## 2 Tool Wear Monitoring Methods
### Optimization of KAN Networks  
A method for replacing the linear structure of ResNet networks with the nonlinear structure of KAN networks was proposed. The KAN algorithm, an intelligent algorithm proposed by [PERSON] et al. [22] in 2024, is based on the Kolmogorov-Arnold representation theorem. This theorem states that the activation function is no longer a fixed Sigmoid or ReLU; instead, it is learnable and applied to the connection rather than the node. The advantages of the KAN algorithm over the traditional MLP algorithm include achieving higher accuracy with fewer parameters and performing curve learning instead of traditional straight-line learning. The network structures of MLP and KAN are shown in Figures 1 and 2. Figure 1: MLP structure. Step I: The raw acquired signals were processed using CWT, converted into CWT time-frequency maps as network inputs, and normalized so that the pixel values satisfied a normal distribution of N(0.5, 0.5). Step II: The CWT time-frequency map is taken as input in the form of a matrix; 64 convolution kernels with a step size of 2 (Reducing the feature map dimension) and a size of 7 \(\times\) 7 (Faster capture of large-scale features) are selected with padding = 0. The ReLU function was selected as the activation function, and the basic features of the images were extracted. Step III: Define two residual blocks: the Basic Block and the Bottleneck Block. The Basic Block contains two 3 \(\times\) 3 (Progressive construction of more complex feature representations) convolutional layers, each followed by a batch normalization layer and the ReLU activation function. The Bottleneck Block contains three convolutional layers: the first reduces dimensionality with a 1c1 convolutional kernel, the middle performs feature extraction with a 3 \(\times\) 3 convolutional kernel, and the last, a 1 \(\times\) 1 convolutional layer, recovers the dimensionality. Each convolutional layer is followed by a batch normalization layer and ReLU activation function. Step IV: Set up four residual packet structures denoted as Layer [0], Layer [1], Layer [2], and Layer [3], where the four packets contain three, four, six, and three residual blocks, \begin{table}
\begin{tabular}{c c c c} \hline
**Layer Name** & **ResNet-34** & **KANS-ResNet-34** & **ResNet-50** \\ \hline Input layer & 7 \(\times\) 7, 64, stride = 2 & 7 \(\times\) 7, 64, stride = 2 & 7 \(\times\) 7, 64, stride = 2 \\  & 3 \(\times\) 3, max pool, stride = 2 & 3 \(\times\) 3, max pool, stride = 2 & 3 \(\times\) 3, max pool, stride = 2 \\ Layer [0] & 3 \(\times\) 3, 64 & \(\left[3\times 3,\ 64\right]\) & \(\left[3\times 3,\ 64\right]\) \\  & 3 \(\times\) 3, 64 & \(\left[3\times 3,\ 64\right]\) & \(\left[3\times 3,\ 64\right]\) \\ Layer [1] & \(\left[3\times 3,\ 128\right]\) & \(\left[3\times 3,\ 128\right]\) & \(\left[3\times 3,\ 128\right]\) \\ Layer [2] & 3 \(\times\) 3, 256 & \(\left[3\times 3,\ 256\right]\) & \(\left[3\times 3,\ 256\right]\) \\ Layer [3] & 3 \(\times\) 3, 512 & \(\left[3\times 3,\ 512\right]\) & \(\left[3\times 3,\ 512\right]\) \\  & 3 \(\times\) 3, 512 & \(\left[3\times 3,\ 512\right]\) & \(\left[3\times 3,\ 512\right]\) \\  & 3 \(\times\) 3, 512 & \(\left[3\times 3,\ 512\right]\) & Average pool, fc, softmax \\  & & & \\  & & & \\ \end{tabular}
\end{table} 
Table 1: Detailed parameters of three models. Figure 3: KANS-ResNet-34 network structure. respectively. In the network structure of KANS-ResNet-34, all four residual packet structures of Layer [0], Layer [1], Layer [2], and Layer [3] were connected using a Basic Block. Step V: The output of Layer [3] is expanded into feature vectors after global average pooling, which is fed into a nonlinear convolutional classifier, including _top_kan, for the classification process. The final output is the three tool wear states. To evaluate the network performance of the KANS-ResNet-34 network, the ResNet-34 and ResNet-50 networks were introduced for comparison.