## 5 Discussion and Conclusions  
The CMBESN model proposed in this paper is validated on the simulation dataset of MSO and Beijing air quality dataset. The classical deep learning network and the broad learning system are compared. The CMBESN-OE model based on the optimization method is also validated on both datasets. The model testing strategy uses a similar approach to K-fold cross-validation, and the results shown in this paper are the better-performing prediction data after the selection.  
From the result curves in Figures 8 and 10, it can be seen that the CMBESN model fits the actual data curve more closely than the BLS, ESN, CMBLS, and GRU models, especially at the peaks. The errors between the predicted and actual values are shown in Figures 10 and 14. The CMBESN model performs better, especially in the air quality dataset. The models can be evaluated quantificationally using the metrics in Tables 6 and 7. For the MSO dataset, the RMSE of CMBESN decreases by 36.1%, 1.7%, 0.98%, and 4.9% relative to ESN, BLS, GRU, and CMBLS, respectively. The \(R^{2}\) improves by 0.01% and 0.07% relative to BLS and CMBLS. For the air quality dataset, the RMSE of CMBESN decreases by 66.8%, 78.1%, 48.2%, and 45.3% relative to ESN, BLS, GRU, and CMBLS, respectively. However, given the time cost, CMBESN takes more time to train the model. The tradeoff between prediction accuracy and the time cost can be studied in the future.  
The CMBESN-OE model derives from the CMBESN based on the optimization algorithms. The concrete structure parameter and hyperparameter of the CMBESN are determined with the algorithms in Section 3.2. The related experiments are shown in Section 4.4. The results in Figures 18 and 20 show that the errors of the CMBESN-OE model are smaller than the CMBESN. Figures 16 and 19 show the RMSE and the regularization coefficient changing along the optimization process. The RMSEs decline obviously with the variation of the optimized parameter. Similar conclusions can be found in Tables 6 and 7. It proves that the optimization proposed in this paper is efficient. The proposed algorithms can improve prediction accuracy by adjusting the structure and training parameters.  
In this paper, a new network is designed and studied for time series prediction. The broad echo state network is improved in the mapping and enhancement layers. It is proved with the experiment that the proposed network can take advantage of the BLS and ESN. Meanwhile, the optimization method is also studied to improve the prediction ability. In the future, the theoretical foundation should be studied to explain the further reason for the combination of the networks. Besides, the combination has taken more time, which can be found in the experiments. Then, the proposed network should be improved from the training time. An ideal network should be explored based on the proposed method in this paper to balance the prediction accuracy and the computational resources.  
Conceptualization, W.-J.L. and [PERSON]-[PERSON]; methodology, [PERSON]-J.L. and [PERSON]-[PERSON]; writing--original draft preparation, [PERSON]-J.L., [PERSON]-T.B. and X.-B.J.; writing--review and editing, [PERSON]-J.L. and [PERSON]-T.B.; funding acquisition, [PERSON]-T.B., [PERSON]-B.J., [PERSON]-L.K. and T.-L.S. All authors have read and agreed to the published version of the manuscript.  
This work was partly supported by the National Key Research and Development Program of China No. 2020 YFC1606801, National Natural Science Foundation of China Nos. 62173007, 61903009, 62006008.  
**Institutional Review Board Statement: Not applicable.**  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline Model & SMAPE & MAE & RMSE & \(R^{2}\) \\ \hline CMBESN & 0.1646 & 11.1110 & 13.5795 & 0.8959 \\ CMBESN-OE & 0.0721 & 6.5313 & 8.3923 & 0.9602 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Evaluation metrics of CMBESN and CMBESN-OE models on air quality dataset.