Parametric dynamic mode decomposition with stagewise coefficient representation for parametric kinetic systems

Efficient prediction of parameter-dependent dynamical systems remains challenging in many-query settings, particularly when the solution exhibits complex temporal behavior. This work develops a reduced-order modeling framework based on dynamic mode decomposition (DMD) for such systems. Unlike conventional DMD-based surrogates that rely on a global reduced representation over the entire time interval, the proposed approach combines space–time reduced bases with low-dimensional parameter-dependent coefficients and represents temporal variations through a stagewise decomposition. This construction provides greater flexibility for capturing evolving dynamical structures while retaining a compact surrogate representation. Based on this framework, localized dynamic mode decomposition (LDMD) is incorporated through time-domain decomposition, leading to the stagewise coefficient representation parametric dynamic mode decomposition (SCR-PDMD) method. LDMD predictions are first constructed at the training parameters, followed by stagewise space–time reduction and parametric approximation of the resulting reduced coefficients. The resulting localized representation can adapt to temporal changes in the dominant dynamics, improving robustness for time-varying or strongly varying systems and enhancing long-term predictive accuracy. Numerical experiments for the Boltzmann equation, the nonlinear Schrödinger equation, and the two-dimensional Gross–Pitaevskii equation demonstrate the accuracy and efficiency of the proposed framework for parametric prediction.