## 2 A Deep Learning Method to Enhance the Deterministic Evaluation of Seismic Responses  
### Artificial Neural Network Methodology  
The ANN methodology is utilizes deep learning models to depict the complex information processing mechanisms of the human brain and nervous system based on the knowledge of a network topology after approximating the biological NN structure and the human brain's stimulus response mechanism [33]. Figure 1 shows a typical neuronal structure in the human body. It can be seen that a single neuron in the human body collects information transmitted by other neurons on dendrites. In the mathematical field, Most conventional ANNs use a multilayer feed-forward backpropagation network, which involves the interconnection of perceptrons with a unidirectional flow of data and computation from inputs to outputs. The objective of applying a multilayer feed-forward backpropagation network in this study is to present a high-precision regression prediction model connecting the seismic intensity (IM) and structural response. As shown in Figure 2, a classical multilayer feed-forward backpropagation network of ANNs is composed of three types of layers: (i) input layers, which receive the input data from the predicted object and are composed of neurons; (ii) hidden layers, which are regarded as a black box with mathematical functions to produce an output specific to the intended results; (iii) output layers, which are the final predicted results of the ANNs. Nodes or neurons make up these layers, representing an activation function for a specific output. Furthermore, a set of weights is used to reflect the connection between each two neuron nodes, providing priority to inputs for the task that the algorithm is seeking to learn. Given input data \(x_{1},x_{2},\ldots,x_{n}\), the output data (\(z_{i}\)) of neuron \(i\) in the hidden layer can be written as  
\[z_{i}=f\left(\sum_{j=1}^{n}w_{ij}\cdot x_{j}+w_{i}\right) \tag{1}\]  
where \(f(\cdot)\) indicates the activation function or transfer function; \(n\) is the number of neuron nodes; k is each neuron node of the given layers; and \(w_{ij}\) represents the synaptic weights. During the deep learning process, the synaptic weights \(w_{ij}\) of each neuron can be written as  
\[w_{ij}=w_{ij}-u\frac{\partial E}{\partial w_{ij}},\ 0\ <u<\ 1 \tag{2}\]  
where \(E\) indicates the loss function, which the ANN aims to minimize. In this study, Matlab 2021a is utilized to implement the ANNs and therefore benefit from building the ANN model in this software package to predict the specifically required dataset, and the following metamodel parameters can enhance the predictive ability. _Number of layers and neurons in hidden layers_: The ANNs map the mathematical relationships between the input and output databases through the connection weight relationships between each layer structure. Evidently, the regression ability of the metamodel largely depends on the complexity of the relationship between the layers of the ANNs. In general, the more hidden layers and neurons there are, the greater the nonlinear fitting ability of the ANNs. At the same time, the calculation cost required for model training calculation will increase significantly, and it will easily lead to the insufficient generalization ability of the ANN model. For the training data, an excellent fitting effect can be obtained from the ANN model. However, the ANN model cannot be well fitted to a database outside the training data, which is called the over-fitting phenomenon. In contrast, too few hidden  
Figure 1: Typical neuron structure in the human body. Figure 2: A classical multilayer feed-forward backpropagation network of ANNs. layers and neurons fails to provide the ANNs with sufficient fitting ability, which will lead to insufficient accuracy in the model training results. Hence, the number of hidden layers and the corresponding number of neurons significantly impact the fitting performance of the ANNs.