## 2 Theoretical Foundation
### Stochastic Kriging Modeling with Limited Samples  
The Kriging model is an unbiased estimation model that predicts responses at unknown test points based on known experimental point information, assuming that all data follow an n-dimensional normal distribution [28]. Essentially, the Kriging model is an interpolation model, and its interpolation result is a linear weighting of the known sample response values. To calculate the weighting coefficients, the Kriging model introduces  
Figure 2: Equivalent average pulse control force.  
a statistical assumption: the unknown function is regarded as a specific realization of a Gaussian stationary random process. Since the stochastic response surface model based on polynomial chaos expansion (PCE) is modeled against the background of the response surface model, its fitting ability for high-order nonlinear problems is slightly insufficient. Thus, deterministic Kriging surrogate models are often considered more attractive than response surface models. Therefore, reference [29] proposes developing the deterministic Kriging surrogate model into the stochastic space, forming the stochastic Kriging surrogate model. Uncertainty research often requires the numerical calculation of a large number of repeated samples. This method effectively reduces the amount of calculation in uncertainty design by introducing a covariance matrix representing the inherent uncertainty of samples into the mathematical model.  
In cases where there are repeated sample points with uncertainty, the DK surrogate model is transformed from the deterministic space to the stochastic space, and random samples in the space are represented in the following form:  
\[\begin{array}{c}Y_{j}(x)=\beta+M(x)+\varepsilon_{j}(x)\\ \eta(x)=\mathrm{var}(\varepsilon(x))\end{array}, \tag{1}\]  
where \(Y_{j}(x)\) represents the \(j\) repeated sample point at position \(x\) (\(j=1,2\cdots,p\), \(p\) representing the response value dimension), \(\beta\) is the global mean, \(M(x)\) represents non-intrinsic uncertainty, and \(n\) is a random variable used to represent the difference between the mathematical expectation and trend terms of data with uncertainty at each position. The term \(\varepsilon_{j}(x)\) is used to represent the perturbation of the repeated sample at \(x\), which is assumed to be a random variable and follows a normal distribution. It is independent of the perturbations of all repeated samples and independent of the non-intrinsic uncertainty. The term \(\eta(x)\) is the variance of the uncertainty at \(x\). In the SK surrogate model constructed in this paper, the prediction of the mean value at unknown locations \(\overline{Y}^{*}(x)\) is estimated using the following form:  
\[\overline{Y}^{*}(x)=\omega_{0}(x)+\sum_{i=1}^{n}\omega_{i}(x)\overline{Y}(x), \tag{2}\]  
where \(\overline{Y}(x)\) represents the mean vector at known locations. The term \(\omega_{i}(x)\) represents the weight assigned to the mean value at known locations, which is used to estimate the mean value at unsampled locations by weighting the known mean values. The construction form of the mean squared error (\(MSE\)) for the SK model with uncertainty can be expressed as the following equation:  
\[MSE=E\bigg{[}\Big{(}Y_{j}(x)-\overline{Y}^{*}(x)\Big{)}^{2}\bigg{]}=E\Big{[} \beta+M(x_{0})+\varepsilon_{j}(x_{0})-\omega_{0}-\omega^{\mathrm{T}}\overline {Y}\Big{]}. \tag{3}\]  
The construction of the mean square error in the SK model considers the inherent uncertainty information of repeated samples during the model construction process. This allows for the representation of the assumed truth and estimation form. Consequently, the estimate of the mean and variance of the SK model for unknown points with known uncertainty can be obtained. By accurately predicting the mean and variance, the SK model effectively addresses the issue of high computational costs associated with traditional methods of uncertainty estimation.