## 4 Scaling Laws for Different Magnetic Configurations  
As reported in [8], a simple analysis indicates that the database is not really homogenous. In this respect, it should be remembered that the International Stellarator Confinement Database includes two main types of magnetic configurations with respect to the rotational transform profile, with and without shear. The first class includes the devices ATF, CHS, HELE, HELJ, and LHD, whereas the shearless devices are W7-A, W7-AS, and TJ-II. Visual inspection of the database reveals that the dependencies of the energy confinement time on the regressors can be different for these two types of configurations. Consequently, it seems natural to particularise the scaling laws for these two different groups of machines. The characteristics of the two corresponding data subsets are reported in Table 2.  
For the magnetic configuration with shear, the scaling law obtained, by applying SR via GP to the corresponding entries of the International Stellarator Confinement Database, is  
\[\tau_{E}^{SR~{}Shear}~{}=\left(7.92^{8.11}_{7.73}\right)\times 10^{-2}\cdot a^{2. 53^{2.58}_{2.48}}R^{0.97^{1.01}_{0.94}}p^{-0.60-0.58}n_{e}^{0.45^{0.48}_{0.43}}B^ {0.67^{2.70}_{0.63}}t_{\frac{2}{3}}^{0.500^{5.51}_{0.29}}\left(1+e^{-(\frac{( \frac{\xi}{b})-0.10^{0.20}_{0.18}}{0.017^{0.028}_{0.015}})}\right) \tag{2}\]  
Equation (2) is not a simple power law, and its performances are slightly better than the ISS04 for this subset of the database, as reported in Table 3.  
The entries of the shearless devices are more inhomogeneous, and indeed for these devices, the _frem_ of ISS04 ranges from 0.25 to 1. Moreover, the dependence from \(t_{2/3}\) is quite difficult to determine, and indeed, as mentioned in [22], it was derived from a different set of data not included in the International Stellarator Confinement DB. It has therefore been decided to provide more freedom to the nonlinear fit of the model, selected by SR via GP, by implementing a non-linear least-square fit with constrained coefficients [23]. To perform the fit, a regularization term has been added to the loss function so that it becomes:  
\[L(f)=\sum_{i}^{n}(y_{i}-f(x_{i}))^{2}\cdot weights+\lambda\cdot\sum w_{i}^{2}\]  
This technique of constrained regularisation complicates the derivation of solid confidence intervals, which have to be obtained with the bootstrap, using the basic percentile technique [24]. In the rest of the paper, the intervals reported have been calculated at 95% confidence level. It is probably worth mentioning that, to the authors' knowledge, this combination of constrained fit and bootstrap is a methodological solution applied for the first time to the investigation of scaling laws for Stellarators. The scaling laws obtained for the shearless magnetic configuration, deploying SR via GP and a constrained non-linear least-square fit, are the following:  
\[\tau_{E}^{\tiny SHEARLESS03}=5.54^{8.61}_{3.70}\times 10^{-2}\cdot a^{2.17^{2. 39}_{1.39}}\cdot R^{0.64}_{0.61}\cdot p^{-0.62^{-0.65}_{-0.66}}\cdot n_{e}^{0. 74^{0.28}_{0.49}}\cdot B^{1.251^{3.30}_{1.21}}\cdot t_{\frac{2}{3}}^{0.30} \cdot\frac{1}{\left(1+1.44^{1.76}_{1.05}\cdot e^{-\frac{2R}{M_{\omega}}} \right)} \tag{3}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Dataset** & **Devices** & **Entries** \\ \hline \multirow{5}{*}{SHEAR} & ATF & 229 \\  & CHS & 196 \\  & HELE & 120 \\  & HELJ & 54 \\  & LHD & 162 \\ \hline \multicolumn{2}{c}{Total} & 761 \\ \hline \multirow{2}{*}{SHEARLESS} & W7-A & 13 \\  & W7-AS & 629 \\  & TJ-II & 316 \\ \hline \multicolumn{2}{c}{Total} & 958 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The two data subsets analysed in the present work.