© Q. Zeng et al., Published by EDP Sciences, 2026

1 Introduction

The electromagnetic gear transmission system is a new type of structure that uses electromagnetic torque to achieve non-contact power transmission. It has significant advantages such as low noise, low wear, high precision, and easy intelligent control. It is widely used in high-end equipment such as precision manufacturing, electric transportation, and aerospace [1,2]. In these complex application scenarios, the system operation state is often affected by factors such as load disturbance and input frequency change. Its dynamic performance is directly related to the stability and response speed of the whole machine [3,4]. Therefore, establishing an efficient and accurate dynamic simulation model has important theoretical significance and engineering value for the design optimization, control strategy formulation, and fault diagnosis of electromagnetic gear systems [5,6].

The current dynamic analysis of electromagnetic gear transmission systems mainly relies on the finite element method, multi-body dynamics model, and coupled multi-physics field modeling technology [7,8]. These traditional methods are highly dependent on accurate geometric modeling and material parameters and require a lot of prior knowledge to ensure the physical authenticity and numerical stability of the model [9,10]. In systems involving strong coupling between electromagnetic fields and mechanical motion, the simulation process is particularly complex, especially in areas with sudden changes in magnetic density (such as tooth surfaces and air gaps), where the direction of magnetic field propagation can be refracted due to differences in magnetic permeability. The bending law of its equivalent trajectory conforms to the principle of magnetic field refraction; that is, the magnetic flux lines are deflected at the boundaries of different media to satisfy the continuity of magnetic flux and conservation of magnetomotive force. If such refraction behavior is not accurately modeled, it can directly affect the prediction accuracy of electromagnetic force density and dynamic load response [11,12]. In addition, such methods usually use numerical solution technology, which consumes a lot of computing resources and takes a long time to simulate, making it difficult to meet the needs of rapid response and real-time prediction of multiple working conditions in engineering [13,14]. More importantly, the electromagnetic gear system exhibits highly nonlinear and strongly coupled characteristics under different working conditions. The traditional model has high parameter tuning and model update costs [15,16], which limits its generalization ability and scalability, making it difficult to adapt to complex working conditions and intelligent design requirements [17,18]. These factors together hinder the application of traditional methods in practical engineering, especially in the promotion and application of intelligent control, adaptive regulation, and rapid design iteration [19,20]. The proposed cGAN (conditional generative adversarial network)-based approach circumvents the need for explicit modeling of such intricate physical phenomena by learning a direct mapping from input conditions to dynamic responses from the high-dimensional finite element dataset.

In recent years, with the development of deep learning technology, data-driven modeling methods have gradually been introduced into the field of engineering system modeling [21,22]. Some scholars have studied the application of data-driven modeling methods in complex nonlinear system modeling and high-dimensional data generation tasks [23,24]. Zhang F et al. proposed a variable-step multi-channel FxLMS (Filtered-x Least Mean Square) algorithm based on a sampling function for active vibration control of gear transmission systems. This algorithm effectively improves the convergence speed and accuracy and enhances the robustness of the system [25]. Lu W et al. established a coupled dynamic model of a motor-multi-stage gear system with tooth root cracks for fault analysis of coal mine machinery. The model can effectively simulate the impact of cracks on the vibration signals of the planetary gear and fixed-axis gear subsystems [26]. However, most of these methods ignore the conditional correlation between system input and response, lack effective constraint mechanisms, and have limited modeling capabilities for time series dynamic behavior [27,28]. In response to the above shortcomings, this paper proposes a method based on conditional generative adversarial networks, which embeds input operating condition information and introduces physical consistency constraints to solve the problem of balancing efficiency and accuracy in the dynamic simulation of electromagnetic gear transmission systems [29,30].

This paper aims to construct an efficient and precision-controlled dynamic response prediction model for electromagnetic gear transmission systems. In terms of methods, based on the cGAN architecture, the multi-condition high-dimensional data generated by finite element simulation is used as training samples, and a deep convolution generator with a fusion residual structure is designed to enhance the feature expression ability. The physical consistency loss function based on the law of conservation of energy is introduced to improve the physical reliability of the prediction results. At the same time, a multi-scale discriminator is used to enhance the ability to discriminate the response timing characteristics. After model training, the dynamic response of the system can be quickly generated under any input condition, significantly reducing the simulation time while maintaining high accuracy. Experimental verification shows that the average relative error of response prediction of this method is controlled within 5.2%, and the prediction time is less than 0.08 s. It effectively realizes the rapid modeling and multi-condition dynamic behavior simulation of electromagnetic gear systems and provides technical support for the intelligent design and efficient performance evaluation of electromagnetic transmission systems.

2 Materials and methods

2.1 Multi-condition data generation

In order to construct a high-quality training data set, this paper establishes a refined finite element model of the electromagnetic gear transmission system based on the joint simulation platform of ANSYS Maxwell and ANSYS Workbench. The model includes the stator, rotor, tooth slot structure, and air gap area and adopts a two-dimensional axisymmetric modeling method to reduce the simulation complexity and retain key physical properties [31,32]. In terms of meshing, an adaptive encryption strategy is used to perform local encryption processing on the tooth surface and air gap area to ensure the spatial accuracy of the electromagnetic force density calculation [33,34].

During the simulation process, key operating parameters that affect the dynamic behavior of the system are selected as variable inputs, including voltage frequency (20—80 Hz), load torque (5—10 Nm), and gear clearance (0.1—0.3 mm, step size 0.1 mm), and a three-factor orthogonal test scheme is constructed. Table 1 presents selected representative condition combinations from this scheme. The batch automation script controls iterative parameter variation and simulation execution across the full factorial space defined by the orthogonal array, generating a total of 1200 sets of system response data under typical operating conditions. Some extreme boundary conditions are also included in the simulation scope to improve the generalization ability of the training model. The operating parameters are shown in Table 1.

In Table 1, the input frequency corresponds to the excitation frequency of the driving electromagnetic field and is an important control variable for the dynamic response of the system. The higher the frequency, the more complex magnetic field coupling and vibration characteristics may be induced. The load torque represents the external mechanical load actually borne by the gear transmission system. Different torques have a significant impact on the rotor magnetic field response and output dynamics. Gear clearance is an important parameter that characterizes the gear meshing accuracy and has a key influence on magnetic force fluctuations, toothing effects, and vibration energy release.

Under each set of working conditions, the transient response information of the system is extracted, including key dynamic indicators such as rotor angular velocity, output torque, electromagnetic force fluctuation, and air gap magnetic flux distribution [35]. In order to obtain stable and effective time domain response characteristics, the simulation time is set to 0.2 s, the time step is set to 1 ms, and the output sequence length is 200. All response data are uniformly normalized and expanded into a one-dimensional tensor according to the time dimension, which is combined with the corresponding working condition parameter vector to form the final data sample. In order to reduce the error caused by boundary condition disturbance and initial state uncertainty in finite element simulation, this paper introduces a multi-cycle filter calibration strategy. In each set of simulation output, the first three cycles are interpreted as the steady-state feature extraction window, the mean and variance of the response within the cycle are calculated as the data benchmark, and the points with obvious initial disturbance influence are eliminated.

In the data structure construction, the sample data is organized in a triple format: input condition vector (frequency, torque, and gap), corresponding time series response, and category label (real/fake) for data annotation in the discriminator training stage. The training set finally constructed contains about 960 groups of samples, and the test set contains about 240 groups of samples. All data are stored in tensor format and passed to the training framework through PyTorch’s data loading interface. The orthogonal design efficiently samples the defined parameter space, while the inclusion of boundary conditions extends the coverage to its extremes. This combined strategy aims to maximize the representativeness of the training dataset for the system’s nonlinear dynamics within the operational envelope, providing a foundation for model generalization. The data generation process ensures the full coverage of the working condition input space and the timing accuracy of the response output, providing a high-dimensional, high-quality, and controllable training foundation for subsequent generative modeling based on cGAN. This effectively replaces the traditional data acquisition method that relies on complex analytical models and step-by-step integration simulation, significantly improving the efficiency and adaptability of dynamic simulation modeling.

2.2 Conditional generator network design

In order to achieve efficient generation of dynamic response sequences of electromagnetic gear transmission systems under multiple working conditions, this paper designs a deep convolutional generator network based on residual structure. The generator takes the input working condition parameters as conditions and outputs a complete dynamic response sequence in an end-to-end mapping manner, avoiding multi-step integration and partial differential solutions in explicit physical modeling and effectively reducing reasoning time and modeling complexity.

The input of the generator consists of two parts: the first part is a standardized operating parameter vector, which contains continuous variables such as voltage frequency, load torque, and gear clearance, with a dimension of 3. The second part is a random Gaussian noise vector with a length of 200, which is used as a sampling representation of the latent variable space. After the two parts are concatenated at the input layer, they are mapped to the initial feature tensor (size 64 × 25) through a set of fully connected layers and then sent to the convolution backbone module.

The backbone structure consists of 6 cascaded one-dimensional convolution residual blocks. This depth was selected through preliminary experiments to balance feature extraction capability with training efficiency for the target sequence length. Each residual block contains two one-dimensional convolution layers with BatchNorm and ReLU activation. The convolution kernel size is 5, the step size is 1, and the padding is 2 to keep the feature dimension unchanged. To avoid the problem of gradient disappearance or degradation during training, each residual block introduces an identity mapping at the output end to short-circuit the input features and enhance the stable transmission of deep network features, as shown in Figure 1.

Figure 1 shows the overall structure of the generator network, from multi-source input to feature mapping, and then extracting dynamic response features step by step through deep residual convolution modules, using residual connections to ensure the stability and performance of deep network training. The entire structural design aims to efficiently generate dynamic response sequences that meet the conditions.

The conditional injection module is embedded in the residual block to improve the representation ability of the working condition in the network. Specific approach: In each residual unit, the working condition vector is copied and extended to the same time-series length as the current feature tensor and fused into the main channel through a channel-by-channel weighted method to achieve layer-by-layer transmission of conditional information throughout the generation process. The weighting coefficients are generated dynamically for each channel by applying a shared fully connected layer to the extended condition vector. This mechanism ensures that the working condition parameters not only exist as initial inputs, but also continuously regulate the intermediate feature representation, improving the consistency of the model’s response to input conditions.

In order to capture the local dynamic details and overall time series characteristics in the system response, the number of network feature channels is adjusted between every two residual blocks. The initial number of channels is set to 64, which is then increased to 128, 256, and then gradually reduced to 64, forming an encoding-decoding bottleneck structure. This design enhances the network's ability to extract high-order features in the intermediate stage while avoiding the loss of response information caused by excessive feature compression in the output stage. At the output end, a set of deconvolution layers (Transposed Conv1D) is set to restore the final feature tensor to the target output dimension (1 × 200). The output activation function uses the tanh function to match the normalized response value interval [–1, 1]. To further improve the smoothness and continuity of the output, a one-dimensional mean filter layer (window width is 3) is added after the output layer to weaken the slight oscillation introduced by random noise.

During the training process, the parameters of the generator are optimized by backpropagation of the joint loss function, with the goal of minimizing the L1 loss between the predicted response and the true finite element data while improving the discriminator’s misleading ability in adversarial training. Due to the strong robustness and training stability of the residual structure, the generator can stably output prediction results that are highly consistent with the true response in terms of waveform structure, amplitude change, and timing trend after multiple rounds of iterations. In addition, in order to alleviate the problem of pattern collapse in the generated results at the beginning of training, a label smoothing strategy is introduced in the early stage of training, setting the true label to 0.9 instead of 1.0, enhancing the generator's ability to utilize the discriminator's gradient information. During the network training process, the Adam optimizer is used for updating, and the initial learning rate is set to 1e−4, β1 = 0.5, and β2 = 0.999. The conditional generator structure fully integrates the input working condition information, time series modeling capabilities, and deep feature expression capabilities. While maintaining the generation stability, it realizes the efficient prediction modeling of the multi-working condition dynamic behavior of the electromagnetic gear transmission system, providing core support for the entire adversarial modeling framework.

Fig. 1 Generator network.

2.3 Discriminator structure design

In order to improve the ability to discriminate the dynamic response of the electromagnetic gear transmission system in adversarial training, this paper constructs a one-dimensional convolutional discriminator network with a multi-scale feature perception mechanism, aiming to accurately distinguish the slight differences between the generated response and the real simulation data from multiple time series resolution levels and enhance the discrimination accuracy and convergence stability of the model in dynamic behavior simulation.

The input data consist of two parts: the first part is a 200-dimensional normalized dynamic response sequence. The second part is a normalized working condition parameter vector. In order to integrate conditional information and improve discrimination accuracy, a conditional splicing strategy is adopted in the input layer of the discriminator. The working condition parameters are repeatedly expanded to the same length as the response sequence and then spliced element by element to form a dual-channel input tensor (size 2 × 200) and sent to the subsequent convolution structure. This direct concatenation strategy provides the discriminator with explicit, pointwise conditional information for matching, contrasting with the generator’s use of weighted feature modulation to continuously regulate the synthesis process. The overall structure of the discriminator consists of three layers of backbone convolution groups, each of which contains two one-dimensional convolution layers, a LeakyReLU activation layer, and an average pooling layer. The convolution kernel sizes are set to 5, 9, and 17, respectively, and the receptive fields correspond to fine-grained, medium-grained, and overall trend-level time windows, respectively, so as to realize multi-scale dynamic response feature extraction. Through the superposition and combination of different convolution kernel sizes, the network can simultaneously identify typical dynamic features such as local mutations, periodic fluctuations, and trend drift, avoiding response misjudgment caused by single-scale modeling.

Each group of backbone convolutions is connected to a group of channel attention modules to further enhance the feature extraction ability of the discriminator. This module first performs global average pooling in the channel dimension to generate a compressed vector and then generates channel weights through two fully connected layers. After sigmoid activation, it is fed back to the original channel to achieve dynamic channel weighting, thereby highlighting the feature channels that are more sensitive to discrimination and suppressing redundant information interference. The output of the intermediate layer is concatenated and passed to a set of fully connected layers to reduce the dimension to a single scalar, and the sigmoid function is used to output the generation probability. In order to suppress overfitting and improve generalization ability, the dropout mechanism is introduced before the fully connected layer of the discriminator, and the dropout rate is set to 0.3. At the same time, spectral normalization is used after all convolutional layers to stabilize the training and control the Lipschitz constant of the weight tensor to ensure the stability of the gradient during adversarial training. In addition, in order to cope with the problem of parameter surge and increased training difficulty caused by multi-scale structures, the number of channels of all convolutional layers is set to gradually increase (32→64→128), and group normalization is used instead of traditional BatchNorm to avoid statistical offset problems during small batch training. In terms of implementation, the entire discriminator structure is built through a PyTorch custom module to keep parameter updates and generators iterated synchronously. The structural configuration and parameter design are shown in Table 2.

Table 2 shows the structural configuration and parameter design of the three groups of backbone convolution modules in the discriminator. The convolution group number indicates the order of the three groups of backbone convolution structures, corresponding to the shallow, middle, and deep feature extraction modules. The convolution kernel sizes are 5, 9, and 17, respectively, which are used to capture short-cycle (fine-grained), medium-cycle (medium-grained), and long-cycle features. By increasing the receptive field, the modeling of dynamic features at different time scales is achieved. The network input is 2 channels (response sequence and working condition information), and then each group of convolution output channels gradually increases (32→64→128) to improve feature extraction capabilities and support higher-dimensional discriminant learning. All convolution groups are set to a step size of 1 to maintain temporal resolution. Padding is set according to half of the convolution kernel (2, 4, 8) to ensure that the output sequence length is consistent with the input so as not to lose edge features.

During the training process, the discriminator loss function uses the standard binary cross-entropy loss and uses smooth labels (real = 0.9, fake = 0.0) to strengthen the adversarial tension and promote the generator output to be closer to the real distribution. In order to improve the stability of the initial training and the learning speed of the discriminator, the generator output is perturbed with Gaussian noise in the first 30 epochs to simulate the boundary fluctuations of the real data to prevent the discriminator from falling into premature convergence.

2.4 Loss function and optimization strategy

In order to ensure that the dynamic response sequence output by the generator has both numerical accuracy and physical rationality and consistency with adversarial training, this paper constructs a set of joint loss functions and uses an optimization algorithm with good stability and convergence for iterative updates during the training process. The designed total loss function consists of three terms: adversarial loss term, L1 response error loss term, and physical consistency constraint term, which constitute the optimization target of the generator in the form of a weighted linear combination.

The adversarial loss term adopts the standard conditional GAN adversarial loss function structure, which is defined as follows:

$$L_{adv }= E_{x,c}\left[logD\left(x,c\right)\right] + E_{z,c}\left[log\left(1 -D\left(G\left(z,c\right),c\right)\right)\right]$$(1)

In formula (1), x represents the real dynamic response, z is the random vector input by the generator, c is the working condition parameter, G is the generator, and D is the discriminator. This loss term is used to drive the generator to improve the authenticity of its generated response so that the generated samples are close to the distribution of real data in the probability of discriminator output, thereby maintaining the training adversarial tension and avoiding mode collapse.

In order to constrain the generated response to be consistent with the finite element simulation results in numerical terms and enhance the accuracy of the model output, this paper introduces the L1 distance loss term as a numerical fitting constraint:

$$L_{L1}= E_{x,z,c}\left[{\left|\left|G\left(z,c\right)- x\right|\right|}_1\right].$$(2)

This term directly measures the point-by-point difference between the generated sequence and the true response and has stronger robustness. L1 loss avoids the situation where large errors dominate the training and effectively improves the generator’s ability to restore high-frequency dynamic characteristics. Considering that the response of the electromagnetic gear system is affected by physical factors such as electromagnetic torque balance, backlash nonlinearity, and load response lag, relying solely on data fitting and adversarial signals may cause the output to violate basic physical laws. To this end, a priori rules derived from the dynamic model are introduced to construct the physical consistency loss term:

$$L_{phy}=E_{z,c}\left[{\displaystyle \sum }_t\left|\frac{d^2{G}_t}{d{t}^2}-F\left(G_t,c\right)\right|\right].$$(3)

In formula (3), G_t is the displacement of the generated response at time t, and F(·) represents the expected acceleration expression function under known system dynamics. The function F(G_t,c) encapsulates the fundamental relationship between electromagnetic torque, mechanical load, and rotational inertia as governed by the system’s equations of motion. It is derived from applying Newton’s second law for rotation to a simplified lumped-parameter dynamic model of the gear system. This term minimizes the residual between the second-order derivative of the generated response and the value derived from the physical model, constrains the physical rationality of the generated sequence during the time series evolution process, and suppresses nonphysical oscillations and phase drift phenomena.

Joint loss function definition and weight: Setting the three losses is to jointly constitute the total loss of the generator through the weight parameters:

$$L_G = {\lambda }_{adv} \times \cdot L_{adv} + {\lambda }_{L1} \times \cdot L_{L1 }+ {\lambda }_{phy} \times \cdot L_{phy. }$$(4)

In formula (4), λ_adv = 1.0, λ_L1 = 10.0, and λ_phy = 2.5. The weights are determined through multiple rounds of cross-validation to ensure that the structural stability and physical interpretability of the response output are effectively improved while maintaining the accuracy of the generated fit. The cross-validation involved a grid search over candidate weight combinations, optimizing for a composite metric of prediction accuracy and physical plausibility on a held-out validation set.

The model training uses the Adam optimizer to update the generator and the discriminator respectively. The generator uses the initial learning rate 1 × 10⁻⁴, the discriminator uses 4 × 10⁻⁴, and the momentum parameters β₁ = 0.5 and β₂ = 0.999 are set. To further improve the convergence speed and model stability in the early stages of training, the following strategy is adopted: in each epoch, the discriminator is updated twice and the generator is updated once, maintaining adversarial gradient balance. The learning rate is dynamically adjusted every 100 rounds of training, and exponential decay (decay rate 0.98) is used to reduce gradient oscillation. Label smoothing (real = 0.9) and Gaussian perturbation are used in the initial 100 rounds to enhance the generalization ability of the discriminator. This choice of hyperparameters follows established practices for stabilizing the early phase of adversarial training. The training data are divided into 80% training set and 20% validation set, and the change of validation loss is used as the early stopping standard to control overfitting. The generated results are saved in each round and the spectrum and energy spectrum are analyzed with the real response to assist in optimizing the parameter adjustment direction. The training is shown in Figure 2.

In Figure 2a, the X-axis represents the number of training rounds, and the Y-axis represents the learning rate. The blue line represents the learning rate of the generator, and the red dotted line represents the learning rate of the discriminator, showing the effect of the exponential decay strategy used in the training process. The learning rate decreases once every 100 rounds. The initial learning rate of the discriminator is slightly lower (0.0005), and the decline rhythm is consistent with that of the generator to ensure balanced gradient changes in adversarial training. The overall curve is smooth, indicating that the learning rate update strategy is stable and can not introduce training shocks. In Figure 2b, the X-axis represents the number of training rounds, and the Y-axis represents the loss value. The generator loss steadily decreases, indicating that the response of the generator output gradually approaches the real data, with small fluctuations, and no mode collapse occurs during the training process. The discriminator loss decreases rapidly in the early stage and remains at a low level in the middle and late stages. The discriminator has strong discrimination ability. The verification loss error fluctuates little, indicating that the model has good generalization ability and no obvious overfitting occurs.

Fig. 2 Model training. a) Learning Rate Decay. b) Generator / Discriminator / Validation Loss.

3 Results

3.1 Prediction performance

Representative multi-operating condition combinations were selected, covering different input frequencies (20, 40, 60, 80 Hz), load torques (5 and 10 Nm), and gear clearances (0.1, 0.2, 0.3 mm). Finite element simulation software was used to simulate the dynamic response of each operating condition and generate high-dimensional time series data as a real reference. The above simulation data were used to train a dynamic response prediction model based on a conditional generative adversarial network. After training is completed, the model is used to generate predicted response data for the same working condition input. For each working condition, the model predicted response is compared point by point with the finite element simulation results, and the mean absolute error and relative error are calculated. MAE (mean absolute error) reflects the absolute deviation size, and MRE reflects the percentage of error relative to the true value.

The X-axis labels on the left subgraph of Figure 3 represent 8 typical working condition combinations, each consisting of input frequency, load torque, and gear clearance, such as 40 Hz–10 Nm–0.3 mm. The Y-axis is the MAE value, which is used to measure the average absolute deviation of the predicted results and the simulation results in numerical terms, and the unit is dimensionless. The MAE values of all working conditions are lower than 0.03, indicating that the overall error is small. The MAE of some working conditions (such as 60 Hz–5 Nm–0.2 mm) is slightly higher, indicating that the model may have a slightly larger error under medium-frequency conditions. Error fluctuations reflect the sensitivity of the model to input changes and have reference value for model robustness assessment. The X-axis of the right subgraph is the working condition frequency segment grouping: low frequency (20–40 Hz), medium frequency (40–60 Hz), and high frequency (60–80 Hz). The Y-axis is the MRE value, which reflects the percentage of the prediction error relative to the true value and measures the relative error size. The average MRE in the low frequency band is about 2.3%; in the medium frequency band, it is 3.8%; and in the high frequency band, it rises to 5.2%, indicating that the relative error of the model prediction is larger under high-frequency conditions. The error fluctuation range increases with the increase of frequency, indicating that the stability of the model decreases in the high-frequency region.

Fig. 3 Prediction performance.

3.2 Response energy difference

Eight representative groups of electromagnetic gear transmission system conditions are selected, covering different combinations of input frequency (20 to 80 Hz), load torque (5 and 10 Nm), and gear clearance (0.1 to 0.3 mm) to ensure that the typical operating range of the system is covered. Finite element software is used to dynamically simulate the above working conditions one by one, and the system response time series signal under each working condition is obtained as the basis of the real signal. The trained conditional generative adversarial network model is applied, and the parameters of each working condition are used as input conditions to generate the corresponding dynamic response prediction signal. The cumulative integral of the square value of the real signal and the predicted signal is calculated, respectively, to obtain the energy accumulation curve over time under each working condition. The cumulative energy difference between the real signal and the predicted signal of each working condition at the end of the simulation is counted as the TED (total energy difference) indicator to quantify the overall energy fidelity. The predicted data is generated repeatedly for each working condition, and the average and standard deviation of TED are calculated to evaluate the prediction stability and error fluctuation.

Figure 4 shows the energy fidelity and error characteristics of the dynamic response of the electromagnetic gear transmission system predicted based on the generative adversarial network. The X-axis of the upper subgraph represents different working condition combinations, covering typical working conditions of different frequencies, loads, and clearances, and the Y-axis is the response energy difference under the corresponding working conditions. The points on the curve represent the average TED value of each working condition, and the error bar shows the standard deviation of TED under the working condition, reflecting the stability and fluctuation range of the prediction error. The TED of some high-frequency or large-load conditions is relatively large, and the fluctuation amplitude is more significant, indicating that such conditions are more difficult for the model to predict and the energy fidelity is slightly poor. The following sub-figure takes the third typical condition (40 Hz frequency, 5 Nm load, 0.2 mm gap) as an example to show the change of the cumulative energy of the model prediction signal and the real simulation signal over time. The blue solid line is the energy accumulation of the real signal, and the red dotted line is the energy accumulation predicted by the generative adversarial network. The overall changes of the two curves are consistent, indicating that the model captures the dynamic energy characteristics of the system well, but the red curve has a slight deviation in some periods, reflecting the existence of a slight energy offset and phase error in the prediction process.

Fig. 4 Energy fidelity and error characteristics.

3.3 Spectral error

The multi-scale discriminator, by extracting dynamic features across different temporal resolutions, underpins the model’s ability to capture frequency-domain characteristics. The dynamic response simulation of the electromagnetic gear transmission system under the set working conditions is carried out using finite element simulation software (ANSYS Maxwell) to obtain the real response signal. A model based on a conditional generative adversarial network is used to generate the predicted response signal under the corresponding working condition. A Fourier transform is performed on the real response signal and the predicted response signal, respectively, to obtain the amplitude spectrum data within the frequency range. The frequency range is set to 0–500 Hz to ensure that the spectrum resolution meets the requirements of characteristic frequency identification. The spectrum amplitude is normalized to eliminate the amplitude dimension difference for quantitative comparison. The spectrum error at each frequency point is calculated, which is defined as the square of the difference between the real spectrum and the predicted spectrum amplitude.

Figure 5 shows the model’s fitting effect on the dynamic response of the electromagnetic gear transmission system in the frequency domain. The overall position of the spectrum amplitude curve of the real response and the spectrum amplitude curve predicted by the model are highly consistent, indicating that the generative adversarial network has better captured the main vibration frequency characteristics and amplitude distribution of the system, verifying the model’s ability to restore the system’s natural frequency response. The mean square error between the real spectrum and the predicted spectrum at different frequency points remains at a low level in most frequency bands, and slight error fluctuations occur at a few peak frequencies, indicating that the model has certain challenges in fitting some high-frequency or complex frequency components, but the overall error control is good. These localized fluctuations may stem from an underrepresentation of the underlying physical mechanisms, such as resonant interactions, within the finite training data for those specific frequencies or reflect inherent limitations in the generator’s capacity to model extremely fine-grained temporal details.

Fig. 5 Spectrum error. a) Real vs. Predicted Spectrum. b) Spectrum Fitting Error over Frequency.

3.4 Prediction time

All timing and throughput evaluations were conducted on a single NVIDIA GeForce RTX 3090 GPU, using PyTorch 1.12.1 with cuDNN acceleration enabled. The finite element simulation method is used to generate the corresponding dynamic response data set for the electromagnetic gear transmission system under eight different working conditions (composed of input frequency, load torque, and gear clearance). A conditional generative adversarial network was constructed, and the generator and discriminator were trained using the operating condition parameters as conditional input. During the training process, a joint loss function and Adam optimizer were used to iteratively optimize the prediction accuracy and stability of the model. For the trained cGAN model, four independent predictions were performed under each set of operating conditions, and the time spent on each prediction was recorded.

Figure 6 shows the distribution of the time required to predict the dynamic response of the electromagnetic gear transmission system in four measurements under eight typical operating conditions. The horizontal axis corresponds to specific working conditions, including different input frequencies, load torques, and gear clearance combinations, and the vertical axis is the prediction time in seconds. Overall, the prediction time remains stable, all within the range of 0.07 to 0.08 s, and the time fluctuation between different measurement times is small, indicating that the model has good calculation efficiency and time stability. There are slight differences in the prediction time between different working conditions. The prediction time is slightly longer under some high-frequency or large clearance conditions, reflecting the impact of complex working conditions on the model calculation load.

Fig. 6 System response performance. (a) 1st measurement. (b) 2nd measurement. (c) 3rd measurement. (d) 4th measurement.

3.5 Physical consistency

Finite element software is used to dynamically simulate the electromagnetic gear transmission system under multiple working conditions (different speeds, torques, and radial displacements) to obtain a high-dimensional dynamic response data set. The data set includes input working condition parameters (such as frequency, torque, and radial displacement) and the corresponding system dynamic response (time series data such as angular velocity and torque). A deep generator based on the residual convolution structure is designed to generate system dynamic response predictions from working condition parameter conditions. Physical consistency constraints (such as angular velocity continuity, torque balance, and residuals of dynamic equations) are introduced as auxiliary losses to promote the generation of results to conform to physical laws. The dynamic equation residual is computed based on the fundamental torque balance for a rotational system: the sum of the electromagnetic torque, the load torque, and the inertial torque equals zero. The residual quantifies the deviation from this equilibrium in the generated response. The conditional generative adversarial network is trained using the preprocessed dataset, and the generator and discriminator are iteratively optimized. Under typical working conditions, the trained cGAN model is used to predict dynamic responses. This paper collects the angular velocity and torque responses output by the model and calculates the prediction error relative to the finite element simulation results. Physical consistency indicators are calculated, including angular velocity error, torque balance error, and dynamic equation residuals, to quantitatively evaluate physical rationality.

Figure 7 shows the results of the physical consistency error analysis of the electromagnetic gear transmission system under eight typical working conditions. The angular velocity error bar chart on the left shows that the relative error of the angular velocity predicted by the model fluctuates between 0.015 and 0.04, indicating that the generative adversarial network can accurately capture the dynamic rotation characteristics of the system. The error increases slightly under the 60 Hz frequency condition, which may be due to the increase in dynamic complexity, which makes prediction difficult. The intermediate torque balance error bar chart shows that the error under each condition is kept within the range of 1.0% to 2.0%, where the error quantifies the relative deviation between the time-averaged torque predicted by the model and the corresponding averaged value obtained from the finite element simulation over the steady-state period. And the torque error is relatively uniform and small, indicating that the model is stable in predicting mechanical balance and has good physical consistency. The box plot of the residuals of the dynamic equation on the right further verifies the physical rationality of the model. The residuals are concentrated between 0.003 and 0.01, and the distribution is relatively compact, indicating that the dynamic response output by the generative model satisfies the system dynamic equation well. Overall, this group of physical consistency index data proves that the dynamic simulation method based on conditional generative adversarial networks can effectively maintain the physical accuracy and stability of the simulation results while ensuring efficient prediction and has good engineering application potential.

Fig. 7 Physical verification. a) Angular velocity error vs. working condition. b) Torque balance error vs. working condition. c) Dynamic equation residual distribution.

3.6 Throughput

This paper selects a representative combination of working conditions, covering different voltage frequencies, load torques, and gear clearance parameters, for a total of eight typical working conditions. Each working condition is standardized and used as model input. The trained generator model is deployed on a fixed hardware platform. Multiple batch-size inputs are prepared, including single-sample and 4-, 8-, 16-, and 32-multi-sample batch inputs, to simulate different application scenarios. This paper runs multiple model inferences for each batch size, records the time taken for each prediction, and calculates the average throughput (the number of samples processed per second). Using single-sample input (batch size = 1), multiple inference tests are performed on eight working conditions, and the average throughput of each working condition is counted and calculated.

The horizontal axis of panel (a) in Figure 8 is the batch size, which is 1, 4, 8, 16, and 32, respectively. The vertical axis is the throughput (the number of samples processed per second). The curve shows that as the batch size increases, the throughput increases significantly, from about 120/s to about 880/s, indicating that batch processing can make full use of hardware resources and improve inference efficiency. The horizontal axis of panel (b) is the name of the specific working condition, involving different frequencies, load torques, and gear clearance combinations. The vertical axis is the throughput of the model under the corresponding working condition. The throughput remains stable under different working conditions, with small fluctuations in values, and is generally maintained in the range of 580 to 630 per s, indicating that the model inference time is less sensitive to changes in working conditions and has good real-time performance and robustness.

Fig. 8 Throughput. a) Throughput vs. Batch Size. b) Throughput under Different Conditions.

4 Discussion

The model exhibits frequency-dependent accuracy, with an MRE of 2.3% in low-frequency conditions versus 5.2% in high-frequency regimes. This degradation reflects the increased complexity of electromagnetic coupling and harmonic content at higher frequencies. Energy fidelity analysis shows total energy differences below 5%, confirming physical plausibility, though minor phase deviations suggest room for temporal alignment improvement.

Unlike conventional data-driven approaches, the conditional architecture enables direct parameter-to-response mapping without retraining, while physical constraints ensure solutions remain within physically feasible space. Model accuracy depends on training data coverage; extrapolation to unseen parameter combinations remains limited. The simplified physics constraints, while effective, do not fully capture multi-physics interactions. Training requires substantial computational resources (∼8 h on RTX 3090), though this one-time cost enables rapid subsequent predictions.

5 Conclusion

This paper proposes a modeling method based on conditional generative adversarial networks to solve the problem of efficient dynamic simulation of electromagnetic gear transmission systems. The model training, conducted on a single NVIDIA RTX 3090 GPU, required approximately 8 h to complete. This paper constructs a multi-condition finite element dataset, designs a generator with a residual structure and a multi-scale discriminator, and combines the joint loss function with physical consistency constraints to achieve a fast and accurate prediction of the system’s dynamic response. Experimental results show that the average relative error of response prediction of this method is controlled within 5.2%, and the prediction time is less than 0.08 s. It effectively realizes the rapid modeling and multi-condition dynamic behavior simulation of electromagnetic gear systems and provides technical support for the intelligent design and efficient performance evaluation of electromagnetic transmission systems. The prediction performance of the method is contingent on the coverage of the training data; its reliability for extreme operating conditions beyond the sampled parameter ranges warrants further investigation.

Funding

This work was supported by the Hunan Provincial Natural Science Foundation Joint Fund Project “Damage Mechanism and Reliability Optimization of High-Speed Spur Gears in Offshore Wind Turbines under Spatiotemporal Uncertainty” (Grant No. 2024JJ7084). Yiyang City’s “Unveiling the List and Taking Command” Major Scientific and Technological Project: Development of Key Technologies for the Processing of High-Performance Finite Element Gears for High-Speed Trains.

Conflicts of interest

The authors declare no conflict of interest.

Data availability statement

This article has no associated data generated and/or analyzed. Data associated with this article cannot be disclosed due to legal/ethical/other reason.

Author contribution statement

Qi Zeng and Junyang Chen designed the research study. Weimin Shi, Yuzhen Liu, and Xuni Yin analyzed the data. Qi Zeng wrote the manuscript. All authors contributed to editorial changes in the manuscript. All authors read and approved the final manuscript.

References

All Tables

All Figures

	Fig. 1 Generator network.
In the text

	Fig. 2 Model training. a) Learning Rate Decay. b) Generator / Discriminator / Validation Loss.
In the text

	Fig. 3 Prediction performance.
In the text

	Fig. 4 Energy fidelity and error characteristics.
In the text

	Fig. 5 Spectrum error. a) Real vs. Predicted Spectrum. b) Spectrum Fitting Error over Frequency.
In the text

	Fig. 6 System response performance. (a) 1st measurement. (b) 2nd measurement. (c) 3rd measurement. (d) 4th measurement.
In the text

	Fig. 7 Physical verification. a) Angular velocity error vs. working condition. b) Torque balance error vs. working condition. c) Dynamic equation residual distribution.
In the text

	Fig. 8 Throughput. a) Throughput vs. Batch Size. b) Throughput under Different Conditions.
In the text