## 3 Methods
### Bidirectional Grid Long Short-Term Memory RNNs  
As shown in Figure 4, BiGridLSTM is a recurrent neural network architecture that combines LSTM with grid architecture and bidirectional architecture. BiGridLSTM consists of two GridLSTM recurrent neural networks in the opposite direction. Each GridLSTM contains two LSTM blocks, Time-LSTM and Depth-LSTM. Each LSTM block is composed of an input gate, a forget gate, an output gate and a memory unit, which are used to control the flow of information, determine whether to allow the flow of information to pass, output or forget, and when to allow reading, writing, or clearing information according to the gate switch. The hidden layer output of the forward GridLSTM is similar to the output of a normal RNN, which consists of the input of the current moment and the output of the previous moment. The outputs of the hidden layer 1 at the time t in the depth direction and the time dimension are affected by the output \(\overset{\rightarrow}{\text{h}}_{\text{t-1,l}}^{\text{T}}\) of the hidden layer 1 at the time t \(-\) 1 in the time dimension and the input x\({}^{\text{D}}(\text{t})\) at time t in the depth direction or output \(\overset{\rightarrow}{\text{h}}_{\text{t-1,l}}^{\text{D}}\) of the hidden layer 1 \(-\) 1 at the time t in the forward GridLSTM. Similarly, the outputs of the hidden layer 1 at the time t in the depth direction and the time dimension are affected by the output \(\overset{\rightarrow}{\text{h}}_{\text{t+1,l}}^{\text{T}}\) of the hidden layer 1 at the time t + 1 in the time dimension and the input x\({}^{\text{D}}(\text{t})\) at time t in the depth direction or output \(\overset{\rightarrow}{\text{h}}_{\text{t,l-1}}^{\text{D}}\) of the hidden layer 1 \(-\) 1 at the time t in the reverse GridLSTM. The outputs of forward and reverse GridLSTM are combined and connected to the output nodes to form the final output of BiGridLSTM. It is worth noting that the GridLSTM does not share the cell state in two different directions. The cell state values of the outputs by the Time-LSTM and Depth-LSTM blocks in forward GridLSTM are only passed to the Time-LSTM and Depth-LSTM blocks of the forward GridLSTM, and the cell status values of the outputs by the Time-LSTM and Depth-LSTM blocks in reverse GridLSTM are only passed to the Time-LSTM and Depth-LSTM blocks of the reverse GridLSTM. Moreover, the forward and reverse GridLSTM are not directly connected. Figure 4: Bidirectional Grid LSTM expands in time. Symbol \({}^{\prime}\) indicates reverse in figure. On the one hand, BiGridLSTM increases the cell state flow in both time and depth, which makes this structure address the problem that RNN can not get remote information and alleviate the vanishing gradient in vertical dimension as mentioned in Section 3.1 above [22]. On the other hand, BiGridLSTM uses the forward and reverse GridLSTM to scan the input sequence together, which makes it overcome the drawbacks of not being able to capture context information at the same time. In this paper, we verify the BiGridLSTM model in the speech recognition task. When compared with other unidirectional models and deep models, the performance can be greatly improved. 1. The simplified formulas for the Time-LSTM and Depth-LSTM in the forward GridLSTM block are defined, as follows: \[ \begin{array}{c}\left(\overset{\rightarrow}{\mathbf{h}}_{t,t},\overset{ \leftarrow}{\mathbf{c}}_{t,t}\right)=\text{TIME-LSTM}(\overset{ \rightarrow}{\mathbf{h}}_{t,t-1},\overset{\rightarrow}{\mathbf{h}}_{t-1,t}, \overset{\leftarrow}{\mathbf{c}}_{t-1,t},\overset{\rightarrow}{\mathbf{c}} \overset{\rightarrow}{\mathbf{c}}_{t-1,t},\overset{\rightarrow}{\mathbf{c}} \overset{\rightarrow}{\mathbf{c}}})\\ \left(\overset{\rightarrow}{\mathbf{h}}_{t,t},\overset{\rightarrow}{ \mathbf{c}}_{t,t}\right)=\text{DEPTH-LSTM}(\overset{\leftarrow}{\mathbf{h}}_{t,t-1}, \overset{\rightarrow}{\mathbf{h}}_{t-1,t},\overset{\rightarrow}{\mathbf{c}} _{t-1,t},\overset{\rightarrow}{\mathbf{\Theta}})\end{array} \] (4)
2.