## 4 Improved Elman Neural Network for Collaboration Performance Evaluation
### Improved Elman Neural Network Algorithm
#### 4.3.2 Adaptive Learning Rate  
The adaptive learning rate refers to the automatic adjustment of the learning rate during the training process. The adjustment formula is as follows:  
\[\eta(t+1)=\left\{\begin{array}{ll}1.05\eta(t),&E(t+1)<E(t)\\ 0.7\eta(t),&E(t+1)>1.05(t)\\ \eta(t),&\textit{other}\end{array}\right. \tag{19}\]  
If the correction of the weights reduces the error, the learning rate should be increased so that it learns at the maximum learning rate. Otherwise, over-tuning has occurred, and the value of the learning rate should be reduced immediately.  
The specific process of the improved Elman neural network algorithm is as follows:  
1. Initialize the connection weight matrix of the Elman neural network \(w_{1},w_{2},w_{3}\)
2. Randomly select 80% of the collected enterprise indicator data as the training set and input it into the network.
3. Calculate the output of the hidden layer, the context layer and the output layer.
4. Error back propagation.  
Elman neural networks use a BP algorithm to revise weights, and the error of the network is:  
\[E=\sum_{k=1}^{l}\left(y_{k}^{{}^{\prime}}-y_{k}\right)^{2}\left(k=1,2,\ldots,l\right) \tag{20}\]  
where \(y_{k}^{{}^{\prime}}\) is the target output and \(l\) is the total number of samples.  
The adjustment of the connection matrix \(w_{1}\) from the context layer to the hidden layer:  
\[\frac{\partial E}{\partial w_{1}}=\frac{\partial E}{\partial y(k)}\frac{ \partial y(k)}{\partial h(k)}\frac{\partial h(k)}{\partial w_{1}(k)}=-e(k)g^ {\prime}(w_{3}h(k))\cdot w_{3}f^{\prime}(w_{2}h(k-1)+w_{1}(x(k-1)))h(k-1) \tag{21}\]  
The adjustment of the connection weight matrix \(w_{2}\) from the input layer to the hidden layer:  
\[\frac{\partial E}{\partial w_{2}}=\frac{\partial E}{\partial y(k)}\frac{ \partial y(k)}{\partial h(k)}\frac{\partial h(k)}{\partial w_{2}(k)}=-e(k)g^ {\prime}(w_{3}h(k))\cdot w_{3}f^{\prime}(w_{2}h(k-1)+w_{1}(x(k-1)))x(k-1) \tag{22}\]  
The adjustment of the connection weight matrix \(w_{3}\) from the hidden layer to the output layer:  
\[\frac{\partial E}{\partial w_{3}}=\frac{\partial E}{\partial y(k)}\frac{ \partial y(k)}{\partial w_{3}}=-e(k)g^{\prime}(w_{3}h(k))h(k) \tag{23}\]  
5. The connection weight matrix \(w_{1}\), \(w_{2}\), \(w_{3}\) are corrected. The correction formula of the network weight is: \[w_{1}(t+1)=w_{1}(t)+\eta e(k)w_{3}h(k-1)\cdot g^{\prime}(w_{3}h(k))f^{\prime}( w_{1}h(k-1))+w_{2}(x(k-1))\] (24) \[w_{2}(t+1)=w_{2}(t)+\eta e(k)w_{3}x(k-1)\cdot g^{\prime}(w_{3}h(k))f^{ \prime}(w_{1}h(k-1)+w_{2}(x(k-1)))\] (25) \[w_{3}(t+1)=w_{3}(t)+\eta e(k)h(k)g^{\prime}(w_{3}h(k))\] (26)  
Aiming at the dynamics and uncertainty of enterprises' collaboration performance evaluation, an Elman neural network is introduced and its network structure is designed based on the constructed evaluation index system. Then, through the study of the existing evaluation information, the importance of each evaluation index is identified and optimized, and a more reasonable and scientific evaluation result is obtained. Finally, the network's learning algorithm is improved by adjusting the relevant parameters to speed up the convergence speed and improve the accuracy of the evaluation results.