To mitigate the impact of thermal errors on the positioning accuracy of the CNC machine tool feed system and improve the consistency of processed products, a thermal error prediction model based on the RIME-optimized BP neural network is introduced. Temperature sensors and a laser interferometer are arranged under various operating conditions to collect temperature and lead screw thermal error data. Fuzzy C-means clustering and grey relational analysis are applied for feature selection from temperature samples, identifying key temperature feature points. The RIME-BP thermal error prediction model is constructed using temperature and screw position coordinates as inputs and screw thermal error as the output. For the H650GA gear grinding machine, the K-fold cross-validation method is used to validate the model′s prediction accuracy, and compared with GA-BP, BP, and SVM models. The results show that the proposed model′s average coefficient of determination (R2) of 0.995, which is 3.54%, 9.58%, and 17.75% higher than the GA-BP, BP, and SVM models, respectively. The proposed method provides theoretical and technical guidance for thermal error compensation and holds significant engineering application potential.