**8. Conclusions**

In this chapter, the higher-order singular value decomposition has been proved to be a flexible and valuable tool in the data-driven reduced-order modeling of solutions of space–time-parameter problems, which are today at the heart of many industrial applications. The methodology has been tested on a problem of fluid– structure interaction of a deformable microcapsule flowing into a microchannel. Stokes equations have been used in the fluid whereas a nonlinear hypereleastic law has been used for the membrane. Different shape solutions computed by the full-order model have been stored into a third-order tensor. First, HOSVD allows us to compute spatial, temporal and parameter principal components and at the same time to compress the data. We get a low-order representation of the solutions with a shared spatial reduced basis. Spatial principal components are observed to provide suitable details in the shape solutions. The modes are arranged in decreasing order of importance according to the relative information content criterion. Next, additional ingredients such as kernel approximation and kernel-based dynamic mode decomposition are used to determine a reduced-order dynamical system for any parameter vector in the admissible parameter domain. The resulting low-dynamical system can be seen as an encoded recurrent neural network set into the PCA space. The approach allows us to explore the different shape solutions and visualize their evolution in the channel in real time.

*Space-Time-Parameter PCA for Data-Driven Modeling with Application to Bioengineering DOI: http://dx.doi.org/10.5772/intechopen.103756*
