**Abstract**

Principal component analysis is a recognized powerful and practical method in statistics and data science. It can also be used in modeling as a dimensionality reduction tool to achieve low-order models of complex multiphysics or engineering systems. Model-order reduction (MOR) methodologies today are an important topic for engineering design and analysis. Design space exploration or accelerated numerical optimization for example are made easier by the use of reduced-order models. In this chapter, we will talk about the use of higher-order singular value decompositions (HOSVD) applied to spatiotemporal problems that are parameterized by a set of design variables or physical parameters. Here we consider a data-driven reduced order modeling based on a design of computer experiment: from high-dimensional computational results returned by high-fidelity solvers (e.g. finite element ones), the HOSVD allows us to determine spatial, time and parameters principal components. The dynamics of the system can then be retrieved by identifying the low-order discrete dynamical system. As application, we will consider the dynamics of deformable capsules flowing into microchannels. The study of such fluid-structure interaction problems is motivated by the use of microcapsules as innovative drug delivery carriers through blood vessels.

**Keywords:** HOSVD, spatio-temporal parametrized problem, approximation, reduced-order model, bioengineering, fluid-structure interaction, deformable capsules, dynamical system, machine learning, artificial intelligence
