Forward and inverse resolvent modelling for instabilities and active control in compressible boundary layers using a Transformer neural operator

Resolvent analysis effectively characterises flow instabilities and guides active control, but repeated singular-value-decomposition-based (SVD-based) calculations remain computationally expensive in high-dimensional parameter studies. Control-oriented workflows also require efficient mappings from prescribed response fields to compatible forcing fields. To address these needs, we introduce ResFormer, a unified Transformer-based neural operator framework for forward resolvent prediction and response-to-forcing reconstruction in compressible boundary layers. The framework is validated across supersonic and hypersonic regimes (

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5.86 5.86 $5.86$

), encompassing laminar, turbulent and controlled base-flow configurations. For forward prediction, ResFormer reconstructs the spatial topology, phase structure and amplitude envelope of both low-frequency Tollmien–Schlichting waves and high-frequency second-mode disturbances over the investigated parameter ranges. For response-to-forcing prediction, ResFormer learns a surrogate mapping from prescribed instability responses to compatible forcing fields. When the complete target response and discrete operator are known, the reference forcing is computed directly as

ModifyingAbove bold italic f With caret Subscript italic ideal Baseline equals bold italic upper A Subscript omega comma beta Baseline ModifyingAbove bold italic q With caret Subscript italic decay f ^ ideal = A ω , β q ^ decay $\kern2.5pt \hat {\boldsymbol{\kern-2.5pt f}}_{\textit{ideal}}=\boldsymbol{A}_{\omega ,\beta }\hat {\boldsymbol{q}}_{\textit{decay}}$

; the physics term therefore acts as an operator consistency regulariser. The reconstructed forcing topology is then reduced to a compact Gaussian wall actuator prior and evaluated using high-fidelity computational fluid dynamics. ResFormer therefore accelerates repeated SVD-based forward and optimal mode evaluations while providing an operator-consistent learned representation for actuator prior identification and surrogate-assisted control design.