## 1 Introduction
### Time-Space Coupling Method of Temperature Field Based on Gauss-Newton Method  
The temporal and spatial evolution law of the temperature field of the surrounding rock of the roadway can be regarded as a binary function of time (\(Fo\)) and space (\(R\)), that is, \(\Theta=f(Fo,R)\). This function needs to fit the unquantified surrounding rock temperature value in time and space. This paper uses the Gauss-Newton method to fit [26; 27]. The principle is as follows:  
Suppose there are \(m\) known temperature data, which are \(\Theta_{1}\), \(\Theta_{2}\), \(\ldots\), \(\Theta_{m}\). There are \(n\) parameters to be solved, which are \(a_{1}\), \(a_{2}\), \(\ldots\), \(a_{n}\), and then the estimated output for each set of independent variables is \(\hat{\Theta}_{1}\), \(\hat{\Theta}_{2}\), \(\ldots\), \(\hat{\Theta}_{m}\). The solution of the time-space coupling characteristics of the temperature field is a process of gradually approaching the estimated output value to the actual output value, that is, \(\hat{\Theta}_{i}-\Theta_{i}\to 0\). Therefore, the solution to the above problem can be transformed into a multidimensional optimization problem:  
\[\min_{a_{1},a_{2},\cdots,a_{n}}\sum_{i=1}^{\text{m}}\left(\hat{\Theta}_{i}- \Theta_{i}\right)^{2} \tag{4}\]  
Let \(r_{i}=\hat{\Theta}_{i}-\Theta_{i}\), then Formula (2) can be transformed into  
\[\min\sum_{i=1}^{\text{m}}\left(r_{i}(a_{1},a_{2},\cdots a_{n})\right)^{2} \tag{5}\] \begin{table}
\begin{tabular}{c c c c} \hline Program & \begin{tabular}{c} **Thermal Conductivity of** \\ **the Surrounding Rock** \\ \end{tabular} & \
\begin{tabular}{c} **Thermal Diffusivity** \\ **Coefficient of the** \\ **Surrounding Rock** \\ \end{tabular} &
\begin{tabular}{c} **Unstable Heat** \\ **Transfer Coefficient** \\ \end{tabular} \\ \hline
1 & \(\lambda=1\) W/(m\(\cdot\)K) & \(a=0.67\)\(\times\) 10\({}^{-6}\) m\({}^{2}\)/s & \(h=26.91\) W/(m\({}^{2}\cdot\)K) \\
2 & \(\lambda=2\) W/(m\(\cdot\)K) & \(a=1.33\)\(\times\) 10\({}^{-6}\) m\({}^{2}\)/s & \(h=15.00\) W/(m\({}^{2}\cdot\)K) \\
3 & \(\lambda=3\) W/(m\(\cdot\)K) & \(a=2.00\)\(\times\) 10\({}^{-6}\) m\({}^{2}\)/s & \(h=12.74\) W/(m\({}^{2}\cdot\)K) \\ \hline \end{tabular}
\end{table} 
Table 1: Simulation scheme. Let \(r=[r_{1}\ r_{2}\cdots r_{m}]^{\mathrm{T}}\), then  
\[\mathrm{min}\sum_{i=1}^{m}(r_{i}(a_{1},a_{2},\cdots a_{n}))^{2}=\mathbf{r}(\mathbf{a})^{ \mathrm{T}}\mathbf{r}(\mathbf{a}) \tag{6}\]  
The Gauss-Newton method is used to solve Formula (4) iteratively, then  
\[\mathbf{a}^{(l+1)}=\mathbf{a}^{(l)}-\big{(}\mathsf{J}\big{(}\mathbf{a}^{(l)}\big{)}^{ \mathrm{T}}\mathsf{J}(\mathbf{a}^{(l)})\big{)}^{-1}\mathsf{J}(\mathbf{a}^{(l)})^{ \mathrm{T}}\mathbf{r}(\mathbf{a}^{(l)}) \tag{7}\]  
where \(l\) is the number of iterations and \(\mathsf{J}(\mathbf{a}^{(l)})\) is the Jacobi matrix of \(\mathbf{r}\) at the \(l\)th iteration. There is a problem of \(\mathsf{J}(\mathbf{a})^{\mathrm{T}}\mathsf{J}(\mathbf{a})\) non-positive definite matrix in the solution process of Formula (5), so the Levenberg-Marquardt algorithm is used to correct it, as shown in Formula (6):  
\[\mathbf{a}^{(l+1)}=\mathbf{a}^{(l)}-\big{(}\mathsf{J}\big{(}\mathbf{a}^{(l)}\big{)}^{ \mathrm{T}}\mathsf{J}(\mathbf{a}^{(l)})+\mu_{l}\mathsf{I}\big{)}^{-1}\mathsf{J}( \mathbf{a}^{(l)})^{\mathrm{T}}\mathbf{r}(\mathbf{a}^{(l)}), \tag{8}\]  
where \(\mu_{l}\) is the damping factor and \(\mathsf{I}\) is the unit matrix. References [20,21] pointed out that the relationship between dimensionless temperature and dimensionless radius conforms to the exponential relationship with e as the base, and the relationship between dimensionless temperature and dimensionless time conforms to the Hill equation relationship, as shown below:  
\[\Theta=\mathrm{I}\times\mathrm{e}^{\mathrm{DR}}+1 \tag{9}\]  
\[\Theta=\frac{\mathrm{E}}{Fo^{\mathrm{G}}+\mathrm{H}}+\mathrm{F} \tag{10}\]  
where \(\mathrm{D}\), \(\mathrm{E}\), \(\mathrm{F}\), \(\mathrm{G}\), \(\mathrm{H}\), and \(\mathrm{I}\) are all undetermined parameters.