3.2. Machine learning model building and optimization By utilizing cross-validation, the dataset is partitioned into smaller subsets and cycled repeatedly to serve as the validation set. This approach can enhance the reliability of outcomes in machine learn- ing problems with a small sample size. Atomic radius, ionic radius, atomic volume, 2nd ionization energy, and 3rd ionization energy were determined to be the descriptors for adsorption energy. The descriptors determined for adsorption distance were atomic number, ionic radius, period, electron configuration (p), and electron affinity.
Various machine learning algorithms were employed to establish mapping relationships between the target variables and input variables.
The machine learning models that were utilized include BPNN, KNN, SVM, and DT. The BPNN model is comprised of three network layers: the input layer, the output layer, and the hidden layer. Back propaga- tion of the output error of the artificial neurons is incorporated into the conventional neural network, and the connection weight coefficients between the artificial neurons are continuously adjusted to make the mapping relationship between input and output more accurate. Dur- ing the process of establishing the mapping relationship, the training parameters play a crucial role in determining the algorithm’s effective- ness. By tuning the model hyperparameters, various algorithmic models with high performance are obtained. Next, the accuracy of these models is tested using a randomly selected validation dataset. To represent the error between the DFT calculated values and the predicted values of different algorithms and to filter out the models with the best fit performance, we introduce the decision factor(R2) and the root mean square error to evaluate the accuracy of the models with the following equation [39]:
 2 = 1−
∑ 
�=1
(
 � −  ̂�
)2
∑ 
�=1
(
 � −  ̄�
)2
(4)
where Y� is the DFT calculated value,  ̂ � is the ML algorithm predicted value,  � is the mean of the DFT calculated value, and n is the number of samples.
The root mean square error (RMSE) was introduced to assess the accuracy of the model with the following equation [39]:
     =
√√
√√1   
∑
�=1
(
 � −  �
)2
(5)
Fig. S4 demonstrate that the BPNN algorithm outperforms other algorithms in predicting the adsorption energy and distance with higher accuracy, and the R2 between the descriptors and the target quantities under this algorithm all exceed 0.8.
From Fig. 3(a) and Fig. 3(b) we can see that the predicted values of the BPNN model for both adsorption energy and adsorption distance show a significant correlation with the data points obtained from the DFT calculations. Therefore, we choose BPNN as the machine learning model for further study.
However, there is a discernible degree of dispersion between the predicted and calculated values, as depicted in Fig. 3(a) and Fig. 3(b).
We aimed to enhance the algorithm’s accuracy by using the minimum squared difference between the output value and the ideal value as the criterion based on BPNN. We optimized the connection weights of the neural network using a genetic algorithm, adjusted the population size and evolution times, and established a GA-BPNN model. The R2 and RMSE of the optimized algorithm are shown in Table 1.
Using the root mean square error (RMSE) as the evaluation variable, the parameters of adsorption energy and adsorption distance input to the machine learning model were optimized for both datasets sepa- rately, with a genetic algorithm population size of 50, 200 generations
of iterative optimization, the following simulation parameters are ob- tained: a BPNN training function of ‘‘trainlm’’, a step size of 0.001, and a training objective of 1.0e−6.
After testing, as shown in Table 2 and Fig. 4(a), the GA-BPNN model was utilized to predict the adsorption energy and distance. The root mean squared error (RMSE) for the adsorption energy model reached a minimum value of 0.1814 with a coefficient of determination (R2) of 0.9874 for a hidden layer of ten nodes. Meanwhile, the optimal number of hidden layer nodes for adsorption distance prediction was three, with an RMSE of 0.1828 and an R2 value of 0.9608. The results from Figs. 4(b), 4(c), and 4(d) indicate that the GA-BPNN model significantly improves the prediction of adsorption energy, reducing the RMSE of the prediction by 64.71% compared with the original BPNN model. The GA-BPNN model also demonstrates a reduction of 54.99% in prediction  翻译成中文