Review for order reduction based on proper orthogonal decomposition and outlooks of applications in mechanical systems

•The applications of the POD method in a variety of research areas are reviewed.•The classification of the POD method is proposed.•The review focuses on the dynamical application of the POD method.•The outlooks of the POD method in actual engineering are discussed. This paper presents a review of proper orthogonal decomposition (POD) methods for order reduction in a variety of research areas. The historical development and basic mathematical formulation of the POD method are introduced. POD for parametric dynamic systems is introduced, and a physical interpretation of the POD approach based on the proper orthogonal modes (POMs) is presented. The equivalence between POD and three other order reduction methods is discussed: the first alternative method is singular value decomposition (SVD), the second is principal component analysis (PCA), and the third is Karhunen-Loeve decomposition (KLD). A classification of POD methods is described based on the parameter adaptation and sampling. Actual applications of POD methods for order reduction in engineering systems are illustrated. Finally, outlooks on the use of POD methods in high-dimensional nonlinear dynamic systems are presented in more detail to provide direct guidance for researchers in various areas of engineering.