# Global Mechanical Response Sensing of Corrugated Compensators Based on Digital Twins
## 2 Method
### Gaussian Process Regression Model  
The irregular shape of corrugated compensators can lead to problems such as high nonlinearity and large data size when calculating the mechanical properties, which leads to high calculation costs. The key to the fast prediction of the structural stress field is to build its surrogate model. In this paper, Gaussian process regression (GPR) is chosen to build the surrogate model of the corrugated compensator. As a machine learning regression method developed in recent years, GPR is well adapted to handle complex problems with high dimensionality, small samples, and nonlinearities [20]. GPR has the advantages of easy implementation, adaptive acquisition of hyperparameters, probabilistic significance of the output, etc. [PERSON] et al. [21] demonstrated how the Gaussian process (GP) works in solving regression problems with a few vivid visual examples. The following is the inference process of GPR in function space [22]. A GP is a collection of random variables, any finite number of which have a joint Gaussian distribution. A GP is completely specified by its mean function and covariance function. The mean function \(m(\mathbf{x})\) and the covariance function \(k(\mathbf{x},\mathbf{x}^{\prime})\) of a real process \(f(\mathbf{x})\) are defined as  
\[ \begin{array}{c}m(\mathbf{x})=E[f(\mathbf{x})]\\ k(\mathbf{x},\mathbf{x}^{\prime})=E[(f(\mathbf{x})-m(\mathbf{x}))(f(\mathbf{x}^{\prime}-m(\mathbf{x}^{ \prime})))]\end{array}  \tag{1}\]  
and the GP can be written as  
\[f(\mathbf{x})\sim\text{GP}\big{(}m(\mathbf{x}),k\big{(}\mathbf{x},\mathbf{x}^{\prime}\big{)}\big{)} \tag{2}\]  
where \(m(\mathbf{x})\) is often assumed to be 0. For the regression problem, consider the following model:  
\[\mathbf{y}=f(\mathbf{x})+\varepsilon \tag{3}\]  
where \(\mathbf{x}\) is the input vector, \(f\) and \(y\) denote the function value and the observation with noise, respectively, and \(\varepsilon\sim N\big{(}0,\sigma_{n}^{2}\big{)}\) is the error function that obeys the Gaussian distribution. The prior distribution of observation \(\mathbf{y}\) and the joint distribution of \(\mathbf{y}\) and predicted value \(f\)- can be obtained as  
\[\mathbf{y}\sim N\Big{(}0,K(X,X)+\sigma_{n}^{2}\;I_{n}\Big{)} \tag{4}\]  
\[ \begin{array}{c}\begin{bmatrix}\mathbf{y}\\ f_{*}\end{bmatrix}\sim N\Big{(}0,\begin{bmatrix}K(X,X)+\sigma_{n}^{2}\;I_{n}&K(X,\mathbf{x}_{*})\\ K(\mathbf{x}_{*},X)&K(\mathbf{x}_{*},\mathbf{x}_{*})\end{bmatrix}\Big{)}\end{array}  \tag{5}\]  
where \(K(X,X)=K_{n}=\big{(}k_{ij}\big{)}\) is an \(n\)-dimensional symmetric and positive covariance matrix, whose matrix elements \(k_{ij}=k\big{(}\mathbf{x}_{i},\mathbf{x}_{j}\big{)}\) are used to measure the correlation between \(\mathbf{x}_{i}\) and \(\mathbf{x}_{j}\); and \(K(\mathbf{x}_{*},X)=K(X,\mathbf{x}_{*})^{\text{T}}\) is an \(n\times 1\) covariance matrix for test points \(\mathbf{x}_{*}\) and inputs of training set \(X\). \(K(\mathbf{x}_{*},\mathbf{x}_{*})\) is the own covariance of the test points \(\mathbf{x}\). \(I_{n}\) is the \(n\)-dimensional unit matrix. From this, the posterior distribution of the predicted value \(f\)- can be calculated as  
\[f_{*}|X,\mathbf{y},\mathbf{x}_{*}\sim N\Big{(}\overline{f}_{*},\text{cov}\Big{(} \overline{f}_{*}\Big{)}\Big{)} \tag{6}\]  
where  
\[\overline{f}_{*}=K(\mathbf{x}_{*},X)\Big{[}K(X,X)+\sigma_{n}^{2}\;I_{n}\Big{]}^{ -1}\mathbf{y} \tag{7}\]  
\[\text{cov}\Big{(}\overline{f}_{*}\Big{)}=K(\mathbf{x}_{*},\mathbf{x}_{*})-K(\mathbf{x}_{* },X)\Big{[}K(X,X)+\sigma_{n}^{2}\;I_{n}\Big{]}^{-1}K(X,\mathbf{x}_{*}) \tag{8}\]  
So \(\bar{f}_{*}=\overline{f}_{*},\sigma^{2}f_{*}=\text{cov}\Big{(}\overline{f}_{* }\Big{)}\) are the mean and variance of the corresponding predicted value \(f\)- of the test points \(\mathbf{x}_{*}\).