## 4 Dense Sequential Networks
### Network Setup  
We tried various NN setups with different designs and input data. After comprehensive experimentation we settled on five setups described in Table 3, which we used to make short- and long-term forecast of \(T_{\text{max}}\) and \(T_{\text{min}}\). Setup X consists of 117 neurons spread over 7 layers (not counting the input layer) and uses only the profile data as input. We experimented with various combinations of the profile variables (interpolated to 118 levels as described in Section 2.1.1) and found that using \(T\),\(T_{d}\) and RH profiles works the best (not shown). Other combinations either produce a larger error or do not improve the error but only increase the network complexity (e.g., if \(p\) or wind profiles are used in addition to \(T\),\(T_{d}\), and RH profiles). Setup Y is the same as Setup X but with the previous day extreme value (either \(T_{\text{max}}\) or \(T_{\text{min}}\)) used as input in addition to the profiles. Setup Z is the same as Setup Y but with the addition of the climatological extreme value as input. Setups Q and R are much simpler (they consist of 15 neurons spread over 5 layers not counting the input layer) and do not rely on the profiles. Setup Q only uses the previous day extreme value as input, while Setup R also uses the climatological extreme value. We also experimented with various NN hyperparameters. Table 4 shows the analysis of the batch size and the number of epochs, which was performed for setup Y for the same-day forecasts of \(T_{\text{max}}\). As can be observed, the batch size does not majorly influence MAE unless the values are really large (e.g., \(\geq\)512) for which the MAE increases. At the \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{2}{c}{\(T_{\text{max}}\)} & \multicolumn{2}{c}{\(T_{\text{min}}\)} \\ \hline  & gradient & value span & gradient & value span \\ avg. T in the lowest 1 km & 1.05 & 1.01 & 0.97 & 0.96 \\
90 th percentile of RH & \(-\)0.16 & 0.16 & 0.17 & 0.18 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The result of the two XAI methods for the same-day forecast of \(T_{\text{max}}\) and \(T_{\text{min}}\) using NN Setup E. The shown values of gradient and value span were averaged over all the test cases and 50 realizations of the training. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Name** & **Neurons in Layers** & **Input Variables** \\ \hline Setup X & 35,35,35,5,3,3,1 & 354 variables: \(T\) profile\((t)\), \(T_{d}\) profile\((t)\), RH profile\((t)\) \\ \hline Setup Y & same as Setup X & 355 variables: same as Setup X + \(T_{\text{max}}(t-1)\) \\ \hline Setup Z & same as Setup X & 356 variables: same as Setup X + \(T_{\text{max}}(t-1)\), \\  & & \(T_{\text{clim}}(t+i)\) \\ \hline Setup Q & 3,3,5,3,1 & 1 variable: \(T_{\text{max}}(t-1)\) \\ \hline Setup R & same as Setup Q & 2 variables: \(T_{\text{max}}(t-1)\), \(T_{\text{clim}}(t+i)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Description of the NN setups used for the short and long-term forecasts of \(T_{\text{max}}\). The second column denotes the number of neurons in consecutive layers with the input layer not shown (the number of neurons in the input layer is always the same as the number of input variables). The time index \(t\) refers to the day of the radiosonde measurement, with \(t-1\) referring to the previous day, and \(t+i\) to the \(i\)-th day in the future. The setups for the \(T_{\text{min}}\) forecast are identical to the setups for \(T_{\text{max}}\) forecast with \(T_{\text{max}}(t-1)\) replaced with \(T_{\text{min}}(t-1)\). In all setups Leaky ReLU was used as activation function for all layers, except for the output layer that used a linear activation function. same time, using a larger batch size offers a very significant reduction in execution time. In the end, we settled for a compromise value of batch size of 256, with a reasonable MAE and a relatively short execution time. On the other hand, the number of epochs does have a significant influence on MAE.