Application of the ANN for the IM Optimization

The ANN implemented in this work is based on a fully connected feed-forward architecture of McCulloch–Pitts neurons and shown in Figure 2a. As a transfer function, a sigmoidal function

$$f(\mathbf{x}) = \frac{1}{1 + \varepsilon^{-x}} \tag{26}$$

is chosen, which is characterized by a simple and fast calculation of the derivative [34]. Moreover, its range of values includes only positive values, which meaningfully describes the estimation of physical quantities, such as the electromagnetic losses. The inputs of the ANN are the optimization parameters that are the geometry parameters of the actual IM design. Within the input selection, which will be described in the following, these will be supplemented by further geometry parameters. Two common problems in the construction of ANN are under-fitting and over-fitting, which, due to insufficiently complex or over-complex architectures, can cause imprecise estimates of the network. This can be counteracted by a multi-stage method in which different network architectures are analyzed by varying the number of neurons and layers. From these possible architectures, the most appropriate one is selected. For this purpose, the following sequential, five-step flow is introduced.


the ANN is essential, since they determine the precision significantly. Therefore, two phenomena must be considered in their selection:


Both aspects can be ensured by the introduced sensitivity analyses. As a training procedure the Levenberg–Marquart method is used in this paper.


The steps for network construction, order and input selection, and characterization are repeated at predetermined intervals once the required number of individuals in the database has been reached. Thus, new individuals added to the database are considered in the ANN, further increasing the estimation accuracy.

The output or solution of the network are the mean loss *P*<sup>L</sup> over the driving cycle of the considered individual. To ensure that the preselection of individuals by the ANN does not discard solutions that would otherwise represent a new optimum, a threshold *f*th is introduced. Here, individuals whose estimated fitness is above the threshold are discarded and otherwise simulated. To determine the threshold, a delta Δ ≥ 0 is introduced. This results from the given estimation reliability *γ* of the ANN as well as the error distribution of the testing error to

$$\int\_{\varepsilon=-\infty}^{\Lambda} p(\varepsilon) \, \mathrm{d}\varepsilon = \gamma. \tag{27}$$

The relationship is visualized in Figure 3a. The threshold, as shown in Figure 3b, is obtained as a function of the current optimum *f*(*x*min) to

$$f\_{\rm th} = f(\vec{x}\_{\rm min}) + \Delta. \tag{28}$$

Estimates above the resulting threshold consequently lie within the *γ* confidence interval with respect to the null hypothesis that these individuals do not represent a new optimum. In addition to the threshold, the relative deviation or distance of the chromosome

of the solution to be estimated from the individuals in the database is also considered. If the distance to the nearest individual exceeds a predefined value, the solution is simulated and not discarded because in that case the estimation of the ANN is based on an extrapolation whose validity is not guaranteed.

**Figure 3.** Representation of the delta based on the distribution of the testing error (**a**) as well as the resulting threshold (**b**) for the preselection of solutions by means of a ANN.
