1. Introduction
Due to the rapid development of the manufacturing industry, traditional processes have become unable to meet higher requirements for product accuracy, variety, and complexity. Therefore, advanced material-adding manufacturing methods have developed rapidly in recent years, especially direct laser deposition, a high-tech process that uses high-energy lasers to melt and deposit metal powder into three-dimensional parts [
1]. It has attracted attention in the automobile, power plant, and aerospace industries [
2], considering the associated good part performance, high manufacturing flexibility, short production cycle, and low cost [
3]. However, the stability of product quality and the repeatability of manufacturing processes are not high, which seriously restricts its development.
As direct laser deposition involves complex interactions between a laser beam, metal powder, substrate material, and process gas [
4], there are many uncertainties. To obtain an ideal deposition layer, numerous process tests are required, which greatly increase the required time and cost. The geometric characteristics during the deposition process are the main factors when measuring whether or not the formation conditions are reasonable. The relationship between the process parameters and the geometric characteristics of the melt track is nonlinear and difficult to be described by a simple mathematical model, so it is of the highest importance to establish an accurate and efficient prediction model using a small amount of experimental data to quickly obtain an ideal deposit.
Prediction of the geometric characteristics of a melt track is a multivariate non-linear regression problem [
5]. In the literature, research efforts have focused on predicting the geometric characteristics of metal deposition using statistical and data mining methods based on prior experimental work. Sun [
6] established a mathematical model by using a central composite design and response surface method. Through an analysis of variance (ANOVA) test of the established model, the relationships between the process parameters and output response and the interaction between the parameters were analyzed and discussed in detail. Mohammad [
7] used a linear regression analysis to research the empirical–statistical relationship between key parameters and the geometric characteristics of the melt path.
Zhang [
8] chose the multiple linear regression model to find the influence rule of the main parameters on the size of single-track 30CrNi
2MoVA steel cladding. The significant factors affecting the width and height of the melt channel and the corresponding regression equation were determined. Davim [
9] established a model using multiple regression analysis (MRA) between processing parameters and the form of single-cladding layer (clad height, clad width, and depth penetration into the substrate); however, the laser cladding process is complicated and nonlinear, so it is difficult to use only an equation to describe this correlation.
Machine learning is a widely used powerful tool that is suitable for determining the non-linear correlations between input variables and output results. In recent years, neural networks have been widely used for predicting direct laser deposition. Xu [
10] designed a learning algorithm based on an evolutionary neural network and established a prediction model between the melt dilution rate, cladding width and height, and process parameters. The model overcomes the local optima problem often observed with neural networks. Acherjee [
11] developed an artificial neural network (ANN) model to predict the quality characteristics of thermoplastic laser welding. In addition, the prediction results of the ANN were compared with those of multiple regression analysis models, demonstrating that its prediction was better than that of the regression model. Mondal [
12] studied multi-objective optimization in the process of CO
2 laser cladding with the width and depth of the cladding layer as the performance indices. Many experiments were carried out using the ANN backpropagation method, and the relationship between the process and response variables was established. Fabrizia [
13] developed an ANN-based machine learning method to determine the correlation between the laser cladding process parameters and the size parameters of a single channel on a 2024 aluminum alloy plate. This method was adjusted several times in order to achieve a high-precision model. Yin [
14] proposed a backpropagation (BP) ANN model to obtain the mathematical relationship between the optimization goals and process parameters and applied a genetic algorithm to optimize the parameters. In order to speed up the convergence and avoid local minimum of the conditional ANN, Zhong [
15] inducted genetic algorithm simulated annealing (GASA) based on the random global optimization into the network training. Meanwhile, the gray correlation model (GCM) was used as a pre-processing tool to simplify the original networks based on obtaining the main influence elements of network inputs. On this basis, a genetic algorithm was applied to optimize the parameters. Although neural networks have been widely used, due to their ability to process multiple and non-linear information in parallel, the problem of poor generalization ability occurs for a small sampling size, such as the one used in this paper.
Through statistical learning theory analysis, it can be seen that the neural network adopted the strategy of minimizing empirical risk, and the data volume and dimensions affected its regression performance. On the other hand, support vector machines(SVM) adopts the strategy of minimizing structural risk; thus, integrating empirical and confidence risk and achieving global optimization more easily. SVM can be used for classification and regression, and has been successfully applied in many practical fields, proving that its generalization ability is better than that of a neural network. As an extension of SVM, the support vector regression (SVR) method can be considered very suitable for direct laser deposition geometric characteristics prediction, and can address the problems of insufficient training data, non-linearity, and strong output coupling, as was the case in this study. To research the transition temperature of NdBa2Cu3O7-u03b4(NBCO) thin films prepared by pulsed-laser deposition, Xiao [
16] proposed a prediction model of pressure, temperature, and laser energy on transition temperature based on SVR. The result showed that the average absolute error of SVR was less than that of multiple non-linear regression. Yang [
17] established an SVR prediction model for the WC volume fraction under different process parameters, which had a smaller prediction error and stronger generalization performance than an ANN model. Ye [
18] took underwater wet shallow-water flux-cored arc welding as a research object, and used SVR to improve the modeling accuracy and prediction speed. Tao [
19] screened out the process parameters that had a strong impact on multiple quality characteristics based on signal-to-noise (S/N) and analysis of variance, established four SVR models to predict each quality characteristic and, finally, proved that this method has high prediction accuracy. Yao [
20] compared the prediction performance of SVR models based on different kernel functions, and found that the model based on a radial basis function (RBF) was more suitable for predicting the geometric characteristics of the deposited melt channel. However, it should be noted that SVR mainly addresses the single-output problem and often uses the construction of multiple SVR models to solve a multi-output problem. This means unequal treatment of each sample point, which not only slows the training speed but also affects the prediction accuracy [
21].
By analyzing the correlation of geometric characteristics in direct laser deposition, the output parameters are shown not to be independent. Single-output support vector regression (S-SVR) ignores the relationships between different output parameters, thus affecting the prediction accuracy. Therefore, this paper proposes a method for predicting laser-deposition geometric characteristics based on multi-output support vector regression (M-SVR). By comparing the predicted results to those of neural network and S-SVR models, the fitting accuracy and generalization ability of the model are detailed.