## 4 Dense Sequential Networks
### Network Interpretation  
Finally, we explored the impact of our predictors on the forecast. Figures 6 and 7 show results of the XAI analysis for forecasts of \(T_{\rm max}\) using Setups X and Z (the results for other setups, some additional forecast lead times and forecasts of \(T_{\rm min}\) are shown in the Supplementary Materials in Figures S1-S6). For Setup X, both the average input gradient and the output value span suggest that the medium-range prediction (up to 10 days, Figure 6a-c) is governed mostly by the lowest 1 km of the temperature profile, and the lowest 200 m of the dew-point temperature profile. Low relative humidity in the bottom 1 km is associated with higher temperature extremes. In the first 10 days, the mid- and upper-tropospheric levels barely affect the temperature prediction, as the values of gradient and span are rather small and noisy. However, the 100-day \(T_{\rm max}\) prediction (Figure 6d) is strongly affected by the entire vertical temperature profile, meaning that the network grasps the (seasonal) info on \(T_{\rm max}\) from the changing atmospheric profile. First, the larger part of the temperature profile (below 1.5 km and above 5 km altitude) shows a negative impact on \(T_{\rm max}\). If the measured temperatures are low (likely winter), then the forecasted \(T_{\rm max}\) will be higher after 100 days (likely spring), and vice versa: high temperatures (likely summer) will lead to lower temperatures after 100 days (likely autumn). Apart from the mean vertical \"signal\", the neural network also learns the vertical profile of the measured temperatures, which define the \(T_{\rm max}\) in 100 days. In the majority of the lower half of the troposphere, the \(T_{\rm max}\) exhibits positive sensitivity to measured temperatures, and the opposite in the upper troposphere. This can be explained by the seasonal differences in the average vertical temperature gradient at the location. The average temperature gradient is largest in the summer and smallest in the winter (see Figure S9 in the Supplementary Materials). The larger the vertical temperature gradient (likely summer), the colder the \(T_{\rm max}\) in 100 days and vice versa. It is also worth noting that the spread of the gradient metric is much larger compared to the spread of the value span metric. For example, the typical standard deviation of the gradient values for the setups shown in Figure 6 is about 0.2 for all input variables at all altitudes (see Figure S7 in the Supplementary Materials). This is significantly larger than the average gradient values (which are limited to the range \([-0.1,0.1]\)). Thus, even though the average gradient value might be zero (indicating a rather small overall influence on the forecasted value), the gradient value for a specific day in the test set might be quite large by size and be either positive or negative. In contrast, the standard deviation of the value span metric is much smaller--typically about 0.02 for the setups shown in Figure 6 (see Figure S8 in the Supplementary Materials). Thus it gives a more dependable measure of the influence of a particular predictor on the forecasted value. Figure 7 show the results of the XAI analysis for forecasts of \(T_{\rm max}\) using Setup Z. The two additional predictors (\(T_{\rm max}(t-1)\) and \(T_{\rm clim}\)) have a large influence on the forecasted value. For the same-day forecasts (Figure 7a), both predictors have a comparable influence on the forecasted value, with the importance of the profiles being smaller; however, with longer forecast times the importance of \(T_{\rm clim}\) increases, while the importance of \(T_{\rm max}\) and the profiles decreases. For the 100-day forecast (Figure 7d) the prediction is almost solely based on \(T_{\rm clim}\).