## 2 Overview of the GA-BP Algorithm
(3) Choose the \(k\)-th input sample at random, along with the intended output:  
\[x(k)=(x_{1}(k),x_{2}(k),\ldots,x_{n}(k))\]  
\[d_{O}(k)=(d_{1}(k),d_{2}(k),\ldots,d_{q}(k))\]  
(4) Determine each neuron's input and output for the buried layer:  
\[hi_{h}(k)=\sum_{i=1}^{n}{w_{ij}x_{i}(k)-b_{h}}\;h=1,2\ldots,p\]  
\[ho_{h}(k)=f(hi_{h}(k))\;h=1,2,\ldots,p\]  
\[yi_{o}(k)=\sum{\frac{p}{h=1}w_{jk}ho_{h}(k)-b_{o}}\;o=1,2,\ldots,q\]  
\[yo_{o}(k)=f(yi_{o}(k))o=1,2,\ldots,q\]  
(5) For every neuron in the output layer \(\delta_{o}(k)\), find the partial derivatives of the error function using the network's actual and desired outputs:  
\[\frac{\partial e}{\partial w_{jk}}=\frac{\partial e}{\partial yi_{o}}\frac{ \partial yi_{o}}{\partial w_{jk}}\]  
\[\frac{\partial yi_{o}(k)}{\partial w_{jk}}=\frac{\partial(\sum_{h}^{p}{w_{ho}ho _{h}(k)-b_{o}})}{\partial w_{jk}}=ho_{h}(k)\]\(\frac{\frac{\partial e}{\partial y_{i}}}{\partial y_{o}}=\frac{\frac{\partial( \frac{1}{\sum_{i=1}^{d}}(d_{o}(k)-y_{o}(k)))^{2}}{\partial y_{i}}}{\partial y_{ o}}\)  
(6) To determine the partial derivatives of the error function for each neuron in the hidden layer, use the connection weights between the hidden layer and the output layer \(\delta_{o}(k)\), as well as the output of the hidden layer \(\delta_{h}(k)\) itself:  
\(\frac{\partial e}{\partial w_{jk}}=\frac{\partial e}{\partial y_{i}}\frac{ \partial y_{i}}{\partial w_{jk}}=-\delta_{o}(k)ho_{h}(k)\)  
\(\frac{\partial e}{\partial w_{ij}}=\frac{\partial e}{\partial hi_{h}(k)} \frac{\partial hi_{h}(k)}{\partial w_{ij}}\)  
(7) Use the \(\delta_{o}(k)\) values of every neuron in the output layer and the outputs of every neuron in the hidden layer \(w_{jk}(k)\) to modify the connection weights:  
\(\Delta w_{jk}(k)=-\mu\frac{\partial e}{\partial w_{jk}}=\mu\delta_{o}(k)ho_{ h}(k)\)  
\(w_{jk}^{N+1}=w_{jk}^{N}+\eta\delta_{o}(k)ho_{h}(k)\)  
(8) Utilize the inputs from each neuron in the input layer and the \(\delta_{h}(k)\) values of each neuron in the hidden layer \(w_{ij}(k)\) to modify the connection weights:  
\(\Delta w_{ij}(k)=-\mu\frac{\partial e}{\partial w_{ij}}=\delta_{h}(k)x_{i}(k)\)  
\(w_{ij}^{N+1}=w_{ij}^{N}+\eta\delta_{h}(k)x_{i}(k)\)  
(9) Compute the global error:  
\(E=\frac{1}{2m}\sum_{k=1}^{m}\sum_{o=1}^{q}\left(d_{o}(k)-y_{o}(k)\right)^{2}\)  
(10) Check whether the error satisfies the specified criteria. When the error exceeds the maximum number of learning iterations or the error reaches the predetermined level of precision, the algorithm stops. Otherwise, select the next sample and continue with a new round of learning. Sufficient training samples are required to ensure the accuracy of BP neural network outputs. They contain input parameters (design parameters) and the expected output (evaluation indicators). The training samples can be obtained from analytical calculations, empirical data, or computational data generated during the design process [20]. However, the gradient descent method is employed to train BP, resulting in local optima and unstable learning and generalization abilities. As new training samples are added, the network's pattern may be altered, leading to deviations in the results [21; 22]. Hence, traditional algorithms cannot fully exploit the advantages of BP neural networks, necessitating the use of GAs for optimization. Genetic algorithms are random global search and optimization methods [23] that describe biological evolution using three operators: replication, crossover, and mutation. The GA generates a set of solutions for each iteration. Initially, the solutions are generated randomly, and in subsequent iterations, new solutions are created through evolution and inheritance. An objective function is used to evaluate the solutions. This process repeats until a convergence criterion is met. The new set of solutions selectively retains high-value old solutions, while incorporating new solutions obtained by combination. Since GAs retain competitive genes in each iteration, they consistently seek the optimal value for the evaluation function [24].