Accelerating Galerkin Reduced-Order Models for Turbulent Flows with Tensor Decomposition

Abstract: Galerkin-based reduced-order models (G-ROMs) offer efficient and accurate approximations for laminar flows but require hundreds to thousands of modes $N$ to capture the complex dynamics of turbulent flows. This makes standard G-ROMs computationally expensive due to the third-order advection tensor contraction, requiring the storage of $N^3$ entries and the computation of $2N^3$ operations per timestep. As a result, such ROMs are impractical for realistic applications like turbulent flow control. In this work, we consider problems that demand large $N$ values for accurate G-ROMs and propose a novel approach that accelerates G-ROMs by utilizing the CANDECOMP/PARAFAC (CP) tensor decomposition to approximate the advection tensor as a sum of $R$ rank-1 tensors. We also leverage the partial skew-symmetry property of the advection tensor and derive two conditions for the CP decomposition to preserve this property. Moreover, we investigate the low-rank structure of the advection tensor using singular value decomposition (SVD) and compare the performance of G-ROMs accelerated by CP (CPD-ROM) and SVD (SVD-ROM). Demonstrated on problems from 2D periodic to 3D turbulent flows, the CPD-ROM achieves at least a $10$-fold speedup and a $16.7$-fold reduction in nonlinear term evaluation costs compared to the standard G-ROM. The skew-symmetry preserving CPD-ROM demonstrates improved stability in both the reproduction and predictive regimes, and enables the use of smaller rank $R$. Singular value analysis reveals a persistent low-rank structure in the $H^1_0$-based advection tensor, and CP decomposition achieves at least an order of magnitude higher compression ratio than SVD.