# A Review of Thermal Error Modeling Methods for Machine Tools
## 2 Thermal Error Modeling Method for Machine Tools
### Neural Network (NN)
The RBF neural network is a feedforward neural network that has good approximation and global optimal performance. However, its key feature functions are more difficult to be extracted than the BP neural network, and its generalization performance is worse [56]. [PERSON] et al. [57] used the RBF neural network to build a more accurate thermal error model. But it took a long time to determine the number of neurons in the hidden layer in the algorithm. In order to improve the prediction accuracy of the RBF model, [PERSON] et al. [58] improved the RBF neural network and constructed the thermal error prediction model by applying the PSO algorithm to optimize the important parameters of the RBF neural network. [PERSON] et al. [59] of Hunan University devoted to solve the problem of low modeling efficiency of the existing thermal error modeling methods and unsatisfactory model prediction accuracy. They proposed a generalized RBF neural network modeling method and applied it to the thermal error modeling of the spindle box of a NC guide rail grinder. Compared with the conventional RBF neural network modeling method, the generalized RBF neural network model had the advantages of high prediction accuracy and generalization ability. [PERSON] and [PERSON] [60] developed a fuzzy RBF neural network prediction model for the milling machine spindle. The experimental results showed that the maximum errors of the Y-axis and Z-axis predictions were reduced to 3.5 \(\mu\)m and 2.9 \(\mu\)m after compensation. [PERSON] et al. [61] of Hua Zhong University of Science and Technology used the fuzzy clustering method and grey relational analysis method to optimize the temperature measurement points, and then established a radial basis function neural network prediction model based on genetic algorithm. Compared with the traditional RBF neural network method, the improved model had higher precision and stronger robustness and the forecasting model residual error ranged from 4.88 to 3.80 \(\mu\)m down to 2.48-2.14 \(\mu\)m.  
Grey system theory has the advantage of dealing with an uncertain system with incomplete information and inaccurate data. It can extract useful information from small samples to study the characteristics of the system. In 2012, [PERSON] et al. [62] combined the gray theory and neural network to establish a parallel gray neural network model (PGNN). After applying it to the spindle on a five-axis machining center, the maximum prediction thermal error was reduced to 3.65 \(\upmu\)m. By using the PSO algorithm to obtain the optimal weight threshold of the gray neural network (GNN), [PERSON] and [PERSON] [63] built the spindle thermal error model and the relative error of the prediction was 1.55%. Similarly, [PERSON] et al. [64] used genetic algorithm to optimize the weight threshold of the grey neural network, and took the average relative error as the evaluation standard of the prediction model, which effectively avoided the problem that the grey neural network fell into the local optimal solution. Considering the influence of cutting conditions on the thermal error of the machine tool, a genetic algorithm gray neural network (GAGN) was used by [PERSON] et al. [65] to optimize the thermal error model of grey neural network. The maximum relative error after compensation was 7.02 \(\upmu\)m. However, the training parameters of the thermal error model established by it need to be determined according to the cutting conditions. When the cutting conditions changed, the original model was no longer be applicable, so its versatility was poor. In addition, there are other improved method based on neural networks such as Fourier neural network, multiple-input multiple-output (MIMO), long short-term memory (LSTM), etc. [PERSON] et al. [66] established a comprehensive mathematical model of milling machine geometric error and thermal error using a Fourier neural network. After compensation, the maximum positioning error was reduced from 20.2 \(\upmu\)m to 4.4 \(\upmu\)m. [PERSON] et al.