A Review of Machine Learning-Based Thermal Error Modeling Methods for CNC Machine Tools

Abstract

Heat source-induced thermal error is a primary element influencing the precision of CNC machine tools. A practical and economical approach to mitigating thermal errors is through thermal error compensation. To provide a comprehensive understanding of thermal error modeling and its advancements, this paper systematically reviews machine learning-based methods for thermal error compensation. Thermal error modeling is the most critical step in thermal error compensation, as it directly influences the effectiveness of the compensation due to its accuracy and robustness. With the rapid development of big data and artificial intelligence, machine learning has emerged as a powerful tool in thermal error modeling, leading to significant research progress in recent years. In this paper, an overview of the thermal error modeling methods based on deep learning that have been researched and applied in recent years is presented. Specifically, two methods for reducing thermal errors, namely, thermal error suppression and thermal error compensation, are introduced and analyzed. Second, machine learning-based thermal error modeling methods are categorized into traditional machine learning-driven and deep learning-driven approaches. The application of these two methods in thermal error modeling and compensation is reviewed and summarized in detail. By synthesizing these studies, this paper identifies key challenges and trends in machine learning-based thermal error modeling. Finally, the thermal error modeling methods discussed in this paper are summarized, and future research directions are proposed to further enhance modeling accuracy and robustness.

1. Introduction

The development trends of the manufacturing industry include intelligence, digitalization, high precision, and environmental sustainability. CNC machine tools, as fundamental equipment for advancing high-end manufacturing, are evaluated primarily by their machining accuracy. Sources of errors affecting machining accuracy include geometric errors, thermal errors, cutting force errors, and other factors [1,2]. Among these, thermally induced errors account for 40% to 70% of the total error [3,4,5].

Methods to eliminate thermal errors can be categorized into two approaches: thermal error suppression and thermal error compensation. The former involves suppressing heat generation through structural design, special material selection, and cooling system optimization [6,7,8]. The latter predicts thermal errors by using high-precision thermal error models and offsets them by generating a counteracting error value through a servo drive system [9]. Due to the high implementation costs, thermal error suppression methods have not been widely adopted. In contrast, thermal error compensation methods based on thermal error modeling have been extensively applied.

Thermal error modeling methods are primarily classified as physics-driven and data-driven approaches. Physics-driven models are typically based on theories of heat conduction and heat convection, utilizing analytical or numerical simulation models to determine the thermal error field of the machine tool. However, these methods are limited by analytical precision, oversimplified boundary conditions, and significant time costs, making them difficult to apply in industrial production [10,11,12]. Benefiting from the recent advancements in big data and artificial intelligence, data-driven thermal error modeling methods have achieved significant progress. Numerous studies have applied machine learning techniques—such as support vector machines, artificial neural networks, ensemble methods, and deep learning models—to predict thermal errors with improved accuracy [13,14,15].

In this review, the latest developments in machine learning-based thermal error modeling and compensation methods for CNC machine tools are summarized comprehensively. The review covers both traditional machine learning methods—such as Ridge Regression, principal component regression (PCR), lasso regression (LASSO), support vector machines (SVMs), backpropagation neural networks (BPNNs), ANFIS, and radial basis function neural networks (RBFNNs)—and deep learning methods, including convolutional neural networks (CNNs), LSTM, GRU, and Transformer networks. Traditional machine learning-driven thermal error modeling methods are limited by their reliance on handcrafted feature extraction and simple model architectures, which makes it challenging to capture complex nonlinear thermal behaviors and dynamic effects such as thermal hysteresis. In addition, these methods often struggle with limited generalization when faced with variable machining conditions. Deep learning-driven thermal error modeling methods, on the other hand, are constrained by their high dependency on large-scale, high-quality datasets, which can be difficult to obtain. They are also prone to overfitting, require significant computational resources, and generally suffer from a lack of interpretability, making it hard to understand and validate the underlying thermal error mechanisms in practical industrial applications. By synthesizing these studies, this review not only highlights the advantages and limitations of different algorithms but also discusses how the literature has addressed the aforementioned challenges and identifies potential future research directions to overcome existing shortcomings. The main abbreviations is shown in Table 1.

The remainder of this paper is organized as follows: Section 2 introduces the two primary methods for eliminating thermal errors in CNC machine tools. Section 3 provides a detailed review of the application of machine learning techniques in thermal error modeling. Section 4 summarizes the findings on machine learning-based thermal error modeling and compensation methods and discusses potential future research directions.

2. Review of Thermal Error Control Approaches

CNC machine tools generate thermal errors during machining and production due to the influence of both internal and external heat sources. Specifically, the high-speed rotation of the spindle generates significant heat, leading to the thermal expansion and deformation of the spindle and surrounding structure. As spindle speed increases, heat generation intensifies, further exacerbating deformation. During the machining process, temperature decreases when the machine is idle, such as during detection or tool changes, causing the spindle to retract and resulting in new error variations. Additionally, fluctuations in thermal errors may arise from changes in the external environment’s temperature and the operator’s body temperature. These thermal errors significantly impact the machining accuracy of CNC machines and are, to some extent, unavoidable. Conventional methods for controlling spindle thermal errors can be classified into suppression and compensation strategies. Suppression methods focus on reducing thermal error generation at the source, employing material selection, structural design, and cooling system installation. Compensation methods, on the other hand, involve developing high-precision thermal error models to predict thermal errors and utilizing a servo drive system to control spindle reverse motion, thereby compensating for the errors.

2.1. Thermal Error Reduction Method Based on Suppression

The core of thermal error suppression methods is to reduce the heat generated during machine operation, thereby minimizing the impact of temperature rise caused by internal heat sources on the machine structure. Existing thermal error suppression methods are generally classified into the following: optimization of the cooling system, selection of special materials for machine construction, and thermal balance control methods. For example, Grama et al. developed a control strategy based on the Cooler Trigger Model to control a traditional bath recirculation cooler unit, improving the effectiveness of cooling [6]. Winiarski et al. divided the column channels into several sections and applied gravity flow for coolant spraying, effectively suppressing the displacement of the vertical fixture’s central column caused by temperature fluctuations [16]. Li et al. proposed a method to suppress machine thermal errors by controlling the temperature at temperature-sensitive points of the cooling system [17]. Also, Ma et al. introduced a multi-objective topology optimization design method to design cooling components for gear grinding machines to control thermal errors. The results show that the heat exchange capacity of the serpentine channels optimized through topology is significantly better than that of traditional serpentine cooling channels [18].

In addition, selecting special materials during the design phase to construct the machine is an effective way to reduce thermal errors. Bae et al. prepared TiC-SUS431 composite materials by using a liquid pressurization impregnation method to suppress thermal deformation [19]. Meanwhile, in another study, Ge et al. used the thermal contraction of carbon fiber-reinforced composites to suppress thermal expansion in metal spindle housings [7]. Experimental and numerical simulation results show that compared with electric spindles without the proposed thermal error control system, the proposed method reduced thermal displacement by 97%.

Thermal balance represents the ideal thermal state of a machine tool, aimed at effectively suppressing the generation of thermal errors. The core principle involves strategically arranging heat sources and cooling mechanisms to offset heat at critical locations, thereby preventing thermal deformation. Thermal symmetry design is commonly employed to achieve thermal balance in machine tools. Weng et al. proposed a new analytical method to characterize the thermal response of machine tool structural components as equivalent rectangular bodies [20]. This method allows for the quick evaluation of the thermal characteristics of machine tool components and guides thermal balance design. Experimental results show that the thermal positioning error and pitch error of the Z-axis were reduced by approximately 35.6% and 73.3%, respectively. Liu et al. proposed a thermal balance control method for the vertical milling machine spindle head structure which involves selectively heating specific areas of the spindle head to adjust the local temperature and suppress thermal bending, improving the spindle’s inclination with respect to the Y-Z plane [8].

The methods mentioned above all require changes to the machine tool structure itself. Therefore, their implementation presents varying degrees of difficulty. Optimizing the cooling system structure and using special materials to build the machine tool both aim to increase the threshold for the heat required to generate thermal errors. However, this threshold cannot be set infinitely high. Therefore, under certain working conditions, the generated heat may still exceed this threshold, leading to thermal errors. Researchers achieve thermal balance by applying thermal symmetry design to make the machine tool reach a thermally balanced state. However, the machine tool cannot always be in thermal balance under all working conditions, as it is impossible for designers to account for every possible working condition. Therefore, the methods mentioned above are considered “front-end” research, which may not adapt well to various real conditions. Additionally, the implementation of these methods involves high economic and modification costs.

2.2. Thermal Error Reduction Method Based on Compensation

The thermal error compensation of machine tools is usually achieved through passive and active compensation. Passive compensation includes the thermal error suppression methods mentioned in the previous section. Active compensation, which is currently the most widely used thermal error control method, is implemented through software compensation and mainly involves error prediction and compensation based on mathematical models. Thermal error compensation methods involve the establishment of a thermal error model to predict the thermal error at the next moment. A servo drive system is then used to control the spindle’s reverse movement to offset the thermal error. High-precision thermal error modeling serves as the foundation for accurate compensation. Conventional thermal error models can be classified into physics-driven and data-driven models. For example, Li proposed an analytical thermal error model for gear grinding machines based on an equivalent temperature field [21]. The transient equivalent temperature field model is derived by the Green function. The Rayleigh–Ritz method is then applied to obtain the thermal error field based on the equivalent temperature field model. However, finite element analysis-based methods are notably labor-intensive and time-consuming. Additionally, these methods often rely on overly simplified machine tool structures and boundary conditions as the basis for simulation analysis. This paper focuses primarily on data-driven thermal error modeling methods.

The essence of data-driven thermal error modeling methods is to establish a nonlinear mapping relationship between thermal error data and temperature data (in most cases). Data-driven methods have been widely applied in the field of thermal error modeling due to the advantage of not requiring the establishment of complex mathematical models. Data-driven modeling methods typically include machine learning, deep learning, and transfer learning. Deep learning is essentially a subset of machine learning. Therefore, this paper divides the methods into traditional machine learning, deep learning, and transfer learning. Traditional machine learning models mainly include SVM [22], BPNN [23], ANFIS [24], RBF [25], etc. Deep learning models mainly include CNN [26], RNN [27], and other neural networks [28]. Transfer learning has limited application in the field of thermal error modeling and thus is not within the scope of this review.

3. Thermal Error Compensation Methods Based on Machine Learning

3.1. Thermal Error Models Based on Traditional Machine Learning

3.1.1. Thermal Error Models Based on Various Regression Methods

The collinearity problem in the thermal error modeling of machine tools mainly arises from the high correlation among the data measured by multiple temperature sensors. This collinearity can affect the model’s prediction performance, stability, and interpretability. Therefore, researchers have introduced linear regression methods such as ridge regression (RR), principal component regression (PCR), and lasso regression (LASSOR) to address the collinearity issue in thermal error modeling. Ridge regression introduces an L2 regularization term to penalize the regression coefficients, preventing the model from overly relying on the values of certain temperature sensors in the presence of collinearity. The regularization term compresses the regression coefficients and reduces their fluctuations, thus improving the stability of the model. Principal component regression (PCR) is a method that combines principal component analysis (PCA) and linear regression (LR). It is commonly used to address multicollinearity issues, improving the stability and generalization of regression models. Principal component analysis reduces the correlation between features by combining highly correlated features into a small number of principal components, effectively lowering collinearity. Lasso regression, similar to ridge regression, introduces an L1 regularization term. Lasso regression compresses some of the regression coefficients to zero, thus automatically performing feature selection and removing temperature sensors that contribute less to the prediction. For features with strong collinearity, lasso regression can select the most important features, helping to reduce model complexity and mitigate overfitting risks.

Liu et al. employed the ridge regression algorithm to construct a thermal error model to inhibit the bad influence of collinearity on the thermal error predicted robustness [29]. The authors validated the effectiveness of the thermal error model based on ridge regression in eliminating the collinearity issue in temperature data by using experimental data from an entire year. Miao et al. established a thermal error model based on the principal component regression algorithm to eliminate the influence of multicollinearity among temperature variables [30]. The results showed that the PCR model significantly reduced the impact of changes in temperature-sensitive points on the model’s accuracy. Liu et al. also established a thermal error model based on the principal component regression method [31]. Furthermore, the authors compared the proposed model with the thermal error compensation model based on ridge regression. The results indicated that the proposed PCR-based thermal error model could effectively control the spindle’s Z-direction thermal error within 5 μm with only two temperature sensors. The autoregressive with exogenous input model is used to describe the relationship between the autoregressive process and external input. In the ARX model, all parameters that describe the effect of an input on the output must be minimized to remove the corresponding input from the model structure. Zimmermann et al. applied Group-LASSO regression to the ARX model for high-precision dynamic thermal error modeling [32].

Unlike linear regression, nonlinear regression aims to fit complex, nonlinear functional relationships. For example, in thermal error modeling, the relationship between the spindle thermal error and time exhibits an exponential form. In such cases, nonlinear regression methods are required for fitting. Common nonlinear regression methods include random forest regression (RFR), extreme gradient boosting regression (XGBoost), and Gaussian process regression (GPR). RFR builds multiple decision trees and averages their predictions to reduce the model’s variance and enhance prediction accuracy. XGBoost employs a gradient boosting technique where the residuals from each iteration’s model are used to train a new weak classifier, and the results of these trees are weighted and combined to form a strong predictive model. In each iteration, XGBoost updates the model by using gradient descent to minimize the loss function. GPR assumes that the function values follow a Gaussian process, which defines a joint distribution of random variables. The correlation between input variables is described by using a covariance function (or kernel function), generating a smooth predictive function in the input space. The output of a Gaussian process is a probability distribution that provides not only the predicted values but also the uncertainty in the predictions.

Zhu et al. developed a thermal error model by using the random tree algorithm, achieving higher prediction accuracy in cases with small-sample data [33]. Lian et al. proposed a two-mode joint thermal error prediction model based on LASSO and random forest regression, which demonstrated maximum prediction errors of 6.08 μm for milling and 1.455 μm for turning in actual machining experiments [34]. Gao et al. established a thermal error model combining extreme gradient boosting with thermal expansion mechanisms to compensate for the thermal error of ball screws [35]. Existing thermal error prediction processes are point-to-point, which poses challenges in understanding the random nature of thermal error predictions and analyzing reliability. Therefore, Wei et al. developed a thermal error interval prediction model based on Gaussian process regression, achieving high-precision thermal error modeling [36].

3.1.2. Thermal Error Models Based on Support Vector Machine Method

In actual machining, thermal errors in machine tools exhibit nonlinearity and time-varying characteristics. The core idea of support vector machine (SVM) is to find a hyperplane that best separates data from different classes. Additionally, SVM can map the data into a higher-dimensional space by using kernel functions, making the data linearly separable in that space. In regression problems, this approach is known as support vector regression (SVR). Unlike traditional regression methods, SVR does not aim to find the best-fit line but instead seeks to find a “boundary” that can tolerate errors for performing regression. SVR can handle nonlinear problems by mapping the data from a low-dimensional space to a high-dimensional space by using kernel functions.

Zhou et al. established a thermal error model for electric spindles by using an improved particle swarm optimization support vector regression algorithm based on simulated annealing, achieving the real-time characterization of thermal error variation, the compensation of thermal errors, and improved machining accuracy [37]. Luo et al. combined support vector regression with feedforward neural networks to create a thermal error prediction model, using a multi-criterion evaluation system to assess prediction performance [38]. Additionally, Dai et al. proposed a thermal error prediction model based on differential evolution, gray wolf optimization, and support vector regression to explore the thermal error variation of electric spindles under different preload conditions [39]. Tan et al. used least squares support vector machine (LSSVM) as the base model [22]. They divided the experimental data into different subsets based on time duration to construct sub-LSSVM models and finally developed a fusion model. This model can simultaneously account for both local and global prediction characteristics. Similarly, Sun et al. also used LSSVM to build a thermal error model but optimized its parameters by using the adaptive boundary Harris hawk algorithm [40]. However, SVM has a high computational cost when handling large-scale datasets. Additionally, the performance of SVM is highly dependent on the choice of the kernel function. An inappropriate kernel function can lead to a decline in model performance, and in some cases, selecting and tuning the kernel function and its parameters require extensive experimentation.

3.1.3. Thermal Error Models Based on Other Traditional Machine Learning Methods

Common methods in traditional machine learning also include Adaptive Neuro-Fuzzy Inference System (ANFIS), BP neural networks (BPNNs), RBF neural networks (RBFNNs), and extreme learning machine (ELM). ANFIS is a hybrid model that combines neural networks and fuzzy inference systems. It utilizes the learning capabilities of neural networks and the reasoning abilities of fuzzy logic for system modeling and control. ANFIS has strong adaptability and is well suited for handling nonlinear problems. Compared with pure neural networks, the ANFIS model is based on fuzzy rules, offering interpretability and providing a basis for decision making. BP neural networks, RBF neural networks, and ELM are all types of feedforward neural networks. The BP neural network uses the backpropagation algorithm for training and is suitable for a wide range of regression problems. The RBF neural network employs radial basis functions as activation functions in the hidden layer. Its training speed is relatively fast, and it can effectively handle nonlinear problems, making it suitable for small-sample learning. Extreme learning machine (ELM) is a novel neural network model with a single-hidden-layer feedforward network structure. The core of ELM is to randomly initialize the weights and biases from the input layer to the hidden layer without training. This avoids the iterative weight updates required in traditional neural networks, making ELM’s training process extremely fast and capable of handling large-scale datasets.

Many researchers have used these methods to establish thermal error models. For example, Abdulshahed et al. proposed a thermal error prediction model based on fuzzy C-means and ANFIS [24]. Figure 1 shows the basic structure of ANFIS with FCM clustering. The results showed that the residuals of the FCM-ANFIS model were less than ±2 μm, indicating a 95% reduction in thermal errors for the machine. Feng et al. established a CNC machine tool thermal error prediction model based on an improved particle swarm optimization algorithm and the radial basis function neural network [41]. Figure 2 shows the prediction results’ comparison and performance. Additionally, Fu et al. used a radial basis function neural network based on the chicken swarm algorithm to establish a thermal error model [25]. Numerous experiments were carried out on the VMC850 machining center, and the results demonstrated that the proposed model exhibited high accuracy and strong robustness. Dai et al. proposed a thermal error model based on a BP neural network optimized by the artificial bee colony algorithm and used experimental data with a bearing preload of 1400 N as input to predict spindle thermal displacement at the preload values of 1450 N, 1550 N, and 1700 N [42]. Zheng et al. established a thermal error prediction model for a multilink high-speed precision press system based on an improved adaptive genetic algorithm and the BP neural network [43]. The proposed model was compared with traditional multivariate linear regression, BP neural networks optimized by genetic algorithms, and BP neural networks optimized by particle swarm optimization, proving the effectiveness of the proposed model. Dai et al. selected temperature-sensitive points by using fuzzy C-means clustering and correlation analysis methods and then established a thermal error prediction model based on kernel extreme learning machine [44]. Li et al. adopted a marine predator algorithm-optimized extreme learning machine to build and predict thermal errors for electric spindles [45]. The proposed model was compared with ELM and ELM optimized by genetic algorithms, validating its effectiveness.

3.2. Thermal Error Models Based on Deep Learning

With the development of big data and artificial intelligence technologies, traditional machine learning methods can no longer meet the higher accuracy requirements. Deep learning methods have brought new opportunities and advancements to machine tool thermal error modeling. This section divides deep learning methods into two categories: those using temperature data as input and those using thermal image data as input.

3.2.1. Thermal Error Models with Temperature Data as Input

Machine tool thermal errors are nonlinear time-series data. Due to the hysteresis between the machine surface temperature and thermal error changes, many researchers choose models with the ability to extract long-sequence time features for modeling. For example, Liu et al. used long short-term memory (LSTM) neural networks to construct a basic thermal error model and optimized the model structure parameters by using the gray wolf optimization (GWO) algorithm [46]. The variational mode decomposition (VMD) algorithm was applied to filter out the impact of high-frequency noise in the data. The proposed VMD-GW-LSTM model was compared with other models, demonstrating its effectiveness. Zeng et al. designed an LSTM network with a sequence-to-sequence structure and attention mechanism to fully extract both long-term and short-term information from thermal error data [47]. The predicted results showed that the SQ-LSTM model outperformed traditional time-series models. Chen et al. also established a thermal error model based on LSTM networks and combined it with working conditions and temperature data [48]. To improve model prediction accuracy, the authors used K-means clustering analysis to obtain the optimal speed classification and established sub-LSTM models. The experiments showed that the K-means segmented LSTM method achieved a prediction accuracy of over 85.0% under different operating conditions. It is worth noting that Sun et al. proposed a spatiotemporal thermal error prediction model for machine tools based on a bidirectional LSTM network structure with axial attention under complex working conditions [49]. After its comparison with BiLSTM and LSTM models, the proposed model demonstrated better prediction accuracy and robustness under complex operating conditions.

The gated recurrent unit (GRU) network and LSTM are both variants of recurrent neural networks. However, GRU is structurally simpler than LSTM, and it also has a faster training speed. Qin et al. introduced the GRU model to handle thermal hysteresis effects and transformed it into a prediction interval-based GRU model to generate spindle axial thermal error prediction intervals [50]. The authors conducted thermal error experiments to validate the superiority of the proposed method. The results showed that compared with Gaussian process regression, the PI-GRU model could better handle thermal hysteresis effects. Additionally, Li et al. proposed a spindle thermal error prediction model combining the sparrow search algorithm and GRU [51]. The proposed model used variable-speed operating condition data for testing and demonstrated that the SSA-GRU model achieved good prediction performance.

Some researchers also build more powerful hybrid models by incorporating other models into the GRU and LSTM framework. For example, Yang et al. constructed a CNN-GRU hybrid model based on a subtraction-based average optimizer [52]. The CNN model is used to extract spatial features from thermal error data, while the GRU model is used to extract temporal features. The authors compared the proposed model’s predictions with those of the SO-ELM model. The SABO-CNN-GRU model outperformed in terms of MAE, RMSE, RPD, MSE, and R². Jia et al. built a 1D CNN-GRU-Attention model by adding an attention mechanism to the CNN-GRU model structure [53]. The attention mechanism allows for the secondary weighting of the learned features, further filtering out the most important features. Chen et al. proposed a spatiotemporal feature fusion network framework to capture long-term relationships and global features from small-sample thermal information data [54]. The network consists of spatiotemporal feature interaction blocks, spatiotemporal feature gated fusion layers, and residual structures. The authors verified the effectiveness of the proposed model through compensation experiments, showing that the thermal error was reduced by 97.91%. Gao et al. developed a convolutional neural network–long short-term memory hybrid neural network model optimized by the Pelican optimization algorithm [55]. Wu et al. built on this work and proposed an attention-based spatiotemporal graph convolution framework with a long short-term memory hybrid network model. Gui et al. innovatively proposed a mist–edge–fog–cloud computing system applied to the geometric and thermal error compensation of a gear hobbing machine based on a graph convolutional neural network [56]. The results demonstrated that the proposed model significantly outperformed traditional time-series models, and the mist–edge–fog–cloud computing system greatly enhanced computational speed.

3.2.2. Thermal Error Models with Thermal Images Data as Input

Many deep learning methods applied in the field of thermal error modeling were initially derived from image processing techniques, such as convolutional neural networks. Due to the limitations of actual machining conditions, many machine tools cannot employ contact-based measurement methods. Therefore, numerous researchers have opted for non-contact measurement instruments, such as thermal imaging cameras. Thermal images captured by these cameras can directly reflect the temperature field of the machine tool, making them a suitable input for thermal error models.

For example, Wu et al. utilized a CNN to develop a thermal error classification model for rotational axes [26]. The authors measured angular positioning errors by using a laser interferometer while collecting thermal image data of the rotational axes by using a thermal imaging camera. Moreover, they calculated thermal image data of temperature rise by subtracting the pixel values of the initial thermal image from those of subsequent images. Finally, a deep CNN was employed to establish a multi-class thermal error model. Based on previous work, Wu et al. used both thermal image data and thermocouple temperature data as input for the model, enabling a more comprehensive representation of the temperature field of the machine tool spindle [57]. To verify the effectiveness of the proposed model, practical cutting compensation experiments were conducted. The results showed that for lathe spindles, the average reduction in compensated radial diameter error was approximately 45%. For milling machine spindles, the average compensation rates for axial and radial errors were 60% and 50%, respectively.

Fu et al. employed a vision-based temperature data acquisition method and developed a thermal error prediction model by using a Vision Transformer (ViT) with a self-attention mechanism [28]. Figure 3 shows the structure schematic of the ViT. An infrared thermal imaging camera was used to capture the temperature field information of the machine tool spindle. The collected thermal images were augmented and expanded through pixel-value differencing and image rotation. A thermal error model integrating self-attention mechanisms and the Vision Transformer was constructed. Comparative analysis with other models demonstrated the superiority of the proposed approach. In prediction performance validation, the ViT model achieved a maximum residual thermal error of 2.678 μm, a mean absolute error (MAE) of 0.105 μm, and a mean squared error (MSE) of 1.197 μm², representing reductions of 39%, 26%, and 67%, respectively, compared with CNN-based models.

Gui et al. pointed out that traditional thermal error modeling has not fully considered the spatial characteristics of machine tool thermal errors [58]. Figure 4 shows the dynamic TS graph of temperature field. Therefore, a sensor network was established based on thermal images of the machine tool to extract sensor nodes and construct a dynamic spatiotemporal graph dataset. The constructed dynamic spatiotemporal graph dataset was used as an adjacency matrix input to the proposed DTSMGCN model. The DTSMGCN model, based on graph convolutional neural networks, incorporates model units capable of processing temporal data. This enables complex network modeling and achieves higher accuracy in thermal error prediction.

Du et al. proposed a thermal error prediction model using indirect thermal images as input [59]. Figure 4 shows thermal error modeling based on a CNN. The original temperature data of the machine tool were transformed into thermal images, which were then used as input for the thermal error model. A deep convolutional network with 10 hidden layers was constructed to enhance the prediction accuracy of the thermal error model. The nonlinear mapping relationship between thermal images and thermal errors was established without preselecting temperature-sensitive points, preserving more relationships between machine tool thermal errors and temperature features. The authors employed a microcontroller as the thermal error compensation controller and conducted application experiments of the thermal error compensation system on a CNC grinding machine, verifying the effectiveness of the system.

3.3. Summary of Machine Learning-Based Thermal Error Compensation Methods

In the above section, machine learning-based modeling methods for machine tool thermal errors are divided into traditional machine learning methods and deep learning methods. Traditional machine learning methods are further categorized into linear regression, nonlinear regression, support vector machine, and other methods. Deep learning methods are classified into those using temperature data as input and those using thermal image data as input.

Linear regression methods, such as ridge regression, principal component regression, and lasso regression, offer advantages including simple principles, high computational efficiency, and strong model interpretability. However, these methods have limitations in fitting accuracy for nonlinear data and are unsuitable for complex working conditions. Nonlinear regression methods, such as random forest regression, extreme gradient boosting regression, and Gaussian process regression, are used to fit complex nonlinear thermal errors and are applicable to complex working conditions. The drawbacks of these methods include a tendency to overfit, the need to select appropriate nonlinear functions, and long training times.

Support vector machines perform well in small-sample scenarios and can map nonlinear data into high-dimensional space by using kernel functions to achieve linear separability. However, support vector machines are not suitable for large-scale data, are highly sensitive to parameters, and have high computational complexity with long training times. ANFIS, BP neural networks, RBF neural networks, and extreme learning machines all effectively handle nonlinear data. However, ANFIS has a complex training process and high computational cost. BP neural networks are prone to falling into local optima and have slow convergence. RBF neural networks are sensitive to parameters such as center location and width. Extreme learning machines have weaker generalization capabilities compared with deep networks, and their internal mechanisms are difficult to interpret.

Deep learning-based thermal error modeling methods primarily utilize various deep neural networks, such as long short-term memory (LSTM) networks, gated recurrent unit (GRU) networks, and convolutional neural networks (CNNs). These three networks can extract information from time-series data and are often used in models with temperature data as input. CNNs can also extract features from image data, making them suitable for models with thermal image data as input. LSTM and GRU can capture temperature information from historical moments, making them effective in addressing the thermal hysteresis effect in machine tools. Traditional recurrent neural networks are prone to gradient vanishing or explosion issues in long time-series modeling, while LSTM mitigates this problem through gating mechanisms, enabling the stable processing of long sequences. GRU has a simpler structure and fewer parameters than LSTM, resulting in faster computation. However, LSTM has complex parameters and requires significant computational resources, while GRU has slightly weaker long-term dependency capabilities. CNNs possess strong feature extraction abilities, support parameter sharing and local connectivity, and can integrate time-series and spatial data for multimodal modeling. However, the original CNN’s ability to model time-series data is inferior to LSTM and GRU, and its performance may be limited with insufficient data, leading to overfitting. Additionally, hyperparameters such as kernel size, stride, and pooling methods significantly impact CNN performance.

4. Summary and Outlook

4.1. Conclusions

Thermal error is one of the critical factors affecting the machining accuracy of machine tools. This paper analyzes and reviews two methods for controlling thermal errors, namely, thermal error suppression and thermal error compensation, as follows:

• (1). Thermal error compensation is divided into physics-driven and data-driven approaches.

• (2). Furthermore, data-driven methods are categorized into two types: traditional machine learning-based and deep learning-based methods.

• (3). Next, three categories of traditional machine learning methods are reviewed, including various regression methods, support vector machine, and other traditional machine learning techniques.

• (4). Additionally, two categories of deep learning methods are reviewed, including models with temperature data as input and those with thermal image data as input. Finally, the advantages, disadvantages, and applicable scenarios of the above methods are summarized.

4.2. Outlook

With the continuous development of industrial capabilities, the increasing demand for machining precision has raised the requirements for machine tool error control. Thermal error modeling and compensation, as effective methods to reduce thermal errors in machine tools, have become a research hot spot. In the era of big data and continuous advancements in artificial intelligence, machine learning technology has become an essential part of thermal error modeling and compensation for machine tools. Emerging large models and deep learning technologies are expected to be more widely applied in this field in the future. Based on the recent research status in machine learning-based thermal error modeling and compensation, this paper proposes the following outlooks:

• (1). Multimodal data fusion. Thermal errors are often influenced by various factors, and single-modal data may not comprehensively reflect the mechanisms behind thermal error generation. By integrating multimodal data, it becomes possible to capture various factors that affect thermal errors, thus improving the prediction accuracy of the model. For example, combining data from temperature sensors, motor speed, and current can be used to model the thermal errors of machine tools under different operating conditions.

• (2). Transfer learning methods. Transfer learning allows knowledge from a pre-trained model to be transferred to a new task, reducing the data requirements and training time for the new task. In thermal error modeling, transfer learning can be used to transfer a model trained on one device to another device. Even if the thermal error characteristics of the two devices are different, fine-tuning the model can adapt it to the new environment. This is particularly useful for multiple similar devices on a production line, significantly reducing the cost and time needed to model each device individually.

• (3). Explainable deep learning models. Currently, deep learning models are often considered “black-box” models, making it difficult to explain their internal working mechanisms. By incorporating explainable neural network structures, such as attention mechanisms, or integrating physical knowledge into deep learning models, the models can not only possess excellent predictive capabilities but also reflect physical laws. This enhances the model’s interpretability, making it more transparent and useful for practical applications.

Author Contributions

S.M.: Investigation and Writing—original draft. C.Y.: Investigation and Writing—review and editing. K.L.: Investigation and Writing—review and editing. C.L.: Conceptualization and Supervision. X.W.: Investigation and Writing—review and editing. T.W.: Supervision and Writing—review and editing. G.F.: Supervision and Writing—review and editing. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

Data are contained within the article.

Conflicts of Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Footnotes

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Figures and Table

Figure 1. Basic structure of ANFIS with FCM clustering [24].

Figure 2. Prediction results’ comparison and performance [41].

Figure 2. Prediction results’ comparison and performance [41].

Figure 3. Structure schematic of ViT [28].

Figure 4. Thermal error modeling based on CNN [59].

Table 1

Main abbreviations.

Full Name	Abbreviation	Full Name	Abbreviation
Ridge regression	RR	Principal component regression	PCR
Lasso regression	LASSOR	Principal component analysis	PCA
Linear regression	LR	Random forest regression	RFR
Extreme gradient boosting regression	XGBoost	Gaussian process regression	GPR
Support vector machine	SVM	Support vector regression	SVR
Least squares support vector machine	LSSVM	Adaptive Neuro-Fuzzy Inference System	ANFIS
Backpropagation neural network	BPNN	Radial basis function neural network	RBFNN
Extreme learning machine	ELM	Fuzzy C-means clustering	FCM
Long short-term memory	LSTM	Gray wolf optimization	GWO
Variational mode decomposition	VMD	Gated recurrent unit	GRU
Convolution neural network	CNN	Vision Transformer	VIT

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