# Prediction of Power Fluctuations of Gaussian Beams After Transmission Through Turbulent Atmosphere
## 4 Prediction Results and Discussion  
The performance of the TSLSTMFCPN in the prediction of laser power fluctuation caused by atmospheric turbulence was initially inspected using the data collected in the turbulence pool experiment. The training epoch and the initial learning rate were set to 50 and 0.01, respectively. A staircase learning rate decaying strategy was adopted, in which the learning rate is reduced to 0.1 times its previous value in each training epoch, and the beta parameter in the Adam optimizer was set to the default value (0.9, 0.999). The TSLSTMFCPN employs the Root-Mean-Square Error (RMSE) as the loss function and adopts Adam as the optimization strategy. The Adam optimizer boasts computational efficiency, low memory requirements, and applicability to large-scale data and parameters. Additionally, its adaptive learning rate adjustment mechanism can significantly enhance training efficiency. This aids us in rapidly converging to optimal network parameters, thereby improving the network's prediction accuracy. Moreover, two benchmarks, the mean absolute error (MAE) and the mean absolute percentage error (MAPE), were used to evaluate the performance of the TSLSTMFCPN in predicting the laser power fluctuation caused by atmospheric turbulence, which give a quantitative indication of the difference between the predicted and measured values. The MAE and MAPE are defined as  
Figure 6: Schematic of TSLSTMFCPN architecture. Figure 7: Prediction error rate of TSLSTMFCPN vs. rounds of training epoch (**a**) using data collected in turbulence pool experiment and (**b**) using data collected in outdoor experiment. \[\text{MAE}=\frac{1}{N}\sum_{i=1}^{N}\left|y_{i}-\overline{y_{i}}\right|^{2} \tag{9}\]  
and  
\[\text{MAPE}=\frac{100\%}{N}\sum_{i=1}^{N}\frac{\left|y_{i}-\overline{y_{i}} \right|^{2}}{y_{i}} \tag{10}\]  
where \(y_{i}\) represents the measured value at time \(i\), and \(\overline{y_{i}}\) represents the predicted value at time \(i\). If the predicted value given by a neural network model perfectly matches the measured value, the MAE equals 0, indicating an ideal model capable of accurate prediction. The MAPE is the mean absolute error obtained by the normalized error of each point, which can effectively mitigate the effect of individual outliers on the overall absolute error. A lower percentage value of the MAPE signifies higher prediction accuracy of a neural network model. As mentioned previously, the laser power and the meteorological parameters, including atmospheric temperature, atmospheric pressure, humidity, wind speed, and ambient illuminance, were acquired simultaneously in the turbulence pool experiment. The data collected by the experiment were divided into a training set, a verification set, and a test set according to proportions of 85%, 10%, and 5%. The data of laser power and meteorological parameters were used as the input and the laser power as the output for the training of the TSLSTMRCPN. Once the trained model was fed with new input data, multiple predicted laser power values for the subsequent time steps were output. Figure 8a illustrates a comparison between the measured laser power values and those predicted by the TSLSTMRCPN using the data obtained in the turbulence pool experiment. It can be seen that the profile of the laser power values predicted by the model matches well with the measured one. The MAE and MAPE were calculated as 0.0598 milliwatt and 0.3%, respectively. These results demonstrate that the TSLSTMRCPN is able to accurately predict laser power fluctuations caused by atmospheric turbulence. Note that the measured ambient illuminance and wind speed remained almost constant in the turbulence pool experiment. In other words, these two meteorological parameters do not contribute to the laser power fluctuation. Then, they were removed from the input data for the network training and prediction, that is, the new input data included laser power, temperature, humidity, and pressure, and the results of the prediction are illustrated in Figure 8b.