## 2 The Parameter Identification Method of Fluid-Solid Coupling of Surrounding Rock Based on GP-DE
### The GP Algorithm  
In the process of back analysis, the operation process of GP is as follows. Assume \(\mathbf{X}=[x_{1},x_{2},\ldots,x_{n}]\) is the \(d\times n\) input matrix, and \(\mathbf{y}=[y_{1},y_{2},\ldots,y_{n}]\) is the output vector, then the training dataset can be expressed as \(\{\mathbf{X},\mathbf{y}\}\); thus, the standard linear regression model with Gaussian white noise can be expressed as:  
\[y_{i}=f(x_{i})+\varepsilon \tag{2}\]  
where \(\varepsilon\) denotes an independent random variable, and \(\varepsilon\sim N\left(0,\sigma_{n}^{2}\right)\), while \(\sigma_{n}^{2}\) represents the variance. The prior distribution of the observed target value \(\mathbf{y}\) can be expressed as:  
\[\mathbf{y}\sim N\left(0,\boldsymbol{C}+\sigma_{n}^{2}\boldsymbol{I}\right) \tag{3}\]  
where \(\boldsymbol{C}=C(\mathbf{X},\mathbf{X})\) denotes a symmetric positive definite covariance matrix of the \(n\)th order. For the test sample \((\mathbf{x}^{\star}\mathbf{y}^{\star})\), where \(\mathbf{x}^{\star}=(x_{1}\mathbf{{}^{\star}}x_{2}\mathbf{{}^{\star}}x_{3} \mathbf{{}^{\star}},\ldots\ x_{n}\mathbf{{}^{\star}})\), \(\mathbf{y}^{\star}=(y_{1}\mathbf{{}^{\star}}y_{2}\mathbf{{}^{\star}}y_{3} \mathbf{{}^{\star}},\ldots\ y_{n}\mathbf{{}^{\star}})\), the joint Gaussian prior distribution of \(\mathbf{y}\) and \(\mathbf{y}^{\star}\) can be obtained and expressed as:  
\[\left[ \begin{array}{c}\mathbf{y}\\ \mathbf{y}^{\star}\end{array} \right]\sim N\left(0,\left[ \begin{array}{cc}C( \mathbf{X},\mathbf{X})+\sigma_{n}^{2}\boldsymbol{I}&C(\mathbf{X},\mathbf{x}^{ \star})\\ C(\mathbf{x}^{\star},\mathbf{X})&C(\mathbf{x}^{\star},\mathbf{x}^{\star})\end{array}  \right]\right) \tag{4}\]  
where \(C(\mathbf{X}\),\(\mathbf{X})\) denotes an \(n\times n\) symmetric positive definite covariance matrix, and \(\boldsymbol{I}\) represents the identity matrix. \(C(\mathbf{X},\mathbf{x}^{\star})=C(\mathbf{x}^{\star},\mathbf{X})^{\mathrm{T}}\) is an \(n\times 1\) covariance matrix consisting of new input test points \(\mathbf{x}^{\star}\) and all input points; \(C(\mathbf{x}^{\star},\mathbf{x}^{\star})\) is the covariance matrix consisting of new input test points \(\mathbf{x}^{\star}\). When the training set \(\boldsymbol{D}\) and the input value \(\mathbf{x}^{\star}\) of a test sample are known, the GP can use the posterior probability formula to calculate the output value \(\mathbf{y}^{\star}\) of the test sample, which can be expressed as:  
\[\mathbf{y}^{\star}|\mathbf{x}^{\star},\boldsymbol{D}\sim N(u_{y^{\star}}, \sigma_{y^{\star}}^{2}) \tag{5}\]  
\[u_{y^{\star}}=C(\mathbf{x}^{\star},\mathbf{X})(C(\mathbf{X},\mathbf{X})+\sigma _{n}^{2}\boldsymbol{I})^{-1}\boldsymbol{y} \tag{6}\]  
where \(u_{y^{\star}}\) and \(\sigma_{y^{\star}  
calculation, the constructed covariance function can meet the requirements of symmetry and positive qualitative. In the early trial calculation process, it is determined to choose the square index covariance function, and its prediction effect is better. It is expressed as:  
\[k_{se}\big{(}x_{p},x_{q}\big{)}=\sigma_{f}^{2}\text{exp}\bigg{(}-\frac{1}{2J^{2} }\big{\|}x_{p}-x_{q}\big{\|}^{2}\bigg{)}+\sigma_{n}^{2}\delta_{pq} \tag{7}\]  
where \(x_{p}\) and \(x_{q}\) can represent the learning samples, prediction samples or combinations of learning and prediction samples depending on a particular situation; \(j\) is the distance correlation between the two data points \(x_{p}\) and \(x_{q}\); \(\sigma_{f}\) is the local correlation; \(\sigma_{n}\) is the standard deviation of the noise; and lastly, \(\delta_{pq}\) is a sign function. When \(p=q\), then \(\delta_{pq}=0\); otherwise, \(\delta_{pq}=1\). The GP-based surface should be trained by representative data samples before it can map the complex nonlinear relation between the jointed parameters and displacements. The data samples can be obtained by model tests, field tests, numerical simulation and other methods. In this study, the data samples were collected using the orthogonal design, uniform design and numerical simulation.