## 2 Materials and Methods
### Experimental Design and Analysis  
The factors influencing hardness (HV\({}_{0.3}\)) and depth (D) of the hardened layer in the laser-induced hybrid hardening process are multifaceted. Notably, induction frequency (IF), laser power (P), scanning speed (V), and induced power (IP) significantly affect both \begin{table}
\begin{tabular}{c c c c c c c c} \hline
**C** & **Si** & **Mn** & **Cr** & **Mo** & **S** & **P** & **Ni** & **Fe** \\ \hline
0.42 & 0.2 & 0.5–0.8 & 0.9–1.2 & 0.2 & \(\leq\)0.035 & \(\leq\)0.035 & 0.03 & Bal. \\ \hline \end{tabular}
\end{table} 
Table 1: Chemical composition of 42 CrMo steel for experiment (wt.%). Figure 1: Schematic diagram of the laser-induction hybrid phase transition process. hardness and depth. RSM and WOA-BPNN were used to analyze the effects of these parameters on hardness and depth. The RSM model is a statistical approach used to address multivariate problems by fitting polynomial functions to the design space, which is constructed using the Box-Behnken design (BBD). This three-level factorial design is ideal for optimizing analytical methods, as it allows the assessment of the effects of multiple factors on response variables while minimizing the need for extensive experimentation. Based on the experimental analysis, we selected the parameters that have a more significant effect on the depth and hardness of the hardened layer. Based on the parameter values in Table 2, BBD experiments were designed using the response surface module in Minitab. For each set of process parameters, 17 data points were generated at induction frequencies of 10 kHz and 15 kHz, resulting in 34 experiments of the laser-induction hybrid hardening process, and some of the experimental parameters are tabulated in Table 3. The 34 samples obtained from the BBD experimental design were insufficient for constructing a neural network prediction model. To enhance the dataset, we used MATLAB to systematically generate an additional 20 sets of experimental parameters within the operational range. We then conducted laser-induced hybrid hardening experiments based on these parameters. These additional data points were evenly distributed between the two induction frequency groups (10 kHz and 15 kHz), similar to the initial 34 experiments. This approach increased the sample size and improved the generalization capability of the model. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Process Parameters** & \multicolumn{3}{c}{**Parameter Selection Values**} \\ \hline Induction frequency & \multicolumn{2}{c}{10 kHz} & \multicolumn{2}{c}{15 kHz} \\ Laser power & 3000 W & 3300 W & 3600 W \\ Scanning speed & 2.5 mm/s & 2.8 mm/s & 3.3 mm/s \\ Induction power & 16 kW & 28 kW & 40 kW \\ \hline \hline \end{tabular}
\end{table} 
Table 2: BBD experimental parameter selection values. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Experiment** & **IF/kHz** & **P/W** & **v/mm\(\cdot\)s\({}^{-1}\)** & **IP/kW** & **D/mm** & **HV\({}_{0.3}\)** \\ \hline (1) & 10 & 3600 & 2.8 & 16 & 5.728 & 704.4 \\ (2) & 10 & 3600 & 3.3 & 28 & 6.307 & 743.2 \\ (3) & 10 & 3300 & 2.8 & 28 & 6.590 & 757.0 \\ (4) & 10 & 3300 & 3.3 & 16 & 4.677 & 708.2 \\ (5) & 10 & 3300 & 2.5 & 16 & 6.173 & 780.1 \\ (6) & 10 & 3300 & 2.5 & 40 & 7.817 & 715.7 \\ (7) & 10 & 3300 & 2.8 & 28 & 6.590 & 757.0 \\ (8) & 10 & 3000 & 3.3 & 28 & 5.674 & 695.4 \\ (9) & 10 & 3600 s & 2.5 & 28 & 7.453 & 742.7 \\ (10) & 10 & 3600 & 2.8 & 40 & 7.520 & 759.0 \\ (11) & 10 & 3000 & 2.8 & 16 & 5.202 & 744.9 \\ (12) & 10 & 3300 & 3.3 & 40 & 6.927 & 736.8 \\ (13) & 10 & 3000 & 2.8 & 40 & 7.008 & 729.5 \\ (14) & 10 & 3000 & 2.5 & 28 & 6.819 & 697.8 \\ (15) & 10 & 3300 & 2.8 & 28 & 6.590 & 757.0 \\ (16) & 15 & 3300 & 2.8 & 28 & 5.863 & 802.5 \\ (17) & 15 & 3300 & 3.3 & 16 & 3.194 & 829.7 \\ (18) & 15 & 3000 & 2.5 & 28 & 6.024 & 763.3 \\ (19) & 15 & 3300 & 2.5 & 16 & 4.380 & 746.8 \\ (20) & 15 & 3600 & 2.8 & 16 & 4.111 & 753.0 \\ \(\vdots\) & \(\vdots\) & \(\vdots\) & \(\vdots\) & \(\vdots\) & \(\vdots\) & \(\vdots\) \\ \hline \hline \end{tabular}
\end{table}