OLTEM: Lumped Thermal and Deep Neural Model for PMSM Temperature

Abstract

Background and Objective: Temperature management is key for reliable operation of permanent magnet synchronous motors (PMSMs). The lumped-parameter thermal network (LPTN) is fast and interpretable but struggles with nonlinear behavior under high power density. We propose OLTEM, a physics-informed deep model that combines LPTN with a thermal neural network (TNN) to improve prediction accuracy while keeping physical meaning. Methods: OLTEM embeds LPTN into a recurrent state-space formulation and learns three parameter sets: thermal conductance, inverse thermal capacitance, and power loss. Two additions are introduced: (i) a state-conditioned squeeze-and-excitation (SC-SE) attention that adapts feature weights using the current temperature state, and (ii) an enhanced power-loss sub-network that uses a deep MLP with SC-SE and non-negativity constraints. The model is trained and evaluated on the public Electric Motor Temperature dataset (Paderborn University/Kaggle). Performance is measured by mean squared error (MSE) and maximum absolute error across permanent-magnet, stator-yoke, stator-tooth, and stator-winding temperatures. Results: OLTEM tracks fast thermal transients and yields lower MSE than both the baseline TNN and a CNN–RNN model for all four components. On a held-out generalization set, MSE remains below 4.0 °C² and the maximum absolute error is about 4.3–8.2 °C. Ablation shows that removing either SC-SE or the enhanced power-loss module degrades accuracy, confirming their complementary roles. Conclusions: By combining physics with learned attention and loss modeling, OLTEM improves PMSM temperature prediction while preserving interpretability. This approach can support motor thermal design and control; future work will study transfer to other machines and further reduce short-term errors during abrupt operating changes.

1. Introduction

In today’s environment, which sets higher requirements for system fidelity and operational efficiency, effective temperature management of permanent magnet synchronous motors has become critical. PMSMs use stator windings to create a rotating magnetic field and rely on permanent magnets for the rotor’s magnetic field. They are widely employed in various high-efficiency and high-performance areas [1], including precision instruments, robotics, medical devices, and aerospace. As demands for PMSM performance and reliability continue to rise [2], the motors often operate under harsher conditions, where heat buildup can significantly affect overall stability and lifespan. Excessive internal temperatures can also weaken performance in multiple ways [3], such as degraded magnetism in permanent magnets [4], aging of insulation materials, and wear of mechanical parts. For instance, operating temperatures exceeding 150 °C can risk the permanent demagnetization of neodymium magnets and accelerate the degradation of insulation materials. These problems not only reduce efficiency but also raise the likelihood of failures and shorten service life. Therefore, real-time and precise monitoring and prediction of temperatures in different parts of the motor are crucial for safe and efficient operation under complex conditions. From both research and application perspectives, precise PMSM temperature prediction aids in the design of effective thermal management systems and advanced thermal materials while also deepening our understanding of motor heat behavior and dynamics. In industrial use, real-time temperature prediction can trigger prompt adjustments (such as speed changes or optimization of control strategies), thereby improving energy efficiency, reducing failure rates, and extending motor lifespan.

In recent years, various temperature detection or estimation methods have been proposed. For instance, Pt100 resistive temperature detectors (RTDs) can measure temperature directly, but they incur additional sensor costs and require mechanical modifications during manufacturing [5]. Another example is the PM-flux method based on PWM voltage and current responses, which can estimate rotor temperature without extra signal injection [6]. Additionally, rotor resistance estimation approaches combine terminal voltage, current, and carrier signal injection with online parameter identification [7,8,9]. LPTNs are simplified models that represent complex thermal systems as multiple interconnected thermal nodes and resistances, providing high computational efficiency and clear physical meaning. Hence, LPTNs are widely used for PMSM thermal analysis [10]. However, they rely on detailed geometric and material information, as well as linear assumptions, making it difficult to model the strong nonlinear thermal behavior and complex losses inside the motor [11]. Recent advances in machine learning provide new approaches to temperature prediction. Data-driven methods employ machine learning models to achieve relatively reliable temperature estimates without requiring a precise machine model [12]. Through the application of deep learning techniques such as recurrent neural networks (RNNs) and temporal convolutional networks (TCNs), researchers can effectively model temperature trends when large amounts of data are available [13,14]. Traditional RNNs are prone to gradient explosion or vanishing problems, and although LSTM can partially address these issues, its more complex structure can lead to overfitting when data are limited [15]. Due to their lack of physical prior knowledge, these data-driven methods are often regarded as “black boxes” and may not meet engineering requirements for interpretability and reliability [16]. To maintain physical interpretability while improving prediction performance, Kirchgässner et al. introduced thermal neural networks (TNN) [17], which combine the thermal resistance and thermal capacitance modeling from LPTNs with supervised learning. They utilize a state-space representation for end-to-end differentiable physics–data fusion. TNN requires fewer parameters and demonstrates lower computational complexity, while achieving prediction performance comparable to or exceeding that of CNN and MLP models, although further improvements remain possible.

In recent years, the integration of physics-based and data-driven models has achieved significant advances in numerous fields. Through the incorporation of physical prior knowledge, these integrated models typically demonstrate enhanced interpretability and improved generalization capabilities [18]. For instance, Yaxin Li et al. proposed a semi-supervised model that embeds prior knowledge, achieving more reliable fault diagnosis even with limited sample data [19]. Tianci Zhang et al. created physical prior features based on failure mechanisms and engineering experience, and implemented a self-supervised learning framework to enhance feature extraction with limited sample data [20]. Tang et al. combined an LPTN with an improved graph neural network for transient temperature field prediction in PMSMs [21]. Similarly, Liu et al. presented a physics-driven iron loss analysis model integrated with CNN, demonstrating excellent results in calculating PMSM iron loss [22].

Although these integrated models demonstrate robust performance, significant challenges persist in processing complex data and identifying key features. The attention mechanism, which dynamically assigns weights, emphasizes the most relevant aspects of input data and significantly enhances the extraction of critical information. It has achieved widespread adoption in deep learning applications. Izaz Raouf et al. employed a cross-attention-based feature aggregation network [23] that merges deep and shallow features in layers to improve diagnostic performance. Thanh-Tung Vo et al. applied a dual-path feature extraction strategy using 1D-CNN and RNN, combined with a multi-head attention mechanism, to integrate spatial and temporal features, leading to higher fidelity and efficiency in induction motor diagnosis [24]. Furthermore, most traditional temperature prediction models rely on LPTN-based, linear approximations for power loss modeling, often overlooking the complex dynamic changes and nonlinear effects under high power density conditions [25,26]. While this simplified approach reduces computational costs, it does not accurately capture nonlinear characteristics or physical constraints. Under high-speed, high-frequency operations, a lack of detailed analysis of nonlinear loss mechanisms can reduce prediction performance, thereby complicating the balance between efficiency, performance, and thermal management [27,28].

In response to the above issues, this paper proposes an OLTEM based on the TNN framework, addressing the highly nonlinear thermal behavior and high power density conditions of PMSMs. Two main improvements are introduced: (1) A novel state-conditioned squeeze-and-excitation (SC-SE) attention mechanism designed to address a key limitation in standard attention approaches. While generic attention can weight features, it often overlooks the system’s internal state. Our SC-SE mechanism explicitly incorporates the motor’s current temperatures as a condition for generating attention, thereby modeling the physically crucial temperature dependency of thermal paths and power loss features. (2) An enhanced power loss module that leverages this SC-SE mechanism in a two-stage architecture. This module combines a deep MLP for initial feature extraction with our physics-guided attention for a final, state-aware refinement, enabling a more accurate characterization of nonlinear loss patterns under diverse operating conditions. The overall framework of the method we proposed is shown in Figure 1, with the research focus on the permanent magnet synchronous motor. We begin by analyzing the temperatures of the permanent magnet, stator teeth, stator yoke, and stator windings, using a hybrid model for thermal analysis. This allows for accurate prediction of the motor’s internal temperature. In addition, comparative experiments validate the effectiveness of the improved module, and hyperparameter experiments identify the optimal combination of model parameters.

The remainder of this paper is organized as follows: Section 2 describes the research target and the LPTN theory. Section 3 explains the baseline TNN structure along with the proposed improvements. Section 4 presents the experimental design, evaluation metrics, and ablation study outcomes. Finally, Section 5 concludes the paper and discusses potential future research directions.

2. Research Object and Modeling

2.1. Research Object

This study focuses on PMSMs. A PMSM uses stator windings to generate a rotating magnetic field, and the permanent magnets on the rotor provide the magnetic field to synchronize the rotor speed with the stator current frequency. Compared to conventional induction motors, these characteristics make PMSMs highly suitable for applications demanding compact size, energy savings, and precise control, such as in electric vehicles and robotics. The stator in a PMSM is similar to that of a traditional induction motor, and it uses a laminated structure to reduce iron losses. The rotor rotates synchronously with the stator’s magnetic field, producing precise and efficient electromagnetic torque. Figure 2 shows a partial view of a PMSM.

When operating at high speeds under heavy loads and demanding conditions, PMSMs experience significant losses, leading to increased internal temperatures. Excessive heat can degrade permanent magnet performance and accelerate the aging and failure of stator windings and insulation materials, reducing overall system reliability. Therefore, real-time temperature prediction inside PMSMs is critical for safe and stable operation in harsh environments.

2.2. Traditional LPTN and Its Limitations

LPTN is a classic approach widely used in thermal analyses of motors and other electromechanical systems. However, as motor systems become more precise and complex, LPTN has certain drawbacks. For example, when the persistent excitation (PE) condition is not satisfied, the identified parameters can have infinite optimal solutions, and sampling noise can cause results to drift toward predefined upper or lower limits [29].

LPTN converts the thermal partial differential equation (PDE) shown in Equation (1) to an ordinary differential equation (ODE) shown in Equation (2). Under certain simplifications, geometric structures with similar thermal properties are merged into simplified thermal nodes, reducing the number of required parameters and the complexity of gradient calculations.

$$\rho c_p{\displaystyle \frac{\partial \vartheta }{\partial t}}=p+\nabla \left(\lambda \nabla \vartheta \right)$$

(1)

$$C_i\left(\zeta (t)\right){\dot{\vartheta }}_i=P_i\left(\zeta (t)\right)+{\displaystyle \sum _{\begin{array}{c}j=1\\ j\ne i\end{array}}^n\frac{{\vartheta }_j-{\vartheta }_i}{R_{ij}\left(\zeta (t)\right)}}+{\displaystyle \sum _{k=1}^m\frac{{\vartheta }_{ex,k}-{\vartheta }_i}{R_{ik}\left(\zeta (t)\right)}}$$

(2)

In Equation (1), the parameters are defined as follows: $\rho$ represents the mass density, $c_p$ denotes the specific heat capacity, $\vartheta$ is the scalar temperature field, $P$ corresponds to the heat generation at a specific point, $\lambda$ signifies the direction-dependent thermal conductivity, and $\nabla$ represents the spatial gradient operator.

In Equation (2), the variables $C_i$, $P_i$, and ${\theta }_i$ correspond to the thermal capacitance, total power dissipation, and average temperature of node i, respectively. The auxiliary temperatures, denoted by ${\theta }_{\mathrm{ex},k}$ (such as ambient air and coolant), serve as boundary conditions for the model. Additionally,$R_{ij}$ and $R_{ik}$ represent the equivalent thermal resistances between nodes, while $\zeta (t)$ is a vector related to operational conditions and time.

LPTN follows basic heat transfer laws and provides clear physical meaning with high computational efficiency [30]. However, traditional LPTN may not adequately describe systems operating at high speeds with rapid state changes and significant nonlinear coupling. Moreover, constructing an LPTN heavily depends on expert knowledge, and parameters like thermal resistance and thermal capacitance are difficult to measure accurately and may vary with external conditions. Figure 3 shows the equivalent thermal network of an LPTN.

Figure 3 illustrates the schematic of the LPTN model. In this model, $T_a$ represents the ambient temperature, which typically serves as a boundary condition. The rotor temperature, denoted by $T_r$, is influenced by the heat source power $P_r$ and the thermal capacitance $C_r$. Similarly, $T_s$ represents the stator temperature, which is associated with the heat source power $P_s$ and the thermal capacitance $C_s$.

The coolant temperature, denoted by $T_c$, acts as a boundary temperature facilitating heat dissipation. The thermal resistance $R_{ij}$ characterizes the thermal performance between nodes $i$ and $j$. Each node is connected to a corresponding thermal capacitance $C_i$, which defines the heat storage capacity of the node $i$. The heat sources $P_s$ and $P_r$ represent the internal heat losses generated within the stator and rotor, respectively.

3. Methodology

3.1. Hyperparameter Optimization

To systematically determine the optimal configuration of the proposed OLTEM model, we conducted an extensive hyperparameter optimization (HPO) process using the Optuna framework, which employs a tree-structured Parzen estimator (TPE) sampling algorithm. The optimization objective was to minimize the MSE on a dedicated validation set over a total of 300 trials. The MSE measures the average squared difference between the predicted values and the actual values and is formally defined as:

$$MSE={\displaystyle \frac{1}{N}}{\displaystyle \sum _{i=1}^N{(y_i-{\widehat{y}}_i)}^2}$$

(3)

where $y_i$ represents the true value of the $i$-th sample, ${\widehat{y}}_i$ denotes the corresponding predicted value, and $N$ is the total number of samples. A lower MSE indicates higher model accuracy.

The search space for the key hyperparameters was defined as follows:

• Learning Rate (lr): A log-uniform distribution between 1 × 10⁻⁴ and 1 × 10⁻².
• Optimizer: A categorical choice from [‘Adam’, ‘RMSprop’, ‘SGD’].
• Hidden Dimension of Power Loss Net (ploss_hidden_dim): An integer value between 64 and 128.
• Slope of Leaky ReLU (leaky_relu_slope): A uniform distribution between 0.01 and 0.3.

The HPO process converged to a robust set of hyperparameters, achieving a minimum validation MSE of 1.39 C². This result surpasses the performance reported for the original TNN model (MSE of 1.9 C² to 2.87 C²), highlighting the efficacy of the architectural enhancements in our OLTEM model. The optimal parameters and corresponding error metrics are detailed in

Table 1. Optimal hyperparameters and corresponding validation error metrics obtained from the HPO process.

Hyperparameter	Optimum
Minimum validation MSE(C2)	1.39
Learning rate	0.001937
Optimizer	RMSprop
ploss_hidden_dim	112
Leaky ReLU slope	0.2104

. All subsequent experiments in this paper utilize this optimized model configuration. The optimization process and parameter importance are further visualized in Figure 4, Figure 5 and Figure 6.

3.2. Baseline Thermal Neural Network (TNN) Architecture

The thermal neural network (TNN) combines physically interpretable thermal networks with neural networks’ nonlinear fitting abilities. This approach enables analyzing complex thermal systems without requiring exact material parameters or prior knowledge. TNN employs a discrete update scheme similar to state equations, where the temperature prediction at the next time step, denoted as $\widehat{\theta }[k+1]$, is expressed as the previous estimate $\widehat{\theta }[k]$ plus an incremental term determined by thermal parameters such as thermal conductivity $\gamma$, power dissipation $\pi$, and inverse thermal capacitance $\kappa$. The ordinary differential equation (ODE) describing the i-th thermal element can be written as:

$$\begin{array}{c}{\widehat{\theta }}_i[k+1]={\widehat{\theta }}_i[k]+T_s{\kappa }_i[k]\cdot ({\pi }_i[k]+\\ {\displaystyle \sum _{j\in \mathcal{M}\setminus i}\left({\widehat{\theta }}_j[k]-{\widehat{\theta }}_i[k]\right)}{\gamma }_{i,j}[k]\\ +{\displaystyle \sum _{j=1}^n\left({\tilde{\theta }}_j[k]-{\widehat{\theta }}_i[k]\right)}{\gamma }_{i,j}[k])\end{array}$$

(4)

These thermal parameters are no longer given by explicit physical functions. Their design is easily affected by various complicated factors, and TNN learns their hidden relationships from measured data with a feedforward neural network, capturing complex internal thermal couplings. Equation (5) shows the general form of this nonlinear mapping.

$$\begin{array}{ll}\hfill h^{(0)}[k]& ={\sigma }^{(0)}\left(W_r\widehat{\varphi }[k]+W_h^{(0)}\varphi [k]+b^{(0)}\right),\hfill \\ \hfill h^{(l)}[k]& ={\sigma }^{(l)}\left(W_h^{(l)}{h}^{(l-1)}[k]+b^{(l)}\right), \forall l>0,\hfill \\ \hfill g_{\theta }[k]& =h^{(L-1)}[k],\hfill \end{array}$$

(5)

In this framework, $h[k]$ represents the output of the neural network, $\varphi [k]$ denotes the input features, ${\sigma }^{(l)}$ is the activation function, $W$ corresponds to the weight matrix, and b signifies the bias term.

TNN is similar to a traditional RNN in that it has a time-based recurrence property, using observations at earlier steps to estimate the temperature distribution at the next step. Figure 7 illustrates the TNN structure.

The proposed model consists of two primary components:

(1)

Thermal Capacitance Estimation Network: Corresponding to the parameter $\kappa$ in Equation (3), this network is responsible for modeling the inverse thermal capacitance using end-to-end trainable constants.

(2)

Thermal Conductivity and Power Dissipation Estimation Network: Corresponding to parameters $\gamma$ and $\pi$ in Equation (3), this sub-network processes the input measurement data and temperature estimates to output the thermal conductivity and power dissipation parameters.

In Equation (4), the TNN carries out nonlinear mapping on these inputs to learn the system’s complex thermal interactions from data. Since the TNN is derived from the LPTN structure, the physical parameters it outputs after training retain interpretability. Examining the output parameters can help identify negligible thermal connections in real systems. Compared with purely black-box models, the TNN has a more precise physical basis, and it can be implemented in real applications by assigning realistic initial temperatures, making its predictions closer to real-world conditions.

3.3. State-Conditioned Squeeze-And-Excitation (SC-SE) Attention Mechanism

Standard attention mechanisms like Squeeze-and-Excitation (SE) provide an efficient approach for feature re-weighting by adaptively learning channel importance [31]. However, in physical systems like a PMSM, thermal parameters such as thermal conductances and power losses are not only dependent on external operational conditions but are also strongly correlated with the component temperatures themselves [32,33]. A generic attention mechanism that only considers input features fails to capture this critical, state-dependent physical prior.

To address this limitation, we propose a novel state-conditioned squeeze-and-excitation (SC-SE) attention mechanism. The core innovation of SC-SE is to make the attention weights conditional on both the input features and the model’s recurrent state vector θ[k] (the estimated component temperatures at the current time step k). This allows the model to learn dynamic, state-dependent relationships, such as how the influence of motor speed on thermal characteristics changes as the motor heats up.

The architecture of the SC-SE module, as illustrated in Figure 8, operates as follows:

• Squeeze: This step is identical to the standard SE block. For a given sub-network’s intermediate feature map U, a global average pooling operation Fsq(·) is applied to compress spatial information into a channel descriptor vector z.
• State-Conditioning: The channel descriptor z is concatenated with the current temperature state vector θ[k] (A). This fused vector, denoted as [z; θ[k]], (B) now contains information about both the input-driven features and the system’s internal thermal state.
• Excitation: The fused vector is fed through a small multi-layer perceptron (MLP), Fex(·, W), to learn a set of channel-wise attention weights s (C).
• Re-weight: The final output of the module is obtained by multiplying the original feature map U with the learned attention weights s.

By integrating this SC-SE module into both the thermal conductance and power loss estimation sub-networks, OLTEM can adaptively prioritize critical heat transfer paths and dominant power loss components based on the real-time operational and thermal state of the PMSM.

3.4. Enhanced Power Loss Estimation Module

In existing hybrid physics-data models, the power loss (PLOSS) sub-network is often simplistic and fails to capture the complex loss dynamics under high power density conditions. To address this, we propose an enhanced power loss estimation module, which comprises a three-stage architecture as illustrated in Figure 9.

• Deep Feature Extraction (MLP): We first employ a deep multi-layer perceptron (MLP) to capture the complex, nonlinear relationships between the input features ξ[k] (including operational conditions) and the current temperatures θ[k]. This stage produces an intermediate feature vector representing a preliminary estimation of the loss components. We utilize LeakyReLU activation functions in the hidden layers to prevent gradient saturation.
• State-Conditioned Attention: The intermediate loss features from the MLP are then fed into our proposed SC-SE attention module. This module applies state-conditioned, adaptive re-weighting to the different loss components. By using the current temperature state θ[k] as a direct conditioning signal, it allows the model to dynamically adjust the contribution of each loss type, yielding a more physically sound, re-weighted feature vector.
• Output Projection and Regularization: Finally, the re-weighted features are passed through a dropout layer for regularization, followed by a final linear layer that projects the features to the desired output dimension. A ReLU activation function is applied to ensure physically plausible, non-negative loss predictions, resulting in the final power loss vector π[k].

This multi-stage design, combining deep feature extraction with physics-guided attention, enables a far more nuanced and precise characterization of power losses.

3.5. OLTEM: A Physics-Informed Recurrent Model

Building upon the TNN baseline and integrating the enhancements from the previous sections, this study proposes the OLTEM. Its complete architecture is presented in Figure 10.

As illustrated, OLTEM operates as a physics-informed recurrent system. At each time step k, the model takes the external input features ξ[k] and the previous temperature state ϑ^[k] as inputs. These are fed into the parameter estimator, which consists of three parallel sub-networks for estimating the key physical parameters:

• Thermal Conductance Network (γ): Estimates thermal conductances, augmented with our SC-SE module.
• Inverse Thermal Capacitance Network (κ): A simpler MLP that learns inverse thermal capacitances.
• Enhanced Power Loss Network (π): Our enhanced, multi-stage module for accurately estimating power losses, also augmented with the SC-SE module.

The estimated parameters γ[k], κ[k], and π[k] are then supplied to the physics-informed state update block. This core component uses the discretized LPTN governing equation ϑ^[k + 1] = f(ϑ^[k], γ, κ, π) to calculate the temperature state for the next time step. The resulting next temp state ϑ^[k + 1] is fed back through a time delay (z⁻¹) to become the previous temp state for the next iteration, completing the recurrent loop. This architecture ensures the model’s predictions are both data-driven and constrained by the fundamental principles of heat transfer.

The entire process of applying this model follows a systematic workflow. The workflow is summarized in Figure 11. This flowchart outlines the process of training and evaluating the motor temperature data model, including data loading, preprocessing, training/testing, and visualization.

(a)

Data Preprocessing: Initially, the raw data from the publicly available “Electric Motor Temperature” dataset on the Kaggle platform is cleaned, outliers are removed, and normalization is performed. To effectively capture the trends in motor operating conditions, two additional features are engineered from time-domain signals during the feature engineering phase: the current vector magnitude ($i_s$) and voltage vector magnitude ($u_s$) constructed from current and voltage signals, respectively. The dataset is then split into training and validation sets using profile id as the splitting criterion, with a ratio of 8:2. Additionally, a generalization set comprising three profile ids is retained within the overall dataset to ultimately assess the model’s ability to generalize.

(b)

Model Structure Setup: The model represents the temperature of each motor component as a node by introducing learnable inverse heat capacity parameters and a thermal conductance network, which simulate the system’s thermal dynamics through thermodynamic equations. The power loss module uses a deep network, physical constraints, and a dynamic load adjustment factor to handle complex conditions. Additionally, the SE module further highlights the significance of different input channels by assigning weights adaptively to each feature channel.

4. Experiments

4.1. Dataset

This study utilizes the electric motor temperature dataset from Kaggle, which contains sensor readings collected at the LEA Laboratory of Paderborn University in Germany. The data were collected from a prototype PMSM produced by a German original equipment manufacturer (OEM). The dataset includes 13 feature columns, including the d/q components of voltage, motor speed, torque, and temperature readings from multiple sensors, covering a total of 185 h of operation. Each measurement session is labeled by a profile id, lasting between 1 and 6 h.

Table 2. Dataset parameter descriptions.

Parameter Name	Description	Unit
u_q	Voltage q-axis component	V
u_d	Voltage d-axis component	V
coolant	Coolant temperature	°C
stator_yoke	Stator yoke temperature	°C
stator_tooth	Stator tooth temperature	°C
stator_winding	Stator winding temperature	°C
motor_speed	Motor speed	rpm
i_d	Current d-axis component	A
i_q	Current q-axis component	A
pm	Permanent magnet temperature	°C
ambient	Ambient temperature	°C
torque	Torque generated by the current	N·m

outlines the main feature definitions:

All training and inference processes were conducted on an Intel 16-core AMD EPYC 9354 processor, Intel, Santa Clara, CA, USA, and an NVIDIA RTX 4090 GPU, NVIDIA, Santa Clara, CA, USA.

4.2. Evaluation Metrics and Baseline

To evaluate the model’s performance, this study employs two key metrics. The primary metric is the MSE, which was previously defined in Equation (3). MSE provides a measure of the overall prediction fidelity. To assess the model’s performance under worst-case scenarios, we also utilize the maximum absolute error (Max.Abs).

The maximum absolute error reflects the largest deviation in model predictions, expressed by:

$$Max.Abs={\max }_{i=1}^n\left|y_i-{\widehat{y}}_i\right|$$

(6)

where $y_i$ denotes the true value of the $i$-th sample, ${\widehat{y}}_i$ represents the predicted value of the $i$-th sample, and $n$ is the total number of samples. A lower $Max.Abs$ value indicates that the model has a smaller maximum prediction error.

These two metrics together provide a comprehensive evaluation of the model’s accuracy.

4.3. Experimental Evaluation and Comparative Analysis

To comprehensively evaluate the performance of the proposed OLTEM model, we conducted a series of comparative experiments. The model was benchmarked against three distinct categories: (1) the original thermal neural network (TNN) as a physics-informed baseline; (2) a representative state-of-the-art deep learning model, a hybrid convolutional neural network–recurrent neural network (CNN-RNN), to assess performance against data-driven approaches; and (3) ablated versions of our own model to validate the contribution of its key components. The CNN-RNN model, a common architecture for time-series forecasting, employs a CNN layer for feature extraction from input signals, followed by an RNN layer (e.g., LSTM or GRU) to capture temporal dependencies. All models were tested on the four main motor components—the permanent magnet, the stator yoke, the stator tooth, and the stator winding—to verify their effectiveness and applicability. The permanent magnet is mounted on the motor’s rotor and supplies the main magnetic field. Excessive temperature can degrade its magnetic properties. The stator yoke is the external structure of the stator, bearing magnetic flux and supporting the stator core. Its temperature serves as an indicator of overall heat generation and cooling efficiency. The stator tooth guides magnetic flux into the stator winding, so its temperature largely depends on current and flux density. The stator winding generates the alternating magnetic field to drive the rotor, and its temperature is an indicator of load levels and operating efficiency.

Figure 12 shows the comparison between predicted temperatures (blue lines) and measured temperatures (green lines) in the generalization set for four key motor components: pm (permanent magnet), stator yoke, stator tooth, and stator winding.

As illustrated in Figure 12, the predicted temperatures closely follow the measured ground truth, even during periods of rapid thermal change. The model’s performance on the generalization set is quantitatively strong. As an overall performance metric, the MSE for all four components is below 4.0 °C². To assess worst-case performance, the maximum absolute error, which represents the largest single-point deviation, remains between 4.34 °C and 8.21 °C. Specifically, the permanent magnet shows an MSE of 3.63 °C² and a maximum absolute error of 5.83 °C. The performance for the other components is as follows: stator yoke (MSE: 0.84 °C², Max.Abs: 4.34 °C), stator tooth (MSE: 1.94 °C², Max.Abs: 6.02 °C), and stator winding (MSE: 3.74 °C², Max.Abs: 8.21 °C). These results demonstrate the model’s robust generalization capability on unseen operating profiles.

Figure 13 displays the distribution of squared errors per sample in both the training and test sets. Most of the single-time-step squared errors remain below 10 °C², which is adequate for typical engineering needs as it corresponds to an error of approximately 3 °C. When switching between operating states, the error occasionally spikes before rapidly returning to a lower level, indicating that the model can quickly correct itself after abrupt changes in the motor’s operating conditions. It is worth noting that the ‘stator winding’ typically exhibits the largest error spikes, especially under transient operating conditions. This is likely due to the fact that its temperature is directly affected by copper losses, which change rapidly and nonlinearly with load variations, making it the most challenging component for dynamic prediction.

Figure 14 further shows the CNN-RNN baseline model predictions. While the CNN-RNN captures the overall trends, its MSE and maximum absolute errors are higher across all four components compared with OLTEM, confirming the superiority of the proposed model.

Figure 15 shows a heatmap of correlations between input features and target temperatures, including voltage ($u_q,u_d,u_s$), current ($i_d,i_q,i_s$), motor speed, torque, coolant, ambient, and core component temperatures (pm), stator yoke, stator tooth, and stator winding. The deeper the red, the stronger the positive correlation, while the deeper blue indicates a stronger negative correlation. The diagonal entries are self-correlations and are thus equal to 1.

From Figure 15, features like u_s (voltage amplitude) and motor speed have a high positive correlation with all temperature points. As voltage and speed increase, the stator and permanent magnet temperatures tend to rise. This aligns with the mechanism of power loss, where higher voltage and speed raise the motor’s thermal load. Additionally, coolant and ambient also demonstrate higher positive correlations with stator and pm temperatures, which is consistent with the fact that changes in coolant and ambient conditions significantly affect the motor’s thermal state. The correlation coefficients among stator yoke, stator tooth, and stator winding exceed 0.9, as heat flow within the stator components is strongly interconnected. This underlines why the thermal network model integrates concepts like “thermal conductance” and “power loss,” which help represent heat paths inside the stator. $i_d$ and $i_s$ exhibit a significant negative correlation, indicating that under actual operating conditions, these two variables may have an inverse relationship.

This heatmap of feature correlations offers valuable insights for feature engineering and model design by highlighting which features have strong relationships with the motor’s thermal dynamics. It also helps analyze how negatively correlated features affect the system’s internal mechanisms, guiding model refinements.

To rigorously evaluate the contribution of each proposed component, we conducted two sets of ablation experiments. The performance of these model variants on the generalization set is detailed in

Table 3. Performance comparison across models.

	pm		stator_yoke		stator_tooth		stator_winding
	$\mathbf{M}\mathbf{S}\mathbf{E}$ (°C2)	$\left|\right|\mathbf{e}\left|\right|$ (°C)	$\mathbf{M}\mathbf{S}\mathbf{E}$ (°C2)	$\left|\right|\mathbf{e}\left|\right|$ (°C)	$\mathbf{M}\mathbf{S}\mathbf{E}$ (°C2)	$\left|\right|\mathbf{e}\left|\right|$ (°C)	$\mathbf{M}\mathbf{S}\mathbf{E}$ (°C2)	$\left|\right|\mathbf{e}\left|\right|$ (°C)
TNN	5.16	6.6	2.32	6.1	3.38	6.9	6.38	9.8
CNN-RNN	4.36	5.3	1.51	6.2	2.15	5.2	5.22	9.0
OLTEM–SC-SE	3.94	6.4	1.52	4.4	2.4	6.4	4.08	7.9
OLTEM–Enhanced Power Loss Estimation Module	4.58	6.3	1.99	5.1	3.04	6.2	4.37	8.6
OLTEM	3.77	5.7	0.91	5.4	2.11	7.3	3.31	9.1

.

First, we analyze the model variant “OLTEM–SE Module,” which in our new framework represents the removal of our proposed SC-SE module, leaving only the enhanced power loss module, as illustrated in Figure 16. Compared to the full OLTEM model, this version exhibits more significant lag and deviation during periods of rapid temperature change. This result validates our core hypothesis: an advanced power loss model alone is insufficient. Without the ability to dynamically adjust the importance of thermal paths and loss features based on the motor’s current thermal state θ[k], the model struggles to accurately capture the complex, temperature-dependent heat transfer processes. This highlights the critical role of our proposed state-conditioned attention.

Second, we examine the “OLTEM–Enhanced Power Loss Estimation Module” variant, where the refined power loss module was removed, and only the SC-SE attention mechanism was retained, as illustrated in Figure 17. While its performance degrades compared to the full OLTEM, it still significantly outperforms the baseline TNN model. This indicates that even with a cruder estimation of the heat sources, our proposed SC-SE module, by virtue of its ability to re-weight features based on the temperature state, can still effectively capture the system’s primary thermal dynamics.

Finally, the complete OLTEM model achieves the best performance (Table 3), demonstrating a powerful synergistic effect between its two key innovations. The enhanced power loss module provides a more accurate heat source input, while the SC-SE module, guided by physical insight, intelligently modulates the influence of these heat sources and thermal paths based on the system’s real-time thermal state. This physics-guided attention mechanism is crucial for achieving the highest prediction performance.

The comprehensive performance metrics for all evaluated models are presented in Table 3. The results clearly indicate that our proposed OLTEM model achieves the best overall performance, recording the lowest MSE across all four monitored components. Notably, when compared to the strong data-driven CNN-RNN baseline, OLTEM demonstrates its superior stability and accuracy, particularly for the stator yoke (0.91 vs. 1.51 °C²) and stator winding (3.31 vs. 5.22 °C²). While the CNN-RNN shows competitive maximum absolute errors in some cases, OLTEM’s consistent advantage in MSE underscores its robustness for reliable, long-term temperature tracking.

Furthermore, the ablation study results within Table 3 validate the effectiveness of our proposed innovations. The removal of the SC-SE attention mechanism or the enhanced power loss module leads to a discernible degradation in performance compared to the full OLTEM model. This confirms that the synergistic effect of physics-guided attention and a more accurate heat source estimation is crucial for achieving the highest prediction fidelity.

5. Conclusions and Future Work

Building on the TNN framework by Kirchgässner et al., this study introduces OLTEM, an improved real-time temperature prediction model. While preserving physical interpretability, OLTEM introduces two significant improvements: a novel state-conditioned squeeze-and-excitation (SC-SE) attention mechanism, and an enhanced power loss estimation component. By making the attention process conditional on the system’s real-time thermal state, our model successfully embeds a crucial physical prior into its LPTN-based architecture, enabling it to effectively handle the nonlinear, temperature-dependent thermal coupling in high power density conditions. In the future, combining more detailed physical prior knowledge with neural network techniques may further improve prediction performance and physical clarity. Although OLTEM has demonstrated promising results for temperature prediction, its generalization to other motor types or cross-domain temperature tasks requires further study. Future work can include applying transfer learning and domain adaptation across more varied datasets and motor designs, aiming to build models with better generalization. Additionally, offering transitional information or incorporating extra state variables during sudden changes may help suppress short-term error spikes in abrupt scenarios. Finally, when deploying the model in industrial environments, balancing computational costs and hardware constraints is a pressing issue so that OLTEM can run reliably on resource-limited embedded devices. By addressing these considerations, the real-time performance and reliability of OLTEM in complex industrial settings can be significantly enhanced, providing more effective support for upcoming thermal management systems.