**7.1 Kolmogorov** *n***-width**

The method is efficient if the spectrum of singular values decays rapidly, leading to a small truncation rank *K*. If the spectrum decays slowly, there are two possible reasons for that: either the entropy (variety) of information in the data is high, or the solutions do not live in a linear space but rather on a nonlinear manifold. To fix the problem, one can proceed by performing a preliminary clustering of the data, scanning the parameter dimension. One can either use standard clustering techniques such as *K*-means, or a multidimensional scaling (MDS) approach. Then for each cluster, one can consider again a HOSVD and reduced-order approach suitable for each element of the cluster.

#### **7.2 Selection of the kernel functions and kernel interpolation points**

As already mentioned, the choice of the kernel function depends on the applications, on the behavior of solutions and/or on the underlying Physics. Without any a priori information, one can use universal approximation kernels like the Gaussian one. The accuracy of the results will also strongly depends on the choice of the kernel interpolation points *a*ð Þ*<sup>j</sup>* . The sampling *a*ð Þ*<sup>j</sup> <sup>m</sup> <sup>j</sup>*¼<sup>1</sup> has to correctly fill in the admissible space, or at least the state-space trajectory of interest. There are different possible strategies. A first candidate is the use of a clustering approach applied to the state-space data. The points *a*ð Þ*<sup>j</sup>* are then the centroids of each cluster. But one can consider more sophisticated approaches like a greedy iterative approach that controls the interpolation error on the data. At each iteration an interpolation point *a*ð Þ*<sup>j</sup>* is added at the location of worst interpolation error, considering all the sample solutions.

**Figure 13.** *Interpretation of the method as a recurrent neural network (RNN) in the PCA space.*

#### **7.3 Interpretation in terms of a recurrent neural network**

Let us remark that the approach can be reinterpreted as a (supervized) two-layer recurrent artificial neural network (RNN) (**Figure 13**) [18]. The first layer consists in generating the features *ki a<sup>μ</sup>* . The second layer is a linear matrix–vector multiplication using the matrix *Aμ*.
