## 4 Modeling Errors of KANs and Discussion
### Vertical (Intra-Method) Comparisons of Modeling Results Among KAN-A/B/C  
Three-dimensional plots of Formulas (3), (6), and (9) are presented in Figure 4. Flank wear physically cannot be negative. When \(x_{1}\) and \(x_{2}\) take small values, both KAN-A and KAN-B predict negative flank wear, while KAN-C only exhibits a small region of negative predictions.  
Figure 4: Three-dimensional plots of mathematical formulas fitted by KAN-A/B/C.  
This indicates that KANs' symbolic conversion process lacks the constraints of physical knowledge. In other words, it currently cannot interpret variable meanings or relationships, and instead only obtains \"white-box\" mathematical expressions between the input features and the output targets through fitting with minimized errors and pruning. Nevertheless, KANs represent an advancement in artificial intelligence interpretability compared to numerous \"black-box\" models.  
Comparisons among the three plots also reveal an insight: KANs' learning process and symbolic formulas are constrained by data characteristics. The final expression forms depend on the dataset scale, dimensionality, and value ranges. Although KAN-A/B/C have distinct architectures in this study, their functional forms exhibit substantial commonalities due to using the same dataset.  
The effective application range of mathematical formulas derived by KANs is also dataset-dependent. For example, if an input variable's sample range during training is [0.5, 20], the formula's validity most likely remains within [0.5, 20]. KANs cannot guarantee the accurate description of input-output relationships outside the range.  
Radar charts plotting the total runtime and error metrics of KAN-A/B/C on training and test sets are shown in Figures 5 and 6. Each endpoint of the radar chart represents an error metric or runtime. The models with higher prediction accuracy and computational efficiency occupy smaller areas.  
KAN-B demonstrates the best overall performance on both training and test sets. While KAN-A performs worst on the training set, its MSE and MaxAE on the test set are lower than those of KAN-C. KAN-B's MaxAE consistently lies between the two, with all other error and runtime metrics being optimal.  
KAN-B has the simplest structure and fewest nodes. It contrasts with typical MLP training experiences. Usually, more complex architectures with more nodes are assumed to have stronger learning capabilities.  
This phenomenon is explained by KART, which states that any multivariate continuous function can be decomposed into a finite combination of univariate functions. Theoretically, KANs approximate complex functions with fewer nodes by adjusting learn  
Figure 5: Comparison of error and runtime among KAN-A, KAN-B, and KAN-C on the training set.  
able activation functions on edges, whereas MLPs rely on increasing layers and nodes to enhance fitting ability. The fundamental difference accounts for their distinct dependencies on node quantity.  
KAN-A/B/C all use B-spline functions as learnable activation functions, in contrast to MLPs' fixed functions like ReLU or Sigmoid. When fitting relationships between features and target variables, MLPs require coordinating numerous parameters to modify global weight matrices, while KANs complete learning through adjustments to B-spline functions, reducing redundant parameters. Node and edge pruning in KANs further amplify the advantage.  
Beyond the above benefits, KAN-B's simple structure acts as a natural regularization. This inherently reduces overfitting risk and stabilizes training performance. These results and discussions infer that model architecture design should prioritize alignment with the problem essence over blindly increasing the complexity.