## 4 Dense Sequential Networks
### Forecast Performance  
Figure 5 shows the performance of the forecasts at different lead times. The NN forecast of \(T_{\text{max}}\) based solely on profiles (Setup X, orange line in Figure 5a) barely beats persistence and becomes worse than the climatological forecast already on day two. This is expected, as the radiosonde observation is local and the info it provides is quickly advected downstream throughout the atmosphere. The forecast is slightly improved when the previous-day \(T_{\text{max}}(t-1)\) is added as an input parameter (Setup Y, green line); however, the best forecast is obtained, when the climatological maximum \(T_{\text{clim}}\) at the forecast valid time is added as an input variable alongside profiles and previous-day maximum (Setup Z, violet line). The MAE for the present-day forecast is 1.8 \({}^{\circ}\)C, thus reduced by 0.5 \({}^{\circ}\)C compared to the persistence forecast. Figure 5c,d show the performance of forecasts of \(T_{\text{min}}\). For all lead times, the predictions of \(T_{\text{min}}\) have a noticeably smaller error compared to predictions of \(T_{\text{max}}\). This indicates that the prediction of the minimum temperatures is easier than the prediction of maximum temperatures (e.g., the MAE of climatological forecasts for \(T_{\text{min}}\) is about 0.8 \({}^{\circ}\)C smaller than for \(T_{\text{max}}\)). For the same-day forecasts, the prediction is also made easier by the shorter time gap between the early-morning radiosonde observation and time of \(T_{\text{min}}\), which typically occurs in the morning. MAE for the present-day forecast of \(T_{\text{min}}\) is 1 \({}^{\circ}\)C for any setup involving measurements of vertical profiles (Setup X, Y, Z). This is approximately 1 \({}^{\circ}\)C smaller than the error for \(T_{\text{max}}\) prediction. The profile-based forecasts beat the persistence forecast at all lead times and become comparable to or worse than the climatological forecast on day three. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline  & & & & \multicolumn{8}{c}{**Batch Size (Number of Epochs = 100)**} \\ \hline  & 1 & 2 & 4 & 8 & 16 & 32 & 64 & 128 & 256 & 512 & 1024 & 2048 \\ \hline MAE avg. & 2.03 & 2.08 & 2.06 & 2.05 & 2.01 & 1.99 & 2.02 & 2.01 & 2.03 & 2.06 & 2.15 & 2.21 \\ MAE 10 th perc. & 1.89 & 1.91 & 1.90 & 1.89 & 1.88 & 1.89 & 1.89 & 1.93 & 1.98 & 2.05 & 2.09 \\ MAE 90 th perc. & 2.31 & 2.42 & 2.28 & 2.33 & 2.11 & 2.13 & 2.22 & 2.14 & 2.22 & 2.15 & 2.31 & 2.35 \\ execution time & 916 s & 504 s & 260 s & 131 s & 67 s & 35 s & 19 s & 11 s & 7.3 s & 5 s & 3.6 s & 2.6 s \\ \hline  & & & & & & & & & & & & & \\ \hline  & 1 & 2 & 5 & 10 & 15 & 20 & 50 & 100 & 150 & 200 & 500 & 1000 \\ \hline MAE avg. & 8.84 & 7.82 & 5.54 & 3.31 & 2.53 & 2.34 & 2.11 & 2.01 & 1.99 & 1.98 & 1.97 & 1.99 \\ MAE 10 th perc. & 7.30 & 6.59 & 3.25 & 2.36 & 2.16 & 2.10 & 1.99 & 1.91 & 1.90 & 1.87 & 1.88 & 1.90 \\ MAE 90 th perc. & 10.00 & 8.91 & 7.36 & 6.05 & 2.91 & 2.65 & 2.28 & 2.22 & 2.14 & 2.17 & 2.11 & 2.07 \\ execution time & 0.4 s & 0.5 s & 0.7 s & 1.0 s & 1.4 s & 1.7 s & 3.8 s & 7.3 s & 11 s & 14 s & 35 s & 70 s \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Influence of batch size and number of epochs on the performance of the NN in setup Y. The MAE values are expressed in \({}^{\circ}\)C. The execution time represents the time it took to train a single NN on a computer using a Nvidia GeForce RTX 3090 GPU. Another interesting and unintuitive feature of Figure 5 is that the error is smaller if the extremes are predicted by NN that uses only previous-day extremes (Setup Q, red line) compared to the persistence forecast (black line). This feature is even more evident in the graphs showing the MAE evolution for the long-range prediction of temperature extremes (Figure 5b,d). While such long-range predictions do not have any practical value, they reveal the underlying features of the NN training. For example, the MAE of forecasts using Setups X, Y and Q appears as a sine wave with twice lesser period than the period of persistence MAE.