## 1 Introduction  
The precision of machining in CNC machine tools has consistently remained a focal point in the manufacturing industry, serving as a crucial indicator of overall machining capabilities within the modern machinery sector. The precision of CNC machine equipment is influenced by a multitude of elements, encompassing dimensional errors, geometric errors, thermal errors, and tool deflection errors [1]. Within the diverse origins of machine tool error, thermal error accounts for the largest proportion, reaching more than 40-70% [2; 3; 4]. The incorporation of high-speed motorized spindles into CNC machine equipment is becoming more and more popular. The attributes of motorized spindles significantly influence the production accuracy, robustness, and dependability of CNC machine tools [5; 6; 7]. Although the motorized spindle rotates at a rapid pace, the primary source of thermal errors affecting machining accuracy is the spindle s thermal deformation resulting from temperature elevation [8]. The thermal expansion of the motorized spindle is the predominant factor that influences a machine tool s accuracy and consistency [9]. The conventional approaches to mitigating thermal error primarily involve error avoidance and error compensation techniques [10]. Error avoidance refers to avoiding the generation of errors or reducing the probability of errors as much as possible by means of design and process control. It usually starts from the system design and engineering point of view to optimize each link and reduce the factors that may lead to errors. For example, in mechanical structure design, usingrigid materials and increasing supporting points can enhance the rigidity of the system and decrease the errors caused by vibration. Error compensation refers to correcting the errors that have occurred by means of measurement and compensation. When the error cannot be completely avoided, it can be corrected by measuring the error accurately and using compensation methods to achieve the target accuracy requirements. Error compensation is usually based on measured data and mathematical models, and the error is corrected by calculating and adjusting control parameters. For example, during machine tool operations, the temperature change of the spindle is recorded simultaneously and compared with the pre-established temperature-thermal error relationship model, and the machining size is corrected by using the compensation algorithm to meet the accuracy requirements. The error avoidance method is usually a one-time investment that is optimized in the design stage. If the system is not properly designed, the need for additional corrections can be costly in terms of time and human and financial resources. Error compensation, on the other hand, is a software-based compensation method that is easier to operate, more economical, and more effective. The error software compensation method has become the current mainstream method to mitigate thermal errors because of its flexible, efficient, cost-effective, and real-time nature [11]. Prior to implementing thermal error compensation techniques, it is essential to accurately forecast the thermal deviation and compensate for the forecasted value within the system via software compensation. Employing a forecasting model for analyzing and compensating for error is a promising approach that offers the potential to diminish machine tool thermal error without incurring additional costs [12]. Therefore, the establishment of a precise model for predicting thermal errors and the exact estimation of predictive results are essential prerequisites and key links to ensuring the effective compensation of thermal error. The model-driven strategy for compensating for thermal errors is also a sustainable and cost-effective compensation method [13]. [PERSON] et al. [14]sequently, the relationship between temperature variations and corresponding thermal error can be investigated. CNC machine tool beds are generally huge.