# Study on Shock Initiation Randomness of Energetic Materials on a Macroscopic Scale
## 4 Applied Example
### Randomness Study
#### 4.3.1 Shock Load Condition Where \(\overline{v}\) Is 1050 m/s
(2) Study on DPR  
(i) Constructing the training sample set and test sample sets  
The sample points that could lead Composition B to be initiated were selected again from the training sample set and test sample sets of the IP, so as to construct the training sample set and test sample sets of the DPR. Therefore, according to the statistical data in Table 4, in addressing the DPR, the training sample set had 1554 sample points, and the 4 test sample sets, with the CVs of the BRVs being respectively 0.005, 0.01, 0.015, and 0.02, had 768, 685, 612, and 585 sample points respectively. (ii) Determining the structure and important parameters of ANNs  
According to the numbers of the BRVs and RRs in the DPR, the neuron numbers of the input layer, hidden layer, and output layer of the GABPNN for the DPR were, respectively, set as 19, 25, and 3. So, the structure of this GABPNN was \(19\times 25\times 3\). In addition, the \(T_{\max}\) was set as 20,000, the \(R_{t}\) was set as 0.05, the \(E_{a}\) was set as 0.001, the \(M_{p}\) was set as 20, the \(G_{\max}\) was set as 30, the \(P_{c}\) was set as 0.75, the \(P_{m}\) was set as 0.1, and the \(C\) was set as 0.5. In addition, the structure and corresponding parameters of the BPNN were the same as the corresponding ones of the GABPNN. The neuron numbers of the input layer and output layer of the RBFNN were the same as the corresponding ones of the GABPNN, and the neuron number of the hidden layer of the RBFNN was equal to the sample size \(q\) of the training sample set. (iii) Performance comparison of ANNs  
Firstly, the GABPNN, BPNN, and RBFNN were trained by the above training sample set. Then, these three trained ANNs were tested by each of the above four test sample sets, and the test results are shown in Table 7 and Figure 8.  
The test results in Table 7 and Figure 8 show that the GABPNN consistently had a Mean Absolute Percentage Errors (MAPE) lower than 2.2% in calculating the \(P\) of each gauge in the numerical model corresponding to these 4 test sample sets, and the MAPEs of the GABPNN were slightly higher than the corresponding ones of the RBFNN only when calculating the \(P\) of the gauge 3 corresponding to the test sample sets where the CVs of BRVs were respectively 0.01 and 0.015, and the MAPEs of the GABPNN were lower than the corresponding ones of the RBFNN and BPNN when calculating the \(P\) of each gauge under other conditions. It is proved that the GABPNN has stronger nonlinear mapping ability than the BPNN and RBFNN, and the GABPNN can accurately calculate the \(P\) of each gauge corresponding to the BRV sample points. Therefore, the GABPNN can be used as the surrogate model of the DPR, and the GABP-MCS can also be used to estimate the statistical characteristics of the \(P\) of each gauge of Composition B. (iv) Estimation of the statistical characteristics of \(P\) based on GABP-MCS  
The sample points that could lead Composition B to be initiated were selected from the prediction sample sets of the IP, so as to construct the prediction sample sets of the DPR. According to Table 6, in addressing the DPR, the 4 prediction sample sets, with the CVs of the BRVs being, respectively, 0.005, 0.01, 0.015, and 0.02, had 78,654, 66,291, 61,281, and 58,711 sample points, respectively. The calculation results of the \(P\)'s of the three gauges in the numerical model corresponding to these four prediction sample sets were obtained by the GABP-MCS, so as to obtain the accurate estimates of the actual statistical characteristics  
Figure 8: The comparisons of the MAPEs that the ANNs calculate the \(P\)’s of the three gauges in the numerical mode in the DPR study under the shock load condition where \(\overline{v}\) is 1050 m/s. (**a**) Gauge 1. (**b**) Gauge 2. (**c**) Gauge 3.  
of the \(P\)'s of these three gauges under the above four randomness conditions, as shown in Table 8 and Figure 9.  
The following are the three observations from Table 8 and Figure 9 and their interpretation.