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.

翻译：针对含时变系数的对流主导问题，我们提出一种动态模态分解（DMD）的变体用于构建降阶模型（ROM）。现有DMD方法（如物理感知DMD和时变DMD）因固有时不变性假设与局部性限制，难以处理此类问题。为克服复合性困难，我们提出将特征线演化作为非自治系统进行学习，进而构建无限维库普曼算子的分段局部时不变近似。通过二维纳维-斯托克斯方程验证了时变DMD算子的精度，并基于拉格朗日方法对一维和二维变系数对流扩散方程进行了测试。最后，我们给出了预测精度与扰动误差的上界估计，以指导秩截断和子区间大小的选择。