## 3 Simplistic Sequential Networks
For Setup A the average value of MAE was 2.32 \({}^{\circ}\)C. The average MAE is almost the same as the 10 th and the 90 th percentile, which means the spread of MAE values is very small and that the realizations have a similar error. The behavior of Setup B is very similar to Setup A (one typical example is shown in Figure 4b). Although there are two neurons, the function is very similar to the one for Setup A and is also mostly linear (at least inside the shown phase space of parameter values). In the majority of realizations, the nonlinear behavior is not evident. The average MAE value is the same as in Setup A while the spread is a bit larger, indicating somewhat larger differences between realizations. Figure 4c-e show three realizations for Setup C which consists of three neurons. Here the nonlinear behavior is observed in the majority of realizations. Figure 4e also shows the 3800 sets of input parameters (indicated by gray dots) that were used for the training, validation, and testing of NNs. As can be observed, most points are on the right side of the graph at intermediate temperatures between \(-\)5 \({}^{\circ}\)C and 20 \({}^{\circ}\)C. As a result, the NN does not need to perform very well in the outlying region as long as it performs well in the region with the most points. This is probably why the behavior in the region with the most points is quite similar for all realizations as well as for different setups. In contrast, the behavior in other regions can be different and can exhibit unusual nonlinearities. The average MAE value in setup C (2.31 \({}^{\circ}\)C) is similar to Setups A and B (2.32 \({}^{\circ}\)C), while the spread is noticeably larger, indicating more significant differences between realizations. Figure 4f shows an example of Setup D with four neurons. Due to an additional neuron, even more nonlinearities can be observed, while the average MAE value and the spread are very similar to Setup C.  
Next, Figure 4g shows an example of the behavior of a somewhat more complex Setup E with 14 neurons distributed over four layers. Since there are considerably more neurons compared to other setups, there are more nonlinearities visible. The higher complexity also results in a somewhat smaller average MAE value (2.27 \({}^{\circ}\)C) while the spread is slightly smaller compared to Setups C and D. We also tried more complex networks with even more neurons but found that the additional complexity does not seem to reduce MAE values (not shown). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Name** & **Neurons in Layers** & **MAE avg. [10 th perc., 90 th perc.]** \\ \hline Setup A & 1 & 2.32 [2.32, 2.33] \({}^{\circ}\)C \\ Setup B & 1,1 & 2.32 [2.29, 2.34] \({}^{\circ}\)C \\ Setup C & 2,1 & 2.31 [2.26, 2.39] \({}^{\circ}\)C \\ Setup D & 3,1 & 2.31 [2.26, 2.38] \({}^{\circ}\)C \\ Setup E & 5,5,3,1 & 2.27 [2.22, 2.31] \({}^{\circ}\)C \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Description of the simplistic neural networks consisting of only a few neurons. All setups used the same two input parameters, the average temperature in the lowest layer between the ground and 1 km and the 90 th percentile of RH in the layer between the ground and 12 km. The second column denotes the number of neurons in consecutive layers: input layer always contains 2 neurons for 2 input parameters and is not included in the table, whereas the output layer always includes a single neuron. Leaky ReLU was used as activation function for all layers in all setups. The shown MAE values represent the error of the same-day forecast of \(T_{\text{max}}\) for the testing set. Since 50 realizations of NN training were performed for each setup, the average value, the 10 th percentile and the 90 th percentile of MAE values are shown. Finally, Figure 4h shows an example of the behavior of a Setup E that is used to forecast \(T_{\text{min}}\) instead of \(T_{\text{max}}\).