## 4 Training and Fine-Tuning of SSDAE-DRN  
It is difficult to obtain the radiated noise of ships, especially the data under different spindle speeds, which limits the training effect and generalization ability of the deep learning algorithm. In Section 1, a mathematical model of ship radiated noise modulation spectrum with arbitrary shaft frequency and blade number based on Gaussian pulse signal is established, which creates conditions for ship modulation spectrum database extension and SSDAE-DRN model training. Based on the analysis of the inherent physical characteristics and the modulation spectrum data of the ship, the modulation signal parameters are summarized (as shown in Table 1). Figure 3: Images of activation functions. (**a**) Relu; (**b**) Sigmoid; (**c**) Elu; (**d**) Tanh. According to the mathematical model of the modulation spectrum and the signal parameters shown in Table 1, the grid search algorithm is used for traversal simulation. The modulation spectrum simulation data is obtained after normalization with shaft frequency as the \"label\", and the number of samples is \(10^{6}\). Among them, the simulated modulation spectrum trace of a 4-blade propeller ship with a shaft frequency of 5 Hz under a typical modulation system is shown in Figure 4. In the simulated envelope spectrum of ship radiated noise shown in Figure 4, the line spectrum at 5 Hz is the shaft frequency line spectrum, the line spectrum at 10 Hz is the second harmonic line spectrum, the line spectrum at 15 Hz is the third harmonic line spectrum, and the line spectrum at 20 Hz is the forth harmonic line spectrum. The envelope spectrum shown in the four subplots all have different degrees of line spectrum missing, and in Figure 4 there is only the 4 th harmonic line spectrum of the shaft frequency. According to the difference between blade modulation, the propeller noise envelope modulation can be divided into uniform modulation and non-uniform modulation. In uniform modulation, the modulation system is the same for each blade, and there is only a leaf frequency line spectrum exists in the modulation spectrum, as shown in Figure 4d. In non-uniform modulation, the modulation system of each blade is different, and in some specific cases, there is a lack of line spectrum, as shown in Figure 4c. The absence of a line spectrum will lead to the poor robustness of the shaft frequency extraction algorithm, the existence of only a leaf frequency line spectrum will lead to the misjudgment of the axis frequency. The actual ship modulation spectrum often has line spectrum components lower than shaft frequency, which are important factors to reduce the accuracy of shaft frequency extraction. The training of SSDAE is divided into two steps: unsupervised pre-training and supervised fine-tuning. Unsupervised pre-training takes the original simulation data after adding noise as the network input, and adopts a greedy algorithm to train layer-by-layer, so that the network learns to reconstruct the intrinsic features of the original data by minimizing the cost function (as shown in Figure 5). \begin{table}
\begin{tabular}{c c} \hline \hline
**Name** & **Parameter Setting** \\ \hline Number of Blades & 3–7 \\ Shaft frequency & 1–20 Hz \\ Frequency sampling interval & 0.1 Hz \\ Modulation system & 0–1 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Modulation signal parameters of ship radiated noise. Next, the SSDAE-DRN model is trained according to Figure 5. The number of input layer nodes is 1500 and the number of hidden layers is 6. The Adma optimizer is used to optimize the network parameters, the learning rate is set to 0.001 and the number of iterations is 200. To evaluate the performance of the model and avoid the overlapping of shaft frequency and blade frequency of different data, which makes the model unable to be generalized, data with shaft frequency as an integer and blade number as 7 in the simulation data are used for training. The training and validation sets are divided according to the proportion of 3:2.