## 1 Introduction
### Echo State Network  
The ESN follows the concept of reservoir computation which is very suitable for modeling complex nonlinear relationships. The ESN contains an input, reservoir, and output layer. The original data are mapped to the high-latitude reserve pool through the input layer, and the model is trained through the pool computation structure. The ESN differs from other network models in terms of training mechanisms. Firstly, there is no error backpropagation in the ESN. Secondly, the ESN need not train the input weight matrix and  
Figure 2: The network structure of CMBLS [33].  
hidden layer weight matrix. Only the output weight matrix needs to be trained [34]. The network structure of the ESN is shown in Figure 3.  
As shown in Figure 3, the input layer of ESN at the time \(t\) is defined as \(\mathbf{u}(t)=[u_{1}(t),u_{2}(t),\ldots,u_{K}(t)]\), the reservoir state is \(\mathbf{x}(t)=[x_{1}(t),x_{2}(t),\ldots,x_{N}(t)]\), and the network output is \(\mathbf{y}(t)=[y_{1}(t),y_{2}(t),\ldots,y_{L}(t)]\), where \(K\), \(N\), \(L\) represents the number of input samples, the number of reservoir neurons, and the output dimension in that order.  
To ensure the echo state property of the ESN, the spectral radius range is usually set to (0,1), and the update and output formulas of the reservoir are shown below.  
\[\mathbf{x}(t+1)=(1-\delta)\mathbf{x}(t)+\delta\cdot f\Big{(}\mathbf{W}^{in}u(t)+\mathbf{W}^{ \text{res}}\mathbf{x}(t)\Big{)} \tag{7}\]  
\[\mathbf{y}(t+1)=g\big{(}\mathbf{W}^{out}\mathbf{x}(t+1)\big{)} \tag{8}\]  
\(\mathbf{x}(t+1)\) denotes the state of the reservoir at the time \(t+1\), \(\mathbf{y}(t+1)\) denotes the network output at the time \(t+1\), \(\delta\) denotes the leakage coefficient, \(\mathbf{W}^{in},\mathbf{W}^{\text{res}}\) denotes the weight matrix generated by the random initialization of the input layer and the reservoir, respectively, \(f(\cdot)\), \(g(\cdot)\) denotes the activation function of the reservoir and the output layer. Formula (7) can continuously update the state of the ESN model after each set of data input to the model until all data input is completed and the output of the network model is obtained by Formula (8). Where \(\mathbf{W}^{out}\) is calculated as follows:  
\[\mathbf{W}^{out}=\mathbf{Y}\mathbf{X}^{T}\Big{(}\mathbf{X}\mathbf{X}^{T}+\lambda\mathbf{I}\Big{)}^{-1} \tag{9}\]  
\(\mathbf{X}\in\mathbb{R}^{N\times K}\) denotes the state matrix of the reservoir pool, \(\mathbf{Y}\in\mathbb{R}^{L\times K}\) means the actual output, \(\lambda\) denotes the regularization coefficient, \(\mathbf{I}\) denotes the unit matrix, and \(\mathbf{W}^{out}\) is computed by many methods, including ridge regression [36], recursive least squares [37], singular value decomposition, and pseudo-inverse solution [38], etc. The training mechanism of the ESN network can be simply training the output weight matrix, which will significantly reduce the difficulty of network training.  
As a clear comparison, the methods of time series prediction above are summarized as follows in Table 1, including the statistical approach, classical deep learning network, and broad learning system.  
Figure 3: ESN model structure [35].  
As shown in Table 1, each model has different characteristics and deficiencies. For the DeepESN in the deep learning networks, the echo state structure brings a new approach to regression modeling. The BLS can reduce the training resources in the typical deep learning networks. Analyzing related works above can provide a solution to combine the different advantages. Then an improved method is proposed in this paper. The new network takes the BLS as the basic framework and integrates the broad structure with the ESN. It is improved both on the learning ability and the network structure scale.