A Dynamic Mode Decomposition Based Reduced-Order Model For Parameterized Time-Dependent Partial Differential Equations

We propose a reduced-order model (ROM) based on dynamic mode decomposition (DMD) for efficient reduced-order modeling of parameterized time-dependent partial differential equations. Given high-fidelity solutions for training time and parameter sets, a proper orthogonal decomposition (POD) is first used to extract the POD modes and the projection coefficients onto the modes. After that, a DMD model is built for each parameter value based on the corresponding projection coefficients. These DMD models are used to predict the projection coefficients over a larger time region than the training time region. We rearrange these predicted coefficients as a new matrix and perform POD again to decompose the matrix into time-dependent and parameter-dependent parts, which are approximated using cubic spline interpolation (CSI) and Gaussian process regression (GPR), respectively. Finally, the ROM solutions at any given time or parameter value are estimated by a simple multiplication of the POD modes and the approximated projection coefficients. The proposed method is tested through several benchmark problems, and the impact of using CSI and GPR to model the time-dependent part on the model accuracy is analyzed. In addition, numerical results for a PDE with 19 parameters demonstrate the great potential of the proposed method in the construction of a ROM for problems with high dimensional parameters.