## 4 Numerical Verification
### Damage Pattern Recognition  
Following damage feature extraction using the AANNs, the LMNNs are used to recognize the damage location and severity. The inputs of LMNNs are the extracted damage features as illustrated in Figure 6c, and the outputs are five dimensional vectors [\(\text{S}_{\text{Ele\,1}}\), \(\text{S}_{\text{Ele\,2}}\), \(\text{S}_{\text{Ele\,3}}\), \(\text{S}_{\text{Ele\,4}}\), \(\text{S}_{\text{Ele\,5}}\)], among which \(\text{S}_{\text{Ele\,k}}\) denotes the \(SRR\) of the \(k\)th element of the bridge. \(\text{S}_{\text{Ele\,k}}\) takes the value of the interval [0, 0.3], suggesting that the maximum severity of damage considered is \(SRR\) = 30%. When the \(SRR\) varies from 1% to 30% for each element, 150 damaged cases are elaborated, as listed in Table 3. These damaged cases and the intact case are divided into a training set and a testing set of samples. The training set consists of 16 cases including 15 damaged cases that the \(SRR\) = 10%, 20%, 30% for each element and the intact case; the testing set comprises 135 damaged cases with \(SRR\)s differing from those in the training set. The training sample set is used to train the LMNNs with the cost function of \(MSE\) to control the training. The LMNNs are considered well trained when the \(MSE\) reaches the stop criterion of \(2\times 10^{-5}\). This criterion is determined by a trial-and-error method to ensure enough accuracy (avoiding excessive training) and high efficiency. At this stage, fed with the damage feature vectors of the test samples, the LMNNs produce the estimated vectors of [\(\text{S}_{\text{Ele\,1}}\), \(\text{S}_{\text{Ele\,2}}\), \(\text{S}_{\text{Ele\,3}}\), \(\text{S}_{\text{Ele\,4}}\), \(\text{S}_{\text{Ele\,5}}\)] for every sample. The difference between the estimated vectors and the target output vectors indicates the error of the damage identification. For instance, for the result corresponding to Ele 1, Figure 8a presents the output of Ele 1 estimated by the LMNNs for all the test samples; Figure 8b presents the associated target outputs; Figure 8c depicts the difference between the estimated values and the target outputs, i.e., the damage identification error. Clearly, the Ele 1 output of the LMNN approximates the target very well. For all the test samples, the damage identification results are given in Figure 9a-c for the estimated values, targets outputs, as well as the identification error. Firstly, the damage location can be detected with great accuracy; there are few incorrect judgments of the damage location. For the aspect of damage severity, some errors obstruct from obtaining a very precise prediction of the severity of damage. However, those errors are within a small range. As seen in Figure 9c, the errors are all below an upper limit of about seven, indicating that the detection may only fail for some minor damage. In a word, the results show that the proposed hierarchical neural network model can effectively detect both the location and severity of damage. \begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Element**} & **Training Set** & **Testing Set** \\ \cline{2-3}  & **SRR (\%)** & \(SRR\) (\%) \\ \hline \multirow{3}{*}{1} & 10 & 1, 2,..., 9 \\  & 20 & 11, 12,..., 19 \\  & 30 & 21, 22,..., 29 \\ \hline \multirow{3}{*}{2} & 10 & 1, 2,..., 9 \\  & 20 & 11, 12,..., 19 \\  & 30 & 21, 22,..., 29 \\ \hline \multirow{3}{*}{3} & 10 & 1, 2,..., 9 \\  & 20 & 11, 12,..., 19 \\  & 30 & 21, 22,..., 29 \\ \hline \multirow{3}{*}{4} & 10 & 1, 2,..., 9 \\  & 20 & 11, 12,..., 19 \\  & 30 & 21, 22,..., 29 \\ \hline \hline \end{tabular} Note: the intact bridge (\(SRR\) = 0) is a case of the training set.  
\end{table} 
Table 3: Training and testing set of samples for damage identification. Notably, the proposed model has considerable generalization capability; it can capture the underlying relations between damage features and damage states on the basis of a fairly limited number of training samples. Figure 8: Identification of damage in Ele 1 of the bridge.