## 2 Methods
### Development of the Meta-Learning-Based Feature Fusion Network
#### 2.2.2 Design of Base Learners  
The P3 SF spectrum exhibits low resolutions and a relatively simple structural composition. To address this, HasNet-5 (Highlight Feature Network-5 for Active Sonar) was developed, as depicted in Figure 8a. In the ETFF spectrum, the density of spectral lines varies as a function of the relative horizontal orientation angle, thereby necessitating multi-resolution feature extraction. Consequently, MRConv (multi-resolution convolutional module) was introduced, as shown in Figure 9. Building upon the MRConv module, TFasNet-7 (Time-Frequency Feature Network-7 for Active Sonar) was designed to further enhance feature extraction capabilities, as illustrated in Figure 8b. The DFSDF spectrum captures the Doppler components of the target echoes, and it is characterized by energy peaks of varying shapes and sizes. To improve the network's feature representation capability, its width was increased, resulting in the development of DasNet-9 (Doppler Feature Network-9 for Active Sonar), as presented in Figure 8c.  
In Figure 8, in denotes the number of input channels, out denotes the number of output channels, k denotes the size of the convolution kernel, * denotes the times sign, s denotes the convolution stride, and p denotes the padding length. Additionally, In indicates the input data, while Out corresponds to the output data. The 2D convolutional layer components in HasNet-5, TFasNet-7, and DasNet-9 serve as the feature extraction networks (FNet), while the fully connected layers function as the classification networks (CNet). The loss function is defined using the cross_entropy function. FM_SI, FM_TFI, and FM_DI represent the feature output vectors of the feature extraction network, acting as data interfaces for fusing target task information.  
In Figure 9, ch_in specifies the number of channels in the input data, h represents the height of the input data, and w denotes its width. Furthermore, ch_out refers to the number of output channels for each of the two processing streams. Finally, the Cut2d_Center module performs cropping centered on either the maximum energy or geometric center of the feature map.  
Figure 7: Network framework and composition.  
The training data for the base learners are derived from datasets associated with n-way classification tasks, and their parameter updates follow Equation (13):  
\[\mathbf{\theta}_{i}=\mathbf{\theta}_{i-1}-\alpha\
abla_{\theta}L(f_{\theta_{i-1}}) \tag{13}\]  
where \(\mathbf{\theta}_{i}\) denotes the parameters of the base learner after the \(i\)-th update, \(\mathbf{\theta}_{i-1}\) denotes the parameters after the \((i-1)\)-th update, \(\alpha\) represents the step size for updating the base learner, and \(\
abla_{\theta}L(f_{\theta_{i-1}})\) is the gradient of the loss function \(L(f_{\theta_{i-1}})\).  
Figure 8: Network structure and parameters of the base learners: (**a**) HasNet-5 is designed for the spatial feature P3 SF. (**b**) TFAsNet-7 is designed for the time–frequency feature ETFF. (**c**) DasNet-9 is designed for the Doppler feature DFSDF.  
Figure 9: Structure of the MRConv module.