# Large Deformation Modeling of Wing-Like Structures Based on Support Vector Regression
## 2 Theory and Method
### Support Vector Regression  
For a given dataset, \(G=\left\{(x_{1},y_{1}),(x_{2},y_{2}),\cdots,(x_{NT},y_{NT})\right\}\), where \(x_{i}=\left(x_{i1},x_{i2},\cdots,x_{id}\right)^{T}\in\mathbb{R}^{d}\) is the \(i\)th training sample with feature dimensionality of \(d\), \(x_{ij}\) is the value of the \(j\)th feature, \(y_{i}\in\mathbb{R}\) is the regression target value of the sample and \(NT\) is the number of training samples. In the linear situation, SVR is used to find a decision function [35]:  
\[f(x)=\langle\omega,x\rangle+b \tag{10}\]  
where \(\omega\in\mathbb{R}^{d}\) is a normal vector of the hyper-plane, \(b\) is the bias parameter and \(\langle,\rangle\) represents the dot product. The optimization objective is to minimize the risk function as follows:  
\[R=\frac{1}{2}\|\omega\|^{2}+c\sum_{i=1}^{NT}\lvert\xi\rvert_{c}(f(x_{i})-y_{i}) \tag{11}\]  
where \(c\) is a regularization constant and \(\varepsilon\) is the deviation from each sample target value \(y_{i}\), which is also the given threshold. \(\lvert\xi\rvert_{c}\) is an \(\varepsilon\)-insensitive loss function, given as [25; 35]:  
\[\lvert\xi\rvert_{c}=\left\{\begin{array}{ll}0,&if\ \lvert\xi\rvert\leq \varepsilon\\ \lvert\xi\rvert-\varepsilon,&\text{otherwise}\end{array}\right. \tag{12}\]  
In order to describe the real deviation from the sample target value to the fitting value, two slack variables, \(\overline{\xi}_{i}\), \(\xi_{i}\), which tackle the infeasible constraints are introduced, and the minimizing problem can be solved as:  
\[R=\frac{1}{2}\|\omega\|^{2}+c\sum_{i=1}^{NT}\left(\xi_{i}+\overline{\xi}_{i}\right) \tag{13}\]  
\[s.t.\left\{\begin{array}{ll}y_{i}-\langle\omega,x_{i}\rangle-b\leq\varepsilon +\xi_{i}\\ \langle\omega,x_{i}\rangle+b-y_{i}\leq\varepsilon+\overline{\xi}_{i}\\ \xi_{i},\overline{\xi}_{i}\geq 0\end{array}\right. \tag{14}\]  
The first term of Equation (13) describes the flatness of the objective function, which is called \"structural risk\", and the second term describes the fitness between the function and actual sample target values, which is called \"empirical risk\". When constant \(c\) equals to zero, the minimizing problem goes back to linear regression, which also means that the rule is too strict [25]. The schematic diagram of SVR has been shown in Figure 1.  
The dual problem of Equation (13) can be derived using the Lagrange function. Additionally, \(\omega\) can be defined as an integration of training patterns of \(x\), linearly. The decision function can be reformulated as [36]:  
\[f(x)=b+\sum_{i}^{NT}\left(\alpha_{i}-\overline{\alpha}_{i}\right)\langle x_{i},x\rangle \tag{15}\]  
where \(\alpha_{i},\overline{\alpha}_{i}\) are the Lagrangian multipliers.  
For the nonlinear problems, it is possible to find an objective function that also means a hyper-plane to satisfy both good flatness and fitness simultaneously. We can obtain a nonlinear SVR model by replacing the dot product with a nonlinear kernel function. The kernel function can be introduced as:  
\[K(x,x^{\prime})=\left\langle\Phi(x),\Phi(x^{\prime})\right\rangle \tag{16}\]  
where \(\Phi(x)\) represents a map from the sample space to the feature space. The principle of kernel mapping in the nonlinear SVR model is shown in Figure 2. With the kernel function, the decision function of nonlinear SVR can be formulated as:  
\[f(x)=b+\sum_{i}^{NT}\left(\alpha_{i}-\overline{\alpha}_{i}\right)K(x,x_{i}) \tag{17}\]  
The Gaussian kernel is widely used in nonlinear SVR, given by:  
\[K(x,x_{i})=\exp(-\gamma||x-x_{i}||^{2}) \tag{18}\]  
Parameters \(\gamma\) and \(c\) are required to be selected in construction of the nonlinear SVR.