# Multiple Thermal Parameter Inversion for Concrete Dams Using an Integrated Surrogate Model
## 2 Methodology
### Experimental Design and ISM Construction  
Section 2.1 determines the design variables and structural response of the thermal parameter inversion problem. An approximate model can be adopted to describe the relationship between the two according to Equation (11), thus replacing the finite element analysis in the optimization iteration. This study uses the surrogate model integration technique to develop the approximate model.  
ISM consists of a linear overlay of multiple individual surrogate models in the hope of counteracting errors in prediction through appropriate weighting. In this paper, among the commonly used surrogate models, Kriging, SVR, and RBF, which have high prediction accuracy in dealing with nonlinear problems [23; 31; 32], are selected to construct the ISM. The prediction result \(f_{s}(x)\) can be expressed as  
\[\left\{\begin{aligned} f_{s}(x)=\sum\limits_{i=1}^{M}\omega_{i}f_{i}( x)\\ & s.t.\ \sum\limits_{i=1}^{M}\omega_{i}=1\end{aligned}\right. \tag{12}\]  
where \(\omega_{i}\) is the weight coefficient, \(f\) is the individual surrogate model, and \(M\) is the number of surrogate models.  
The key to constructing an ISM is the weight coefficient determination. In general, the weights of the surrogate model should reflect our confidence in the model and filter out adverse effects. The current approach to determining the weight coefficients is mainly based on the generalized mean square error (\(GMSE\)). Assuming there are \(N\) training samples, a surrogate model is constructed \(N\) times, each time leaving out one of the training samples. Subsequently, the global error is evaluated using the difference between the exact response at the omitted sample and the predicted value:  
\[GMSE= \frac{1}{N}\sum\limits_{k=1}^{N}\left[f(\mathbf{x}_{k})-f\ ^{(-k)}(\mathbf{x}_{k}) \right]^{2} \tag{13}\]  
where \(f(\mathbf{x}_{k})\) is the exact response at \(\mathbf{x}_{k}\), and \(f\ ^{(-k)}(\mathbf{x}_{k})\) is the corresponding predicted response from the surrogate model constructed using all except \((\mathbf{x}_{k},f(\mathbf{x}_{k}))\).  
In this point, a surrogate model with a small \(GMSE\) will have a large weight in the ISM. The Ref [33] provides a heuristic method for obtaining weights with the following equation:  
\[\left\{\begin{aligned} \omega_{i}&=\omega_{i}^{*}/ \sum\limits_{j=1}^{M}\omega_{j}^{*}\\ \omega_{i}^{*}&=(E_{i}+\alpha\overline{E})^{\eta}\\ \overline{E}&=\frac{1}{M}\sum\limits_{i=1}^{M}E_{i} \end{aligned}\right. \tag{14}\]  
where \(E_{i}\) is the \(GMSE\) of the \(i\)th surrogate model. \(\alpha\) and \(\eta\), two parameters that require the user to specify, describe the importance of \(\overline{E}\) and \(E_{i}\), respectively, \(\alpha~{}<~{}1\) and \(\eta~{}<~{}0\). Small values of \(\alpha\) and large negative values of \(\eta\) assign high weights to the best surrogate model. In contrast, large values of \(\alpha\) and small negative values of \(\eta\) assign relatively balanced weights to individual surrogate models. The recommended values are \(\alpha=0.05\), \(\eta=-1\).  
The experimental points used for model training and testing should be sufficiently representative to ensure the satisfactory generalization performance of the ISM. In contrast, if not properly constructed, the reliability of both the integrated and individual surrogate models can hardly be guaranteed. As a result, it is essential to design appropriate datasets for model training and testing.  
The uniform design method is used in this research to generate training and test samples for ISM. Uniform design [34; 35] is one of the most widely used sampling methods, which can effectively capture the critical information in the design space without sufficient a priori information. It also has the advantages of simple operation and efficient sampling and is widely used in complex systems modeling.  
In addition, the performance of the constructed model is evaluated by the determination coefficient [36; 37].