## 5 Conclusions  
In order to quantitatively detect the depth of the natural cracks in the metallic material, especially in the area-aluminum alloy, the natural cracks are generated by the three-point bending test under the cyclic applied loads, and then the depths of the natural cracks are calibrated. Based on the specimens with the natural fatigue crack, PECT platform are constructed, and a series of experiments are carried out to investigate the time domain signal above the cracks with different depth and different detection points. To improve the quantitative characterization accuracy, a GA-based BPNN algorithm is proposed in which the Latin Hypercube is explored to obtain the more uniform population distribution in GA and, consequently, to optimize the weight for BPNN. The investigation indicated that:  
1. Compared with the artificial crack, the natural cracks has the complex geometric distribution, especially in crack depth and the direction.
2. PEC can sense the weak change of the natural crack depth even if there is a prefabricated opening.
3. The proposed GA-based BPNN algorithm can improve the characterization accuracy of the natural crack with the relative error as 2.19%.  
In this paper, the number of specimens with the natural cracks are limited. Therefore, in future, more samples for the natural cracks are acquired or generated by the intelligence algorithm, such as Principal Component Analysis [16] or Generative Adversarial Network methods for data enhancement [24], to improve the characterization accuracy further.  
Conceptualization, Y.Y. and C.S.; Formal analysis, C.S. and H.L.; Investigation, C.S. and H.L.; Methodology, Y.Y. and C.S.; Software, F.W. and D.L.; Validation, C.S. and H.L.; Writing--original draft, C.S. and F.W.; Writing--review and editing, all authors. All authors have read and agreed to the published version of the manuscript.  
This work was supported by the National Nature Science Foundation of China [grant numbers 52275522 and 61960206010], the Sichuan Science and Technology Program [grant number 2022 NSFSC0323], and the Open Fund of Key Laboratory of Traction Power [grant number TPL2206].  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multicolumn{2}{c}{**Probability Parameters**} & \multirow{2}{*}{**Baseline**} & \multirow{2}{*}{**Prediction**} & \multirow{2}{*}{**Relative Error (\%)**} \\ \(\textbf{P}_{m}\) & & & & \\ \hline \multirow{4}{*}{0.4} & 0.001 & & 1.4783 & 7.61 \\  & 0.01 & & 1.5062 & 5.86 \\  & 0.1 & & 1.4904 & 6.85 \\  & 0.001 & & 1.5104 & 5.60 \\
0.65 & 0.01 & 1.6 & 1.5327 & 4.21 \\  & 0.1 & & 1.5153 & 5.29 \\  & 0.001 & & 1.4649 & 8.44 \\
0.9 & 0.01 & & 1.4921 & 6.74 \\  & 0.1 & & 1.5130 & 5.44 \\ \hline \multirow{4}{*}{0.4} & 0.001 & & 1.0945 & 8.79 \\  & 0.01 & & 1.1363 & 5.31 \\  & 0.1 & & 1.1098 & 7.52 \\  & 0.001 & & 1.1319 & 5.68 \\
0.65 & 0.01 & 1.8 & 1.1637 & 3.03 \\  & 0.1 & & 1.1471 & 4.41 \\  & 0.001 & & 1.0741 & 10.50 \\
0.9 & 0.01 & & 1.1071 & 7.74 \\  & 0.1 & & 1.1325 & 5.63 \\ \hline \hline \end{tabular}
\end{table}
Table 7: \(Cont.\)  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Comparison Type** & **Baseline Value (mm)** & **Predicted Value (mm)** & **Relative Error (\%)** \\ \hline Experimental Test Data & 2.4905 & 2.4360 & 2.19 \\ \hline \hline \end{tabular}
\end{table}
Table 8: Validation of fatigue natural crack prediction after weight optimization.