Thermo–Electromagnetic Co-Simulation of Electric Motor Cooling Systems Using Particle Swarm Optimization

Driven by the rapid development of new energy vehicles and intelligent industrial manufacturing, electric motors have been advancing toward higher efficiency and increased compactness [1, 2], resulting in continuously rising power density and increasingly pronounced thermo–electromagnetic coupling effects. The elevation of winding temperature induces an increase in electrical resistivity [3], which in turn intensifies electromagnetic losses and forms a positive thermo–electromagnetic feedback loop [4, 5], substantially elevating the risk of thermal failure. Consequently, an efficient thermal management system has become a fundamental prerequisite for ensuring operational reliability and service life of electric motors [6]. As the primary carrier of thermal management, the design quality of the cooling system directly determines motor power density, operational efficiency, and durability [7, 8]. Conventional experience-based design approaches or optimization methods relying on a single physical field are unable to simultaneously satisfy multi-objective requirements such as temperature regulation, energy efficiency, and structural compactness [9-11], and therefore no longer meet the demands of modern electric motor design.

Thermo–electromagnetic co-simulation techniques provide an effective means for accurately characterizing coupling effects. High-fidelity simulations are capable of precisely resolving internal loss distributions and temperature field characteristics within electric motors; however, their prohibitive computational cost severely limits applicability in multi-parameter, multi-objective optimization scenarios requiring extensive iterations. Intelligent optimization algorithms—particularly PSO—have been widely adopted in motor design optimization due to their simple structure, rapid convergence, and ease of engineering implementation [12, 13]. Nevertheless, existing PSO variants fail to sufficiently accommodate the complex physical characteristics inherent to thermo–electromagnetic coupling. As a result, optimization processes lack explicit physical guidance, are prone to convergence toward designs exhibiting local thermal concentration, and struggle to achieve a coordinated improvement in both accuracy and computational efficiency [14, 15]. Accordingly, the development of a synergistic design methodology that integrates the physical fidelity of simulation with the efficiency of intelligent optimization has emerged as a critical pathway for addressing thermo–electromagnetic co-optimization challenges in electric motor cooling systems.

In the field of thermo–electromagnetic co-simulation of electric motors, extensive research efforts have been undertaken. With respect to high-fidelity simulation, multiphysics coupling techniques based on the finite element method (FEM) and finite volume method (FVM)—including electromagnetic–fluid–thermal coupled simulations—have become the mainstream technical approach for accurately analyzing thermo–electromagnetic coupling phenomena [16, 17]. However, the inherent limitation of low computational efficiency has remained largely unresolved. To balance accuracy and efficiency, surrogate model–assisted simulation techniques have been rapidly developed. Models such as artificial neural networks (ANNs), Gaussian process regression (GPR), and response surface methods have been widely employed for fast prediction of motor losses or temperature fields [18, 19]. Nevertheless, most existing studies rely on a single surrogate model architecture, which is insufficient to simultaneously accommodate the dual-path coupling characteristics of electrical losses and thermal temperature fields. As a result, predictive accuracy and generalization capability are limited, rendering such approaches inadequate for high-precision optimization requirements.

The application of PSO and its variants in electric motor design optimization has continued to expand. Conventional PSO has been successfully applied to single-objective optimization problems, such as winding parameters and cooling structure dimensions. To alleviate premature convergence, strategies including chaotic mapping and adaptive parameter regulation have been incorporated, leading to enhanced population diversity and improved convergence behavior [20]. However, most existing PSO variants primarily focus on mathematical or algorithmic enhancements [21, 22], while insufficient attention has been paid to the underlying physical mechanisms of thermo–electromagnetic coupling. Consequently, optimization processes lack sensitivity and responsiveness to temperature field states, often steering solutions toward designs with localized thermal concentration and failing to achieve a globally optimal balance between thermal and electromagnetic performance.

Regarding simulation–optimization co-design, existing studies predominantly adopt a serial paradigm of “simulation followed by optimization,” in which simulation data are first generated to train surrogate models, and optimization is subsequently performed based on these models. Under this paradigm, surrogate model accuracy remains static and cannot adapt to the dynamically evolving design space during optimization, leading to accuracy degradation in later optimization stages. Moreover, the separation between simulation and optimization prevents simulation data from dynamically feeding back into model refinement, thereby hindering coordinated improvements in accuracy and efficiency. Although intermittent simulation-based calibration mechanisms have been explored in some studies, a dynamic interactive closed-loop framework has not yet been established. As a result, the fundamental imbalance between accuracy and efficiency remains unresolved, and the lack of a dynamic interaction–driven co-design framework constitutes a critical research gap.

In response to the core limitations identified in existing studies—including insufficient surrogate model adaptability, the absence of physical guidance in optimization processes, and weak coordination between simulation and optimization—the primary objective of this study is to develop a thermo–electromagnetic co-design methodology for electric motor cooling systems that integrates physical simulation with intelligent optimization. Through this integration, simultaneous improvements in optimization efficiency, predictive accuracy, and engineering applicability are targeted.

To achieve this objective, the principal innovations are summarized below. First, a dual-path surrogate model architecture is proposed, in which dedicated surrogate submodels are constructed for electrical loss prediction and thermal temperature field prediction, respectively. Coupled parameters are introduced to enable bidirectional data interaction between the two paths, allowing accurate representation of the dual-path characteristics inherent to thermo–electromagnetic coupling and significantly enhancing both predictive accuracy and generalization capability. Second, a thermal-state feedback–driven adaptive PSO strategy is developed, in which temperature field uniformity metrics are embedded into the parameter regulation mechanism. As a result, the optimization process is endowed with real-time awareness and responsiveness to physical field states, providing explicit physical guidance and effectively mitigating designs prone to localized thermal concentration. Third, a dynamic optimization–high-fidelity simulation interactive verification loop is established. Through periodic simulation–based calibration of optimal solutions during optimization, surrogate model bias is dynamically corrected, while newly generated high-fidelity simulation data are continuously incorporated into the training sample database, enabling the co-evolution of surrogate model accuracy and optimization reliability.

The remainder of this study is organized below. Section 2 presents the fundamental mechanisms of thermo–electromagnetic coupling in electric motors, the basis of PSO, and the core theories of the surrogate model, followed by the mathematical modeling of the thermo–electromagnetic co-optimization problem. Section 3 details the proposed adaptive chaotic PSO–based thermo–electromagnetic co-simulation framework, including dual-path surrogate model construction, adaptive optimizer design, and implementation of the dynamic interactive closed-loop process. Section 4 conducts comparative experimental validation using a 200 kW permanent magnet synchronous drive motor with a liquid-cooled system as the case study, with emphasis on optimization efficiency, accuracy, and engineering feasibility. Section 5 provides an in-depth discussion of the physical implications of the results, the engineering applicability and limitations of the proposed method, and potential directions for future research. Section 6 concludes with a summary of the main findings and clarifies the academic contributions and engineering significance of the study.

3.1 Overall framework architecture

The proposed thermo–electromagnetic co-simulation framework based on adaptive chaotic PSO is designed to overcome the efficiency bottleneck inherent in traditional serial “simulation–optimization” workflows by establishing a parallel, interactive, and mutually driven system that deeply integrates the physical fidelity of simulation with the efficiency of intelligent optimization. Through the coordinated operation of five core modules, the framework achieves thermo–electromagnetic co-optimization. These modules follow a hierarchical and closed-loop logic of problem formulation → efficient prediction → intelligent optimization → accuracy calibration → decision output, progressing sequentially while forming a closed loop. First, coupled problem definition and solution space mapping provide the foundational boundaries of the framework by explicitly specifying the mathematical representations of design variables, objective functions, and constraints, as well as the admissible solution space. Second, the dual-path surrogate model, serving as the core intermediate layer, enables rapid and accurate prediction of electrical losses and thermal temperature fields, thereby providing efficient evaluation support for iterative optimization. Third, the adaptive chaotic particle swarm optimizer performs multi-objective search based on surrogate model outputs, while a thermal-state feedback mechanism ensures explicit physical guidance throughout the optimization process. Fourth, the optimization–simulation interactive verification closed loop dynamically calibrates surrogate model accuracy by feeding high-fidelity simulation results back into model updates, thereby enhancing predictive reliability. Finally, a multi-criteria decision-making (MCDM) system selects engineering-preferred solutions from the Pareto-optimal set. An overview of the thermo–electromagnetic co-simulation framework based on adaptive chaotic PSO is illustrated in Figure 1.

Figure 1. Thermo–electromagnetic co-simulation framework based on adaptive chaotic PSO

The data flow and control flow within the framework jointly constitute a mutually driven closed-loop chain. Design variable boundaries and objective function representations generated by the coupled problem definition module are used to guide sample acquisition and training of the dual-path surrogate model. The surrogate-predicted electrical losses and temperature fields are then supplied to the optimizer as fitness evaluation inputs, driving iterative position updates of the particle swarm. During optimization, selected candidate optimal solutions are fed into the high-fidelity thermo–electromagnetic co-simulation module for accuracy calibration. The resulting high-precision data are subsequently utilized to update surrogate model parameters and, in parallel, to enrich the training sample database, thereby enhancing model generalization capability. The updated surrogate model is then fed back to the optimizer, enabling the coordinated improvement of optimization accuracy and efficiency. Upon completion of the optimization iterations, the generated Pareto-optimal solution set is passed to the MCDM system, from which an engineering-feasible optimal design is selected. The decision outcome may further be used to retrospectively assess the rationality of the coupled problem definition, thereby establishing a full-process closed-loop regulation mechanism.

3.2 Dual-path surrogate model construction

The core design principle of the dual-path surrogate model is to accommodate the intrinsic dual-path characteristics of thermo–electromagnetic coupling. This is achieved by independently constructing an electrical surrogate model and a thermal surrogate model, while enabling bidirectional data interaction through coupling parameters. In this manner, the nonlinear mapping relationships between electrical losses and thermal temperature fields are accurately captured, while high predictive efficiency is maintained. A schematic illustration of the proposed architecture is provided in Figure 2. Sample acquisition constitutes the foundation of surrogate model construction. Latin hypercube sampling (LHS) is employed to achieve uniform coverage of the full design variable space, thereby ensuring sample representativeness and diversity and preventing degradation of model generalization caused by insufficient local-space information. All sample data are generated through high-fidelity thermo–electromagnetic co-simulation, with the full-dimensional design variables serving as inputs. The corresponding outputs include loss distributions of motor components, temperature fields of critical components, and pressure drop across the cooling system. Data preprocessing is conducted sequentially, including normalization, outlier elimination, and correlation analysis. Normalization is applied to eliminate dimensional discrepancies among variables and to enhance training efficiency. Outlier removal is performed using the three-sigma (3σ) criterion to ensure data quality. Correlation analysis is subsequently employed to identify input variables exhibiting strong relevance to the output responses, thereby effectively reducing model complexity and improving generalization capability.

Figure 2. Schematic diagram of the dual-path surrogate model

The electrical surrogate model is implemented using a DNN architecture to accommodate the strong nonlinear relationships between electrical losses and electrical path variables. The model inputs consist of electrical path variables, while the outputs correspond to the distributions of copper losses and iron losses within the motor. The network architecture is designed such that the number of input-layer neurons matches the dimensionality of the electrical variables, three fully connected hidden layers are employed, and the number of output-layer neurons corresponds to the loss categories. The rectified linear unit (ReLU) activation function is adopted to enhance nonlinear fitting capability. Model training is performed using the Adam optimizer to minimize the mean squared error (MSE) loss function. The dataset is partitioned into training, validation, and test subsets with a ratio of 7:2:1, which are used for parameter learning, hyperparameter tuning, and generalization assessment, respectively. The thermal surrogate model adopts a hybrid architecture combining a CNN and a gradient boosting tree (GBT). The inputs include thermal path variables and the loss distributions predicted by the electrical surrogate model, while the outputs comprise the maximum temperature, average temperature, temperature difference of key components, and the cooling system pressure drop. Within this hybrid structure, the CNN is responsible for capturing the spatial distribution characteristics of the temperature field, whereas the GBT is employed to accommodate the predominantly linear relationships between pressure drop and the input variables. The training strategy remains consistent with that of the electrical surrogate model, with particular emphasis placed on validating the predictive accuracy of temperature field distributions to ensure reliable thermal state assessment.

The dual-path surrogate model achieves close alignment with the thermo–electromagnetic coupling mechanism through a bidirectional coupling mechanism. The loss distributions predicted by the electrical surrogate model are provided as thermal source inputs to the thermal surrogate model, serving as the primary driving factors for temperature field prediction. Conversely, the winding temperature feedback coefficients predicted by the thermal surrogate model are fed back into the electrical surrogate model to correct the temperature-dependent resistivity calculation, thereby accurately representing the coupled feedback loop of “temperature–resistivity–loss.” Overall model accuracy is evaluated using R², RMSE, and MAE. It is required that the RMSE of key output parameters does not exceed 3%, ensuring that the surrogate model provides both efficient and accurate performance evaluation for subsequent optimization iterations and achieves a balanced trade-off between optimization efficiency and precision.

3.3 Adaptive chaotic PSO optimizer driven by thermal state feedback

Parameter self-adaptation is achieved through thermal-state feedback, enabling a balanced enhancement of global exploration and local exploitation capabilities. Particle encoding is implemented using a real-valued representation, whereby the position vector of each particle directly corresponds to a complete cooling system design scheme, expressed as x_i= [x_i,elec,x_i,therm]^T, where x_i,elec denotes the subset of electrical path variables and x_i,therm represents the subset of thermal path variables. The particle dimensionality is consistent with the total number of design variables. Population initialization is performed using a Tent chaotic mapping to generate a uniformly distributed initial swarm, thereby mitigating premature convergence caused by uneven initial distributions. The Tent map is defined as:

$z_{k+1}= \begin{cases}2 z_k & 0 \leq z_k<0.5 \\ 2\left(1-z_k\right) & 0.5 \leq z_k \leq 1\end{cases}$                  (11)

where, z_k denotes the chaotic variable at the k-th iteration. The initialization procedure consists of first generating the chaotic sequence z_k, followed by a linear mapping into the feasible domain of the design variables [x_min,x_max], given by x = x_min+ z_k(x_max−x_min).

A chaotic perturbation mechanism is introduced to balance global exploration and local exploitation during the iterative process. The perturbation is triggered either at later stages of iteration or when population diversity falls below a predefined threshold. Population diversity is quantified using the standard deviation of fitness values, defined as:

$D=\sqrt{\frac{1}{N} \sum_{i=1}^N\left(f_i-\bar{f}\right)^2}$                       (12)

where, N is the population size, f_i denotes the fitness value of the i-th particle, and $\bar{f}$ represents the mean fitness of the population. When it drops below the diversity threshold D_th, the perturbation mechanism is activated. The perturbation strategy applies a small-amplitude chaotic disturbance to the particle’s best positions, expressed as:

$p_{ {i,new }}=p_{ {i,old }}+\delta\left(z_k-0.5\right)\left(x_{\max }-x_{\min }\right)$                     (13)

where, p_i,old denotes the historical best position of the particle, and δ=0.08 is the perturbation coefficient, selected to balance disturbance intensity and convergence stability. The principal innovation lies in the thermal-state–feedback–driven adaptive parameter adjustment mechanism, in which the thermal state is quantified using a temperature field uniformity index, defined as:

$\theta=\frac{T_{\max }-T_{a v g}}{T_{a v g}}$                      (14)

where, T_max is the maximum motor temperature, and T_avg is the average temperature. A larger value of θ indicates a more non-uniform temperature field and a higher risk of thermal concentration.

Adaptive parameter regulation is driven by θ, through which the inertia weight w, cognitive learning factor c₁, and social learning factor c₂ are dynamically updated according to:

$\begin{aligned} & w=w_{\max }-\frac{w_{\max }-w_{\min }}{1+\exp \left(-k\left(\theta-\theta_{t h}\right)\right.}, \\ & c_1=c_{1, \text { base }}+k_1 \theta, c_2=c_{2, \text { base }}-k_2 \theta\end{aligned}$                   (15)

where, w_max = 0.9 and w_min = 0.4 define the upper and lower bounds of the inertia weight, respectively; θ_th = 0.15 is the thermal-state threshold; k = 5 is the decay coefficient; c_1,base = 1.5 and c_2,base = 1.5 are the baseline learning factors; and k₁ = 0.8 and k₂ = 0.8 are adaptive coefficients. When θ > θ_th, both w and c₁ are increased while c₂ is reduced, thereby encouraging global exploration to escape regions associated with thermal concentration. Conversely, when θ ≤ θ_th, w and c₁ are decreased and c₂ is increased, promoting refined local exploitation. The multi-objective optimization procedure is conducted using an ε-dominance strategy to identify Pareto-front solutions. An external archive is employed to store non-dominated solutions, while crowding-distance sorting is applied to preserve solution diversity. Through this mechanism, convergence toward a uniformly distributed Pareto front is ensured.

3.4 Optimization–high-fidelity simulation interactive verification closed loop

The core design logic of the optimization–high-fidelity simulation interactive verification closed loop is to dynamically calibrate surrogate model accuracy, thereby establishing a virtuous cycle in which optimization drives model learning, and improved models enhance optimization accuracy. This mechanism addresses the degradation of predictive accuracy that may occur during long-term iterative optimization and enables coordinated improvements in both optimization efficiency and precision. A schematic illustration is provided in Figure 3. Although the surrogate model substantially reduces computational cost, prediction bias may increase when optimization iterations concentrate on regions near optimal solutions, where sample information is often sparse. In contrast, high-fidelity simulation is capable of accurately resolving the underlying thermo–electromagnetic coupling physics. Dynamic interaction between these two components effectively compensates for the surrogate model’s accuracy deficiencies. The triggering mechanism of the closed loop adopts a dual-criterion strategy, combining periodic triggering and accuracy-based triggering, to ensure both timeliness and necessity of calibration. Periodic triggering is configured to execute once every five optimization iterations, thereby balancing calibration frequency against computational overhead. Accuracy-based triggering is determined by the surrogate model prediction error evaluated on solutions stored in the external archive, quantified using the RMSE:

$R M S E_{{val }}=\sqrt{\frac{1}{M} \sum_{j=1}^M\left(y_{j, { sim }}-y_{j, p r e}\right)^2}$                        (16)

where, M denotes the number of archive solutions selected for validation, y_j,sim represents the reference values obtained from high-fidelity simulation, and y_j,pre denotes the corresponding surrogate model predictions. When RMSE_val > 5%, calibration is forcibly triggered to prevent accuracy deterioration from adversely influencing the optimization trajectory.

Figure 3. Schematic of the optimization–high-fidelity simulation interactive verification closed loop

The calibration procedure strictly follows a progressive sequence of sample selection → high-fidelity simulation → model update, thereby ensuring both calibration efficiency and effectiveness. During sample selection, the top 10% of non-dominated solutions from the external archive are chosen as calibration samples. These solutions represent the current optimal design region; calibrating against them enables targeted improvement of model accuracy in critical regions. High-fidelity simulation is performed using electromagnetic–fluid–thermal coupled simulations. The selected samples are supplied to the high-fidelity models to compute accurate benchmark data for model calibration, including motor loss distributions, temperature fields of key components, and cooling system pressure drop. Model updating is conducted using an incremental training strategy, whereby complete retraining of the surrogate model is avoided. Instead, newly acquired high-fidelity samples are appended to the training dataset, and model parameters are fine-tuned to achieve accuracy correction. The loss function for incremental training is defined as:

$Loss_{{incr}}=\alpha \cdot M S E_{{new }}+(1-\alpha) \cdot M S E_{ {old }}$                    (17)

where, MSE_new denotes the MSE associated with newly added samples, MSE_old represents the MSE over the original training dataset, and α = 0.6 is a weighting coefficient selected to balance calibration toward new samples while preserving the existing generalization capability of the model.

Termination of the closed loop is required to satisfy the dual criteria of model accuracy and optimization convergence, thereby ensuring the completeness and reliability of the optimization process. The model accuracy criterion requires that the validation error of the surrogate model remain below 3% for two consecutive calibration cycles, indicating that predictive accuracy in the vicinity of the optimal solution region has stabilized and meets the required precision. The optimization convergence criterion is determined based on the variation of solutions stored in the external archive. The variation is defined as the mean Euclidean distance between non-dominated solutions in the archive across two consecutive generations and is expressed as:

$\Delta F=\frac{1}{M} \sum_{j=1}^M \sqrt{\sum_{d=1}^D\left(f_{j, d}^{t+1}-f_{j, d}^{t}\right)^2}$                           (18)

where, D denotes the dimensionality of the objective function space, and f_j,dt represents the value of the d-th objective for the j-th archived solution at iteration t. When f_j,dt < 10^-3, the optimization process is considered to have converged. Once both criteria are satisfied, the interactive verification closed loop is terminated, and the optimizer proceeds to complete the remaining iterations based on the updated high-accuracy surrogate model. A stable Pareto-optimal solution set is then obtained as the final output. Through the dynamic complementarity between high-fidelity simulation and the surrogate model, the closed-loop framework preserves the efficiency of the surrogate model while ensuring the reliability of the optimization results, thereby providing essential support for achieving a balanced trade-off between accuracy and efficiency in thermo–electromagnetic co-optimization.

3.5 MCDM system

The primary objective of the MCDM system is to integrate practical engineering requirements and to select a unique, feasible optimal design from the Pareto-optimal solution set produced by the optimization process, thereby ensuring effective linkage between multi-objective optimization outcomes and engineering application. Because solutions on the Pareto front are mutually non-dominated, optimality cannot be determined based on any single objective alone; consequently, a decision mechanism incorporating engineering preferences is required. In this study, the Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) is adopted to construct the decision model. This method ranks candidate solutions by evaluating their relative closeness to the ideal solution, offering advantages of conceptual simplicity, computational efficiency, and suitability for multi-objective, multi-attribute decision scenarios. As a result, the influence of engineering preferences on decision outcomes can be objectively reflected.

Reasonable allocation of decision weights is critical to ensuring the scientific validity of the decision process. The Analytic Hierarchy Process (AHP) is employed to determine the weights of individual objective functions, enabling quantitative representation of engineering preferences. Within AHP, a hierarchical structure is constructed, and pairwise comparisons of objective importance are performed based on expert judgment, forming a judgment matrix A = (a_ij)_n×n, where n denotes the number of objective functions and a_ij represents the relative importance of the i-th objective with respect to the j-th objective. The maximum eigenvalue λ_max of the judgment matrix and its corresponding eigenvector are computed, and the normalized eigenvector yields the weight vector w = [w₁, w₂, w₃]^T. To ensure consistency of the weight assignment, a consistency check is performed using the consistency ratio CR = CI/RI, where CI = (λ_max−n) / (n−1) is the consistency index and RI is the random consistency index. When CR < 0.1, the judgment matrix is considered to satisfy the consistency requirement. For different application scenarios, decision weights can be dynamically adjusted to reflect specific engineering priorities. For example, in new energy vehicle drive motor applications, higher weights are typically assigned to cooling power consumption and maximum motor temperature, whereas in industrial motor applications, greater emphasis is placed on cooling system volume and reliability.

The TOPSIS decision-making procedure strictly follows a progressive sequence of normalization → weighting → ideal-solution determination → closeness-based ranking. The first step involves standardization of objective functions. Since all objectives considered in this study are of the minimization type, range normalization is employed to eliminate dimensional inconsistencies, expressed as:

$z_{i j}=\frac{x_{j, \max }-x_{i j}}{x_{j, \max }-x_{j, \min }}$                  (19)

where, x_ij denotes the value of the j-th objective function for the i-th Pareto solution, x_j,max and x_j,min represent the maximum and minimum values of the j-th objective, respectively, and z_ij is the normalized objective value. In the second step, a weighted decision matrix is constructed by combining the normalized matrix with the weight vector, i.e., Z = z_ij⋅w_j, thereby explicitly reflecting the relative importance of each objective. The third step involves computation of the positive and negative ideal solutions. The positive ideal solution is defined as $Z^{+}=\left[\max \left(z_{i 1} w_1\right), \max \left(z_{i 2} w_2\right), \max \left(z_{i 3} w_3\right)\right]$, representing the optimal weighted values of each objective, whereas the negative ideal solution is defined as $Z^{-}=\left[\min \left(z_1 w_1\right), \min \left(z_2 w_2\right), \min \left(z_3 w_3\right]\right.$, corresponding to the least favorable weighted values. In the fourth step, the Euclidean distances between each solution and the ideal solutions, as well as the corresponding closeness coefficients, are computed. The distance metric is defined as:

$d_i^{+}=\sqrt{\sum_{j=1}^3\left(Z_{i j}-Z_j^{+}\right)^2}, d_i^{-}=\sqrt{\sum_{j=1}^3\left(Z_{i j}-Z_j\right)^2}$                  (20)

The closeness coefficient is defined as $C_i=d_i^{+} /\left(d_i^{+}+d_i^{-}\right)$, with C_i in the range of [0, 1]. A larger value of C_i indicates that the corresponding solution is closer to the positive ideal solution. Finally, all Pareto-optimal solutions are ranked according to C_i, and the solution with the highest closeness coefficient is selected as the recommended optimal design. The corresponding design parameters and predicted performance indicators are then reported, providing a direct basis for engineering decision-making and validation.

The proposed method demonstrates pronounced advantages in both efficiency and accuracy over comparative approaches, primarily attributable to the precise alignment of three core modules with the intrinsic characteristics of thermo–electromagnetic coupled optimization. The thermal-state feedback mechanism incorporates physical field information into the optimization process via the temperature field uniformity index θ. Therefore, parameter adaptation is no longer governed solely by algorithmic heuristics but is endowed with explicit thermal management guidance. This design effectively mitigates the stagnation of solutions in regions of thermal concentration that commonly arises from indiscriminate search strategies in conventional optimization. The experimentally observed reduction of θ from 0.21 to 0.09 exhibits a strong correlation with the improvement in T_max, thereby validating the rationality of employing this index as the basis for adaptive regulation. The dual-path surrogate model architecture accurately matches the dual-path nature of thermo–electromagnetic coupling. The electrical surrogate model leverages a DNN to accommodate the strong nonlinearity inherent in loss prediction, while the thermal surrogate model employs a CNN to capture the spatial characteristics of temperature fields. Relative to a single DNN surrogate, the average RMSE is reduced by 50.8%. Compared with GPR, which incurs high computational cost, and support vector machines, which often exhibit limited generalization capability, the DNN + CNN dual-path configuration achieves an effective balance between accuracy and efficiency, thereby providing reliable and rapid performance evaluation for iterative optimization. The optimization–high-fidelity simulation interactive verification closed loop further resolves the degradation of surrogate accuracy caused by insufficient sampling in the vicinity of optimal solutions through a dynamic calibration mechanism. As a result, the final validation error is controlled within 2.1%, enabling coordinated attainment of computational efficiency and predictive accuracy.

The proposed method exhibits substantial engineering applicability. With a 91.1% reduction in computational time relative to pure high-fidelity simulation-based optimization—representing an improvement exceeding one order of magnitude—the framework can be directly integrated into engineering design workflows for motor cooling systems, substantially shortening development cycles and reducing research and development costs. In application scenarios with stringent requirements on thermal management and power density, such as new energy vehicle drive motors and high-efficiency industrial motors, the method enables rapid generation of multi-objective balanced optimal designs, thereby providing direct support for engineering decision-making. Nevertheless, several limitations remain. First, initial sample acquisition relies on high-fidelity simulations; although LHS maximizes sample representativeness, constructing an initial dataset that adequately covers the full design space still incurs non-negligible computational expense. Second, the threshold value of the temperature-field uniformity index θ is derived from sample statistics specific to a given motor type; motors with different power ratings or structural configurations require re-calibration of this threshold, which weakens methodological generality. Third, the present study focuses on steady-state thermo–electromagnetic coupling, whereas real motor operation involves transient conditions such as start-up and acceleration. The dynamic coupling between losses and temperature under transient regimes has not yet been addressed, thereby limiting applicability under complex operating conditions.

Compared with existing studies, the proposed method exhibits three principal advantages. First, relative to conventional PSO variants, prior work has predominantly relied on mathematical adjustments of inertia weights and learning factors to improve convergence, with limited consideration of the underlying thermo–electromagnetic coupling physics. As a result, optimization trajectories may deviate from engineering requirements. By introducing thermal-state feedback, physical guidance is imparted into the optimization process, effectively addressing this limitation; experimentally, convergence efficiency is improved by 47.1% relative to conventional PSO. Second, compared with existing surrogate-assisted optimization approaches, most studies adopt a single surrogate architecture and lack a dynamic interaction mechanism between simulation and optimization, making it difficult to balance accuracy and efficiency. Through the coordinated design of a dual-path surrogate model and an interactive verification closed loop, the average prediction RMSE is reduced to 1.75%, while computational efficiency is markedly enhanced. Third, in contrast to the serial simulation–optimization paradigm, in which simulation and optimization are decoupled and simulation data cannot dynamically inform model refinement, the proposed parallel interactive closed-loop framework establishes a virtuous cycle in which optimization drives model learning and improved models, in turn, enhance optimization accuracy. This mechanism substantially reduces computational cost and yields a pronounced efficiency advantage.

Future research may extend the applicability and comprehensiveness of the method along four directions. (i) Extension to transient thermo–electromagnetic co-optimization, through the development of dynamic surrogate models capable of capturing time-varying loss and temperature characteristics, thereby accommodating transient operating conditions. (ii) Incorporation of active learning strategies, whereby samples with the highest information gain—identified via uncertainty quantification—are selected for high-fidelity simulation, optimizing the initial sample acquisition process and further reducing the computational cost of sample construction. (iii) Integration of thermo–electromagnetic–structural multiphysics coupling, by incorporating structural stress and vibration constraints into the optimization objectives, thereby enhancing design completeness and avoiding structural reliability issues arising from purely thermo–electromagnetic optimization. (iv) Establishment of a motor cooling system experimental test bench, enabling experimental validation of optimized designs, comparison between simulation predictions and measured data, and further model calibration to strengthen engineering credibility.