## 3 Materials and Methods
### U-Lossian Deep Learning Network
When we detect over-suppression of this custom loss, it typically means that the model is near convergence. We tested with high learning rates and found that the learning rate of \(1\times 10^{7}\) produced the best results; therefore we trained all experiments with it. To prevent loss function over-suppression, we trained intermediate layers with the usual U-net strategy but supervised the final layer with the loss function to reduce sub-optimal convergence. The input of our U-lossian model is an image representing the voice spectrogram, which is denoted as \(X\). The network learns to map the input to the prediction \(Y\) according to a convolutional feature given by Equation (1):  
\[Y=F_{n}(X_{n}|\Theta_{n})=h(WX_{n}+b),\Theta_{n}=[W,b] \tag{1}\]  
here \(X_{n}\) is a one-dimensional input matrix of \(N\) feature maps, \(W\) is a set of \(N\) one-dimensional kernels that is used to extract a set of features from the input values, \(h\) is an activation function, and \(b\) is a bias vector. Two maximum pooling layers are included in the U-lossian model. The maximum pooling approach is used by the pooling layer in the contraction route to extract prominent characteristics in a region. The pooling size is \(2\times 2\) and the step size is 2, which is the same as downsampling twice. Two-layer fusion layers are also included in the model. Combining the features retrieved by the contraction path's convolutional layer with the output results of the expansion path's upsampling layer, and merging the information lost in feature extraction into the expansion path, can improve the output result. One-layer Dropout layers are included in the model. We improve the model's prediction performance on the test set, and employ the Dropout layer in the U-lossian model to speed up the convergence. Two layers of upsampling are also included in the model. In the expansion path, the features of the same size retrieved by the respective contraction layer are combined, and up-sampling is performed twice to generate an output of the same size as the original input. A loss function, denoted by \(\ell:H\times Z\rightarrow\mathbb{R}_{+}:=[0,+\infty)\), measures the difference between a predicted label and a true label, e.g., \(L^{2}\) loss: \(\ell(f_{\theta},z)=\frac{1}{2}(f_{\theta}(x)-y)^{2}\), where \(z=(x,y)\). Training loss for a set \(S=(x_{i},y_{i})_{i=1}^{n}\) is denoted by \(L_{S}(\theta)\) or \(L_{n}(\theta)\) or \(R_{n}(\theta)\) or \(R_{S}(\theta)\),  
\[L_{S}(\theta)=\frac{1}{n}\sum_{i=1}^{n}\ell(f_{\theta}(x_{i}),y_{i})+\lambda W. \tag{2}\]  
here \(\lambda W\) is a regularization term that penalizes the complexity of the model. The loss function used in the study is the cross-entropy loss function, which is defined as follows:  
\[ \begin{split} L(Y,P(Y/X))&=-logP(Y/X)\\ &=-\frac{1}{N}\sum_{i=1}^{N}\sum_{j=1}^{M}y_{ij}log(p_{ij}).\end{split}  \tag{3}\]  
The loss function is \(L\), the output variable is \(Y\), and the input variable is \(X\). \(y_{ij}\) is a binary index showing whether category \(j\) is the real category of input instance \(xi\). \(N\) is the input sample size, \(M\) is the number of potential categories, and \(y_{ij}\) is a binary index indicating whether category \(j\) is the real category of input instance \(x_{i}\). The probability that the model prediction input instance \(x_{i}\) corresponds to category \(j\) is represented by \(p_{ij}\). The optimal model weights and bias settings are determined by minimizing the loss function, allowing the model to better correctly forecast the PD. A good optimization strategy may speed up model convergence, improve data feature learning, fine-tune the neural network's weight and bias parameters, and reduce the loss function to the maximum degree possible. The strategy of applying an adaptive learning rate will be more favorable to model training and prediction accuracy; also, the model network will be deeper and converge faster. Therefore, the Adaptive Moment Estimation (Adam) [81] algorithm is used to minimize the loss function of the U-lossian model.