$\texttt{CRLS}$: Convolutional Regularized Least Squares Framework for Reduced Order Modeling of Transonic Flows

TL;DR

The paper introduces CRLS, a novel convolutional regularized least squares framework for reduced-order modeling of transonic flows with shocks, improving accuracy and efficiency over traditional methods.

Contribution

CRLS combines smoothing, Bayesian hyperparameter selection, and deconvolution to enhance shock capturing in reduced-order models of transonic flows.

Findings

CRLS significantly improves shock location and strength predictions.

It reduces the number of POD modes needed by 42%.

CRLS outperforms standard POD and smoothed-POD baselines.

Abstract

We develop a convolutional regularized least squares ($\texttt{CRLS}$) framework for reduced-order modeling of transonic flows with shocks. Conventional proper orthogonal decomposition (POD) based reduced models are attractive because of their optimality and low online cost; however, but they perform poorly when snapshots contain parameter-dependent discontinuities, leading to smeared shocks, stair-stepping, or non-physical oscillations. In $\texttt{CRLS}$, we first map each full-order snapshot to a smoother representation by applying a one-dimensional Gaussian convolution with reflect padding along the flow field coordinates. The convolution hyperparameters (kernel width and support) are selected automatically by Bayesian optimization on a held-out set of snapshots. POD bases are then extracted from the smoothed data, and the parametric dependence of the POD coefficients is learned via radial basis function interpolation. To recover sharp shock structures, we introduce an efficient deconvolution step formulated as a regularized least squares problem, where the regularization centers the reconstruction around a nearest-neighbor reference snapshot in parameter space. The resulting $\texttt{CRLS}$ surrogate is evaluated on inviscid transonic flow over the RAE2822 airfoil, modeled by the steady compressible Euler equations solved with SU2 over a Latin hypercube sample of Mach number and angle of attack. Compared with standard POD and smoothed-POD baselines, $\texttt{CRLS}$ yields markedly improved shock location and strength, lower surface-pressure and field-level errors, and a $42$\% reduction in the number of POD modes required to capture a fixed fraction of snapshot energy. These results demonstrate that $\texttt{CRLS}$ provides an accurate, data-efficient, and largely automated route to shock-aware reduced order models for high-speed aerodynamic design.