## 3 LSTM Network
### Forward Pass of LSTM Network  
As shown in Figure 6, the LSTM cell acts like a gated leaky neuron, where \(x_{t}\) represents the input at time \(t\); \(h_{t}\) means the cell output; \(C_{t}\) and \(\widetilde{C}_{t}\) denote the cell state and internal hidden state, respectively; \(f_{t}\), \(i_{t}\) and \(o_{t}\) are the forget gate, input gate and output gate, respectively; and \(\sigma(x)\) and \(\tanh(x)\) are the activation functions described using logistic  
Figure 4: Steady state amplitudes for LCO. (**a**) Vertical amplitude. (**b**) Torsional amplitude.  
Figure 5: Time histories of CFD-based nonlinear aeroelastic forces. (**a**) Lift force. (**b**) Torsional moment.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**B (m)** & **H (m)** & **m (kg/m)** & \begin{tabular}{c} **\(\mathbf{I_{m}}\)** \\ **(kg\(\cdot\)m\({}^{2}\)/m)** \\ \end{tabular} & \begin{tabular}{c} \(\mathbf{f_{h}}\) (Hz)** \\ \end{tabular} & \
\begin{tabular}{c} \(\mathbf{f_{\alpha}}\) (Hz)** \\ \end{tabular} &
\begin{tabular}{c} \(\mathbf{\xi_{h}}\) = \(\mathbf{\xi_{\alpha}}\) \\ \end{tabular} \\ \hline
0.8 & 0.08 & 12.00 & 0.44 & 2.01 & 3.79 & 0.005 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Structural parameters of bridge deck system with various leading edges.  
sigmoid \(\frac{1}{1+e^{-x}}\) (in the range of \([0\,1]\)) and hyperbolic tangent sigmoid \(\frac{1-e^{-2x}}{1+e^{-2x}}\) (in the range of \([-1\,1]\)), respectively. The forget gate essentially empowers the memory cell to forget or remove outdated information from previous time steps by using the nonlinear activation function and the multiplication operation. The input gate is used to add new information to the cell state together with the internal hidden state, and the output gate generates the output information from the cell state.  
Accordingly, the forward pass of LSTM in Figure 6 can be expressed as:  
\[f_{t}=\sigma\Big{(}W_{f}\cdot[h_{t-1},x_{t}]+b_{f}\Big{)} \tag{5}\]  
\[i_{t}=\sigma(W_{i}\cdot[h_{t-1},x_{t}]+b_{i}) \tag{6}\]  
\[\widetilde{C}_{t}=\tanh(W_{C}\cdot[h_{t-1},x_{t}]+b_{C}) \tag{7}\]  
\[C_{t}=f_{t}*C_{t-1}+i_{t}*\widetilde{C}_{t} \tag{8}\]  
\[o_{t}=\sigma(W_{o}\cdot[h_{t-1},x_{t}]+b_{o}) \tag{9}\]  
\[h_{t}=o_{t}*\tanh C_{t} \tag{10}\]  
where the symbol \(*\) denotes the dot product operation, and \(W\)- and \(b\)- are the corresponding weight and biases matrices, whose dimensions are the hyper parameters usually determined by trial and error. During the calculation, the current input and hidden state generate the output and new hidden state to be used (together with input) for the calculation at the next time step. As shown in Figure 7, an LSTM network usually consists of an input layer, several hidden layers of LSTM cells and an output layer. It is noted that each LSTM layer has its own initial state that is usually set to zero. For each LSTM layer, its input is the output of the previous layer and its output will be the input of the next layer. The last LSTM layer will be the input for the output layer (usually a linear layer) to generate the final output data \(\widetilde{y}_{j}\) (where \(j\) denotes the \(j\)th training sample).