The geometric precision of machined gears is reduced by thermal errors. So the prediction and control of thermal errors are essential. But the prediction and control are a process involving the processing of a large-volume thermal data, and then the processing efficiency is low, which severely hinders the geometric precision improvement. To solve this problem, a new mist-edge-fog-cloud system (MEFCS) architecture is proposed for the error prediction and control. A finite element model is established to prove the applicability of bidirectional long short-term memory (Bi-LSTM) network. A cosine and sine gray wolf optimization (CSGWO) algorithm is proposed to optimize the batch size. Then the CSGWO-Bi-LSTM network error model is proposed. The predictive accuracy is 90.80%, 94.57%, 95.77%, 96.79%, 97.51%, 98.45%, and 98.92% for the multiple linear regression model, recurrent neural network, LSTM network, Bi-LSTM network, CSGWO1-Bi-LSTM network, CSGWO2-Bi-LSTM network, and CSGWO3-Bi-LSTM network, respectively. The volume of the transferred data is reduced by 11/16 with the data-based model, and the volume of the transferred thermal data is reduced to 1/10 with the designed system. A precision threshold is set, and the predictive accuracy is improved by 8.31% by the system with the precision threshold compared with the system without the precision threshold. With the proposed MEFCS, the accuracy level of the tooth profile deviation fHα is increased from ISO level 5 to ISO level 3. The total execution time of the mist-cloud structure, mist-edge-cloud structure, mist-fog-cloud structure, and mist-edge-fog-cloud structure is 206 s, 200 s, 186 s, and 167 s, respectively.