Correction: Gao et al. A Thermal Error Prediction Method of High-Speed Motorized Spindle Based on Pelican Optimization Algorithm and CNN-LSTM. Appl. Sci. 2024, 14, 381

In the original publication [1], there was a mistake in Figure 25 and some texts related to the calculated results of Figure 25.

At the end of the article, while calculating four evaluation metrics, we made a careless mistake due to an incorrect function call in the software toolkit. As a result, there was a slight deviation in the calculated results. We have rectified the programming error and recalculated all the metrics in Figure 25. The corrected Figure 25 appears below.

A correction has been made to Section 4 Prediction Performance Analysis, Paragraph 8:

Figure 25 displays the evaluation index values of each model across various speeds. When the spindle speed was 1000 r/min, the RMSE values for the POA-CNN-LSTMNN, CSOA-CNN-LSTMNN, CNN-LSTMNN, and LSTMNN models were 0.3127, 0.5434, 0.6803, and 0.8952, respectively; the MAE values were 0.2581, 0.4403, 0.5662, and 0.7331, respectively; the $R^2$ values were 0.9931, 0.9791, 0.9672, and 0.9432, respectively; and the MSE values were 0.0978, 0.2953, 0.4628, and 0.8014, respectively. When the spindle rate was 2500 r/min, the RMSE values of the POA-CNN-LSTMNN, CSOA-CNN-LSTMNN, CNN-LSTMNN, and LSTMNN models were 0.4079, 0.5941, 0.684, and 1.0052, respectively; the MAE values were 0.3495, 0.4907, 0.5861, and 0.8512, respectively; and the $R^2$ and MSE values were 0.9925 and 0.1664, 0.9841 and 0.353, 0.979 and 0.4679, and 0.9544 and 1.0104, respectively. When the spindle rate was 4000 r/min, the RMSE values of the POA-CNN-LSTMNN, CSOA-CNN-LSTMNN, CNN-LSTMNN, and LSTMNN models were 0.6378, 0.7783, 1.0908, and 1.6581, respectively; the MAE values were 0.5344, 0.6353, 0.9253, and 1.3835, respectively; the $R^2$ values were 0.992, 0.988, 0.9765, and 0.9458, respectively; and the MSE values were 0.4067, 0.6058, 1.1897, and 2.7491, respectively. When the spindle rate was 7000 r/min, the RMSE values of the POA-CNN-LSTMNN, CSOA-CNN-LSTMNN, CNN-LSTMNN, and LSTMNN models were 0.7553, 1.1819, 1.3975, and 2.4868, respectively; the MAE values were 0.6158, 0.9717, 1.1699, and 2.0364, respectively; the $R^2$ values were 0.9947, 0.987, 0.9818, and 0.9424, respectively; and the MSE values were 0.5704, 1.3969, 1.9529, and 6.184, respectively. When the spindle speed was 8500 r/min, the RMSE and MAE values of the POA-CNN-LSTMNN, CSOA-CNN-LSTMNN, CNN-LSTMNN, and LSTMNN models were 0.9083 and 0.7241, 1.2877 and 0.9867, 1.5078 and 1.2611, and 2.0011 and 1.6803, respectively; and the $R^2$ and MSE values were 0.9938 and 0.825, 0.9875 and 1.6582, 0.9828 and 2.2734, and 0.9697 and 4.0042, respectively. When the spindle speed was 10,000 r/min, the RMSE values of the POA-CNN-LSTMNN, CSOA-CNN-LSTMNN, CNN-LSTMNN, and LSTMNN models were 0.8078, 1.6685, 2.176, and 3.5479, respectively; the MAE values were 0.6737, 1. 2743, 1.6481, and 2.8598, respectively; the $R^2$ values were 0.9957, 0.9814, 0.9685, and 0.9162, respectively; and the MSE values were 0.6525, 2.7839, 4.7349, and 12.5877, respectively. The average RMSE, MAE, $R^2$, and MSE values of the POA-CNN-LSTMNN model at different speeds were 0.6383, 0.5259, 0.9936, and 0.4531, respectively. The mean RMSE, MAE, $R^2$, and MSE values of the CSOA-CNN-LSTMNN model at different speeds were 1.009, 0.7998, 0.9845, and 1.1822, respectively. The average RMSE, MAE, $R^2$, and MSE values of the CNN-LSTMNN model at different speeds were 1.2561, 1.0261, 0.976, and 1.8469, respectively. The mean RMSE, MAE, $R^2$, and MSE values of the LSTMNN model at different speeds were 1.9324, 1.5907, 0.9453, and 4.5561, respectively. The POA-CNN-LSTMNN model had 36.7%, 49.2%, and 67% lower average RMSE values than the CSOA-CNN-LSTMNN, CNN-LSTMNN, and LSTMNN models, respectively. The average MAE values of the POA-CNN-LSTMNN model were 34.2%, 48.7%, and 66.9% lower than those of the CSOA-CNN-LSTMNN, CNN-LSTMNN, and LSTMNN models, respectively. Compared to the CSOA-CNN-LSTMNN, CNN-LSTMNN, and LSTMNN models, the average $R^2$ and MSE values of the POA-CNN-LSTMNN model increased by 0.9%, 1.8%, and 5.1% and decreased by 61.7%, 75.5%, and 90%, respectively.

A correction has been made to Section 5 Conclusions, Point 3:

On average, the POA-CNN-LSTMNN model exhibited a reduction of 51% and 49.9% in the RMSE and MAE, respectively. Additionally, the $R^2$ was 2.6% higher on average compared to the other models, and MSE was reduced by 75.7% on average.

The authors state that the scientific conclusions are unaffected. This correction was approved by the Academic Editor. The original publication has also been updated.