## 4 The Development of the Prediction Models  
As demonstrated in Figure 1, the workflow for developing models to predict the thermal conductivity of EG-Al2O3 nanofluids can be divided into several parts: data process, hyperparameter optimization, and model evaluation. After that, SHAP analysis is conducted to understand the importance of each feature in the model. This structured workflow ensures a systematic approach to building and evaluating predictive models, leading to more reliable and interpretable outcomes. 1. Data process: In this study, 94 data points collected from four published experimental studies are divided into two sub-datasets: (a) the training set and (b) the test set. Generally, the training set contains more data than the test set to validate the model generalization ability. Thus, the data allocation ratio of training set to test set is 7:3, aligning with the number of samples in the training set used in a similar work [61]. To reduce the prediction error caused by the differences in parameter dimensions, all data samples are normalized into the range of 0 to 1. 2. Hyperparameter optimization: The process of hyperparameter optimization is critical to enhance model performance across various indices, including accuracy and efficiency. Inadequate hyperparameter settings can precipitate either underfitting or overfitting, thereby degrading the effectiveness of the model. To that end, the Optuna framework [62] is employed, a Python-based library renowned for its robust and efficient capabilities in hyperparameter optimization. In the optimization process, each trial represents a single evaluation of a set of hyperparameters. The objective is to find the optimal combination of hyperparameters that maximizes the prediction accuracy. The framework of Optuna can be divided into several steps as follows: (a) defining the search space, (b) suggesting hyperparameters, (c) evaluating model performance, (d) updating the search space based on evaluation results, and (e) repeating these steps for a predetermined number of trials or until a stopping criterion is met. To determine the optimal number of iterations and data allocation methods, three versions of the ratio of training set to test set (60:40, 70:30, and 80:20) are used to generate the prediction models through 2000 iterations. The optimization process is executed on a laptop equipped with a 13 th Gen Intel(R) Core (TM) i9-13900 HX processor (2.20 GHz), 16.0 GB RAM, and an NVIDIA 4060 graphics card. After identifying the optimal number of iterations, the model is reoptimized using the optimal iteration number to determine the best hyperparameters. Furthermore, RMSE is utilized to establish the objective function to evaluate optimization performance. The setting ranges of hyperparameters for the different models are shown in Table 2.
3. Model evaluation: In this step, several statistical indices, including MSE, RMSE, MAE, MAPE, and R\({}^{2}\), are adopted to evaluate model performance. These indices provide a comprehensive evaluation of each model's accuracy and predictive capability. The definitions of these performance indices can be found in the literature [63; 64; 65; 66]. The mathematical expressions of all indices are shown as follows: \[\text{MSE}=\frac{1}{V}\sum_{v=1}^{V}\left(y_{true}-y_{predict}\right)^{2}\] (7) \[\text{RMSE}=\sqrt{\sum_{v=1}^{V}\left(y_{true}-y_{predict}\right)^{2}}\] (8) \[\text{MAE}=\frac{1}{V}\sum_{v=1}^{V}\left|y_{true}-y_{predict}\right|\] (9) \[\text{MAPE}=\frac{1}{V}\sum_{v=1}^{V}\left|\frac{y_{true}-y_{predict}}{y_{ true}}\right|\times 100\%\] (10) \[\text{R}^{2}=1-\frac{\sum\limits_{v=1}^{V}\left(y_{true}-y_{predict}\right)^{2} }{\sum\limits_{v=1}^{V}\left(y_{true}-\overline{y}_{true}\right)^{2}}\] (11)  
where \(V\) is the number of samples, and \(\overline{y}_{true}\) is the mean of the actual values. These evaluation indices collectively provide a robust framework for assessing the performance of our models.