## 3 Adaptive Data Fusion Method Based on ADCNN for Fault Diagnosis
### Feature Fusion  
The core architecture of feature fusion is a deep convolutional neural network (DCNN), performing a feature-level fusion of input multi-channel image features, further optimizing the data structure, and then using adaptive learning rate Adam to perform gradient descent optimization training. Lastly, the normalized Softmax function is used for fault diagnosis and classification.  
Figure 5: Vibration signal scale.  
Figure 6: Temperature signal two-dimensional GASF image.  
The DCNN architecture diagram is shown in Figure 7. The feature fusion includes three layers of convolution layers and pooling layers. The convolution layer uses convolution kernels to extract high-level features of multi-source input. ReLu enhances the nonlinear capability of the model. The pooling layer uses 2 \(\times\) 2 maximum pooling to extract the important features of the feature map to enhance the model's anti-noise ability, and at the same time compresses the feature dimension to reduce the computational complexity, so that the training speed can be accelerated and converged. In order to avoid overfitting, the model adds a Dropout between the convolutional layer and the pooling layer and randomly discards 30% of the neurons during each training. The gradient of the discarded neurons is 0, so each training reduces dependencies between random neurons. After the three-level convolution and pooling layers for feature fusion, the fused feature map is flattened into one-dimensional vector data and input to the fully connected layer classifier for fault prediction and classification. For the loss function, categorical cross-entropy is selected with the activation function Softmax to re-adjust the model output, which has a good effect on multi-classification problems. Categorical cross-entropy can effectively measure the distinguishability between discrete probability distributions. The loss of the sample can be calculated by the following formula:  
\[\text{Loss}=-\sum_{i=1}^{\text{outer}}y_{i}\cdot\log\hat{y}_{i} \tag{16}\]  
Output size is the number of scalar values in the model output. \(y_{i}\) is the corresponding target value, \(\hat{y}_{i}\) is the i-th scalar value.  
The Softmax function maps the probability distribution of each classification result and compresses the K-dimensional vector containing any real number into another K-dimensional vector, so that the range of the classification results is between (0, 1), and the sample vector X belongs to the \"j\". The probability of classification is:  
\[\text{P}(y=j|X)=\frac{e^{X^{\text{TW}}y_{i}}}{\sum_{k=1}^{K}e^{X^{\text{TW}}y_ {k}}} \tag{17}\]  
The stochastic gradient descent optimization method of the model is adaptive moment estimation (Adam). Traditionally, the network weights use a fixed learning rate to update the weights and biases during training, which is prone to low computational efficiency and the inability to effectively find the global minimum. For the global minimum value issue, [PERSON] combines the advantages of the adaptive gradient algorithm (AdaGrad) and root mean square propagation (RMSProp) to dynamically adjust the learning rate according to the gradient to improve sparse gradients and uses an exponential moving average to maintain the average value of the weight gradient magnitudes, which can adaptively  
Figure 7: DCNN core architecture diagram.  
reduce the effect of noise in non-stationary signal classification models. [PERSON]'s formula for updating weights and biases is:  
\[\left\{\begin{aligned} \text{w}^{\prime}&=\text{w}-\eta \frac{\partial\left(\text{w},\text{b}\right)}{\partial\text{w}}\\ \text{b}^{\prime}&=\text{b}-\eta\frac{\partial\left( \text{w},\text{b}\right)}{\partial\text{b}}\end{aligned}\right. \tag{18}\]