## 3 Proposed Method
### Meta-Learning for Rapid Training of Multi-Machine Models for Anomaly Detection and Prediction  
The rapid training of meta-learning multi-machine models for anomaly detection and prediction used in the PQUM-DNS was introduced in a previous study [19]. This method utilizes abundant data from numerous machines, trains the Metatrain model using the AE + meta-learning approach, and fine-tunes the Metatrain model with a small amount of data from new machines. This process yields a model adapted to a new machine, facilitating the inference for anomaly detection. In this study, the anomaly detection results were combined with prediction models to forecast future anomalies based on past anomaly detection outcomes. The following provides a brief introduction to the techniques employed. (1) Meta-learning  
Meta-learning is a technique aimed at enabling machine-learning systems to swiftly adapt to new tasks or environments [4]. Traditional machine-learning algorithms often require large amounts of labeled data to train models. In addition, it is necessary to collect and label substantial data for retraining when faced with new tasks. In contrast, the goal of meta-learning is to train a \"learner\" that is capable of rapidly learning new tasks from a small amount of labeled data. This approach typically relies on prior experience with numerous similar tasks and applies this experience to new tasks. These tasks can be expressed as:  
\[\text{Lear }\theta\text{ such that }\Phi_{i}=f_{\theta}\big{(}D_{i}^{tr}\big{)} \text{ is good for }D_{i}^{ls} \tag{1}\]  
\[\theta^{*}\!=\underset{\theta}{argmax}\!\!\sum_{i=1}^{n}logP\big{(}\phi_{i}|D_ {i}^{ls}\big{)} \tag{2}\]  
where  
\[\Phi_{i}=f_{\theta}\big{(}D_{i}^{tr}\big{)} \tag{3}\]  
\[D_{meta-train}=\big{\{}\big{(}D_{1}^{tr}\,\ D_{1}^{ls}\big{)},\ \big{(}D_{n}^{tr}\,\ D_{n}^{ls}\big{)}\} \tag{4}\]  
\[\text{Ti} \begin{cases}D_{i}^{tr}=\big{\{}\big{(}\chi_{1}^{t},y_{1}^{t}\big{)},\cdots,\big{(}x_{i}^{t},y_{i}^{t}\big{)}\big{\}}\\ D_{i}^{ls}=\big{\{}\big{(}x_{i}^{t},y_{1}^{t}\big{)},\ldots,\big{(}\chi_{i}^{t },y_{i}^{t}\big{)}\big{\}}\end{cases}  \tag{5}\]  
where _Ti_ is a (meta-learning) task. An illustration of the meta-learning method is shown in Figure 9 [31]. Various meta-learning models have been proposed for deep learning that are generally categorized as follows: learning good weight initializations, metamodels that generate the parameters of other models, and learning transferable optimizers. Model-agnostic meta-learning belongs in the first category and learns a good initial weight initialization to achieve fast adaptation to new tasks, enabling rapid convergence and fine-tuning of small-scale training samples [32]. (2) Anomaly Detection  
Figure 9: Illustration of the meta-learning method. The AI degradation level index was used to detect anomalies and the model was established using an unsupervised AE algorithm, as shown in Figure 10. This method involves constructing a model with normal data and applying an AE to compute the root mean square error (RMSE) between the input and output data, referred to as the reconstruction error. Here, the reconstruction error was defined as the AI degradation level index. A smaller value indicates a closer alignment between the model input and output values, leading to a better data reconstruction capability and a higher likelihood of normal equipment or processes. In practical applications, suitable threshold values are defined based on the conditions of the equipment or processes used for anomaly determination. These are expressed as:  
\[h=g(W_{1}X+b_{1}) \tag{6}\]  
\[X^{\prime}=f(W_{2}h+b_{2}) \tag{7}\]  
\[Minimize\ RMSE\Big{(}X,X^{{}^{\prime}}\Big{)} \tag{8}\]  
(3) Anomaly Prediction  
Time-series algorithms can use historical data from the past to predict future trends. In this study, historical records of the health status of semiconductor manufacturing processes and machine equipment in the past can be used to predict future health status using time-series algorithms.