A reduced order method for nonlinear parameterized partial differential equations using dynamic mode decomposition coupled with k-nearest-neighbors regression

Accurately constructing a reduced order model (ROM) of nonlinear parameterized partial differential equations (PDEs) has always been a challenging problem in engineering and applied sciences. Dynamic mode decomposition (DMD) is a popular and efficient data-driven method for ROM, however, it is proposed for the model order reduction of time-dependent problems that it is inapplicable for the parameterized problems. In this paper, a new ROM is proposed based on the k-nearest-neighbors (KNN) regression and DMD, namely, KNN-DMD. The KNN can approximate the solution at any given parameter value by choosing and averaging the nearest k DMD solutions based on the distance between the given parameter value and other parameter values, leading to the applicability of DMD to parameterized problems. We apply the proposed method to various nonlinear parameterized PDEs, i.e., heat equations, reaction-diffusion equations, Burgers equations, and Navier-Stokes equations for the two-dimensional fluid flow over a cylinder. The results demonstrate the applicability and efficiency of the proposed KNN-DMD as a real-time ROM for parameterized PDEs. Furthermore, KNN-DMD shows better predictive ability than the POD-based ROMs outside of the training time region.