3.1 Reduced Order Thermal Modeling

The thermal behavior of the electric machine is represented using a lumped parameter thermal network in which the stator and rotor cores are modeled as distinct thermal nodes. Each node is characterized by an equivalent thermal capacitance that represents its heat storage capability, while thermal resistances describe heat transfer between components and to the ambient environment. This formulation allows the complex three dimensional heat conduction problem to be approximated by a low order dynamic system, significantly reducing computational complexity compared to full FEM analysis.

The stator and rotor nodes (T_s and T_r) are thermally coupled through an inter component thermal resistance (R_sr), accounting for conductive heat transfer across structural interfaces. In addition, heat exchange between each core node (R_sg, R_rg, C_s, C_r) and the ambient environment (T_amb) is modeled through equivalent thermal resistances (R_ra) representing convection and overall cooling effects present in the dynamometer setup. This structure ensures that the principal heat flow paths influencing core temperatures are explicitly reflected in the model.

Fig. 1. Lumped parameter thermal network

3.2 ODE-Based Thermal Dynamics Formulation

In dynamometer operation, the dominant heat sources within the machine are closely related to electrical loading conditions. To incorporate this effect without relying on detailed electromagnetic loss calculations, the internal heat generation is represented by a current dependent loss formulation. The loss term is parameterized as a function of the measured RMS current, allowing the model to capture the primary influence of copper losses as well as additional loss components such as core and stray losses in an aggregated manner. This approach enables the thermal model to remain compact while preserving the essential coupling between electrical excitation and thermal response.

where I(t) is the RMS phase current [A], a is the quadratic loss coefficient representing copper (Joule) losses [W/A²], b is the linear loss coefficient representing iron and stray load losses [W/A] and c is the constant loss term representing no-load/core losses [W].

The motor losses caused by the current are distributed to the stator and rotor nodes as shown in Equations (2) and (3), respectively.

where P_s is the power dissipated in the stator core [W] and P_r is the power dissipated in the rotor core [W].

Based on the thermal network and loss modeling described above, the core thermal dynamics are expressed as a set of coupled ordinary differential equations. These equations describe the temporal evolution of stator and rotor core temperatures as functions of inter node heat transfer, ambient cooling, and internal heat generation. Instead of solving the equations numerically in a conventional manner, the proposed approach embeds these governing dynamics into a physics-informed neural network framework. The neural network is trained to represent the temperature trajectories, while the thermal resistances, capacitances, and loss related coefficients are treated as learnable physical parameters.

where T_s is the stator core temperature [°C], T_r is the rotor core temperature [°C], T_amb is the ambient temperature [°C], C_s is the thermal capacitance of the stator core [J/K], C_r is the thermal capacitance of the rotor core [J/K], R_sr is the thermal resistance between stator and rotor cores [K/W], R_ss is the thermal resistance between stator core and ambient [K/W], R_ra is the thermal resistance between rotor core and ambient [K/W], $\eta_s$ is the fraction of total losses dissipated in the stator core [-], $\eta_r$ is the fraction of total losses dissipated in the rotor core [-], P_loss(I) is the total electrical loss power generated in the machine [W], dT_s/dt denotes the time derivative of stator temperature [K/s] and dT_r/dt denotes the time derivative of rotor temperature [K/s]. Temperatures are expressed in °C, while thermal capacitances and thermal resistances retain the conventional units of J/K and K/W, respectively, since temperature differences are identical in °C and K.

By enforcing the thermal dynamic residuals during training, the model maintains consistency with the underlying heat transfer physics while benefiting from the approximation flexibility of neural networks. This integration forms a reduced order thermal digital twin capable of reproducing the essential behavior of high fidelity FEM models at a fraction of the computational cost.

where R_s, R_r are the physics residuals of stator and rotor thermal ODEs [-], T_s ^NN, T_r ^NN are neural network predicted stator and rotor temperatures [°C], all other parameters are identical to those defined in equations (1)-(5).

3.3 Physics-Informed Neural Network Integration

A fully connected feedforward neural network is employed to represent the temporal evolution of stator and rotor core temperatures. The network takes time, RMS current, and ambient temperature as inputs and outputs the predicted stator and rotor core temperatures. By using time as a continuous input, the model captures the dynamic nature of thermal behavior rather than relying on discrete step prediction. The network architecture consists of multiple hidden layers with nonlinear activation functions, providing sufficient capacity to approximate the nonlinear relationship between electrical loading and thermal response.

Unlike purely data driven models, the proposed framework incorporates physical knowledge through a physics-informed loss function as shown in Fig. 2. The total loss consists of two primary components: a data loss and a physics loss. The data loss measures the discrepancy between the predicted core temperatures and reference temperatures obtained from FEM simulations. The physics loss enforces consistency with the governing thermal dynamics by penalizing the residuals of the reduced order thermal equations. These residuals are computed using automatic differentiation to obtain temperature time derivatives from the neural network outputs.

This dual loss formulation ensures that the model not only fits the observed temperature trajectories but also adheres to the physical principles of heat transfer embedded in the thermal network. As a result, the model exhibits improved robustness and interpretability compared to conventional black-box neural networks.

where T_NN is the predicted temperature vector, T_meas is the measured temperature vector, $\lambda_{data}$ is the data fitting weight, $\lambda_{phys}$ is the physics constraint weight, $\lambda_{reg}$ is the regularization weight and $\theta$ denotes neural network parameters, including neural network weights and the physical coefficients {C_s, C_r, R_sr, R_sa, R_ra, $\eta_s$, $\eta_r$, a, b, c}. In addition to neural network weights, key thermal parameters including thermal resistances, thermal capacitances, and coefficients of the current dependent loss model are treated as trainable variables. To maintain physical plausibility, these parameters are constrained to positive values through appropriate parameterization. Joint optimization of these parameters with the neural network enables the model to perform system identification of the underlying thermal characteristics directly from data.

To ensure physical consistency, all learnable physical parameters (e.g., thermal capacitances and resistances) are constrained to be positive. Instead of imposing hard constraints, we adopt a soft constraint using the Softplus function, which guarantees positivity while maintaining smooth gradients for stable optimization. Specifically, each physical parameter $\theta$ is reparameterized as:

where $\tilde{\theta}$ is an unconstrained variable learned during training. This formulation ensures that $\theta$>0 at all times, which is consistent with the physical interpretation of thermal capacitance and resistance. Compared to hard clipping or ReLU based constraints, the Softplus function provides smoother gradients near zero, improving convergence stability and preventing vanishing gradient issues.

As a result, both neural network parameters and thermal physical coefficients are reliably identified while ensuring accurate satisfaction of the physics constraints.

Fig. 2. Schematic diagram of physics-informed neural network

4.1 FEM-Based Thermal Simulation of the Motor System

High fidelity thermal simulations based on the finite element method (FEM) are conducted to obtain reference core temperature data. The FEM model represents the detailed geometry and material properties of the motor, including conductive heat transfer within the stator and rotor cores, as well as boundary conditions associated with cooling and ambient interaction. Electrical loading conditions are varied to reflect different dynamometer operating states, resulting in time series temperature profiles for both stator and rotor cores. These FEM derived temperature trajectories serve as ground truth data for training and validation.

This study utilized electromagnetic analysis data from a 7.5 kW motor, with its parameters summarized in Table 2. For ease of analysis, components unrelated to electromagnetics and heat were omitted. The external geometry of motor are shown in Fig. 3.

Fig. 3. 3D model of a 7.5 kW class motor

Thermal analysis was conducted by assigning material properties and boundary conditions to the motor components, including the armature windings, field body, and armature core body. To replicate the actual experimental environment of the motor, simulations were performed under various operating conditions, including different rotational speeds and external load levels. For the motor thermal analysis, electrical heat sources generated in the electromagnetic windings and losses in each core component were modeled and applied as heat inputs to the corresponding motor parts. Heat transfer mechanisms, including conduction and radiation, were considered in the simulation. The resulting cross sectional temperature distribution of the motor is shown in Fig. 4.

Fig. 4. Motor thermal analysis results (temperature distribution)

To validate the motor FEM analysis, measurement data from an actual dynamometer system were utilized as shown in Fig. 5. By comparing the FEM simulation results with the measured data from the real system, the reliability of the FEM analysis can be verified. The motor dynamometer system consists of the test motor, a power supply unit, thermocouples, various sensors for noise and vibration measurement, a CompactRIO system for data acquisition, a test controller, and a load motor controller.

Fig. 5. Configuration of the motor-generator(M-G) set

To employ the simulation results of the motor electromagnetic and thermal analysis models as training datasets for the PINN under various operating conditions, the FEM analysis results were compared with measurement data obtained from a motor dynamometer test that emulated real operating conditions as shown in Fig. 6 and 7.

Fig. 6. Comparison of experimental and FEM-simulated stator housing temperature

Fig. 7. Comparison of experimental and FEM-simulated rotor housing temperature

In this comparison, the simulation parameters and dynamometer operating conditions were configured identically. Accordingly, the stator and rotor core temperatures were estimated through thermal analysis, enabling the acquisition of internal core temperature data that cannot be directly measured in practice. These otherwise inaccessible temperature distributions were obtained using FEM based simulations.

The thermal responses are generated under representative dynamometer scenarios that include changes in electrical loading and associated current levels. The RMS current is used as the primary electrical input variable influencing internal heat generation. By varying current profiles over time, the dataset captures both steady-state and transient thermal behaviors. Ambient temperature is assumed constant in the present study to focus on internal thermal dynamics, while the modeling framework remains applicable to varying ambient conditions.

The FEM generated time series data are organized into structured datasets containing time, RMS current, stator core temperature, and rotor core temperature. To ensure consistency in training, input variables are normalized, while temperature values retain their physical scale. The dataset is partitioned chronologically into training and validation subsets, enabling evaluation of model generalization to unseen temporal segments. This setup reflects realistic deployment scenarios in which the model must predict future thermal behavior based on learned dynamics.

Fig. 8. Dataset for the PINN-based thermal model

4.2 PINN-Based Core Temperature Prediction

The inputs to the PINN-based core temperature prediction model are time, RMS current, and ambient temperature, while the outputs are the stator and rotor core temperatures. The detailed network architecture is summarized in Table 3. The neural network is designed based on a multi layer perceptron (MLP) architecture. The hyperparameters related to the network structure, including the number of hidden layers, the number of hidden units, and the activation function, were selected to adequately capture the smooth and continuous temporal response characteristics of the thermal system. To ensure numerical stability in the calculation of ODE residuals involving differential operations, the tanh activation function which is continuously differentiable was adopted.

For physics-informed learning, the ODE based PINN physical parameters were incorporated into the neural network training process. Since this study focuses on designing a physics-informed neural network for the thermal model, thermal related parameters were utilized during training, as summarized in Table 4. The identified parameters were compared with typical ranges reported in the literature for lumped parameter thermal and electrical models of induction machines. Thermal capacitances (C_s, C_r) are generally determined based on nodal heat capacities in lumped thermal networks, while thermal coupling coefficients ($\eta_s$, $\eta_r$) represent convection and conduction efficiency between nodes ^[23]. Electrical resistances (R_sr, R_sa, R_ra) fall within commonly reported ranges for stator and rotor windings and are known to vary with temperature ^[24]. Furthermore, FEM based electrothermal studies confirm that such parameters can be reduced to equivalent lumped representations while preserving physical fidelity.

The hyperparameters and optimization parameters used for training the designed neural network were selected as summarized in Table 5. Optimization was performed using the Adam algorithm, a stochastic first order gradient based method, to promote stable convergence in the multi scale problem where both physical parameters and neural network weights are learned simultaneously. The learning rate $\alpha$ in Adam, which controls the magnitude of parameter updates, was set to enable gradual convergence without excessively distorting the smooth temporal response characteristics of the thermal system. The batch size, which influences the variance of gradient estimation in mini-batch training, was chosen to balance learning stability and generalization performance. Excessively small batch sizes can introduce noise, whereas overly large batch sizes may degrade generalization. Therefore, the selected value preserves the temporal continuity of thermal dynamic data while ensuring stable training. Furthermore, the weighting coefficients of the loss function, $\lambda_{data}$ and $\lambda_{phys}$, were adjusted to maintain a balance between data fitting and physical equation enforcement, alleviating the common issue of scale imbalance among loss terms in PINN training. The regularization coefficient $\lambda_{reg}$ was introduced to prevent excessive growth of network weights, thereby reducing overfitting and constraining the thermal physical parameters from diverging to nonphysical values. These optimization settings were designed to enable stable learning of the slow, low-frequency dynamics of the thermal system while ensuring that the solution satisfies the underlying physical constraints.

The predicted stator and rotor core temperatures are compared with reference results obtained from FEM simulations as shown in Fig. 9 and 10. The proposed model accurately reproduces both transient and steady-state thermal responses under varying current profiles. The temperature trajectories exhibit agreement with FEM results, with low prediction errors across the validation dataset.

Fig. 9. Comparison of predicted and FEM-simulated stator core temperature

Fig. 10. Comparison of predicted and FEM-simulated rotor core temperature

The training loss history in Fig. 11 shows smooth convergence without oscillatory behavior.

Fig. 11. Training performance of the PINN-based thermal model

The model learned to satisfy the governing thermal ODEs while fitting the reference data. Furthermore, The model maintains stable performance when evaluated on unseen temporal segments of the dataset, indicating good generalization to operating conditions not directly encountered during training. The inclusion of physics-based constraints prevents overfitting and ensures that predictions remain physically plausible even beyond the training domain. This robustness is essential for deployment in practical dynamometer environments where operating conditions may vary.

Table 6 presents the quantitative evaluation of the proposed PINN model. The training dataset yielded an RMSE of 2.99 °C and an MAE of 1.00 °C, corresponding to a normalized RMSE (NRMSE) of 6.0% of the temperature range. In particular, the majority of prediction errors fall within a ±5 °C band, indicating reliable temperature tracking performance for practical thermal monitoring applications. These results confirm that the model successfully captures the dominant thermal dynamics of the motor core under the training conditions.

In the validation dataset, the RMSE of the T_s ^NN and T_r ^NN increased moderately to 5.416 °C and 6.061 °C, which is primarily attributed to the limited reproduction of thermal inertia effects in the predicted temperature rise curves. Because motor core temperature dynamics are governed by thermal capacitance and delayed heat diffusion, minor phase discrepancies in the transient response can produce larger instantaneous errors during rapid temperature transitions. Nevertheless, most predictions remain within an acceptable temperature error range for engineering-level monitoring. When considering the overall dataset, the T_s ^NN model achieved an RMSE of 3.503 °C, MAE of 1.102 °C, and NRMSE of 7.06%, while the TrNN model achieved an RMSE of 3.615 °C, MAE of 1.216 °C, and NRMSE of 7.76%. These results demonstrate that the proposed PINN-based models effectively capture the dominant thermal behavior of both stator and rotor, while also highlighting the need for improved transient thermal inertia modeling to further enhance generalization performance.

To further assess the generalization capability of the proposed model, additional experiments were conducted using different training/validation split ratios of 9:1, 8:2, and 7:3. The corresponding evaluation results are summarized in Table 7.

For the 8:2 and 7:3 split cases, the validation RMSE consistently remained higher than the training RMSE, while exhibiting similar performance levels and consistent error trends across both configurations. In contrast, the 9:1 split showed a relatively lower validation RMSE compared to the training RMSE, which may indicate a biased evaluation due to the high similarity between training and validation datasets. Therefore, the 9:1 split was excluded from further consideration. Based on these results, the 8:2 split was selected as a representative configuration, providing a balanced trade-off between training data availability and validation reliability. Nevertheless, the validation error remained relatively higher than the training error. This discrepancy is attributed to the thermal inertia induced dynamic response delay and nonlinear behavior during operating condition transitions, as discussed in the manuscript. In particular, the validation dataset contains a higher proportion of transient regimes with larger temperature variation rates compared to the training dataset, which leads to increased prediction error.

From a practical perspective, the acceptability of the prediction error should be evaluated with respect to industrial thermal protection criteria. According to IEC 60034-1, temperature limits of electrical machines are defined based on insulation classes (e.g., Class B: 130 °C, Class F: 155 °C, Class H: 180 °C), and thermal protection systems are typically designed with safety margins of approximately 10-15 °C to account for sensor uncertainty, environmental variations, and modeling inaccuracies. In this context, the observed validation RMSE of 5.29 °C remains within a practically acceptable range, as it is sufficiently smaller than the typical thermal protection margins and does not significantly affect the detection of critical temperature thresholds. Therefore, the proposed model is considered to provide adequate accuracy for real-time thermal monitoring and over temperature protection applications. Future work may further improve prediction accuracy by incorporating additional thermal states and environmental variations.

To quantitatively evaluate the effectiveness of the proposed PINN framework, a comparative analysis was conducted against a purely data-driven neural network called vanilla MLP and a conventional LPTN model under identical datasets and training conditions. However, training the vanilla MLP with the same number of epochs as the PINN resulted in severe overfitting. To mitigate this issue and ensure a fair comparison, the vanilla MLP was instead trained with 50 epochs and a batch size of 10.

Table 8 summarizes the performance metrics, including RMSE, MAE, and NRMSE, for all models. The results clearly show that the proposed PINN model achieves the lowest error across all evaluation metrics.

The vanilla MLP model, while flexible in capturing nonlinear relationships, exhibits relatively higher prediction errors due to the lack of physical constraints, which can lead to overfitting and reduced generalization performance. On the other hand, the LPTN model reflects physical interpretability but shows limited accuracy due to its simplified structure and inability to fully capture complex nonlinear thermal dynamics. In contrast, the proposed PINN framework effectively integrates physical laws into the learning process, enabling both improved accuracy and enhanced generalization. The incorporation of physics-based constraints reduces the feasible solution space and guides the model toward physically consistent solutions, thereby improving prediction performance. These results demonstrate that embedding physical knowledge into neural networks provides a clear advantage over both purely data-driven and purely physics-based approaches.

4.3 Discussion

A major advantage of the proposed framework is the substantial reduction in computational cost. While FEM based thermal simulations require intensive numerical computation and are unsuitable for real-time use, the trained physics-informed neural network performs temperature prediction through direct function evaluation. This leads to a drastic decrease in computation time, enabling near instantaneous inference. The speed-up achieved makes the model suitable for real-time thermal monitoring, online performance assessment, and potential integration into control or protection systems.

Table 9 presents the quantitative comparison of computation time between the FEM simulation and the proposed PINN model. The FEM simulation requires approximately 3600 seconds, whereas the PINN inference takes only 0.012 seconds. This corresponds to a speedup of approximately 3×10⁵ times, demonstrating a significant computational advantage of the proposed approach.

An important feature of the proposed approach is the identification of physically interpretable thermal parameters. The learned thermal resistances and capacitances exhibit consistent trends with expected thermal characteristics of motor components. Similarly, the coefficients associated with the current dependent loss model reflect the relationship between electrical loading and internal heat generation. This interpretability distinguishes the proposed framework from purely black box neural network models and supports its use as a digital thermal twin.

When input conditions change, FEM based models typically require reanalysis under the new operating conditions, which limits their applicability in fast response scenarios. In contrast, the proposed ODE-based PINN model embeds the governing thermal dynamics directly into the model structure, enabling immediate temperature prediction without additional simulation or retraining. Unlike purely data driven empirical models, the physics-informed nature of the proposed approach provides improved generalization capability, making it particularly advantageous for rapid prediction and computational efficiency under varying operating conditions.

The proposed model employs a simplified 2 node LPTN consisting of stator and rotor nodes, along with the assumption of constant ambient temperature. While this simplification enables efficient parameter identification and integration with the PINN framework, it inevitably introduces certain limitations in representing detailed thermal dynamics. First, the exclusion of additional thermal nodes, such as winding and permanent magnet temperatures, is primarily motivated by model tractability and data availability. Incorporating these components would significantly increase model complexity and the number of parameters to be identified, which may reduce training stability and require additional sensing or high fidelity simulation data. Therefore, the current model focuses on capturing the dominant thermal behavior through stator–rotor interactions. The 2 node LPTN is generally valid for scenarios where global thermal dynamics are of primary interest, such as overall temperature trends and system level thermal responses. However, it may have limited accuracy in capturing localized temperature variations, especially in components with strong thermal gradients, such as windings or permanent magnets. In addition, the assumption of constant ambient temperature simplifies the boundary conditions but does not fully reflect realistic operating environments where ambient conditions may vary over time. Variations in ambient temperature can affect heat dissipation and, consequently, prediction accuracy. Despite these limitations, the proposed PINN framework can be readily extended to incorporate additional thermal nodes and time varying boundary conditions. Future work will focus on extending the model to multi node thermal networks and incorporating dynamic ambient temperature profiles to improve modeling fidelity and robustness.

Although the proposed method has been validated using a single 7.5 kW IPM motor, this study primarily aims to demonstrate the feasibility and effectiveness of the proposed PINN based framework rather than providing exhaustive validation across all motor types. The proposed approach is fundamentally based on general physical principles, such as thermal dynamics and loss mechanisms, which are commonly shared across different motor ratings and types. Therefore, the framework itself is not inherently limited to a specific motor configuration. However, it is acknowledged that the identified parameters and model performance may vary depending on motor capacity, topology (e.g., IPM, SPMSM, induction motor), and cooling conditions. As such, additional validation across diverse motor types and operating conditions is necessary to fully establish the general applicability of the proposed method. We will focus on extending the proposed framework to a broader range of motor systems, including different power ratings and motor topologies, as well as incorporating transfer learning or parameter adaptation techniques to improve scalability and generalization performance.

Also, the ambient temperature is assumed to be constant to focus on the internal thermal dynamics of the motor and to simplify the model structure. While this assumption is reasonable under controlled experimental conditions, variations in ambient temperature can influence heat dissipation through convection, thereby affecting both the thermal equilibrium and transient response of the system. Consequently, neglecting ambient temperature fluctuations may lead to deviations in temperature prediction, particularly during long term operation or under dynamically changing environmental conditions.

Despite this limitation, the proposed PINN framework can be readily extended to incorporate time-varying ambient temperature as an additional input or boundary condition. For instance, the ambient temperature can be modeled as a time-dependent variable $T_{amb}(t)$, enabling the model to capture dynamic environmental effects. Future work will focus on integrating real-time ambient temperature measurements and extending the framework to account for environmental variability, thereby enhancing prediction accuracy and robustness in practical applications.