## 3 Prediction of Pump Amplitude Based on the Support Vector Regression Algorithm  
To accurately determine the relationship between the mud pump's amplitude and input parameters, it was necessary to conduct a predictive analysis of the pump's amplitude. In this section, the support vector regression (SVR) algorithm is used to predict the amplitude of the mud pump, enabling the prediction of amplitude conditions using a small amount of data on speed, torque, and amplitude. When using support vector machines for the regression analysis, nonlinear mapping \(\Phi(x)\) was employed to map the input vectors to a high-dimensional feature space. Subsequently, linear regression was performed in the high-dimensional feature space to find the optimal hyperplane that minimized the errors of all samples from the optimal hyperplane. Assuming the training sample set is \(\{(x_{i},y_{i}),i=1,2,\cdots,N\}\), where \(N\) is the number of samples, \(x_{i}\) represents the input values, and \(y_{i}\) represents the desired output values. The regression model of the SVM can be expressed as the following equation [40]:  
\[f(x)=w\times\Phi(x)+b \tag{6}\]  
where \(w\) and \(b\) are the weight vector and bias, respectively. Introducing the penalty factor \(c\) and slack variables \(\xi_{i}(1=1,2,\cdots,N)\) under the insensitive loss function \(\varepsilon\), the solution of the SVR is an optimization problem. \[\left\{ \begin{array}{l}\min\limits_{w,b}\frac{1}{\|w\|^{2}}+c\sum\limits_{i=1 }^{N}{(\xi_{i}+\xi_{i}^{*})}\\ s.t\left\{\begin{array}{l}y_{i}-w\cdot\Phi(x)-b\leq\varepsilon+\xi_{i}\\ y_{i}-w\cdot\Phi(x)-b\geq-\varepsilon-{\xi_{i}}^{*}\\ \xi_{i}\xi_{i}^{*}\geq 0,(i=1,2,\cdots,N)\end{array}\right.\\ \end{array} \right. \tag{7}\]  
Figure 13: Variation of rotor amplitude with rotational speed under rated voltage. By introducing the Lagrange multiplier \(\left\{\alpha_{i},\alpha_{i}^{*},\beta_{i},\beta_{i}^{*},\left(i=1,2,\cdots,N \right)\right\}\), Equation (7) can be rewritten as Equation (8):  
\[ \begin{array}{l}\underset{w,b}{\text{min}}L\left(w,b,\xi,\xi^{*}\right)=\frac{ 1}{2}\|w\|^{2}+c\sum\limits_{i=1}^{N}\left(\xi_{i}+\xi_{i}^{*}\right)\\ -\sum\limits_{i=1}^{N}\alpha_{i}[\epsilon+\xi_{i}-y_{i}+w\cdot\Phi(x_{i})+b]\\ -\sum\limits_{i=1}^{N}\alpha_{i}^{*}\Big{[}\epsilon+\xi_{i}^{*}+y_{i}-w\cdot \Phi(x_{i})-b\Big{]}\\ -\sum\limits_{i=1}^{N}\left(\beta_{i}\xi_{i}+\beta_{i}^{*}\xi_{i}^{*}\right) \end{array}  \tag{8}\]  
By taking the partial derivatives of \(W\), \(b\), \(\xi_{i}\), and \(\xi_{i}^{*}\) in Equation (8), the following can be obtained:  
\[\left\{ \begin{array}{l}\frac{\partial L}{\partial w}=w-\sum\limits_{i=1}^{N }\left(\alpha_{i}-\alpha_{i}^{*}\right)\cdot\Phi(x)=0\\ \frac{\partial L}{\partial b}=-\sum\limits_{i=1}^{N}\left(\alpha_{i}-\alpha_{i }^{*}\right)=0\\ \frac{\partial L}{\partial\xi_{i}}=c-\alpha_{i}-\beta_{i}=0\\ \frac{\partial L}{\partial\xi_{i}^{*}}=c-\alpha_{i}^{*}-\beta_{i}^{*}=0\end{array} \right. \tag{9}\]  
By substituting Equation (9) into Equation (8), the dual form of Equation (7) can be obtained:  
\[ \begin{array}{l}\underset{\alpha,\alpha^{*}}{\text{max}}\sum\limits_{i=1}^{ N}\left(\alpha_{i}-\alpha_{i}^{*}\right)y_{i}-\sum\limits_{i=1}^{N}\left(\alpha_{i}+ \alpha_{i}^{*}\right)\epsilon\\ -\frac{1}{2}\sum\limits_{i=1}^{N}\sum\limits_{j=1}^{N}\left(\alpha_{i}-\alpha_ {i}^{*}\right)\left(\alpha_{j}-\alpha_{j}^{*}\right)\Phi(x_{i})\Phi(x_{j})\\ s.t.\sum\limits_{i=1}^{N}\left(\alpha_{i}-\alpha_{i}^{*}\right)=0\\ \alpha_{i}\geq 0,\alpha_{i}^{*}\leq c,\left(i=1,2,\cdots,N\right)\end{array}  \tag{10}\]  
By introducing the kernel function \(K(x_{i},x_{j})=\Phi(x_{i})\Phi(x_{j})\), the data are mapped from a low-dimensional space to a high-dimensional space, thereby transforming the linearly inseparable problem into a linearly separable problem. This allows for the calculation of inner products in the high-dimensional space to be performed as function calculations in the low-dimensional space.