# Back Analysis of Geotechnical Engineering Based on Data-Driven Model and Grey Wolf Optimization
## 2 Reduced-Order Model  
### Constructing the ROM Model  
The ROM was used to predict the system response using a low-order model based on numerical methods and the proper orthogonal decomposition in engineering. For any \(x_{i}\), \(i=1,2,\ldots,I\), and \(\theta_{j},j=1,2,\ldots,J\), the proper orthogonal decomposition was utilized to obtain the following equation [24].  
\[\tilde{u}^{h}\big{(}x_{i},\theta_{j}\big{)}=\sum_{k=1}^{K}\beta_{k}\big{(} \theta_{j}\big{)}\varphi^{k}(x_{i})+\widetilde{g}\big{(}x_{i},\theta_{j}\big{)} \tag{1}\]where \(\tilde{u}^{h}\) denotes the solution of field variables for geotechnical structure, \(\theta_{j}\) and \(x_{i}\) denote the parameter and design variables of a geotechnical model, \(\varphi\) and \(\beta\) note the unknown coefficient of ROM, and \(\widetilde{g}(x,\theta)\) is an extended boundary condition in the entire domain.  
\[\widetilde{g}(x,\theta)=\left\{\begin{array}{c}g(x,\theta)\text{ {on }}\partial\Omega\\ 0\text{ {elsewhere}}\end{array}\right. \tag{2}\]  
Equation (1) can be rewritten as follows:  
\[\tilde{u}^{h}=\varphi\beta+\widetilde{g} \tag{3}\]  
By utilizing the Latin hypercube sampling (LHS), the set of design variables \(\theta_{j},\ j=1,2,\ldots,J\) was constructed for determining the unknown coefficient \(\varphi\). Then, the corresponding discrete solutions (snapshots) of the numerical model \(w_{j}=u^{h}\big{(}\theta_{j}\big{)}-\widetilde{g}(\theta_{j}),\ j=1,2,\ldots,J\), were acquired based on numerical techniques such as finite element, discrete element, boundary element, etc. The spatial Gram matrix by \(\mathbf{M}^{x}\) can be obtained as follows:  
\[M^{x}_{ij}=\big{(}w_{i}\cdot w_{j}\big{)},\ i,j=1,2,\ldots,J \tag{4}\]  
where \((w_{i}\cdot w_{j})\) notes the inner product between \(w_{i}\) and \(w_{j}\).  
The descending order of the positive eigenvalues of \(\mathbf{M}^{x}\) is listed in the following form.  
\[\lambda_{1}\geq\lambda_{2}\geq\ldots\geq\lambda_{J}\geq 0 \tag{5}\]  
The first K eigenfunctions \(\varphi^{k}(x),\ k=1,2,\ldots K\) corresponds to the first \(K\) eigenvalues, providing the orthogonal principal direction of snapshots. If \(r^{k}=\big{(}r^{k}_{j}\big{)}_{j=i,i,\ldots,J}\) denotes the \(k\)th eigenvector of \(\mathbf{M}^{x}\), then its dual \(k\)th eigenfunctions \(\varphi^{k}(x)\) can be determined according to the following form.  
\[\varphi^{k}(x)=\sum_{j=1}^{K}r^{k}_{j}w_{j}(x) \tag{6}\]  
where K notes the basis size of the proper orthogonal decomposition and can be solved in the following inequation.  
\[\frac{\sum_{i=1}^{K}\lambda_{i}}{\sum_{i=1}^{J}\lambda_{i}}>k \tag{7}\]  
where \(k\) is the user-specified tolerance and equals 0.9999 in this study.  
The following penalized minimization problem can solve the unknown coefficient \(\beta\).  
\[\min_{\beta_{j}\in R^{K}}\|u^{h,j}-\varphi\beta_{j}-\widetilde{g}_{j}\|^{2}+ \mu\|\beta_{j}\|^{2} \tag{8}\]  
\(\beta_{j}\) can be solved by the following normal equation.  
\[\Big{(}\varphi^{T}\varphi+\mu I_{K}\Big{)}\beta_{j}=\varphi^{T}\Big{(}u^{h,j }-\widetilde{g}_{j}\Big{)},\ j=1,2,\ldots,J \tag{9}\]  
where \(\mu\) notes a small regularization parameter.