 2.

Ablation of the operator consistency weight in Case PI. The interior momentum channel relative error is evaluated in decoded physical variables using the same spatial and variable support as the reported physics loss comparison. Values denote the mean and sample standard deviation over three training seeds. Improvements are calculated by paired seedwise comparison with data-only training.

Table 2. Long description

The table presents data on the impact of different training strategies on relative L2 error and improvement percentage. It has four rows and three columns. The columns are labeled ‘Training strategy’, ‘Forcing relative L2’, and ‘Improvement (%)’. The rows are labeled with different training strategies: Data-only, lambda p equals 0; Physics-informed, lambda p equals 0.01; Physics-informed, lambda p equals 0.1; and Physics-informed, lambda p equals 1. The values in the ‘Forcing relative L2’ column are presented with their respective standard deviations. The ‘Improvement (%)’ column shows the percentage improvement for each strategy. Notable trends include increasing improvement percentages with higher lambda p values in the physics-informed strategies.

6.1. Workflow from inverse forcing to realisable wall actuation

The complete workflow is summarised in figure 12. A naturally growing TS response is first generated in CFD simulations by an upstream single-frequency wall-normal blowing/suction actuator. The target frequency component is extracted by lock-in processing and projected onto the theoretical TS mode to obtain a CFD-consistent coherent response. A smooth attenuation target is then constructed from this extracted response and supplied to the inverse ResFormer. The inverse predicted forcing field is subsequently reduced to a Gaussian wall actuator prior, including the actuator location, width, frequency and phase reference. Finally, controlled CFD simulations are performed using a limited phase and amplitude scan around this actuator prior, and the resulting responses are evaluated using projection-based modal metrics.

Figure 12.

Compact workflow for realisable flow control guidance. A CFD-consistent natural TS response is obtained from CFD simulations through lock-in extraction and modal projection. A smooth attenuation target is then constructed and fed into the inverse ResFormer to predict the ideal forcing distribution. The predicted forcing topology is reduced to a Gaussian wall actuator prior, which is calibrated through CFD phase and amplitude scans. The controlled responses are finally evaluated using modal projection metrics.

Figure 12. Long description

Panel A: Operator-guided actuator construction. This section includes steps for obtaining a CFD-consistent natural TS response through CFD simulations, lock-in extraction, and modal projection. It then constructs a smooth attenuation target and feeds it into the inverse ResFormer to predict the ideal forcing distribution. The predicted forcing topology is reduced to a Gaussian wall actuator prior. Panel B: Validation and control evaluation. This section involves calibrating the actuator prior through CFD phase and amplitude scans. The controlled responses are evaluated using modal projection metrics to provide realisable actuator guidance and coherent TS suppression evidence.

The two-stage construction is not a physical necessity but a modular design choice. The resolvent database naturally provides paired response and volumetric forcing fields, so retaining the intermediate forcing preserves actuator-independent information about the dynamically relevant location, spatial support, frequency and phase. Its realisation as wall blowing and suction is not unique because volumetric forcing and wall-normal velocity act in different input spaces, and different wall actuator parameters may produce similar attenuated responses. A direct neural mapping $\mathcal{G}_{{wall}}: \hat {\boldsymbol{q}}_{\textit{decay}}\rightarrow \boldsymbol \mu _c$ is possible in principle, but it would be specific to the selected actuator family, control objective and actuation constraints. It would also require paired target responses and CFD-optimised actuator parameters, which are absent from the present resolvent database. The present study therefore uses the predicted forcing as an actuator prior and restricts the CFD stage to low-dimensional calibration and sensitivity analysis.

Detailed definitions of the lock-in extraction, modal projection, operator clean target construction, actuator parameter extraction, wall boundary condition and interval projection metrics are provided in Appendix D.

6.2. CFD-consistent TS extraction and clean attenuation target

A wall-normal blowing/suction disturbance imposed in a nonlinear CFD solver does not generate a perfectly isolated theoretical TS mode. The extracted target frequency response may contain the coherent TS wavepacket, the actuator near-field response, weak nonlinear distortions, numerical reflections and components outside the selected modal subspace. Therefore, the target response used for inverse forcing prediction is not taken directly from the raw CFD perturbation field. Instead, the upstream forced CFD solution is processed by lock-in extraction at the prescribed frequency and then projected onto the theoretical TS density mode in the near-wall region.

Figure 13 illustrates this extraction procedure. The raw lock-in density field contains a near-wall wavepacket at the imposed frequency, but it also includes structures that are not represented by the selected theoretical TS mode. After modal projection, a cleaner locally coherent TS component is obtained. Applying a streamwise attenuation mask to this locally projected component provides a diagnostic illustration of how the extracted TS response can be modified. However, the local projection coefficient may still contain streamwise oscillations caused by residual non-modal structures. Therefore, this locally projected target is used only to demonstrate the extraction procedure; the smoother operator clean target used for inverse prediction is constructed separately.

Figure 13.

Extraction of the CFD-consistent coherent TS response from the upstream forced CFD simulation. The raw lock-in density response, ${Re}(\hat {\rho }_{\textit{CFD}})$ , contains the target frequency near-wall wavepacket together with components outside the selected TS modal subspace. Projection onto the theoretical TS density mode yields a cleaner locally coherent component, ${Re}(\hat {\boldsymbol{q}}_{{nat}}^{\textit{proj}})$ . A streamwise attenuation mask is then applied to construct a locally attenuated diagnostic target, ${Re}(\hat {\boldsymbol{q}}_{\textit{decay}})$ .

For inverse prediction, the locally fluctuating projection is replaced by an operator clean target. A constant complex calibration coefficient $c_{\textit{cal}}$ is first fitted over an upstream calibration interval to match the global amplitude and phase of the CFD-extracted coherent TS response. The clean natural response is represented by $c_{\textit{cal}}\hat {\boldsymbol{q}}_{\textit{TS}}$ , where $\hat {\boldsymbol{q}}_{\textit{TS}}$ denotes the theoretical TS state in conservative variable form. The attenuated target is then constructed as

(6.1) \begin{align} \hat {\boldsymbol{q}}_{\textit{decay}}(x,y) = W(x)c_{\textit{cal}}\hat {\boldsymbol{q}}_{\textit{TS}}(x,y), \end{align}

where $W(x)$ is a smooth streamwise attenuation mask. This representation removes small-scale streamwise fluctuations while retaining the CFD-consistent incoming response level. It is therefore more suitable for inverse forcing prediction and for subsequent reduction to a compact wall actuator representation.

Figure 14 shows the operator clean target used as the actual input to the inverse ResFormer. Compared with the locally projected field in figure 13, the constant $c_{\textit{cal}}$ representation gives a smoother and more interpretable target response.

Figure 14.

Construction of the operator clean attenuation target for inverse forcing prediction. (a) Envelope level construction. The grey region denotes the interval used to estimate the calibration coefficient $c_{\textit{cal}}$ , and the densely spaced black markers appear as a continuous black bar and indicate the selected valid points. The clean natural response is represented as $\hat {\boldsymbol{q}}_{{nat}}^{{clean}}=c_{\textit{cal}}\hat {\boldsymbol{q}}_{\textit{TS}}$ , while the attenuated target is obtained by applying the smooth mask $W(x)$ . (b) Two-dimensional density fields of the clean natural response and the constructed decay target. The decay target preserves the upstream coherent TS structure and smoothly suppresses the downstream response according to $W(x)$ .

6.3. Reduction of inverse forcing to a Gaussian wall actuator

Based on the operator clean attenuation target, the inverse ResFormer predicts a complex volumetric forcing field whose near-wall spatial and phase features are reduced to a low-dimensional Gaussian wall actuator prior.

The reduction is based on the dominant near-wall forcing envelope. The envelope peak determines the actuator centre, its full width at half-maximum (FWHM) determines the Gaussian width, and the complex forcing phase in the dominant near-wall region provides the initial phase reference. In this way, the inverse predicted forcing topology is compressed into a compact actuator description while preserving its main spatial and phase information.

Figure 15 shows the reduction from the inverse predicted forcing field to the Gaussian actuator. The forcing field is localised near the target attenuation region, and its streamwise envelope is well represented by a finite-width Gaussian profile. This localisation indicates that the inverse operator provides a physically interpretable actuator prior rather than an arbitrary distributed forcing field.

Table 3.

Actuator parameters extracted from the inverse predicted forcing envelope.

Table 3. Long description

A table with 8 rows and 3 columns comparing different parameters and their values for Case S. The columns are labeled Quantity, Symbol, and Value. The table includes the following data: Row 1: Quantity, Control frequency; Symbol, omega; Value, 0.2958. Row 2: Quantity, Actuator centre; Symbol, x_c; Value, 499.535 millimeters. Row 3: Quantity, Actuator centre; Symbol, Re_x,c; Value, 1.6984 times 10 to the power of 6. Row 4: Quantity, Gaussian width; Symbol, sigma_c; Value, 25.512 millimeters. Row 5: Quantity, FWHM; Symbol, FWHM; Value, 42.480 millimeters. Row 6: Quantity, Reference cosine coefficient; Symbol, b_c(0); Value, 0.6408. Row 7: Quantity, Reference sine coefficient; Symbol, b_s(0); Value, 0.7677. Row 8: Quantity, Control amplitude; Symbol, A_c; Value, Calibrated in CFD simulations.

Figure 15.

Extraction of a realisable wall actuator from the inverse predicted forcing field. (a) Near-wall inverse forcing amplitude. (b) Streamwise envelope and Gaussian approximation. The inverse forcing is first reduced to a near-wall streamwise envelope. The peak of this envelope determines the actuator centre $x_c$ , and its FWHM determines the Gaussian width $\sigma _c$ . The complex forcing inside the dominant near-wall region provides the phase reference used for the subsequent CFD phase scan.

The extracted actuator parameters are summarised in table 3. The values of $b_c(0)$ and $b_s(0)$ correspond to the reference phase $\Delta \phi =0$ . The conversion from the operator coordinate $ \textit{Re}_x$ to the millimetre coordinate used by the CFD boundary condition is given in Appendix A.4. Because the volumetric forcing magnitude does not uniquely determine the wall velocity amplitude, $A_c$ is calibrated in the CFD simulations.

6.4. CFD validation and modal evaluation metrics

The extracted Gaussian actuator is implemented in CFD simulations as a wall-normal blowing/suction boundary condition. The upstream single-frequency generator is kept fixed in all simulations, while the downstream control actuator is varied in relative phase and amplitude. In the phase scan study, four phase offsets, $\Delta \phi =0,\ \pi /2,\ \pi$ and $3\pi /2$ , are considered at the fixed control amplitude $A_c=10^{-6}$ . After identifying the most effective phase, an amplitude scan is performed at $\Delta \phi =\pi /2$ using $A_c=10^{-7}$ , $10^{-6}$ and $10^{-5}$ .

To provide a direct visualisation of the control induced modification, the raw lock-in density fields of the baseline and the best-performing tested case are compared in figure 16. The displayed streamwise range is restricted to $ \textit{Re}_x\in [1.0\times 10^6,\,2.0264\times 10^6]$ , with $0\leqslant y\leq 5$ , so that the developed near-wall wavepacket, actuator region and downstream response can be examined more clearly. The green dashed line denotes the actuator centre $x_c$ , while the shaded region indicates its spatial support $x_c\pm \sigma _c$ . Both panels use the same spatial coordinates and colour scale to enable a direct comparison.

As shown in figure 16(a), the baseline case exhibits a relatively regular and spatially continuous near-wall TS-like wavepacket. In the best-performing tested case with $\Delta \phi =\pi /2$ and $A_c=10^{-6}$ , shown in figure 16(b), the regular wavepacket is visibly disrupted across the actuator support region, and its downstream continuation becomes weaker and less coherent than that of the baseline. This qualitative comparison indicates that the inverse guided wall actuator interacts effectively with the target frequency disturbance field.

Figure 16.

Raw lock-in density fields for (a) the uncontrolled baseline and (b) the best-performing tested case with $\Delta \phi =\pi /2$ and $A_c=10^{-6}$ . The green dashed line denotes the actuator centre $x_c$ , and the shaded region indicates the actuator support $x_c\pm \sigma _c$ . Both panels use the same spatial coordinates and colour scale.

The coherent target mode content is quantified by the downstream interval projection coefficient $c_{I_d}$ . The control effectiveness is measured by

(6.2) \begin{align} R_{\textit{coh}}= \dfrac { |c_{I_d,ctrl}| }{ |c_{I_d,base}| } . \end{align}

Here, $R_{\textit{coh}}\lt 1$ indicates that the controlled case contains a smaller globally coherent TS component than the baseline case over the same downstream evaluation interval.

Because attenuation of the selected coherent mode may be accompanied by amplification or redistribution of other disturbance components, a complementary measure of the total Chu disturbance energy over the downstream evaluation region $\varOmega _E$ is introduced:

(6.3) \begin{align} R_{E_{\textit{all}}}^{\textit{Chu}} = \dfrac { E_{{all,ctrl}}^{\textit{Chu}} }{ E_{{all,base}}^{\textit{Chu}} }. \end{align}

Here, $E_{\textit{all}}^{\textit{Chu}}$ denotes the time-averaged Chu disturbance energy integrated over $\varOmega _E$ and computed from all fluctuation content resolved by the saved CFD snapshots. No lock-in selection or temporal bandpass filtering is applied. The subscript ‘all’ therefore refers to the retained temporal fluctuation content rather than to integration over the complete CFD domain. Thus, $R_{E_{\textit{all}}}^{\textit{Chu}}\lt 1$ denotes a reduction of the total Chu disturbance energy over $\varOmega _E$ , whereas $R_{E_{\textit{all}}}^{\textit{Chu}}\gt 1$ denotes an increase. The complete energy density definition, temporal averaging procedure and spatial integration region are provided in Appendix D.6.

Figure 17 shows the phase scan results at $A_c=10^{-6}$ . All tested phase offsets reduce the coherent target mode component relative to the baseline, but the suppression is strongly phase-dependent. The smallest coherent ratio is obtained for $\Delta \phi =\pi /2$ , with $R_{\textit{coh}}=0.140$ , corresponding to an approximately $86.0\,\%$ reduction of the globally coherent target TS content in the downstream interval. The other phase offsets give larger coherent ratios: $0.531$ for $\Delta \phi =0$ , $0.608$ for $\Delta \phi =\pi$ , and $0.369$ for $\Delta \phi =3\pi /2$ . Therefore, the inverse guided actuator location and frequency are dynamically relevant, while the realisable wall actuation phase still needs to be calibrated in CFD simulations.

Figure 17.

Phase scan of the coherent TS content at $A_c=10^{-6}$ based on downstream interval constant projection. The ratio $R_{\textit{coh}}=|c_{I_d,{ctrl}}|/|c_{I_d,{base}}|$ is used to quantify the globally coherent target mode content. The most effective phase among the tested cases is $\Delta \phi =\pi /2$ , which reduces the coherent TS component to approximately $14.0\,\%$ of the baseline value.

After identifying $\Delta \phi =\pi /2$ as the most effective phase, an amplitude scan is performed at this phase. Figure 18 shows the downstream coherent TS amplitude for $A_c=10^{-7}$ , $10^{-6}$ and $10^{-5}$ . The corresponding coherent ratios are $0.455$ , $0.140$ and $0.592$ , respectively. The suppression of the selected TS component is therefore non-monotonic with the wall control amplitude. The weakest actuator does not provide sufficient control authority, whereas the strongest actuator no longer provides the phase-matched cancellation obtained at the intermediate amplitude. The corresponding total Chu disturbance energy ratios over the downstream evaluation region are $0.995$ , $0.995$ and $0.987$ , respectively. These different trends confirm that the actuator amplitude must be selected according to the prescribed modal control objective rather than by the total Chu disturbance energy over $\varOmega _E$ alone. Among the tested amplitudes, $A_c=10^{-6}$ gives the strongest attenuation of the coherent TS component.

Table 4.

Phase and amplitude sensitivity results for the inverse prior actuator geometry. The coherent amplitude ratio $R_{\textit{coh}}$ quantifies the prescribed TS control objective, whereas $R_{E_{\textit{all}}}^{\textit{Chu}}$ measures the total Chu disturbance energy over the downstream evaluation region $\varOmega _E$ . Both quantities are normalised by the uncontrolled baseline.

Table 4. Long description

A table with six rows and five columns comparing different cases of alignment, global phase error, and peak localization error. The columns are labeled Case, Δφ, A_c, R_coh, and R_Chu_Eall. The rows are labeled Baseline, Phase scan, Best tested phase/amplitude, Phase scan, Phase scan, Amplitude scan, and Amplitude scan. Each row provides specific values for each case. The table includes various cases with different phase shifts and amplitudes, showing their corresponding alignment, coherence, and peak localization error values.

Figure 18.

Amplitude scan at the favourable phase $\Delta \phi =\pi /2$ , downstream zoom. The coherent interval projected TS component is minimised at $A_c=10^{-6}$ among the tested amplitudes. The non-monotonic trend indicates that the wall control amplitude must be calibrated in the nonlinear CFD setting rather than directly inferred from the magnitude of the inverse predicted volumetric forcing.

The quantitative results are summarised in table 4. The phase scan shows that the control effect depends strongly on the relative phase between the incoming TS wave and the downstream wall actuator. The amplitude scan further shows that the realisable wall actuation strength has an effective range. These observations support the use of the inverse operator as a physics-informed actuator prior: it provides the actuator location, spatial support, target frequency and phase reference, while the final relative phase and amplitude are calibrated through a small number of CFD simulations.

6.5. Actuator prior sensitivity and downstream disturbance energy response

To assess the robustness of the inverse derived actuator prior, additional CFD simulations vary the actuator centre and Gaussian width while keeping the calibrated phase $\Delta \phi =\pi /2$ and peak amplitude $A_c=10^{-6}$ fixed.

For the location sensitivity, the inverse prior centre $x_c=499.535\,\mathrm{mm}$ is compared with two simple placement rules. The first places the actuator at the prescribed attenuation target location, $x_c=465.302\,\mathrm{mm}$ , and the second places it at the peak of the uncontrolled TS response, $x_c=477.115\,\mathrm{mm}$ . All three location cases use $\sigma _c=25.512\,\mathrm{mm}$ . For the width sensitivity, the inverse prior width is compared with $\sigma _c=25.512 / 2 =12.756\,\mathrm{mm}$ and $\sigma _c=25.512+ 25.512/2=38.268\,\mathrm{mm}$ at the fixed inverse prior centre.

Figure 19 compares the downstream interval projected TS responses. The inverse prior actuator produces the smallest coherent ratio among the three locations, $R_{\textit{coh}}=0.140$ . Moving the actuator to the attenuation target location increases this ratio to $0.438$ , while placement at the uncontrolled response peak gives $R_{\textit{coh}}=0.796$ . Thus, the peak of the uncontrolled response alone does not provide an effective actuator placement criterion; among the tested locations, the inverse prior location yields substantially stronger coherent suppression.

Figure 19.

Downstream coherent TS content for the actuator location and width sensitivity cases at $\Delta \phi =\pi /2$ and $A_c=10^{-6}$ . The curves are normalised by the interval projected coherent amplitude of the uncontrolled baseline. The inverse prior and narrow cases provide the strongest attenuation of the selected coherent TS component, whereas the response peak and wide actuator cases provide substantially weaker modal suppression.

The location and width cases are evaluated using the two metrics defined in equations (6.2) and (6.3). The coherent amplitude ratio $R_{\textit{coh}}$ measures attenuation of the prescribed TS component, whereas $R_{E_{\textit{all}}}^{\textit{Chu}}$ measures the net change in the total Chu disturbance energy over the downstream evaluation region. These complementary metrics distinguish achievement of the modal control objective from possible redistribution into other resolved disturbance components.

As summarised in table 5, the inverse prior actuator reduces the coherent TS component by approximately $86.0\,\%$ , while the downstream window disturbance energy remains nearly unchanged $(R_{E_{\textit{all}}}^{\textit{Chu}}=0.995)$ . The two alternative placement rules provide weaker modal attenuation and increase the same downstream window energy metric by approximately $4.8\,\%$ and $9.9\,\%$ , respectively. This result supports the physical relevance of the control region identified by the inverse model and shows that the actuator cannot be placed arbitrarily according to the uncontrolled response alone.

Table 5.

Sensitivity of the coherent TS component and total Chu disturbance energy to actuator location and width. All controlled cases use $\Delta \phi =\pi /2$ and $A_c=10^{-6}$ .

Table 5. Long description

A table comparing different cases with various parameters and their values. The table has six rows and five columns. The columns are labeled as Case, x subscript c (mm), sigma subscript c (mm), R subscript coh, and R subscript Chu E subscript all. The rows are labeled as Baseline, Inverse prior, Attenuation target location, Uncontrolled response peak, Narrow actuator, and Wide actuator. Each row provides specific values for each parameter. For example, the Inverse prior case has × subscript c value of 499.535 mm, sigma subscript c value of 25.512 mm, R subscript coh value of 0.140, and R subscript Chu E subscript all value of 0.995. The table presents a detailed comparison of these parameters across different cases.

The narrow actuator gives almost the same coherent ratio as the inverse prior, $R_{\textit{coh}}=0.143$ , but increases the downstream window disturbance energy by approximately $12.4\,\%$ . It is therefore not superior to the inverse prior width when both metrics are considered. Conversely, the wide actuator reduces the downstream window disturbance energy by approximately $9.8\,\%$ but leaves approximately $79.8\,\%$ of the coherent TS amplitude. These opposite rankings demonstrate that attenuation of the selected coherent mode and reduction of the window integrated disturbance energy are distinct control objectives.

6.6. Implications for realisable control guidance

The combined phase, amplitude, location, width and disturbance energy results demonstrate that the inverse predicted forcing topology provides useful but deliberately incomplete guidance for realisable wall control. The inverse model substantially narrows the actuator search space by identifying a dynamically relevant centre, spatial support, target frequency and initial phase reference. The realisable phase, peak amplitude and detailed wall actuator width must nevertheless be calibrated according to the prescribed CFD level control objective.

This distinction arises because the inverse problem is formulated in a volumetric forcing space, whereas the CFD simulations impose a wall-normal velocity boundary condition. The high-dimensional inverse forcing topology is therefore compressed into the low-dimensional actuator parametrisation

(6.4) \begin{align} \boldsymbol{\mu }_{c}= \left ( x_c,\sigma _c,\omega ,\theta _{\textit{in}v},\Delta \phi ,A_c \right ), \end{align}

after which a limited number of CFD simulations are used for realisability calibration. The present results show that this prior is particularly useful for identifying an effective actuator region, while avoiding the stronger modal and energetic penalties produced by the two simple alternative placement rules tested here.

The workflow also provides a foundation for surrogate-assisted control. In a future implementation, the current target frequency response could be estimated from sensor data, high-fidelity simulation or a forward neural operator; a desired attenuation target could then be prescribed; and the inverse operator could generate an updated actuator prior. A forward surrogate or a limited number of CFD evaluations may then be used to refine the final phase and amplitude. Thus, the present CFD validation closes the methodological loop of this study by connecting operator-based response-to-forcing prediction with practical wall actuation guidance.