Physics-Aware Reduced-Order Modeling of Nonautonomous Advection-Dominated Problems

We present a variant of dynamic mode decomposition (DMD) for constructing a reduced-order model (ROM) of advection-dominated problems with time-dependent coefficients. Existing DMD strategies, such as the physics-aware DMD and the time-varying DMD, struggle to tackle such problems due to their inherent assumptions of time-invariance and locality. To overcome the compounded difficulty, we propose to learn the evolution of characteristic lines as a nonautonomous system. A piecewise locally time-invariant approximation to the infinite-dimensional Koopman operator is then constructed. We test the accuracy of time-dependent DMD operator on 2d Navier-Stokes equations, and test the Lagrangian-based method on 1- and 2-dimensional advection-diffusion with variable coefficients. Finally, we provide predictive accuracy and perturbation error upper bounds to guide the selection of rank truncation and subinterval sizes.