## 3 Public Data and Modeling Experiments
### Modeling Experiment Based on KAN-B  
KANs' learning capability does not correlate positively with the number of network nodes or layers. KART states that any multivariate continuous function can be decomposed into a combination of finite univariate functions. KANs can directly learn to approximate these functions efficiently. This reduces dependence on network depth. Therefore, KAN-B was designed as a KAN with an extremely simple structure. Its purpose was to investigate KAN characteristics through analysis of its test results. We aimed to analyze the similarities and differences between the mathematical relationships of KAN-B and those of KAN-A. Most parameters of KAN-B are consistent with KAN-A, with modified parameters listed in Table 4.  
The initial, post-training, and pruned structures of KAN-B are shown in Figure 2. After first loading the training data, KAN-B identified speed (\(x_{3}\)) as the most important variable, again influenced by data dimensionality. After training, the increased weights of the feed rate (\(x_{1}\)) and depth of cut (\(x_{2}\)) demonstrate that the sparse connections from the input layer to the first hidden layer prioritize these two features. After pruning, the nodes and edges of low modeling importance were removed, which makes the relationship between input vectors and output targets intuitive. Mathematical relationships between the input variables (feed rate, depth of cut, and speed) and the output variable (flank wear) were derived based on KAN-B. The formulas with coefficients retained to 1, 2, and 4 decimal places are provided, as follows:  
\[y_{fw}=0.1|10.7x_{1}+2.2x_{2}-6.0| \tag{6}\]  
\[y_{fw}=0.13|10.72x_{1}+2.23x_{2}-5.98|+0.04 \tag{7}\]  
\[y_{fw}=0.1324|10.7228x_{1}+2.2276x_{2}-5.9784|+0.0438 \tag{8}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Number** & **Training Stage** & **Procedures and Parameter Settings** \\ \hline \hline \multirow{4}{*}{1} & \multirow{4}{*}{Model Pruning} & Node attribution score threshold \\  & & (node\_th): \(1\times 10^{-2}\) \\  & & Edge attribution score threshold \\  & & (edge\_th): \(1\times 10^{-2}\) \\ \hline \hline \multirow{4}{*}{2} & \multirow{4}{*}{Second Training} & Optimization algorithm: LBFGS \\  & & Number of iterations (steps): 300 \\ \cline{1-1}  & & Regularization coefficient (\(\lambda\)): 0 \\ \hline \hline \multirow{4}{*}{3} & \multirow{4}{*}{Symbolic KAN Conversion} & R\({}^{2}\) threshold for goodness-of-fit: 0.4 \\ \cline{1-1}  & & Preference weight for model simplicity (weight\_simple): 0.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: KAN-B training protocols and parameters. Formula (6) highlights a core term: \(|10.7x_{1}+2.2x_{2}-6.0|\). The absolute value describes wear asymmetry: wear accelerates when the linear combination exceeds the threshold, while the wear rate remains lower below the threshold. The coefficient 0.1 before the absolute value acts as a scaling factor. The linear terms \(10.7x_{1}\) and \(2.2x_{2}\) indicate that the feed rate (\(x_{1}\)) contributes more to wear than the depth of cut (\(x_{2}\)). Speed (\(x_{3}\)) does not appear in the formula, suggesting that within the experimental parameter range, its impact on flank wear is insignificant. The constant term \(-6.0\) represents a critical threshold: when \(10.7x_{1}+2.2x_{2}>6.0\), the wear begins to increase. This constant may relate to turning conditions or material properties. Figure 2: The network structure and evolution of KAN-B. (**a**) The input layer’s \(x_{3}\) connects linearly to two nodes in the hidden layer, while the output layer performs linear signal aggregation. (**b**) The input variables \(x_{1}\) and \(x_{2}\) establish both linear and nonlinear connections with the second node in the hidden layer, while \(x_{3}\)’s connection weakens. The output layer now performs nonlinear signal aggregation. (**c**) The two irrelevant nodes in the hidden layer are removed, \(x_{3}\)’s connection disappears entirely, and \(x_{1}/x_{2}\) form nonlinear connections with the hidden layer.