A sparse dynamic mode decomposition for time-dependent parameterized partial differential equations based on generalized empirical interpolation

Reduced-order modeling of parameterized partial differential equations has emerged as a prominent research focus in computational mathematics. As an efficient data-driven approach, dynamic mode decomposition (DMD) has demonstrated remarkable advantages. However, existing improved DMD methods still exhibit systematic limitations when addressing complex problems where spatial dimensions significantly exceed temporal dimensions. To mitigate this problem, we innovatively propose a GEIM (generalized empirical interpolation method)-DMD reduced order modeling framework that synergistically integrates GEIM with DMD. By employing GEIM to construct a deterministic spatial collocation point selection strategy for dimensionality reduction, the GEIM-DMD maintains accuracy comparable to standard DMD while significantly reducing computational costs. To further extend parametric applications, a K-nearest neighbors (KNN)-based algorithm, termed KNN-GEIM-DMD, which constructs an efficient mapping from the parameter space to the solution space, is proposed. Numerical results demonstrate the high efficiency and accuracy of the proposed method, which matches that of the standard DMD and exceeds that of the compressed DMD.