## 1 Introduction
If there are multiple locations for measuring temperature, on the one hand, it means that there are too many sensors and that complicated wiring will cause a lot of human and financial costs. On the other hand, it means that the temperatures of various sections within the machine tool are collinear, which will affect the model s identification and stability. Therefore, the accurate selection of the thermal critical temperature points plays a pivotal role in predicting and compensating for thermal errors [22]. [PERSON] et al. [23] proposed an iterative elimination-based approach for opting for key temperature points and finally determined the temperatures of three key points as the inputs to the model from eight points, thereby establishing a thermal error forecasting model built upon the random forest algorithm and making a comparison with the traditional active learning model, which proved that the precision of this model was higher. [PERSON] et al. [24] developed two thermal sensitive point selection methods--the method for assessing thermal error sensitivity and the method for conducting thermal modal analysis--and used the cantilever beam as an example for experimental analysis, finally proving that modeling the thermal errors utilizing the key temperature point data chosen through these two new methods resulted in better robustness of the models. [PERSON] et al. [25] downsized the quantity of temperature measurement locations from eight to four by utilizing the selection method of the analysis of statistical correlations and a self-organizing feature map, which efficiently eliminated the multicollinearity between temperature measuring points, and put forward the IPSO-BP model, concluding that the proposed IPSO-BP model displayed a heightened capability for predicting the thermal elongation of a spindle. [PERSON] et al. [26] used the gray model combined with the fuzzy C-mean to select key thermal measurement locations and used the adaptive fuzzy inference system of the fuzzy C-mean to design and develop a model for predicting thermal errors, and the experimental results substantiated that the absolute residuals of the predicted thermal error from the model were below 2 \(\upmu\)m. [PERSON] et al. [27] put forward a model for forecasting a spindle s thermal elongation by utilizing the principles of BiLSTM. For model training, eight key temperature measurement points were selected, including seven on the spindle and one representing the ambient temperature, and it was finally proven that the constructed model achieved a forecasting precision exceeding 85%. [PERSON] et al. [28] reduced the number of temperature sensors at the center of a measured object from three to one by building an improved binary polynomial regression model: finally, it was proven that the model accuracy reached 85.99% with only one sensor. [PERSON] et al. [29] presented a methodology for selecting critical temperature points based on sensitivity analysis and then combined a multiple regression algorithm error forecasting model [33]. The utilization of neural networks in modeling the thermal errors of machine equipment has been steadily growing due to their powerful nonlinear processing capability, massively parallel computation, strong computational capability, high fault-tolerant rate, etc. The long short-term memory neural network (LSTMNN) falls under a distinct category of recurrent neural networks (RNNs). LSTMNN can better capture and remember long-term dependencies by introducing memory units and gating mechanisms. Compared with the standard RNNs, LSTMNN displays improved capability in handling long-term dependencies within time series data, thus improving the memory ability of the model. Thermal retention effects and the cumulative thermal deformation not only influence the thermal error dependent on the temperature properties at the present moment but are also inextricably linked to the temperature characteristics at previous consecutive times [34]. Therefore, LSTMNN has a unique advantage in dealing with CNC machine tool thermal errors.