## 4 Comparison of Predicted Values and Experimental Results  
Firstly, as discussed, experimental data (training data and the corresponding thermal errors) should be measured. The ability of feature learning is an inherent characteristic of the proposed model, and one of the most observable outward manifestations is model's performance on predicting outputs based on given inputs. In the experiment, two sets of thermal error data and temperature data for the heavy-duty machine tool-foundation system were collected. The first set of data was used to establish the thermal error predictive model. The second was used to verify the accuracy of the predictive model. During the experiment, the collected data is divided into two parts: the first part is 11:30-19:30 data (training data), the second part is 19:30-20:30 data (test data). 32,400 experimental data are obtained for each sensor by setting 1 s sampling period of temperature and displacement data. We used 18 temperature values and one displacement value at the same time constitute a sample, so a total of 32,400 experimental samples during the 9-h data collection process. Figures 6-8 show the temperature changes at the measurement locations on the columns of the machine tool and the concrete foundation. Figure 5: Data acquisition of thermal error. Figure 6: The temperature changes at the measurement locations on the left column. It can be seen from Figures 6 and 7 that the highest temperature occurs at around 7 PM, and significantly lags behind the highest outdoor temperature (the highest outdoor temperature appears around 2 PM in this season). The temperature difference between 11:30 and 20:30 is 2.5 \({}^{\circ}\)C. There is a temperature gradient in the vertical direction of the machine tool and the temperature decreases with height. Figure 8 shows that temperature differences the concrete foundation are larger than those of the superstructure. The variation of temperature with time can clearly be observed. It should be noted that Figures 6-8 shows data collected by several sensors. In an ideal scenario, the temperature difference between the two sensors is always a constant (if so, the coordinate offset could be easily obtained using Figure 9). However, the external environment (opening and closing doors of workshop, etc.) makes it inconstant. Besides, the change of seasons also has effect on temperature difference. As a result, temperature measured by a sensor at a certain moment could be independent and changeable, and the thermal error cannot be obtained simply using Figure 9. Therefore, in order to obtain accuracy values in reality, people have to find another way to model the accuracy value. That is why deep neural network is proposed as an alternative. In the experiment, in order to train a model with high robustness, all the data in training a neural network model should be obtained one by one by human-labor. Figure 8: The temperature changes at the measurement locations the concrete foundation. Figure 7: The temperature changes at the measurement locations on the right column. This paper focuses on the thermal error caused by the environment temperature. The results show that the thermal error caused by the ambient temperature is mainly concentrated in the Z direction, and the error in other directions is rather small (the largest error in the other direction occurs in the x direction, which is only 7.6% of the error in the Z direction). Therefore, thermal error in the Z direction is predicted by temperature value as input. Because there are some noises in the collected data, this paper uses the integral differential Jacobian polynomial method to reduce the noise of the collected displacement data [25]. Actually, the training data is not related to time, because the temperature is changed with the change of sensors' location, and that is how we collect a new set of training data.