## 2 Reliability Analysis Based on the AdaBoost Surrogate Model
### Reliability Analysis Model Based on the Adaptive AdaBoost Algorithm  
The basic idea of a reliability analysis based on the adaptive AdaBoost algorithm involves employing an adaptive iterative method to find the optimal training sample set, as elaborated in Section 2.2.1. Then, a high-precision surrogate model for failure discrimination of complex structures under multiple failure modes is obtained through simple binary classifier iteration based on the integration and iteration characteristics to the AdaBoost algorithm, as discussed in Section 2.2.2. Finally, the Monte Carlo method is used to determine the failure probability. The detailed analytical procedures are delineated as follows:  
1. Use the adaptive method to find the optimal sample center.
2. According to the sample center, use Latin hypercube sampling to generate training samples.
3. Initialize the weight distribution of the training data. Assign the same weight to each training sample as follows: \(\mathbf{D_{1}}=(w_{11},w_{12},\ldots,w_{1i},\ldots,w_{1N}),w_{1i}=\frac{1}{N}\), \(i=1,2,\ldots,N\).
4. In the \(m\)-\(th\) iterations, obtain \(G_{m}(\mathbf{x})\), which represents the current \(m\)-round iterative classifier; obtain \(e_{m}\), denoting the present classification error; and obtain \(\alpha_{m}\), which signifies the cumulative coefficient as elaborated below: 1. Obtain the basic classifier \(G_{m}(\mathbf{x}):x_{i}\rightarrow\{-1,+1\}\) by learning the training dataset with a weight distribution of \(\mathbf{D}_{m}\).  
2. Calculate the classification error rate of \(e_{m}\) for the training dataset: \[e_{m}=P(G_{m}(x_{i})\
eq y_{i})=\sum\limits_{i=1}^{N}w_{mi}I(G_{m}(x_{i})\
eq y_ {i})\] (3)
3. Calculate the coefficient for \(G_{m}(x)\), where \(\alpha_{m}\) represents the importance of \(G_{m}(x)\) in the final classifier: \[\alpha_{m}=\frac{1}{2}\log\frac{1-e_{m}}{e_{m}}\] (4) It is evident from the preceding equation that \(e_{m}\leq\frac{1}{2}\) and \(\alpha_{m}\geq 0\), and that \(\alpha_{m}\) increases as \(e_{m}\) decreases, thereby indicating that basic classifiers with lower classification error rates have a greater impact on the final strong classifier.
4. Update the weight distribution of the training dataset to obtain a new weight distribution for the subsequent interaction: \[\mathbf{D}_{m+1}=(w_{m+1,1},w_{m+1,2},\ldots,w_{m+1,i},\ldots,w_{m+1,N})\] (5) \[w_{m+1,i}=\frac{w_{mi}}{Z_{m}}\exp(-\alpha_{m}y_{i}G_{m}(x_{i})),i=1,2, \ldots,N\] (6) where \(Z_{m}=\sum\limits_{i=1}^{N}w_{mi}\exp(-\alpha_{m}y_{i}G_{m}(x_{i}))\). As a result, the weights of samples misclassified by the fundamental classifier, \(G_{m}(x)\), are augmented, whereas the weights of accurately classified samples are diminished. In this manner, the AdaBoost algorithm can concentrate on samples that are more challenging to differentiate.
5. Integrate the weak classifiers: \[f(\mathbf{x})=\sum\limits_{m=1}^{M}\alpha_{m}G_{m}(\mathbf{x})\] (7) The ultimate strong classifier is denoted by \[G(\mathbf{x})=sign(f(\mathbf{x}))=sign(\sum\limits_{m=1}^{M}\alpha_{m}G_{m}(\mathbf{x}))\] (8)
6. Use the Monte Carlo method to calculate the failure probability based on the final strong classifier.  
Figure 2 illustrates the flowchart of steps 1-6 for the reliability analysis of complex structures utilizing the adaptive AdaBoost algorithm proposed in this paper.