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).

至于 GMDH 模型设计方法，本研究利用 416 组仿真数据计算 f 和 j，进而以此数据训练 GMDH 模型，以直接预测 f 和 j。在对比表 7 中所示两种模型的训练集与测试集的性能指标时，我们发现 GMDH 模型在预测 f 方面表现优异，训练集和测试集均具有高 R2 值，表明直接使用 GMDH 模型进行 f 的预测既合理又高效。然而，在预测 j 方面，GMDH 模型的 R2 值相对较低，尽管略逊于热传递 Q 的性能，但依然处于良好的预测水平。均方根误差（RMSE）和平均绝对百分比误差（MAPE）的值表明，模型的预测误差相对较小，尤其是测试集上的 MAPE（1.85%），显示出较低的百分比误差，优于 Q 的性能，并展现出强大的泛化能力。模型清晰的网络结构以及每个神经元的计算公式已在补充材料中呈现（参见补充材料 S1）。GMDH 模型的训练数据亦作为补充材料提供（参见补充材料 S2）。