## 3 Methods
### Grid Long Short-Term Memory RNNs  
GridLSTM was first proposed in [9] in 2015, which is an improved structure of the LSTM recurrent neural network. Subsequently, [PERSON] and [PERSON] et al. [20] applied GridLSTM to the field of speech recognition and achieved better recognition results. GridLSTM adopts a method of arranging LSTM blocks into a multidimensional grid, providing a unified framework for depth and timing calculations, and making each dimension in each grid correspond to a series of LSTM blocks. GridLSTM is linearly related to the gating between adjacent cells in each dimension, which makes it possible to mitigate the vanishing gradient in each dimension. As shown in Figure 2, this article uses two-dimensional GridLSTM models for acoustic modeling, in which each grid contains a Depth-LSTM block for depth calculations and a Time-LSTM block for time calculations, respectively.  
Figure 2: GridLSTM (GridLSTM) blocks. A GridLSTM block contains two LSTM blocks in different dimensions. Time-LSTM denotes the LSTM in the time dimension; Depth-LSTM denotes the LSTM block in the depth dimension.  
The calculation process of Time-LSTM block is as follows:  
\[\begin{split}\mathrm{i}_{t,l}^{T}&=\sigma(\mathrm{W}_{i,l }^{D}\mathrm{h}_{t,l-1}^{D}+\mathrm{V}_{i,l}^{T}\mathrm{h}_{t-1,l}^{T}+\mathrm{ U}_{i,l}^{T}\mathrm{c}_{t-1,l}^{T}+\mathrm{b}_{i,l}^{T})\\ \mathrm{f}_{t,l}^{T}&=\sigma(\mathrm{W}_{f,l}^{D} \mathrm{h}_{t,l-1}^{D}+\mathrm{V}_{f,l}^{T}\mathrm{h}_{t-1,l}^{T}+\mathrm{U}_{ f,l}^{T}\mathrm{c}_{t-1,l}^{T}+\mathrm{b}_{f,l}^{T})\\ \mathrm{\hat{c}}_{t,l}^{T}&=\tanh(\mathrm{W}_{c,l }^{D}\mathrm{h}_{t,l-1}^{D}+\mathrm{V}_{c,l}^{T}\mathrm{h}_{t-1,l}^{T}+\mathrm{ b}_{c,l}^{T})\\ \mathrm{c}_{t,l}^{T}&=\mathrm{f}_{t,l}^{T}\circ \mathrm{c}_{t-1,l}^{T}+\mathrm{i}_{t,l}^{T}\circ\mathrm{\hat{c}}_{t,l}^{T}\\ \mathrm{o}_{t,l}^{T}&=\sigma(\mathrm{W}_{o,l}^{D} \mathrm{h}_{t,l-1}^{D}+\mathrm{V}_{o,l}^{T}\mathrm{h}_{t-1,l}^{T}+\mathrm{U}_{ o,l}^{T}\mathrm{c}_{t,l}^{T}+\mathrm{b}_{o}^{T})\\ \mathrm{h}_{t,l}^{T}&=\mathrm{W}_{\mathrm{proj},l}^{ T}(\mathrm{o}_{t,l}^{T}\circ\tanh(\mathrm{c}_{t,l}^{T}))\end{split} \tag{2}\]  
The calculation process of Depth-LSTM block is as follows:  
\[\begin{split}\mathrm{i}_{t,l}^{D}&=\sigma(\mathrm{W }_{i,l}^{D}\mathrm{h}_{t,l-1}^{D}+\mathrm{V}_{i,l}^{T}\mathrm{h}_{t-1,l}^{T}+ \mathrm{U}_{i,l}^{D}\mathrm{c}_{t,l-1}^{D}+\mathrm{b}_{i,l}^{D})\\ \mathrm{\hat{f}}_{t,l}^{D}&=\sigma(\mathrm{W}_{f,l }^{D}\mathrm{h}_{t,l-1}^{D}+\mathrm{V}_{f,l}^{T}\mathrm{h}_{t-1,l}^{T}+ \mathrm{U}_{f,l}^{D}\mathrm{c}_{t,l-1}^{D}+\mathrm{b}_{f,l}^{D})\\ \mathrm{\hat{c}}_{t,l}^{D}&=\tanh(\mathrm{W}_{c,l}^{ D}\mathrm{h}_{t,l-1}^{D}+\mathrm{V}_{c,l}^{T}\mathrm{h}_{t-1,l}^{T}+\mathrm{b}_{c,l}^{ D})\\ \mathrm{c}_{t,l}^{D}&=\mathrm{f}_{t,l}^{D}\circ \mathrm{c}_{t,l-1}^{D}+\mathrm{i}_{t,l}^{D}\circ\mathrm{\hat{c}}_{t,l}^{D}\\ \mathrm{o}_{t,l}^{D}&=\sigma(\mathrm{W}_{o,l}^{D} \mathrm{h}_{t,l-1}^{D}+\mathrm{V}_{o,l}^{T}\mathrm{h}_{t-1,l}^{T}+\mathrm{U}_{ o,l}^{D}\mathrm{c}_{t,l}^{D}+\mathrm{b}_{o,l}^{D})\\ \mathrm{h}_{t,l}^{D}&=\mathrm{W}_{\mathrm{proj},l}^{ D}(\mathrm{o}_{t,l}^{D}\circ\tanh(\mathrm{c}_{t,l}^{D}))\end{split} \tag{3}\]  
In Equations (2) and (3), we use the superscript T and D to represent the Time-LSTM block in the time dimension and the Depth-LSTM block in the depth dimension; the subscript t represents time, and the l represents the number of layers or depth. For example, \(\mathrm{c}_{t,l}^{D}\) denotes the cell state of Depth-LSTM in the depth dimension of the first layer at the time t and \(\mathrm{h}_{t,l}^{T}\) denotes the cell state of Time-LSTM in the time dimension of the first layer at time t. W represents the weight between the output of the Depth-LSTM block in the upper layer grid and each gating unit, and V represents the weight between the output of Time-LSTM at the previous moment and the gating unit in the grid. U represents the weight between the cell states of the LSTM blocks in each dimension of the neighboring grid. In this paper, the initial value of \(\mathrm{c}_{t,0}^{D}\) is set to zero.