## 4 Simulation Results
### Preprocessing of Data and Parameters
#### 4.2.2 Analysis of Predictive Performance  
The predictive results \(W_{cf}\) and \(T_{o}\) of pulverizing system are shown in Figure 14, including the curves of actual value, mechanism model with nominal parameters (MMNP), LSTM-based empirical model (EM-LSTM), GRU-based empirical model (EM-GRU) and hybrid model with parameter identification only (HMPIO). The upper subgraphs in Figure 14 depict the overall trend of outputs. As a supplement, the lower subgraphs show the details of the corresponding output as well as the difference between models. Because the time-varying parameters in MMNP are the initial nominal values and have not been updated, there are differences between them and the true values. Undoubtedly, this leads to unsatisfactory predictive accuracy. The EM-LSTM and EM-GRU are data-driven models, which learn the dynamic characteristic information of the system through the input and output data. The HMPIO can be said to be a mechanism model in the form of neural networks, which not only has interpretability, but also makes parameters easy to train through BPTT algorithm. Compared to MMNP, HMPIO achieves higher predictive accuracy by training the parameters to approximate the true values gradually. The iterative training  
Figure 13: The iterative process of parameters in three blocks and error compensation models. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Parameter** & **Std. Value** & **Con. Value** & **Actual Value** & **Parameter** & **Std. Value** & **Con. Value** & **Actual Value** \\ \hline \(w_{p2\|1}^{1\ (1)}\) & 0.4666 & 96.9988 & 97 & \(w_{p1\|}^{1\ (2)}\) & 0.3694 & 0.2477 & 0.245 \\ \(w_{p2\|2}^{2\ (2)}\) & 0.8617 & 2.0617 & 1.988 & \(w_{p1\|}^{2\ (4)}\) & 0.8076 & \(8\times 10^{-4}\) & \(7.93\times 10^{-4}\) \\ \(w_{p2\|4}^{2\ (2)}\) & 0.1774 & 0.1774 & / & \(b_{p2\|}^{2\ (2)}\) & 0.1836 & 0.3672 & / \\ \(w_{p1\|}^{1\ (2)}\) & 0.3447 & \(6.21\times 10^{-5}\) & \(6.13\times 10^{-5}\) & \(w_{p0\|4}^{2\ (1)}\) & 0.5169 & 1.3101 & 1.31 \\ \(w_{p0\|1}^{2\ (2)}\) & 0.1650 & \(2.99\times 10^{-5}\) & \(3.38\times 10^{-5}\) & \(w_{p0\|1}^{3\ (2)}\) & 0.4461 & 0.0523 & 0.05 \\ \(w_{p0\|1}^{4\ (1)}\) & 0.6089 & 5.4134 & 5.4135 & \(w_{p0\|1}^{4\ (3)}\) & 0.2476 & 0.0298 & 0.0287 \\ \(w_{p0\|1}^{5\ (1)}\) & \(-0.3812\) & \(-1.91\times 10^{-10}\) & / & \(w_{p0\|1}^{5\ (4)}\) & 0.9328 & 0.9798 & 0.99 \\ \(b_{b0\|}^{5\ (1)}\) & 0.9738 & \(5.84\times 10^{-7}\) & / & \(w_{p0\|3}^{0\ (3)}\) & 0.3334 & 1285.01 & 1283.85 \\ \(w_{p14}^{2\ (3)}\) & 0.3048 & 429144.41 & 429150 & \(w_{p1\|}^{4\ (3)}\) & 0.8271 & \(3.862\times 10^{-5}\) & \(3.85\times 10^{-5}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 9: The convergence value, normalized value and actual value of all parameters. results of the parameters are shown in Figure 13. According to the evaluation indexes proposed by [PERSON] [59], the performance of digital twin model can be analyzed comprehensively. The results are listed in Table 10. The indexes include the accuracy of each output parameter (AOP), global DT accuracy (GDTA), the accuracy variance of each output parameter (ACVAR), the global accuracy variance (GAVAR), the response time (ART) and the response time variance (RTVAR). From the results in Table 10, for pulverizing system, the HMPIO has the highest predictive accuracy and fastest response time. In addition, for this block, the predictive accuracy of HMPIO can meet the requirements, so no error compensation model (ECM) is added. The predictive results of furnace, including \(\rho_{b}\), \(T_{gs}\), \(P_{b}\), \(O_{cp}\) and \(Q_{sl}\), are shown in Figures 15 and 16. The predictive error of MMNP is much larger than that of other methods, especially for \(O_{cp}\) and \(Q_{sl}\). As for the results of HMPIO for variables \(\rho_{b}\) and \(P_{b}\), there are predictive errors in both steady and dynamic processes.