## 2 Experimental Methods
### Machine Learning  
#### 2.5.1 Back Propagation and Support Vector Machine  
There is a complex nonlinear mapping relationship between spraying-process parameters and the microstructure and service life of the coating. It is necessary to establish a quantitative relationship between spraying-process parameters (powder feeding rate, distance between two powder feeders, and powder type) and microstructure (MAP content) and thermal cycle life. Machine learning is expected to provide a new solution to this problem, considering the problem of multi-input and multi-output, which is more suitable for small sample data. In this study, back propagation (BP) neural network and support vector machine (SVM) are selected to solve this problem.  
The BP neural network is a multi-layer feedforward neural network based on an error back propagation algorithm, which has good robustness. The neural network trains the network by establishing a set of error functions with the goal of minimizing the error between the expected outputs. The algorithm has been widely used in various nonlinear modeling prediction problems, and a lot of research has been carried out. The literature can be used for reference [27; 28].  
The SVM algorithm is one of the most influential machine learning algorithms based on supervised learning, which is more suitable for small sample prediction. The SVM regression prediction algorithm is a binary classification model. The goal is to find a hyperplane in the high-dimensional space to segment the data. The principle of segmentation is to ensure the optimization of data classification. By constructing functions, finding the optimal hyperplane, and optimizing the objective function, the prediction task is realized [29].  
The principle of SVM algorithm prediction is as follows:  
1. Constructing hyperplanes: The separated hyperplane is a linear equation in two-dimensional space, and it becomes a hyperplane equation when it is extended to high-dimensional space. The specific manifestations are as follows: \[f(x)=w^{l}x+b\] (1) where \(w\) is the weight, and \(b\) is the bias.
2. SVM algorithm soft interval: The use of a soft interval minimizes the error and improves the prediction accuracy of the algorithm. A slack variable \(\varepsilon\) is introduced in the algorithm to allow for errors at certain data points, while the minimization error and maximization interval are balanced by the penalty parameter C. The objective of optimization is formulated as follows: \[\min_{w,b,\varepsilon}=\frac{1}{2}\parallel w\parallel^{2}+C\sum_{i=1}^{m} \left(\varepsilon_{i+\varepsilon_{i}^{*}}\right)\] (2) where C is a penalty parameter, and \(\varepsilon_{i}\) and \(\varepsilon_{i}^{*}\) represent a slack variable.
3. Conditional constraints: In order to ensure that the result error of the SVM prediction algorithm is within a certain range, we use conditional constraints. The specific content is shown in the following formula: \[y_{i}-\langle w,x_{i}\rangle-b\leq\varepsilon+\varepsilon_{i}\] (3) \[\varepsilon_{i},\varepsilon_{i}^{*}\geq 0\] (4) where \(x_{i}\) is the input value in the data set, and \(y_{i}\) is the corresponding output value.