An Efficient Reduced-Order Model Based on Dynamic Mode Decomposition for Parameterized Spatial High-Dimensional PDEs (Communications in Computational Physics, 2025)
Dynamic mode decomposition (DMD), as a data-driven method, has beenfrequently used to construct reduced-order models (ROMs) due to its good performancein time extrapolation. However, existing DMD-based ROMs suffer from high storageand computational costs for high-dimensional problems. To mitigate this problem,we develop a new DMD-based ROM, i.e., TDMD-GPR, by combining tensor traindecomposition (TTD) and Gaussian process regression (GPR), where TTD is used todecompose the high-dimensional tensor into multiple factors, including parameter-dependent and time-dependent factors. Parameter-dependent factor is fed into GPRto build the map between parameter value and factor vector. For any parametervalue, multiplying the corresponding parameter-dependent factor vector and the time-dependent factor matrix, the result describes the temporal behavior of the spatialbasis for this parameter value and is then used to train the DMD model. In addition,incremental singular value decomposition is adopted to acquire a collection of importantinstants, which can further reduce the computational and storage costs of TDMD-GPR.The comparison TDMD and standard DMD in terms of computational and storagecomplexities shows that TDMD is more advantageous. The performance of the TDMDand TDMD-GPR is assessed through several cases, and the numerical results confirmthe effectiveness of them.
Keywords: Parameterized time-dependent PDEs; tensor train decomposition; dynamic mode decomposition; incremental singular value decomposition.