1. Data imbalance problem.

Traditional fault diagnosis methods face the challenge of data imbalance, where normal data vastly outnumbers fault data. While some existing research has addressed this issue by oversampling, undersampling, or generating synthetic data, these traditional methods often fail to capture complex fault patterns accurately. Although techniques such as SMOTE (Synthetic Minority Over-sampling Technique) and other data augmentation methods have made progress in some areas, their ability to generate complex fault data is limited [13]. In contrast, this paper innovatively uses Generative Adversarial Networks (GAN) to generate diverse fault data, expanding the dataset and realistically simulating various fault patterns, effectively solving the data imbalance issue and enhancing the model’s learning and generalization capabilities.

2. Intelligence level and limitations of data-driven fault diagnosis models.

Existing digital twin and machine learning methods often rely on large amounts of historical data for training. However, in certain cases (e.g., data scarcity or extreme operational conditions), traditional purely data-driven methods face significant challenges. Most existing methods rely on expert knowledge and heuristic rules, which are not effective in detecting latent faults in complex systems. Although PINNs (Physics-Informed Neural Networks) have made progress in combining physical models with data-driven methods, they still struggle with generalization when faced with complex and limited datasets [14]. This paper innovatively introduces the inverse PINN technique, which combines the physical constraints of the equipment with sensor data for fault inference, greatly improving the accuracy and lead time for fault prediction and addressing the challenges of traditional methods in handling complex equipment faults.

3. Integration of physical constraints and data-driven models.

Although some existing studies have attempted to combine digital twin technology with physical constraints, they often lack sufficient data support or the precision of the physical models. For example, Fritzsche et al. (2020) proposed a digital twin-based approach for lifecycle management of medical equipment, primarily focusing on operational monitoring and predictive maintenance but did not address how to accurately diagnose faults when data is scarce [15]. Moreover, most studies rely heavily on a large number of sensors to simulate the equipment’s operational state, but such data is often incomplete or noisy, leading to lower diagnostic accuracy and stability. This paper combines GAN-generated fault data with the inverse PINN model, leveraging the physical model to infer the internal state of the equipment, significantly improving fault diagnosis accuracy and robustness.

4. Fault prediction capabilities and real-time performance.

Most existing fault prediction models are based on traditional time-series models like LSTM and ARIMA, which have limitations in capturing complex equipment behaviors and processing high-dimensional, multi-source data [16]. For example, while LSTM excels at time-series data processing, it suffers from gradient vanishing problems when dealing with long-term dependencies in complex equipment, affecting prediction accuracy and real-time performance. In contrast, this paper innovatively uses the Transformer model, which overcomes the limitations of LSTM through its self-attention mechanism, significantly improving fault prediction accuracy and timeliness. Additionally, by combining GAN-generated fault data, the model benefits from a larger and more diverse training set, enhancing robustness and generalization.

5. Scalability and multi-scenario applications.

Most existing research focuses on small-scale single equipment systems and lacks validation on the scalability of large medical equipment systems or cross-scenario applications. While some studies have explored equipment scheduling and resource optimization, such as Gao et al. (2019), they are usually limited to single hospital or small-scale scenarios and do not effectively address collaborative equipment scheduling in complex, multi-hospital environments [17]. The model framework proposed in this paper, combining digital twin, inverse PINN, and GAN-generated fault data, not only performs well in fault diagnosis for single equipment but also exhibits strong scalability and can be extended to multi-device, multi-hospital collaborative scheduling and fault prediction.

6. Integration of physical constraints with deep learning models.

Traditional digital twin models primarily focus on monitoring and simulating equipment states, lacking the integration of deep learning algorithms, which limits their adaptability in processing complex environment data [18]. While some studies have combined physical models with neural networks, they mainly focus on data-driven optimization and prediction without fully considering how to integrate physical laws with deep learning for complex equipment system fault inference. This paper integrates the physical model with deep learning techniques through the inverse PINN, ensuring physical consistency in the model and greatly improving fault prediction and reasoning capabilities, overcoming the limitations of traditional purely data-driven models.

1. Principles and implementation of inverse PINNs.

Inverse Physics-Informed Neural Networks (Inverse PINNs) are a technique that deduces the initial conditions and potential physical states of equipment by analyzing its monitoring data and operating status. Compared to traditional fault diagnosis methods, the core advantage of Inverse PINNs lies in their ability to integrate the equipment’s physical model with real-time data, capturing state variables that are difficult to observe directly in complex systems, thereby diagnosing the root cause of equipment failures with greater precision. The key idea of Inverse PINNs is to solve the inverse problem through neural networks, meaning that, starting from the equipment’s monitoring data and combining it with the physical model, the initial conditions and sources of faults are deduced in reverse. Traditional PINN models focus on solving the forward problem of physical equations via neural networks, where given the initial and boundary conditions, the model predicts the future behavior of the system. In contrast, Inverse PINN models take the observed monitoring data and, under the constraints of the physical model, deduce the system’s initial state and potential fault locations. In this paper’s application, Inverse PINNs are used for fault diagnosis in medical imaging equipment.

The principle of PINNs involves using neural networks to solve partial differential equations (PDEs) in physical systems, incorporating physical constraints (such as physical laws, boundary conditions, etc.) into the loss function. Unlike traditional PINNs, which solve forward problems, Inverse PINNs focus on deducing the system’s initial conditions and potential fault sources from observed outcome data. First, Inverse PINNs use equipment monitoring data and operational status, combined with physical information, to transform this data into neural network inputs. Through the process of minimizing the loss function, the model deduces the root cause of the equipment failure in reverse.

The goal of Inverse PINNs is to adjust the neural network parameters by minimizing the loss function, ensuring that the model both adheres to physical laws and most accurately explains the monitoring data. Mathematically, this can be expressed as:

Where represents the physical state predicted by the Inverse PINN model, and uuu is the actual physical state of the equipment. By incorporating the equipment’s physical model, the Inverse PINN bridges the gap between data and physical laws, allowing it to accurately infer faults in complex physical scenarios.

The loss function of this model generally consists of the following components:

1. Data Loss): Measures the error between the equipment’s monitoring data and the model’s predictions.
2. Physics Loss(: Ensures that the predicted physical state complies with the physical equations governing the equipment’s operation.
3. Boundary Condition Loss() and Initial Condition Loss(): Ensures that the solution satisfies the initial and boundary conditions of the equipment.

By combining digital twin data with the Inverse PINN model, we can not only monitor the equipment’s current operational state but also deduce its internal physical state through reverse inference, pinpointing potential fault sources. This process involves integrating the physical model of the equipment during operation, monitoring data, and boundary conditions to form a comprehensive diagnostic procedure. This physical model provides critical constraints for the Inverse PINN, allowing the model to reverse-deduce equipment faults using real-time data and perform data-driven fault diagnosis in combination with the physical model.

2. Inverse PINN process for fault diagnosis in medical imaging equipment.

In the fault diagnosis of medical imaging equipment, the internal physical states of the device (such as temperature distribution, vibration patterns, etc.) are often difficult to observe directly. However, Inverse PINNs, by leveraging monitoring data (e.g., readings from external temperature sensors, vibration data from the device surface) and integrating the device’s physical model, can infer the internal physical state of the equipment.

For instance, consider a localized overheating issue within a piece of equipment. Using the heat conduction equation, we can construct a temperature model of the device. Combining this with external temperature monitoring data, Inverse PINNs can reverse-infer the internal temperature distribution of the device. In this way, when a specific component experiences abnormal heat generation, the Inverse PINN can promptly identify and localize the fault point.

The operation of medical imaging equipment involves various complex physical phenomena, such as heat conduction, electromagnetic fields, and vibrations. The application of Inverse PINNs in such scenarios follows several key steps:

Data Acquisition and Preprocessing: Real-time data is collected from the medical imaging equipment during its operation. This data includes parameters such as internal temperature, pressure, and vibration, along with historical monitoring data when faults occur.

Physical Model Definition: The physical model of the equipment is defined, encompassing the heat conduction equation, vibration equation, and electromagnetic field equation.These models provide constraints for the Inverse PINN, allowing the neural network to make predictions while adhering to physical laws. Suppose the physical model of the device can be represented as follows:

where represents the device’s physical equations (such as the heat conduction or vibration equation), and denotes the physical state of the device over space and time.

Neural Network Architecture Design: A deep neural network model is constructed , taking spatial position x and time t as inputs to predict the physical state of the equipment. Through the framework of the physics-informed neural network, physical equations are embedded into the loss function as constraints. Typical network architectures include multiple fully connected layers, with activation functions such as ReLU or tanh to enhance the model’s non-linear expression capability.

Loss Function Definition and Optimization: The loss function comprises data , physics constraint loss , boundary condition loss , and initial condition loss . The objective of the loss function is to minimize the discrepancy between the monitoring data and the physical model’s predicted results while ensuring the solution satisfies the equipment’s physical constraints.

Reverse Deduction of Fault Source: Through the training of the neural network model, the Inverse PINN iteratively updates the network weights until the loss function is minimized. This process ultimately outputs the initial state of the device and the potential fault source. For instance, in a heat conduction problem, by backtracking the temperature distribution, the model can identify the root cause of abnormal heat generation within the device and deduce the potential fault location.

Result Interpretation and Visualization: The final output of the Inverse PINN is the initial physical state of the equipment and the location of the fault source. To facilitate interpretation and analysis, this paper utilizes digital twin technology to visually represent the state of the equipment, aiding maintenance personnel in quickly locating the issue and addressing it.

By following these steps, Inverse PINNs enable more accurate fault diagnosis in complex medical imaging equipment, enhancing the precision and efficiency of identifying and resolving issues.

1. Importance of the physical model.

A physical model is a mathematical representation that describes the operational behavior of a piece of equipment. It includes the fundamental physical characteristics and governing principles of the device. In Inverse PINNs, these physical models provide direction and constraints for the model’s learning process, ensuring that the network outputs results that are physically meaningful. For example, in medical imaging equipment, the operational state may be affected by factors such as temperature, pressure, and vibration. The physical model can describe how these factors influence equipment performance through corresponding equations.

2. Learning mechanism of inverse PINN.

Inverse PINNs utilize the constraints from physical models by embedding physical equations into the neural network’s loss function. By minimizing the loss function, the Inverse PINN can simultaneously optimize the network parameters while satisfying the physical constraints. This method allows the neural network’s learning process to be not only data-driven but also to fully consider the physical laws. For instance, when there is a change in the operational state of the equipment, the Inverse PINN can quickly adapt to this change and make adjustments based on the physical model, thereby achieving more accurate fault diagnosis.

3. Advantages of synergistic interaction.

The synergistic interaction between Inverse PINNs and physical models has several advantages:

Improved Accuracy: By introducing physical constraints, Inverse PINNs can reduce errors caused by sparse data or noise, thus improving the accuracy of fault diagnosis.

Enhanced Robustness: When dealing with complex and dynamic physical systems, Inverse PINNs can effectively manage uncertainty, providing more stable and reliable results.

Providing Physical Explanations: Inverse PINNs not only provide fault diagnosis results but also explain the causes of faults, offering more valuable references for maintenance personnel.

Overall, the close integration of Inverse PINNs and physical models demonstrates superior performance in fault diagnosis for precision equipment like medical imaging devices.

4. Advantages and limitations.

This paper presents a unified fault diagnosis framework that, through the application of the Inverse PINN model, combines physical models with real-time monitoring data to achieve a complete closed-loop process from fault detection to root cause analysis. This framework not only improves the accuracy of fault diagnosis but also enhances the intelligence level of equipment management, providing robust support for the safe and efficient operation of medical equipment.

Inverse PINNs have significant advantages in fault diagnosis of complex systems, particularly in the fault diagnosis of medical imaging equipment. By integrating digital twin technology, they achieve high-precision predictions of the equipment’s state. However, Inverse PINNs also have their limitations, mainly in the following aspects:

Strong Dependence on the Initial Model: If the physical model or boundary conditions are improperly set, the accuracy of the diagnosis may be affected.

High Computational Resource Requirements: The training process of Inverse PINNs typically requires large amounts of data and computational resources.

Through the application of the aforementioned processes and techniques, Inverse PINNs can diagnose potential faults in complex medical equipment in real-time and with precision, providing strong technical support for equipment maintenance. The synergistic interaction between Inverse PINNs and physical models, coupled with result validation and system integration to form a comprehensive fault diagnosis framework, not only highlights the innovation and practicality of this research but also offers a feasible solution for intelligent maintenance and fault prediction of medical equipment. This research provides a new perspective and direction for future equipment management and maintenance, bearing significant theoretical and practical value.

1. Impact of data imbalance on model performance.

In machine learning model training, particularly in classification tasks, the issue of data imbalance can have the following effects:

Model Bias Toward Normal Data: Due to the large amount of normal data, the model is more likely to learn patterns from normal data, ignoring fault data. This results in the model’s poor predictive ability for fault data during testing.

Shift in Decision Boundary: Data imbalance may cause the decision boundary of the classifier to shift towards the minority class (fault data), leading to a lower recall rate for fault detection, meaning faults are more likely to be misclassified as normal conditions.

Loss Function Bias Toward Normal Data: Standard loss functions (such as mean squared error or cross-entropy) may focus more on optimizing normal data in cases of data imbalance, further exacerbating the model’s neglect of fault data.

To solve these issues, this paper combines the Inverse PINN approach with data synthesis and augmentation techniques to balance the proportion of fault and normal data in the dataset, thereby improving the model’s generalization ability and accuracy in fault diagnosis.

2. Data synthesis and augmentation techniques.

To address the problem of data imbalance, this paper adopts two strategies: synthetic fault data and data augmentation techniques, each designed to increase the quantity of fault data and enrich the data distribution from different perspectives.

Synthetic fault data: Based on the physical model of the equipment, we can simulate the physical response of the equipment under various fault conditions using Inverse PINNs. Since the physical state of the equipment is closely related to its fault modes, Inverse PINNs can not only be used for fault diagnosis but also for generating realistic fault data. Specifically, we can artificially set different fault conditions (e.g., abnormal temperature rise, excessive vibration) based on the physical model and use Inverse PINNs to generate operational data of the equipment under these fault conditions.

For example, suppose we have constructed a thermal conduction model of the equipment, such as the previously discussed heat conduction equation:

In normal working conditions, the heat source term q is known and fixed, making the temperature changes within the equipment predictable. However, under fault conditions, the heat source term q may exhibit anomalies, such as overheating caused by equipment failure. By introducing different fault conditions (such as increasing a local heat source term) into the physical model, we can generate temperature distribution data of the equipment under various fault modes.

The synthetic fault data generated in this way not only maintain physical consistency but also effectively expand the quantity of fault data, helping to balance the dataset. Compared to traditional data synthesis techniques, such as oversampling-based SMOTE algorithms, this method has higher physical plausibility and realism.

Data augmentation techniques: Data augmentation techniques involve applying certain transformations to existing data to generate new samples, thereby enhancing the diversity of the dataset. To address the scarcity of fault data, this paper adopts several common data augmentation techniques to expand the fault data:

Noise Injection: By injecting small amounts of random noise into the fault data, we can generate more samples. Noise injection can be done by adding Gaussian noise or noise from other distributions, helping the model improve its robustness to different fault signals.

Data Rotation and Mirror Transformation: Although the physical state of medical imaging equipment does not change with spatial transformations, rotating or mirroring the data collected by sensors can generate samples with different spatial characteristics.

Temporal Perturbation: Faults in medical imaging equipment typically occur over specific time intervals. By perturbing the time series of fault events, more samples can be generated that show slight temporal differences but still represent fault conditions.

The core idea behind these augmentation techniques is to generate new variations of fault data through reasonable transformations, allowing the model to learn from a greater variety of samples, thereby improving its performance on the minority class (fault class) data.

3. Evaluation of the effectiveness of balancing the dataset.

To assess the improvements in model performance resulting from handling data imbalance, this paper designs a set of experiments. The model’s performance is evaluated using the following metrics:

Accuracy: Measures the overall performance of the model on all test data.

Recall: Specifically focuses on the model’s performance on fault data, measuring the model’s ability to detect fault samples.

Precision: Evaluates the accuracy of the model in classifying fault data.

F1 Score: A comprehensive evaluation metric that combines recall and precision, providing a more holistic reflection of the model’s performance on imbalanced datasets.

By combining the Inverse PINN model with synthetic fault data and data augmentation techniques, the experimental results show:

Significant improvement in recall, indicating that the model becomes more sensitive in detecting faults.

Increase in precision, showing that the model’s misclassification rate for fault data decreases.

Notable improvement in F1 score after introducing the balanced dataset, demonstrating the effectiveness of data augmentation and synthesis techniques.

4. Other methods to address data imbalance.

In addition to the aforementioned methods, several other common techniques can be used to handle data imbalance, such as:Weighted Loss Function: Assigns higher weights to minority classes (fault data), making the model pay more attention to fault samples during training. This method encourages the model to learn better on fault samples by modifying the weight parameters in the loss function.

where represents the weight of the sample. The weight for fault data can be set higher than for normal data, thus balancing the error during the training process.Downsampling Normal Data: In cases of extreme imbalance, the number of normal data samples can be reduced to achieve balance. However, this method may lead to loss of information in some cases and should be used cautiously.

1. Introduction to generative adversarial networks (GANs).

A Generative Adversarial Network (GAN) consists of a generator and a discriminator, both of which are trained through an adversarial process. The generator’s task is to generate realistic data samples from random noise, while the discriminator must distinguish between the generated data and real data. Through this adversarial training process, both the generator and discriminator improve, resulting in increasingly realistic generated data [12].

2. Working principle of GAN for generating fault data.

A GAN is composed of two components: the generator and the discriminator. The generator is responsible for creating new fault data, while the discriminator determines whether the generated data is real or fake. Through adversarial training, the quality of the generated data improves [18]. The training process of GAN follows these steps:

Generator: The generator takes random noise z (such as from a standard normal distribution) as input and generates fault data .

Discriminator: The discriminatorreceives input data and judges whether it is real fault data. Feedback from the discriminator helps improve the generator’s ability to produce realistic data.

Loss Function: The goal of the GAN is to optimize the following loss function to guide the adversarial process between the generator and the discriminator:

where:

1. represents the samples generated by the generator using random noise.
2. is the discriminator’s assessment of the realness of the data.
3. represents the distribution of real data, and represents the distribution of noise.

Training Process: The generator and discriminator are optimized alternately, gradually improving the generator’s ability to produce samples that are highly similar to real fault data.

Throughout the training process, the generator continuously learns how to generate more realistic data, while the discriminator improves its ability to distinguish between generated and real data. When the GAN reaches convergence, the distribution of the samples generated by the generator will closely resemble that of real fault data.

1. Loss function of inverse PINN.

The loss function for the Inverse PINN can be expressed by the following formula:

where:

: Observation data loss, which reflects the discrepancy between the model’s output and the actual observed data. The formula is typically as follows:

where is the model’s predicted result, is the actual observed data, and N is the number of observation samples.

: Physical constraint loss, ensuring that the model’s output adheres to physical equations (e.g., the heat conduction equation). It is usually represented in residual form:

where represents the residual of the physical equation, and Nis the number of physical constraint samples.

: Boundary condition loss, ensuring that the model’s output meets the expected results under boundary conditions. It can be represented as:

where is the number of boundary condition samples.

: Initial condition loss, ensuring that the model’s state is correct under initial conditions:

where is the actual initial value, and is the model’s predicted initial state.

2. GAN-generated data loss.

When integrating GAN with Inverse PINN, the loss function for the GAN-generated data is typically expressed as:

where:

represents the data generated by the GAN.

:is the real fault data.

is the number of generated data samples.

3. Combined loss function.

By combining the GAN-generated data loss with the Inverse PINN loss function, we obtain a new comprehensive loss function:

where:

• λ: A weight factor used to balance the influence between physical constraints and the generated data.

How physical constraints are incorporated into the Inverse PINN framework, emphasizing their impact on model training and fault prediction

Mathematical Representation of Physical Constraints

The governing physical equations (e.g., heat conduction, wave propagation) are embedded into the neural network through physics-informed loss functions.

The inverse PINN optimizes a multi-objective loss function

Neural Network Architecture Enhancement:

We now explicitly describe how the PINN-based architecture is adapted for inverse problem-solving:

Input: Sensor readings (e.g., surface temperature, vibration patterns).Hidden Layers: Modified fully connected layers (MLP) with physics-based activation constraints.

Output: Estimated internal temperature distribution and vibration modes, used for fault localization.

The regularization mechanism ensures that the model predictions remain physically plausible, improving generalization and robustness.

With these refinements, we provide a clearer and more structured explanation of how physics-based constraints guide deep learning model optimization.

1. Experimental dataset.

The experimental dataset includes real operational data and known fault data from medical imaging equipment, specifically from the United Imaging uCT760 CT scanner. The generator in the GAN utilizes the distribution of real fault data collected from this specific device to generate virtual fault data that closely resembles the real data. These synthetic data not only help the model balance the fault samples in the training dataset but also enrich fault scenarios, which aids in improving the robustness of the model. To facilitate data testing, we simulated the following two typical fault scenarios using GAN:

Temperature Abnormality: The internal sensor of the uCT760 detects a continuous rise in temperature, a common issue in medical imaging equipment. The generated fault data simulate the equipment’s operational status under different temperature conditions, which can impact the system’s performance and safety.

Sensor Fault: Fault data are generated for scenarios where the sensor itself is damaged or provides inaccurate readings. This issue is common in medical imaging equipment like the uCT760, where sensor malfunctions can result in misleading or incomplete diagnostic information.

The operational data of the equipment cover several key monitoring indicators, such as temperature, pressure, and vibration. The dataset can be divided into the following two categories:

Normal Operational Data: Data of the equipment operating under normal conditions, accounting for the majority of the experimental data. These data reflect the equipment’s working status without faults.

Known Fault Data: Monitoring data when the equipment experiences various known faults (such as temperature abnormalities, mechanical failures, and sensor faults). These data are used for fault detection tasks by the model and are relatively scarce.

The experimental dataset consists of \(N = 10000\) normal data samples and \(M = 500\) fault data samples, with a data dimension of \(d = 12\) (corresponding to the readings of 12 sensors on the equipment). The dataset undergoes standardization preprocessing to ensure that data from all dimensions are within the same numerical range, improving the efficiency and accuracy of model training.

2. Hyperparameters for GAN and inverse PINN.

GAN Hyperparameters: Generator and Discriminator Architecture: Both the generator and discriminator are multi - layer perceptrons (MLPs). The generator has 3 hidden layers with 256, 128, and 64 neurons respectively, followed by an output layer of size 12 (matching the data dimension). The discriminator also has 3 hidden layers with 128, 64, and 32 neurons respectively, and a single - neuron output layer for binary classification.

Learning Rate: The learning rate for both the generator and discriminator is set to \(0.0002\).

Batch Size: The batch size during training is 64.

Number of Epochs: The GAN is trained for 500 epochs.

Optimizer: We use the Adam optimizer with a beta1 value of 0.5 and a beta2 value of 0.999.

Inverse PINN Hyperparameters: Neural Network Architecture: The Inverse PINN uses a feed - forward neural network with 5 hidden layers, each having 100 neurons.

Learning Rate: The learning rate for the Inverse PINN is set to \(0.001\).

Batch Size: A batch size of 32 is used during training.

Number of Epochs: The Inverse PINN is trained for 1000 epochs.

Optimizer: The Adam optimizer is also used for the Inverse PINN, with default beta values (\(\beta_1 = 0.9\), \(\beta_2 = 0.999\)).

Weight of Physical Loss: In the loss function of the Inverse PINN, the weight of the physical loss term is set to 0.5, while the weight of the data - fitting loss term is 0.5.

3. Experiment setup.

The experiment is divided into two parts: Traditional Fault Diagnosis Methods vs. Inverse PINN Model.

This section aims to compare the effectiveness of traditional fault diagnosis methods based on statistical and machine learning approaches with the Inverse PINN model proposed in this paper. Traditional methods include:

Support Vector Machine (SVM): A classic binary classification model used to distinguish between normal data and fault data. The kernel function is set to the radial basis function (RBF), and the regularization parameter C is set to 1.

Random Forest: A model that integrates multiple decision trees and is suitable for handling high - dimensional equipment data. The number of decision trees in the forest is set to 100.

Logistic Regression: A binary classification method based on a linear model, used to establish relationships between normal data and fault data. The regularization strength C is set to 1.

The Inverse PINN model proposed in this paper combines physical models of the equipment with data - driven methods to infer faults using monitoring data from the equipment.

Impact of Fault Data Generation on Model Performance: Due to the scarcity of fault data in real - world operations, we used GAN to generate synthetic fault data and combined these with real fault data for model training. The impact of the generated data on improving model performance was evaluated by comparing the performance of the following models:

Model trained on both real and generated data: The model is trained on both real data and synthetic fault data generated by GAN, and its performance is then evaluated.

By providing these detailed hyperparameters, it becomes easier for other researchers to reproduce the experimental results.

1. Discriminator design.

The discriminator receives either real data or generated data samples and classifies them through a neural network, outputting the probability that the data is real. The goal of the discriminator is to enhance its ability to distinguish between real data and generated data.

The network structure of the discriminator is as follows:

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2. Training GAN.

The generator and discriminator undergo adversarial training through the following steps:Training the Discriminator: For given real data and data generated by the generator, the discriminator is optimized to correctly distinguish between these two types of data. The loss function for the discriminator is:

Training the Generator: The objective of the generator is to produce data that the discriminator classifies as “real,” so the loss function for the generator is:

Both components are optimized alternately, allowing the generator to gradually learn how to create realistic fault data while the discriminator becomes increasingly adept at differentiating between real data and synthetic data.Generating Fault Data: Once the GAN training is complete, the generator can produce a large volume of device operation data under various fault conditions. This synthetic fault data, when combined with real fault data, forms a more balanced dataset.

Compared to traditional data augmentation methods, the fault data generated by GAN has the following advantages:

Diversity: GAN is capable of generating operational data for devices under various fault conditions, enriching the diversity of the dataset.

Realism: Through adversarial training, the data samples generated by GAN closely resemble the distribution of real data, ensuring the reliability and usability of the generated data.

Dynamic Fault Simulation: By adjusting the noise zzz input to the generator, GAN can generate operational data for devices under different fault modes, helping the model adapt to various potential fault scenarios.

1. Performance evaluation metrics.

To comprehensively assess the performance of each model, several common classification metrics were employed:

Accuracy: The proportion of correctly classified samples to the total number of samples. The formula is as follows:

where TP is the true positive, TN is the true negative, FP is the false positive, and FN is the false negative.

Precision: The proportion of correctly classified fault samples to all samples predicted as faults. The formula is:

Recall: The proportion of actual faults correctly identified as faults, measuring the model’s ability to capture faults. The formula is:

• F1 Score: The harmonic mean of precision and recall, used to balance the classification capabilities of the model. The formula is:

2. Traditional methods vs. inverse PINN.

First, the performance of traditional fault diagnosis methods (support vector machine, random forest, logistic regression) was compared with that of the inverse PINN model. The experimental results show that the inverse PINN model outperforms traditional methods in both the accuracy of fault identification and the lead time for fault prediction. Particularly in high-dimensional complex data scenarios, the inverse PINN effectively captures subtle changes preceding a fault’s occurrence by combining physical constraints with data-driven approaches.

The S1 Fig below illustrates the comparison of the performance of traditional fault diagnosis models with the proposed fault diagnosis model that integrates GAN and inverse PINN across different performance metrics. By comparing the following four key performance indicators (accuracy, recall, F1 score, and precision), it is evident that:

Accuracy: The improved model achieved an accuracy of 92%, which is a 7 percentage point increase over the traditional models.

Recall: The improved model reached a recall of 88%, showing a significant enhancement compared to the 78% of the traditional models.

F1 Score: The F1 score of the proposed model is 90%, while the traditional model scored 81%, indicating an overall improvement in performance.

Precision: The precision of the improved model increased to 89%, while the traditional model was at 82%.

3. Handling Data Imbalance and the Effectiveness of GAN-Generated Fault Data.

To address the issue of data imbalance [19], this study utilized Generative Adversarial Networks (GANs) to generate fault data, enhancing the proportion of fault samples in the training set. This strategy significantly improved the robustness and performance of the inverse PINN model under small sample data conditions. Experiments showed that the inverse PINN model using synthetic data outperformed the model that did not use synthetic data in terms of accuracy, recall, and F1 score for fault detection, demonstrating the positive impact of GAN-generated fault data on model performance.

The Table 1 below shows the changes in model performance before and after addressing the imbalance in data:

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Table 1. The changes in model performance.

https://doi.org/10.1371/journal.pone.0324180.t001

4. Reference basis for maintenance and repair.

The application of the inverse PINN model not only enhances the fault diagnosis capability of medical imaging equipment but also provides important reference data for equipment maintenance and repair. Specifically, by combining the physical model of the equipment with real-time monitoring data, the inverse PINN can achieve a comprehensive derivation from data to physical state. This process not only facilitates early identification of faults but also provides a scientific basis for formulating targeted maintenance plans [20].

In this paper, we utilize digital twin technology to visualize the diagnostic results of the inverse PINN model, ensuring that maintenance personnel can intuitively understand the equipment’s status. The key elements include:

Data Fusion: Integrating information from equipment sensors, monitoring systems, and historical fault data to provide comprehensive input for the inverse PINN model [21].

Visualization Tools: Developing user-friendly interfaces that  display real-time equipment status, fault predictions, and historical data, aiding maintenance personnel in quickly identifying problems and making informed decisions [24].

We have expanded the comparative analysis section to contrast our approach with alternative hybrid fault diagnosis models. Table 2 summarizes the key differences between our proposed method and other state-of-the-art techniques, highlighting the advantages of integrating digital twin, inverse PINN, and GAN-based fault augmentation [25,26].

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Table 2. Comparison with Other State-of-the-Art Hybrid Methods.

https://doi.org/10.1371/journal.pone.0324180.t002

Key Findings from Experiments:

Our approach outperforms purely data-driven methods in fault localization accuracy and generalization on unseen failures.GAN-based fault augmentation significantly improves model robustness in low-data regimes.Inverse PINN constraints improve fault prediction precision by 18% compared to standard PINN models.These additions strengthen our work’s comparative validity and highlight its unique advantages over existing methodologies.