## 3 Online Processing Parameters Prediction and Reverse Design Framework
### The Reverse Design of Processing Parameters Based on Bayesian Optimization  
The adaptability of online machine learning allows models to quickly adjust to changing data distributions, providing more accurate predictions. Bayesian optimization [30] efficiently selects the most promising experimental configurations during experiments, thereby reducing costs and time. Combining the strengths of both, the reverse design of Bayesian optimization using high-precision online machine learning models that continually adapt to new data is highly beneficial for rapid iteration in industrial settings. In Bayesian optimization, the choice of the acquisition function is crucial as it directly impacts the performance and efficiency of the Bayesian optimization algorithm. The primary role of the acquisition function is to help Bayesian optimization select the next point in the candidate point set for experimentation or evaluation, with the aim of maximizing the expected improvement in the objective function while striking a balance between exploring unknown regions and exploiting known optimal regions. Currently, an effective strategy [30] for achieving this balance is to encourage the selection of data points in unknown regions during the early stages of model iterations when there are limited observed data and the model has greater uncertainty. As the model undergoes multiple iterations, the focus shifts toward selecting points in regions where the optimal solution is likely to exist, given the increased amount of observed data and that the model has converged toward the optimal solution. We select the maximum confidence upper bound function with an adaptive learning rate as the acquisition function, and its formula is as follows:  
\[\alpha_{GP-UCB}=\mu_{t}(x)+\sqrt{\beta_{t}}\cdot\sigma_{t}(x) \tag{8}\]  
where \(t\) is the time step, \(\mu_{t}\) denotes the mean, \(\sigma_{t}\) denotes the variance and \(\sqrt{\beta_{t}}\) is the weight, according to [PERSON] et al. [28], who formulated a schedule for \(\mu(\centerdot)\). For the weights of this framework, we adopt the following calculation method:\[\beta_{t}=2\log\left(\frac{t^{2}2\pi^{2}}{3\delta}\right)+2d\log\left(t^{2}dbr \sqrt{\log\left(\frac{4 da}{\delta}\right)}\right) \tag{9}\]  
where \(\delta\) is a constant from 0 to 1, \(a\), \(b\) and \(r\) are constants greater than 0, \(d\) is the dimension of the optimization objective and \(t\) is the time step. In the maximum confidence upper bound function, the mean represents exploitation (knowledge of the model) and the variance represents exploration (selection of data in an unknown region). The adaptive learning rate decreases from large to small over time. The larger the weighting of variance, the greater the model uncertainty and the more likely it is to explore areas not observed by the model; conversely, the smaller the weighting of variance, the more the model focuses on stability and favors existing data. The adaptive learning rate can be weighed against the importance of the mean versus the uncertainty. The reverse design process for process parameters is as follows: set the candidate range for process parameters and the objective function as the minimum difference between the predicted value and the target performance of material. Update the Gaussian process regression model based on the acquired streaming data. The Gaussian process regression model can predict the mean and variance of material performance based on the input process parameters. Use the acquisition function, Equation (8), to select the most likely candidate values in the candidate range of process parameters and input them into the objective function. Iterate until the maximum number of rounds, selecting the candidate point that best satisfies the objective function as the recommendation for the next round of process parameters.