## 7 KAN-Based Tool Wear Modeling Using Simulated Turning Test Data
### Modeling Results of KAN-A/B/C and Discussion of Their Physical Significance  
Subsequently, KAN-A/B/C were, respectively, employed to model and predict tool wear in simulated turning operations. All formulas retained four decimal places for precision. The formulas demonstrate the following:  
\[\text{KAN-A}:0.0814(-0.0019x_{1}-1)^{2}-0.0775 \tag{19}\]  
\[\text{KAN-B}:0.0358(0.0034x_{1}+1)^{2}-0.0561+\frac{0.1317}{7.2778-0.0087x_{3}} \tag{20}\]  
\[\text{KAN-C}:0.0004x_{1}-0.0021 \tag{21}\]  
where \(x_{1}\) (step) was identified by all three KAN architectures as a significant variable influencing tool wear. KAN-B specifically indicated that increasing \(x_{3}\) (temperature) accelerates tool wear progression. It is readily apparent that KANs' computational approach differs from conventional physics-based analysis. In other words, physics-driven formula derivation focuses on clarifying physical meanings and analyzing physical processes, providing researchers and engineers with broad reference values. In contrast, KAN-based modeling primarily depends on the sample data itself. It can establish explicit 'white-box' formulas within given sample ranges. These formulas yield accurate modeling and prediction results, though they do not guarantee physical accuracy. Thus, KANs can be characterized as a data-driven method that produces explicit formulas with potential physical interpretability. Compared to traditional \"black-box\" methods, KANs offer superior physical interpretability, though their derived formulas may not always align with physical assumptions. Tool wear is a dynamic process involving multi-physics coupling (thermal, mechanical, and material deformation effects). In the current prediction results, KANs achieve explicit modeling through univariate or bivariate functions, specifically by establishing functional relationships between the time step ('step') and contact zone temperature ('temperature'). The variable'step' was identified by all KAN architectures as strongly correlated with the wear amount, consistent with the cumulative nature of tool wear. The role of 'temperature' reflects its direct influence on the hardness and chemical stability of tool materials under elevated temperatures. However, KANs do not account for the influence of cutting forces and cutting speeds on tool wear, which are typically critical factors in physics-based modeling. It suggests that, as a data-driven regression method, KANs may struggle to incorporate the assumptions and constraints inherent in physical principles. On the other hand, KANs' characteristics offer new insights for advancing tool wear research. While many traditional models are highly accurate and meaningful, they often rely heavily on empirical formulas or physics-based derivations. Such physics-based models require extensive experimental calibration of parameters. Conversely, KANs adopt a data-driven approach to directly learn the implicit relationships among machining parameters, thereby reducing manual intervention. Consequently, KAN-based regression and derivation are more outcome-oriented. The formulas provided by KAN-A/B/C achieve minimal errors and high R\({}^{2}\) values during both training and prediction phases, as demonstrated in Table 12.  
Unlike conventional deep learning methods, which typically require large, labeled datasets, KANs achieve stable predictions in the experiment using only 350 samples (with a 1:1 training-test split). Provided that machining conditions remain unchanged, the formulas derived by KANs hold significant reference value for tool wear prediction. Moreover, although the physical interpretability of KAN-A/B/C's formulas is limited, they are explicit functions, enabling intuitive analysis of the relationship between input parameters and tool wear.