Proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD) are two complementary singular-value decomposition (SVD) techniques that are widely used to construct reduced-order models (ROMs) in a variety of fields of science and engineering. Despite their popularity, both DMD and POD struggle to formulate accurate ROMs for advection-dominated problems because of the nature of SVD-based methods. We investigate this shortcoming of conventional POD and DMD methods formulated within the Eulerian framework. Then we propose a Lagrangian-based DMD method to overcome this so-called translational problem. Our approach is consistent with the spirit of physics-aware DMD since it accounts for the evolution of characteristic lines. Several numerical tests are presented to demonstrate the accuracy and efficiency of the proposed Lagrangian DMD method.

适当的正交分解（POD）和动态模式分解（DMD）是两种互补的奇异值分解（SVD）技术，在各种科学和工程领域中，它们广泛用于构造降阶模型（ROM）。尽管它们很受欢迎，但由于基于SVD的方法的性质，DMD和POD都难以为对流主导的问题制定精确的ROM。我们调查了在欧拉框架内制定的常规POD和DMD方法的缺点。然后，我们提出了一种基于拉格朗日的DMD方法来克服这一所谓的平移问题。我们的方法符合物理感知DMD的精神，因为它考虑了特征线的演变。