## 1 Introduction  
High manufacturing quality grants manufacturers a competitive edge. Modern manufacturing processes exhibit new characteristics, such as multi-stage and multi-parameter inputs, both of which influence quality. High-quality products tend to stem from the design of appropriate manufacturing parameters. Hence, designing favorable parameters hinges on predicting manufacturing quality at an early production stage, which provides reference for set high-quality products at lower costs. Several approaches and methods have been adopted to predict manufacturing quality. Statistical quality control [1], a traditional method, has been widely used to assess the quality and performance of manufacturing processes. Based on this method, other techniques have been developed, e.g., linear regression [2, 3], nonlinear regression [4], inference learning [5], fuzzy theory [6], and graph theory [7]. These approaches have successfully been applied to manufacturing quality prediction, but only in situations in which the factors (e.g., materials, equipment, and technological parameters) maintain a certain level of stability. This narrow application limits the use of these methods. In particular, these methods act as a bottleneck when handling multi-parameter manufacturing. Recently, the development of computer technologies, such as artificial intelligence, has caught the attention of researchers, because of their minimal information requirements, which do not need to take into account the manufacturing processes [8; 9; 10; 11]. Artificial neural networks (ANNs) are among the most famous representations, alongside self-organizing neural network [12], back propagation neural network (BPNN) [13], radial basis function neural network [14], and probability neural network [15]. As it is affected by multi-parameter inputs from the multi-stage manufacturing processes, the ANN modeling exhibits feature learning difficulties and network calculation complexities. Therefore, to ensure prediction accuracy, it is imperative to develop and apply new techniques using multivariate analysis, as well as an intensive network to enable feature extraction and learning. Recently, various multivariate analysis approaches have been widely applied to manufacturing processes for different industries, for example, principal component analysis (PCA) [16], linear discriminant analysis [17], rough-set theory [18], independent component analysis [19], and feature reinforcement approach [20]. However, these approaches have only affected the datasets of linear structures and gauss distributions; they are not suitable for use with real data with nonlinear features. To overcome this problem, manifold learning (ML), seeking to find a low-dimensional parameterization of high-dimensional data, has been proposed [21]. As a result of its nonlinear computational efficiency, global optimality, and asymptotic convergence guarantees, ML has also been employed in applications such as dimension reduction and feature extraction [22; 23; 24]. These studies indicated that the ML technique is capable of better restoring intrinsic dimension from high-dimensional data. Thus, it reduces the multi-parameter disturbances in feature learning. To enhance the feature learning of ANN, a deep learning (DL) technique has been proposed [25; 26], and, combined with the ANN framework, forms a deep neural network (DNN). Unlike a traditional network (shallow learning), the foundation of a DNN is its hierarchy; that is, the information representation is delivered from lower to higher levels, which makes the higher levels more abstract and nonlinear. Through representation by the hierarchical levels, the \"deeper\" feature of the multi-parameter inputs and the manufacturing quality can be sufficiently fitted by the regression models [27]. Therefore, the DNN has been widely used in applications such as classification [28] and time series forecasting [29]. The DL techniques are also utilized in the manufacturing field [30; 31; 32].