## 3 Public Data and Modeling Experiments
### Modeling Experiment Based on KAN-C  
KAN-C, featuring the largest number of neural nodes, offers greater degrees of freedom in modeling input-output relationships compared to KAN-A and KAN-B. Figure 3 illustrates its three network structures.  
Figure 3: The network structure and evolution of KAN-C. (**a**) The input layer’s \(x_{3}\) forms strong connections with 5 hidden nodes and weak connections with 3 hidden nodes, while the output layer linearly aggregates signals. (**b**) The input features \(x_{1}\) and \(x_{2}\) connect to a small subset of hidden nodes, while \(x_{3}\)’s connections are significantly weakened. The output layer remains a linear aggregator. (**c**) The hidden layer is pruned to retain only 3 nodes (17 removed). The inputs \(x_{1}\), \(x_{2}\), and \(x_{3}\) nonlinearly connect to hidden nodes, and the output layer performs nonlinear signal aggregation.  
The initial structure displays the neurons and connections in KAN-C. The post-training structure shows dynamic parameter adjustments, such as reduced weights (observed as lighter colors) for some neurons and connections and strengthened weights for frequently activated neurons (observed as darker colors). Despite its complex initial structure, numerous nodes and edges were pruned away in the final pruned structure.  
Similar to KAN-A and KAN-B, KAN-C retains KANs' common characteristics: upon data loading, input features with larger numerical values were deemed more important. After training and pruning, information transmission in KAN-C shifted to prioritize the feed rate (\(x_{1}\)) and depth of cut (\(x_{2}\)) as primary features.  
Mathematical relationships between the input variables (feed rate, depth of cut, and speed) and the output variable (flank wear) were established using KAN-C. Formulas with coefficients retained to 1, 2, and 4 decimal places are provided, as follows:  
\[y_{fw}=0.3x_{2}+0.2|6.9x_{1}-2.2|-0.2 \tag{9}\]  
\[y_{fw}=0.29x_{2}+0.21|6.89x_{1}-2.23|-0.29 \tag{10}\]  
\[y_{fw}=0.2877x_{2}+0.2058|6.8862x_{1}-2.2267|-0.1393-0.1451e^{-0.0004x_{3}} \tag{11}\]  
The piecewise linear structure of Formula (9) aligns well with the general physical characteristics of tool wear, where wear rates often exhibit abrupt changes as process parameters exceed critical thresholds in practical machining. First, focusing on its linear term: \(0.3x_{2}\) indicates a positive correlation between the depth of cut (\(x_{2}\)) and flank wear (\(y_{fw}\)), with the coefficient 0.3 quantifying the intensity of this effect.  
The nonlinear term \(0.2|6.9x_{1}-2.2|\) reveals a critical effect of the feed rate (\(x_{1}\)). The turning point occurs when \(6.9x_{1}=2.2\), where the impact of the feed rate on wear is minimized. The presence of the absolute value function validates KANs' capability to automatically identify nonlinear features. Speed (\(x_{3}\)) does not appear in the formula, indicating its contribution is lower than other features in this dataset. An alternative possibility is that \(x_{3}\) implicitly influences other parameters, thereby affecting flank wear (\(y_{fw}\)).  
Furthermore, Formula (9) contains both linear and absolute value terms, showing similarities with Formulas (3) and (6). KAN-A, KAN-B, and KAN-C exhibit commonalities in structure, training processes, and modeling results. This demonstrates that KANs' modeling of physical processes is less susceptible to network design and possesses generalizability.