**3. Reduced-order modeling of capsule dynamics**

Eq. (6) provides a summarization of the family of spatio-temporal capsule shape solutions in the time interval *t* 0, *t <sup>N</sup>* � �. Unfortunately, this algebraic model has no predictability capability for time *t* >*t <sup>N</sup>*. To derive a predictable time-dependent model from the data tensor T *<sup>x</sup>*, one has to derive a differential system that approximates the FSI system of Eqs. (2) and (3). The HOSVD reduction can thus be valued in the context of model-order reduction.

Consider of parameter vector of interest *μ*∈ *D*. The capsule position approximate solution (5) can be rewritten as

$$\bar{\mathfrak{x}}(\mathbf{X}, \boldsymbol{\mu}, t) = \mathbf{X} + \sum\_{k=1}^{K} a\_{\boldsymbol{\mu}, k}(t) \boldsymbol{\varrho}\_k(\mathbf{X}) \tag{10}$$

with

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

$$a\_{\mu,k}(t) = \sum\_{\ell=1}^{L} \sum\_{m=1}^{M} a\_{k\ell m} \boldsymbol{\nu}\_{\ell}(\boldsymbol{\mu}) a\_m(t). \tag{11}$$

Let *aμ*ð Þ*t* be the vector-valued function

$$\mathfrak{a}\_{\mu}(t) = \begin{pmatrix} a\_{\mu,1}(t), \dots, a\_{\mu,K}(t) \end{pmatrix}^T \in \mathbb{R}^K. \tag{12}$$

We would like to derive a differential system of unknowns *aμ*ð Þ*t* . Since *x X*ð Þ¼ , *μ*, 0 *X*, we have the natural initial condition *aμ*ð Þ¼ 0 **0**. Practically, from expansion (8), we can compute coefficients *a<sup>μ</sup>*,*k*ð Þ*t* at discrete times *t <sup>n</sup>*, and have thus access to the list of coefficient vectors

$$\mathfrak{a}\_{\mu}^{n} = \mathfrak{a}\_{\mu}(t^{n}), \quad n = 0, \dots, N. \tag{13}$$

A dynamical reduced-order model consists in determining (or approximating) a Lipschitz continuous mapping *<sup>F</sup><sup>μ</sup>* : *<sup>K</sup>* ! *<sup>K</sup>* such that

$$a^0\_{\mu} = \mathbf{0}, \quad a^{n+1}\_{\mu} \approx \mathbf{F}\_{\mu} \left( a^n\_{\mu} \right) \quad \forall n \in \mathbf{0}, \ldots, N-1 \tag{14}$$

from the data *a<sup>n</sup> μ* n o*<sup>N</sup> n*¼0 . We get a low-order discrete dynamical system. Note that, here, we do not search for parameters of a model, but for the equations of the model themselves. Since the problem of finding such a mapping is infinite dimensional, one has to restrict the search to a mapping in a (suitable) finite dimensional functional space.
