## 3 Public Data and Modeling Experiments
### Modeling Experiment Based on KAN-A
Mathematical relationships between the input variables (feed rate, depth of cut, and speed) and the output variable (flank wear) were derived via the KAN. Formulas with coefficients retained to 1, 2, and 4 decimal places are provided, as follows:  
\[y_{fw}=0.1|9.8x_{1}-3.2|-0.5\sin(0.6x_{2}+2.2)-0.62 \tag{3}\]  
\[y_{fw}=-0.5\sin(0.61x_{2}+2.19)+0.14|9.81x_{1}-3.22|-0.05|0.02x_{3}-5.5|-0.1 \tag{4}\]  
\[y_{fw}=0.0011x_{3}-0.495\sin(0.6052x_{2}+2.1927)+0.1438|9.815x_{1}-3.2184| \tag{5}\]  
\[-0.053|0.0213x_{3}-5.5006|-0.103\]  
A higher coefficient precision can enhance modeling accuracy, but it might also bring in noise due to the retention of some negligible terms. Lower-precision formulas facilitate clearer interpretation of mathematical connections between inputs and outputs. Taking Formula (3) as an example, the effects of the feed rate (\(x_{1}\)), depth of cut (\(x_{2}\)), and speed (\(x_{3}\)) on flank wear (\(y_{fw}\)) are discussed. The physical meaning of \(0.1|9.8x_{1}-3.2|\) is that when \(9.8x_{1}=3.2\), the relationship between the feed rate and flank wear undergoes a sudden change. At lower feed rates, flank wear decreases linearly with an increasing feed rate, possibly due to insufficient cutting force prolonging tool-material friction time, where frictional effects dominate tool wear. Conversely, at higher feed rates, flank wear increases linearly with the feed rate. Sharp increases in cutting force elevate mechanical stress and thermal load on the tool. Consequently, the tool wear is accelerated. Unlike the feed rate (\(x_{1}\)), the second term \(-0.5 sin(0.6x_{2}+2.2)\) in the formula indicates that flank wear fluctuates periodically with the depth of cut. It may arise from periodic changes in tool force distribution caused by varying cutting depths. Due to the phase shift of the sine function, as the depth of cut increases, flank wear first decreases and then increases overall. It is hypothesized that specific cutting depths may correspond to the tool's resonance frequency, exacerbating wear; at other depths, stress dispersion or non-resonant conditions lead to less severe wear. Speed (\(x_{3}\)) does not explicitly appear in the formula for two reasons: firstly, KANs' pruning process removed the speed-related connections deemed insignificant to the output. Secondly, terms with \(x_{3}\) as the coefficients were too small to contribute to the flank wear modeling, and were thus eliminated when retaining 1-digit decimal precision. Compared with machine learning and deep learning approaches, KAN-based symbolic modeling not only clarifies mathematical relationships between variables (as a white-box model) but also provides precise guidance for machining process optimization. For example, to minimize flank wear, the feed rate (\(x_{1}\)) and depth of cut (\(x_{2}\)) can be configuredbased on \(0.1|9.8x_{1}-3.2|\) and \(-0.5 sin(0.6x_{2}+2.2)\), while the speed (\(x_{3}\)) can be determined according to roughing or finishing requirements. This advantage is generalizable: the established mathematical relationships adapt to different datasets. For researchers, changes in physical relationships become focal points of discussion. For engineers, mathematical connections between variables under diverse machining demands and conditions can be obtained to optimize processes toward specific goals. It transcends reliance on empirical intuition.