## 2 Materials and Methods
When selecting the hyperparameters for the subsequent experiment, any of the acquisition functions can be used, such as _expected improvement per second plus, expected improvement, expected improvement plus, expected improvement per second, lower confidence bound_, or _probability of improvement_, as the optimization option. In supervised learning, a fundamental concern revolves around the accuracy of the resulting model. A critical challenge in this context is the phenomenon of overfitting [43], where a model fits perfectly on the dataset on which it was trained, but has difficulty generalizing effectively to new, unseen data. Conversely, simpler models, such as a least-squares line, are less susceptible to being influenced by inherent noise but may inadequately capture the relationship between variables, leading to underfitting. It is unlikely that both overfitting and underfitting models will generalize well, demonstrating the importance of finding a balance between the two models [44]. Assessing the generalization ability of a model raises an important question: How can we assess its performance on new data? The ideal approach is to evaluate the model with new data from the same population as the training dataset [45]. However, practical constraints often limit the ability to conduct independent validation studies. Therefore, it is advisable to first estimate the predictive performance before investing resources in external validation studies. Usually, this estimation is performed using resampling techniques, such as cross-validation [44]. There are many alternative CV schemes, and the k-fold CV is one of the most popular. The k-fold CV is a widely used resampling method in the practical application of statistical learning methods. The set of observations are randomly divided into k groups (folds) of approximately equal size. In each iteration, the first fold is designated as the validation set, and the model is trained on the remaining k - 1 folds. The mean square error (MSE) is then calculated for the observations in the remaining group. This process is repeated k times, resulting in k estimates of the test error (MSE\({}_{1}\), MSE\({}_{2}\),...., MSE\({}_{k}\)). The final k-fold CV estimate is obtained by averaging these values, as in the following Equation (3) [33]:  
\[CV_{k}=\frac{1}{k}\sum_{i=1}^{k}MSE_{i} \tag{3}\]  
Finally, the observations of the test set are used as an independent dataset to evaluate the model's performance and to select the model. The model with the smallest test RMSE and the smallest mean absolute error (MAE) is selected as the final model, as these metrics ensure that the selected model generalizes well to a new dataset. In addition to RMSE, mean square error (MSE), mean absolute error (MAE), and coefficient of determination (R\({}^{2}\)are used to evaluate the model during the test performance analysis, as in the following Equations (4)-(7):  
\[MSE=\frac{1}{n}\sum_{i=1}^{n}(y_{i}-\hat{y}_{i})^{2} \tag{4}\]  
\[RMSE=\sqrt{\frac{1}{n}\sum_{i=1}^{n}(y_{i}-\hat{y}_{i})^{2}} \tag{5}\]  
\[R^{2}=1-\frac{\sum_{i=1}^{n}(y_{i}-\hat{y}_{i})^{2}}{\sum_{i=1}^{n}(y_{i}- \overline{y})^{2}} \tag{6}\]  
\[MAE=\frac{1}{n}\sum_{i=1}^{n}\lvert y_{i}-\hat{y}_{i}\rvert \tag{7}\]  
where n is the total number of experimental observations, \(y_{i}\) is the experimental values for each of the runs, \(\hat{y}_{i}\) is the predicted value for each of the observed values, and \(\overline{y}\) is the mean of the experimental values of the runs. We assessed the cross-validation loss of neural network regression models by examining various hyperparameter settings, including regularization strengths (lambda), activation functions ('relu', 'tanh','sigmoid', and 'none'), hidden layer sizes, and hidden node sizes. We then fine-tuned the hyperparameters using Bayesian optimization to find the optimal configuration that minimizes model error and improves performance.