*1.1. Model Order Reduction*

Advances in applied mathematics, computer science (high-performance computing) and computational mechanics met to give rise to a diversity of Model Order Reduction (MOR) techniques [1]. These techniques do not reduce or modify the model, they simply reduce the complexity of its resolution and thus transform a complex and time-consuming calculation, into a real-time response while maintaining precision. These new techniques have completely altered traditional approaches of simulation, optimization, inverse analysis, control and uncertainty propagation, all them operating under the stringent real-time constraint.

In a few words, when approximating the solution *u*(**x**, *t*) of a given Partial Differential Equation (PDE), the multipurpose finite element method assumes an approximation

$$\mu(\mathbf{x},t) = \sum\_{i=1}^{\mathsf{M}} \mathsf{U}\_{i}(t)\mathsf{N}\_{i}(\mathbf{x})\_{\mathsf{M}} \tag{1}$$

where *U<sup>i</sup>* represents the value of the unknown field at node *i* and *Ni*(**x**) is the associated shape function. When N (the number of nodes) increases the solution process becomes cumbersome.

POD-based model order reduction learns offline the most adequate (in a given sense) reduced approximation basis {*φ*1(**x**), · · · , *φ*R(**x**)}, and project the solution in it

$$u(\mathbf{x}, t) \approx \sum\_{i=1}^{R} \xi\_i(t) \phi\_i(\mathbf{x}),\tag{2}$$

where now, the complexity scales with R instead of N, with R ≪ N in general.

The so-called Proper Generalized Decomposition (PGD from now on) goes a step forward and assume a general approximation

$$u(\mathbf{x},t) \approx \sum\_{i=1}^{\mathbb{M}} T\_i(t)X\_i(\mathbf{x}),\tag{3}$$

where now both the space and time functions, *Xi*(**x**) and *Ti*(*t*) respectively, are computed during the solution process.

A particularly appealing extension of the just introduced space-time separated representation consists of the space-time-parameter separated representation leading to the a so-called *computational vademecum* that expresses the solution of a parametrized PDE from [2,3]

$$\mu(\mathbf{x}, t, \mu\_1, \dots, \mu\_{\mathbf{Q}}) \approx \sum\_{i=1}^{\mathbb{M}} X\_i(\mathbf{x}) T\_i(t) \prod\_{j=1}^{\mathbb{Q}} M\_i^j(\mu\_j), \tag{4}$$

where *µ<sup>j</sup>* , *j* = 1, . . . , Q, represent the model parameters. Once constructed off-line that parametric solution (4), it offers under very stringent real-time constraints—in the order of milliseconds—simulation, optimization, inverse analysis, uncertainty propagation and simulation-based control, to cite a few. Thus, at the beginning of the third millennium a real-time dialogue with physics no longer seemed to be the domain of the impossible.

PGD-based techniques have been widely considered for the real-time simulation and decision-making in a variety of problems of industrial relevance. However, prior to use it, one must extract the parameters to be included as extra-coordinates in the problem statement, and then included in the parametric representation of its solution. In the case of morphological and topological descriptions, as considered later in the present work, the extraction of the adequate parametrization represents the most difficult task. Some attempts of combing PGD-based MOR and manifold learning [4] were addressed in [5–8].
