*3.1. Evaluation of the Number of Hidden Layer Nodes*

The BP network has a varied number of nodes in the hidden layer, and the hidden nodes affect the error of the output connected neurons [41]. If the number of neurons in the hidden layer is too small, the network's ability to learn is limited, resulting in the need for more training to decrease its fault tolerance. On the other hand, network iterations will increase with too many neurons, thereby extending the training time of the network, and reducing the generalization capacity of the network, resulting in a decrease in predictive ability. The optimal number of nodes needs to be explored to confirm the effect of different nodes on network performance. In practical situations, the number of nodes in the hidden layer is selected by first determining the approximate range of the number of nodes using the empirical formula before using the step-wise test strategy to establish the best number of nodes with the smallest error by training and comparing the networks with different neurons. The best number of hidden layer nodes can be derived from the following formula [42,43]:

$$l = \sqrt{(m+n)} + a \tag{8}$$

where *l* represents the number of neurons in the hidden layer, *n* denotes the number of neurons in the input layer, *m* is the number of neurons in the output layer, *a* is the constant, and 1 < *a* < 10. According to this formula, the value range of the hidden layer nodes of the network was 4–12, and the performance of the artificial neural network under different numbers of nodes is shown in Figure 7. When the number of hidden layers was 5, the minimum *MSE* was 0.00080796, indicating superior model performance.

**Figure 7.** Performance of artificial neural network models with different hidden layer nodes (learning rate = 0.1 and training goal = 0.001).

Table 5 summarizes the predictive performance of the optimal neural network. The findings showed a validation set of *R* = 0.979, *RMSE* of 0.138, and *MAE* of 0.153. The ANN model with a 4-5-3-3 structure performed effectively. Table 5 further shows the results of the model, which were generally consistent with those obtained during training and testing, indicating that the model can generalize within the range of data used for training.

**Table 5.** Artificial Neural Network Results.


Based on the data shown in Figure 8, the error curves of the model training sample, the corrected sample, and the test sample were well correlated. The curve trend slowly decreased, indicating that the network was trained on the training data. To avoid overfitting with the validation data, the *MSE* between the initial fitting and validation will become

smaller and smaller, but as the network begins to overfit the training data, the *MSE* will become larger. In the default setting, the training ends when the validation error is added six consecutive times, and the best performance is obtained from the lowest validation error period (drawing circle). Finally, the obtained best artificial neural network parameters are shown in Table 6. Figure 9 shows the training state of the model training phase.

**Figure 9.** The neural network training state (epoch 18, validation stop).


**Table 6.** Optimum ANN parameters for the design of the model.

### *3.2. Evaluation of Prediction Results*

The regression curves for assessing the accuracy of the ANN estimation are shown in Figure 10. Estimates of the threshing performance of the ANN were evaluated by regression analysis between the predicted and experimental data. To validate the ANN model, we applied the estimation and regression methods. The regression value for the threshing characteristics was calculated as 0.9525. Figure 10 displays the optimal curve resulting from multiple iterations of the R2 curve.

**Figure 10.** The predicted and the measured values of threshing performance.

#### *3.3. Sensitivity Analysis*

Sensitivity analysis was performed to examine the sensitivity of the various factors influencing the threshing characteristics, and Table 7 shows the effects of the various input factors. As shown, different input variable values, i.e., the size of different sensitivity, reflected the effect of the input variable on the output variable. RS affected the predicted threshing performance when the network values had distinct input variables. However, the relative importance of the remaining input variables varied based on changes in the input variables. RS was the most important input in all trials followed by TC, SC, and FQ. Sensitivity analysis revealed that RS, TC, and SC were the most vital factors affecting threshing performance, with an average relative importance of 15.00%, 14.89%, and 14.32%, respectively. The results further showed that FQ had a minimal effect on threshing performance, with an average relative importance of 11.65%.


**Table 7.** Sensitivity Analyses of the Relative Importance of Artificial Neural Network Input Variables.

#### **4. Discussion**

Threshing is one of the most critical operations of combine harvesters during grain production, which is a complex, nonlinear, multi-parameter physical process. The working performance index of the threshing device has a significant on the separation, cleaning, and other parts and the working quality of the whole machine and has always been one of the main concerns of the engineered design. A flexible threshing device has the advantage of reducing the crushing rate of rice grain. Therefore, a comprehensive and accurate design of a flexible threshing performance evaluation model has important theoretical value and practical significance. In this study, the BP artificial neural network was used to model the threshing performance factors based on four factors: RS, TC, SC, and FQ. Determining the optimal network architecture is related to the number of hidden layers and neurons. The optimum network geometry was found to be 4-5-3-3 by evaluating different number of hidden layer nodes in this study. The performance of the ANN model was verified by comparing the predicted dataset with the experimental results (measured data). The sensitivity analysis performed for the described ANN model indicated that four working variables of the flexible threshing device had the greatest contribution to threshing performance attributes compared to FQ. These results can guide the optimal design of a flexible threshing cylinder to achieve the maximum performance of the device.
