Thermal error has become a key reason hindering machine tool’s thermal stability improvement. The error compensation is carried out from the view of error mechanism of spindle systems to increase the thermal stability of machine tools. The hysteresis effect of the thermal expansion is revealed with theoretical modeling of error mechanism, and long-term memory characteristics of thermal error on historical thermal information are demonstrated. Then the applicability of long short term memory (LSTM) neural networks for the training of the error model is proved. The variational mode decomposition (VMD) decomposes error data into several inherent modal function (IMF) components to remove the coupling effect of high- and low-frequency data, improving the robustness and generalization capability of the error model. Moreover, the hyper-parameters of LSTM neural networks are optimized with grey wolf (GW) algorithms to remove the sensitivity of the predictive performance to its hyper-parameters. Finally, error models are trained with VMD-GW-LSTM neural networks, VMD-LSTM neural networks, and recurrent neural networks (RNNs). To verify the effectiveness of compensation methods, the error compensation and machining were performed at different working conditions. The results show that compensation rates of the VMD-GW-LSTM network model are 77.78%, 75.00%, and 77.78% for Sizes 1, 2, and 3, respectively. Moreover, the predictive performance and compensation performance of the VMD-GW-LSTM network model is far better than that of VMD-LSTM network and RNN models.

热误差已成为阻碍机床热稳定性提高的关键原因。从主轴系统的误差机制的角度进行误差补偿，以提高机床的热稳定性。通过误差机理的理论模型揭示了热膨胀的滞后效应，并证明了热误差对历史热信息的长期记忆特性。然后证明了长期短期记忆（LSTM）神经网络在误差模型训练中的适用性。变分模式分解（VMD）将误差数据分解为几个固有的模态函数（IMF）组件，以消除高频和低频数据的耦合效应，从而提高了误差模型的鲁棒性和泛化能力。此外，LSTM神经网络的超参数已通过灰狼（GW）算法进行了优化，以消除预测性能对其超参数的敏感性。最后，使用VMD-GW-LSTM神经网络，VMD-LSTM神经网络和递归神经网络（RNN）训练错误模型。为了验证补偿方法的有效性，在不同的工作条件下进行了误差补偿和机加工。结果表明，尺寸1、2和3的VMD-GW-LSTM网络模型的补偿率分别为77.78％，75.00％和77.78％。而且，VMD-GW-LSTM网络模型的预测性能和补偿性能远远优于VMD-LSTM网络和RNN模型。