Supplementary Material S1-ATE-D-24-04532

As for the GMDH model design method, we used 416 sets of simulation data to calculate f and j, and then trained the GMDH model with this data to directly predict f and j. When comparing the performance metrics of the training and test sets between the two models shown in Table 7, we found that the GMDH model performs very well in predicting f, with high R2 values for both the training and test sets, indicating that directly using the GMDH model to predict f is reasonable and efficient. However, in predicting j, the R2 value of the GMDH model is lower. Although slightly worse than the performance for heat transfer Q, it is still at a good prediction level. The RMSE and MAPE values indicate that the model's prediction error is relatively small, especially the MAPE (1.85% on the test set), which shows a relatively low percentage error, better performance than for Q, and strong generalization ability. The clear model network structure and the calculation formulas for each neuron are presented in supplementary material (see Supplementary Material S1). The training data for the GMDH model is provided as supplementary material (see Supplementary Material S2).

针对数据分组处理方法（Group Method of Data Handling, GMDH）的模型设计流程，我们采用416组仿真数据用于计算参数f与j，并基于该数据集训练GMDH模型以直接预测f和j。对比表7中两款模型的训练集与测试集性能指标后可知，GMDH模型在f的预测任务中表现优异，训练集与测试集的决定系数R²均处于较高水平，这表明直接采用GMDH模型预测f具备合理性与高效性。不过在j的预测任务中，GMDH模型的R²值相对更低，尽管其表现略逊于换热量Q的预测性能，但仍处于优秀的预测精度区间。均方根误差（Root Mean Square Error, RMSE）与平均绝对百分比误差（Mean Absolute Percentage Error, MAPE）指标显示，该模型的预测误差整体较小；尤其测试集上的MAPE仅为1.85%，百分比误差极低，其预测性能优于换热量Q的预测结果，且泛化能力较强。模型清晰的网络结构与各神经元的计算公式已在补充材料中给出（详见补充材料S1）。GMDH模型的训练数据集同样以补充材料形式提供（详见补充材料S2）。