Physics-guided neural network in gearbox fault diagnosis

Abstract

Intelligent fault diagnosis involves deep learning (DL) algorithms. A hybrid model that integrates both physics-based and data-driven approaches can offer effective health monitoring solutions. In this study, a physics-guided framework is developed, in comparison with classical feedforward neural network (FNN) and convolutional neural network (CNN) models, to be employed in gearbox fault detection. The proposed framework integrates a main vibration-data-driven model, that is called CNN-A trained on either envelope or fast Fourier transform (FFT) signals, with a complementary CNN-B that specifically learns from the gear mesh frequency (GMF) harmonics and their sidebands. The accuracy of fault detection is improved in the framework through a physics-and-data-driven (PDD) parameter fusion mechanism in which the model updates the weights in CNN-A by considering the weights of CNN-B, based on the healthy and faulty classes. The physics-guided DL approach is evaluated using experimental vibration measurements of a gearbox test rig. Experimental data with varying speeds and loads from another gearbox test bench is utilized for validation. It is concluded that the framework is robust under varying operating conditions, achieving higher and more stable accuracy, that is, higher performance reliability, compared to the CNN and FNN models.

Introduction

Gearboxes are widely used in industry, for example, in wind turbines and transmissions, due to their versatile functions. Their reliable operation is crucial in minimizing downtime. Modern machinery is becoming increasingly complex, while operating conditions are typically harsh and uncertain. An unanticipated machine failure requires unplanned maintenance, and maintenance delays often affect other related components in the system. Monitoring the health condition of a machine can help increase machine availability and reduce the cost of downtime. ¹

In condition monitoring, feature extraction maps the original signals into statistical parameters that convey machine status information. The design of the feature extractor plays a very important role in obtaining high accuracy. Feature extraction methods can be divided into two types: physics-based and data-driven. Intelligent fault diagnosis in rotating machinery utilizes domain expertise and data-driven approaches and a capability to process mechanical data. ² Several machine learning (ML)-based models can be utilized to achieve fault detection, such as k-nearest neighbors (kNNs),^3,4 artificial neural networks (ANNs),^5,6 support vector machine,^7,8 and Bayesian networks.^9–11 Dynamic Bayesian networks-based method was proposed by Cai et al. ⁹ to detect faults in electronic systems. This approach aimed to reduce the overall complexities of Bayesian networks in fault diagnosis. Cai et al. ¹¹ introduced a method that employs object-oriented Bayesian networks for real-time fault diagnosis within a subsea production system. Recently, convolutional neural networks (CNNs) have been widely used for machinery, for example, bearing fault detection, due to their ability to automatically extract and learn intricate hierarchical features from raw data.^12,13 For instance, a CNN can automatically extract local features through the use of convolution and pooling operations, eliminating the need for manual feature extraction. Once the features have been extracted, they are fed into the fault detection model to assess the health status of the machine. Signal features can be extracted from the time domain, ¹⁴ frequency domain, ¹⁵ and the time-frequency domain. ¹⁶ Features of the time-frequency domain can be extracted using the wavelet packet transform,^17,18 Morlet wavelet transform, ¹⁹ and short-time Fourier transform. ²⁰

Xing et al. ²¹ introduced a new loss function to enhance the robustness of an autoencoder in bearing fault detection. Binanzer and Dazer ²² explored an autoencoder-based technique for early-stage damage detection in gearboxes, including various operating conditions, without labeled training data. Liu et al. ²³ also employed an autoencoder-based model for early-stage gearbox fault detection. Stacked and convolutional autoencoders can learn healthy gearbox representations from vibration data and detect anomalies via reconstruction error analysis.^23,24 However, these approaches are largely data-driven and do not explicitly embed physical fault mechanisms such as gear-mesh modulation or sideband structures, which may limit robustness under varying operating conditions.^23,25 Liu et al. ²⁵ developed a long short-term memory (LSTM) autoencoder model to identify anomalies in multivariate time series data. This approach aims to detect white etching cracks (WECs) in gearbox bearings, a critical issue in wind turbine technology.

ML models can achieve acceptable accuracy in fault detection; however, they are limited in their ability to fully analyze and extract value from large datasets, and integrating physical knowledge into these models, by focusing on relevant fault information and reducing misclassifications, remains a challenge. Advanced deep learning (DL) models have been created that incorporate physical principles. Raissi et al. ²⁶ presented a DL approach to address complex problems governed by nonlinear partial differential equations (PDEs).

Physics-informed machine learning (PIML) integrates ANNs with physical equations, such as differential equations, during training. Physics-informed neural networks (PINNs) use a loss function that includes both data-driven and physics-based components. Karniadakis et al. ²⁷ reviewed trends in PIML methodology, as an innovative approach applied to various domains, such as lubrication by solving the Reynolds differential equation. Marian and Tremmel ²⁸ explored the recent advancements and applications of PIML in tribology. Sheng Shen et al. ²⁹ proposed a method integrating a physics-informed threshold model with a deep CNN for bearing fault detection. The model uses features such as vibration amplitude and frequency response, which are indicative of different fault conditions. Ni et al. ³⁰ explored a novel diagnostic framework for bearing fault diagnosis under varying operational conditions. They proposed a physics-informed residual network (PIResNet) that includes three main components, that is, modal-property-dominant-generated layer, domain-conversion layer, and parallel bi-channel residual learning architecture. Zhu et al. ³¹ provided a comprehensive overview of how DL techniques are utilized to enhance the fault diagnosis of rotating machinery. Qin et al. ³² explored inverse PINNs to recognize dynamic model parameters for bearing fault diagnosis, as it leverages digital twin technology to handle sample imbalance issues. Krupp et al. ³³ introduced an approach that combines the strengths of physics-based modeling with data-driven neural networks to diagnose faults in rolling element bearings. Their methodology integrates physics-guided features into a neural network architecture, enhancing the model’s ability to interpret and learn from vibration data. Sadoughi and Hu ³⁴ integrated the fault characteristic information into a CNN model by proposing a physics-based methodology that employs spectral kurtosis and envelope analysis to isolate sidebands from raw sensor data while reducing non-transient elements of the signals. Their physics-based convolutional neural network (PCNN) method can simultaneously monitor multiple bearings and identify faults. Chen et al. ³⁵ developed an innovative approach using a physical knowledge-based fault signature to optimize LSTM hyperparameters in gearbox fault detection. Their method leverages physical knowledge of gear faults to enhance the effectiveness of the LSTM model in identifying gearbox anomalies. He et al. ³⁶ proposed a solution for dynamics modeling based on gearbox vibration measurements by introducing a physics-guided ordinary differential equation (ODE) neural network that uses vibration data to improve the accuracy of the model.

Simulation-driven digital twin expands the fault sample space, but highlights challenges due to the differences between simulations and real-world data in complex systems. ³⁷ Cai et al. ³⁸ utilized simulations of a rigid-flexible coupled dynamics model fused with real sensor measurements in a digital twin framework for gearbox fault diagnosis. In the literature on fault diagnosis, several techniques are applied to reduce the simulation–reality gap, for example, transfer learning, ³⁹ Gaussian process regression and deep residual shrinkage network on simulation and experimental data, ⁴⁰ and Boltzmann machines merging features from simulated and real data via a feedforward network with an adaptive mechanism. ³⁸

Beyond vibration analysis, thermography has been explored as an alternative approach for gearbox condition monitoring, for example, using the temperature field simulation image combined with deep learning. ⁴¹ Nevertheless, thermography requires specialized instrumentation and controlled environmental conditions, whereas vibration sensing remains widely deployed and provides direct access to fault-specific spectral characteristics.^22,37

The downsides of the existing algorithms reviewed in the literature include the dependence on data quality and the relatively time-consuming feature extraction in traditional fault diagnosis with classical signal processing or hybrid techniques. ⁴² In this regard, the differences between simulation and reality in vibration patterns and noise, ³⁹ time-consuming inference (because of finite element analysis and deep models) and lack of reliability in the outputs, ⁴⁰ and unsuitability for real-time deployment due to high computational cost and not easily explainable outputs ³⁸ are the challenges of simulation-driven fault diagnosis. The online computational complexity and storage space needed for searching neighbors are the main challenges associated with kNNs. ³ Recent studies show that CNN-based fault detection models typically go through time-consuming trial searches to optimize hyperparameters. ¹⁰ LSTM networks, as a variation of recurrent neural networks (RNNs), are powerful for sequential data; however, they suffer from computational costs, hyperparameter tuning difficulties, and potential overfitting. Less interpretability compared to simpler models and struggling with very long sequences are also their possible disadvantages. ²⁵ Autoencoders, having similar downsides, for example, interpretability and generalization issues, require significant training data to perform well. ²⁵ Bayesian networks also have their pros and cons. Although powerful, they can be computationally complex and challenging to interpret, especially when dealing with complex networks, with a lack of ability to model feedback loops. ¹¹

As explored, methods that use time-domain signals often suffer from high computational costs. When dealing with noisy experimental data, purely data-driven models are usually struggling to perform a reliable fault detection with high accuracy and robustness, particularly when the system is under varying operating conditions.

Taking into account the literature on physics-guided and physics-informed methodologies and the drawbacks of the algorithms explored above, the key contributions of this paper, developing a novel physics-and-data-driven (PDD) concept in gearbox fault detection, are as follows:

• A data-driven DL model trained in the frequency domain employing fast Fourier transform (FFT) and envelope signals.

• A methodology based on physical knowledge of faults extracted to achieve more stable detection accuracy compared to purely data-driven approaches. As input features, gear mesh frequencies (GMFs) and their harmonics, as well as the vibration amplitude readings in the sidebands are utilized to improve the learning capacity for robust fault detection, enabling improved sensitivity to incipient damage.

• A dual-CNN architecture consisting of a data-driven CNN (CNN-A) trained on FFT or envelope signals and a physics-guided CNN (CNN-B) trained only once using the GMF harmonics and modulation sidebands, as described above.

• A PDD parameter fusion mechanism designed to integrate the gearbox unhealthy response into a physics-guided framework. The integration is performed at the weight-level by adaptively combining the learned parameters of CNN-A and CNN-B, developing the methodology beyond feature-level physics-guided methods.

• A computationally efficient physics-guided framework with fast training and strong generalization capability, demonstrating higher accuracy and more stable performance under varying operating conditions compared to classical CNN and FNN models.

• Effectiveness, robustness, and generalization of the proposed method evaluated using two independent experimental gearbox test rigs with different fault types and operating conditions.

Methodologies for gearbox fault detection

The methodologies include implementations of data-driven neural networks, development of a physics-based approach, and their integration in gearbox fault detection.

Classical convolutional neural network (CNN)

CNNs are traditionally designed for tasks such as image recognition, but have recently been successfully applied to fault diagnosis. In this study, gearbox faults are identified using a 1D CNN-based fault diagnosis model. The model comprises two CNN layers with two kernels in each layer, and one fully connected layer, as illustrated in Figure 1. The rectified linear unit (ReLU) activation functions and max-pooling layers are incorporated between each CNN layer. The CNN layer group in this network extracts a feature map containing faulty information from the input data. The network is trained using the adaptive moment estimation algorithm (Adam) to minimize cross-entropy loss, diagnosing gearbox faults from short time intervals of vibration data under various speed and load conditions. The probabilities of different gearbox conditions are estimated by applying Softmax to the output values of three nodes, with the gearbox condition of the input data recognized as the one with the highest probability.

Figure 1.

Architecture of the CNN model.

The architecture of the hidden layers and neurons in Figure 1 is implemented using a growing and pruning technique, based on the variation of weights in consecutive epochs. Neurons are grown when there is a fluctuation; otherwise, if there is no change in some weights in a few epochs, pruning of the respective neurons is performed.

Artificial neural network (ANN)

ANNs are supervised ML algorithms that can be employed in classification problems. A feedforward neural network (FNN) is developed using Keras with four hidden layers (each containing 90 neurons) followed by an output layer for multiclass classification (3 classes). The model uses ReLU activations in hidden layers and Softmax in the output layer. It is compiled with Adam optimizer and categorical cross-entropy loss function suitable for multiclass classification tasks.

Physics-guided neural network (PGNN)

Physics-guided machine learning approaches, such as PGNNs, utilize known physical laws and equations to guide the learning process. This leads to the development of models that are more accurate, interpretable, and transferable. In recent years, DL has been recognized as a valuable tool for gearbox condition monitoring and fault detection. However, many DL approaches rely solely on data and do not incorporate physical knowledge into the learning and prediction processes. Without integrating physical knowledge, purely data-driven approaches may exhibit limited adherence to physics principles, resulting in poorer model performance. In this study, a physics-guided methodology is developed for gearbox fault detection, consisting of two components. Let $\mathcal{D}=\{(x_i,y_i){\}}_{i=1}^N$ denote the labeled vibration dataset, where $x_i$ and $y_i$ are the input signal and corresponding fault label, respectively. The proposed method consists of two independently trained neural networks:

• A data-driven CNN (CNN-A), denoted by $f_A(·;w_A)$ , which is a classical CNN trained in the frequency domain^29,30 on FFT and envelope signals.

$$w_A^{*}=\arg \underset{w_A}{\min }\frac{1}{N}\sum _{i=1}^N\ell (f_A(x_i;w_A),y_i),$$

(1)

where $\ell (·,·)$ is the categorical cross-entropy loss.

• A complementary physics-guided CNN (CNN-B), denoted by $f_B(·;w_B)$ , which serves as a supporting model trained only once using physics-based features extracted from the unhealthy envelope and FFT signals, that is, six GMFs and their harmonics as well as their corresponding first and second sidebands, to input fault-related information into gearbox health classification.

$$w_B^{*}=\arg \underset{w_B}{\min }\frac{1}{N}\sum _{i=1}^N\ell (f_B(\varphi \left(x_i\right);w_B),y_i),$$

(2)

where $\varphi (·)$ denotes the physics-based feature extraction operator constructed from GMF and its sidebands, and $\ell (·,·)$ is the categorical cross-entropy loss. The two networks are optimized independently using stochastic gradient descent, and no parameter sharing is performed during this stage.

Feature extraction

GMF is characteristic of a gear assembly that appears in the frequency spectrum. It is defined as follows:

$$\mathrm{GMF}={\mathrm{\Omega }}_{\mathrm{Gear}}\times N_{\mathrm{Gear}},$$

(3)

where ${\mathrm{\Omega }}_{\mathrm{Gear}}$ is the gear rotational speed and $N_{\mathrm{Gear}}$ is its number of teeth. A schematic diagram showing the feature extraction methodology using GMF, its harmonics, and their sidebands is presented in Figure 2. The same methodology is applied to the GMF’s second and third harmonics. As shown, two sidebands, from each side of GMF, are extracted having appropriate physical information of the gearbox, including rotational speed and number of teeth, to improve the fault detection accuracy.

Figure 2.

The proposed physics-guided methodology with the feature extraction using GMF, its harmonics, and their sidebands; integration of data-driven learning and physics-based modeling.

Physics-and-data-driven (PDD) parameter fusion mechanism

A purely data-driven algorithm struggles to generalize well to unseen samples, as it lacks the capability to effectively uncover and utilize hidden information. The core concept of the proposed approach is to integrate known fault-related knowledge, embedded in the envelope and FFT signals, as prior information and constraints into the data-driven training process. This aims to create a DL model with physical significance of gearbox faults. The methodology incorporates unhealthy information, provided by CNN-B, into the training process of CNN-A using a PDD parameter fusion mechanism. When the predictions of CNN-A differ from that of CNN-B with respect to healthy and faulty classes, the weights in CNN-A are updated by amplifying the effect of the weights of CNN-B and making combined weights as obtained in equation (8). This penalty steers the model toward the established physical knowledge and refines potentially erroneous data-driven health classifications. Figure 2 presents the physics-guided methodology with an assumed terminating loss level $ϵ$ .

• Feedforward operations in CNN-A and CNN-B

The procedure of the feedforward operations in the components of the physics-guided framework, that is, CNN-A and CNN-B, is outlined below:

•  1. Initialization: The weights and biases for each connection between the input and output neurons are randomly initialized. If there are $n$ inputs, $n+1$ weights (including one for bias) are initialized.

•  2. Feedforward operation: For each input $x_i$ , the weighted sum is calculated as

$$z=\sum _{i=1}^n{w}_i{x}_i+b,$$

(4)

where $n$ is the number of inputs, $w_i$ are the weights, and $b$ is the bias. Activation functions, ReLU and Softmax, are applied to determine the output $y=f\left(z\right)$ , where f is the activation function.

•  3. Loss calculation: The error $e$ is obtained using the categorical cross-entropy loss function.

•  4. Weight update: Weights are adjusted based on the error using a learning rate $\eta$ :

$$w_i^{\prime }=w_i+\eta {\mathrm{ex}}_i.$$

(5)

The bias $b$ is updated similarly:

$$b^{\prime }=b+\eta e.$$

(6)

•  5. Iteration: Steps 2–4 are repeated for a specified number of epochs or until convergence (when the error is minimized).

• Combined PDD weights

Given the independently trained parameters $w_A^{*}$ and $w_B^{*}$ , the PDD parameter fusion is formulated as the solution of the following regularized optimization problem:

$$\begin{array}{c}{w}_{\mathrm{PDD}}\left(\alpha \right)=\arg \underset{w}{\min }\hfill \\ [\alpha \parallel w-w_A^{*}{\parallel }_2^2+(1-\alpha )\parallel w-w_B^{*}{\parallel }_2^2],\hfill \end{array}$$

(7)

where $\alpha \in [0,1]$ is the fusion coefficient. The closed-form solution of equation (7) is given by

$$w_{\mathrm{PDD}}\left(\alpha \right)=\alpha w_A^{*}+(1-\alpha )w_B^{*}.$$

(8)

This equation defines a continuous family of fused models parameterized by $\alpha$ . The fused parameters define the final PDD classifier, whose prediction for input $x_i$ is

$${\hat{y}}_i\left(\alpha \right)=f_{\mathrm{PDD}}(x_i;w_{\mathrm{PDD}}\left(\alpha \right)).$$

(9)

In this work, the weight-combination factor $\alpha$ is treated as a learnable hyperparameter and is determined by minimizing the empirical loss of the fused model:

$${\alpha }^{*}=\arg \underset{\alpha \in [0,1]}{\min }\frac{1}{N}\sum _{i=1}^N\ell (f_{\mathrm{PDD}}(x_i;w_{\mathrm{PDD}}\left(\alpha \right)),y_i).$$

(10)

The final PDD model used for fault diagnosis is therefore obtained as

$$w_{\mathrm{PDD}}^{*}=w_{\mathrm{PDD}}\left({\alpha }^{*}\right)={\alpha }^{*}{w}_A^{*}+(1-{\alpha }^{*})w_B^{*}.$$

(11)

The overall process of updating the weights of CNN-A based on the fault information provided with CNN-B can be summarized as follows:

• − Retrieving weights: The learned parameters (weights) from two separate CNN models are obtained, that is, $w_A^{*}$ and $w_B^{*}$ .

• − Setting fusion coefficient: A weight-combination factor ( $\alpha$ ), which adjusts the influence of each model’s weights in the combined framework, is defined as presented in equation (8).

• − Combining weights: A new set of weights is calculated by taking a weighted average of each corresponding layer’s weights from both models.

• − Compiling model: A new model with the combined weights, specifying how to optimize and evaluate performance during training, is iteratively compiled until the loss is lower than an assumed level ( $ϵ$ ).

The proposed method follows a bi-level optimization structure: the inner level trains data-driven and physics-guided models independently, while the outer level learns an optimal fusion coefficient by minimizing the classification loss of the fused model. The PDD process is particularly useful for leveraging the effectiveness of multiple models to improve overall performance. By combining their learned parameters, the process creates a more robust model that can potentially perform better than either individual model alone.

Physics-guided framework and architecture

Figure 3 shows the framework and architecture of the physics-guided methodology. The threshold model CNN-B provides initial health categorization based on physical information, guiding the learning process of the classical CNN-A. The CNN architecture includes convolutional and pooling layers to automatically extract high-level features from raw vibration data. Consideration of the PDD parameter fusion ensures that the model aligns with the system physical behavior, making fault detection more accurate and reliable.

Figure 3.

Framework of the physics-guided methodology.

Gearbox test rig for case study – wear-type defect artificially created by pitting

Experimental datasets, for training and testing the models, are obtained using Sharif University of Technology (Sharif)’s gearbox test rig (Figure 4) designed for laboratory testing and fault detection through vibration analysis. The test bench is composed of a drive motor, a two-stage gearbox, and a dynamometer. A rough surface-level defect ( $\mathrm{length}\cong 15$  mm) is artificially created in the output gear tooth of this gearbox by pitting using a hammer and a wearing object in the laboratory (Figure 4, area on the gear tooth shown by dashes). Creating the artificial wear-type defect by pitting causes the gear to develop some small pits on the tooth surface. Based on this, the defect is treated as a proxy for incipient pitting damage (referring to a form of contact fatigue, characterized by small pits on the gear surface). Artificially induced pits do not reproduce the full microstructural or subsurface evolution and fatigue history of service-formed pitting; however, they capture the essential surface-level excitation effects on vibration signals. When there is no load, the intensity of the vibrations decreases, which can be mistakenly identified as a healthy condition. Therefore, data collection is performed under both with- and without-load conditions, which not only addresses this issue but also allows for a comparison and better understanding of the system operating conditions.

Figure 4.

Sharif’s gearbox test rig.

Figure 4 illustrates the layout of the gearbox. Details such as the number of gears and shafts, input and output speeds, and their GMFs are shown in Figure 5.

Figure 5.

Schematic diagram of the gearbox (Sharif); information in with- and without-load conditions.

The gearbox specifications are given in Table 1. The gears are helical, with a helix angle of 30° and a module of 2.5 mm, which have smoother operation and more evenly distributed loading due to the gradual engagement of the teeth compared to the spur gears. Vibration data acquisition is performed under the following conditions:

• The accelerometer is placed vertically on the bearing which is closest to the gearbox to detect vibration acceleration.

• Measurements are conducted in two conditions, that is, with and without load. In both conditions, the data acquisition is carried out with the properties mentioned in Table 2.

• The method of applying the load involves wrapping a belt around a pulley and tightening the hook to stretch the belt, thereby applying force. During this process, the output shaft speed is recorded, measured at 6.25 Hz with load, and 7.05 Hz without load using a tachometer. Based on the output speed and the number of teeth, GMF can be calculated, as shown in Table 3.

Table 1.

Specifications of the gearbox components (Sharif).

Component	Number of teeth
Pinion, input	31
Gear, input side	63
Pinion, output side	23
Gear, output	39

Table 2.

Vibration data acquisition properties under with- and without-load conditions (Sharif).

Parameter	Value
Number of samples	6144
Sample length (signal duration in Figure 6)	0.60 s
Number of repetitions	100

Table 3.

GMFs of the gearbox test rig (Sharif).

Operating condition	GMF 1	GMF 2	Input speed	Output speed
With load	667.8 Hz	243.75 Hz	21.55 Hz	6.25 Hz
Without load	705.6 Hz	279.95 Hz	22.79 Hz	7.05 Hz

Fault diagnosis using vibration analysis in time and frequency domains

Generally, in fault diagnosis using vibration analysis, the aim is to separate the vibration signals of the defective gearbox from the healthy one. The algorithms developed in this study (see Section Methodologies for gearbox fault detection) are used for this purpose. Healthy and unhealthy time-domain signals are displayed in Figure 6. As shown, the unhealthy signals are impulsive in the with- and without-load conditions.

Figure 6.

Time signals (Sharif); (a) healthy, (b) unhealthy without load, and (c) unhealthy with load.

The frequency spectrum of the gearbox unhealthy signals in the loaded and without-load conditions are obtained using envelope and FFT, shown in Figure 7. The frequencies 243.75 Hz, along with their harmonics, 478.5 and 732.5 Hz, are observable in the acceleration frequency spectrum in the loaded condition. Notably, the frequency 243.75 Hz indicates GMF of the output gear. Correspondingly, in the signals of the without-load condition, the frequencies can be seen at 275, 550, and 825 Hz. The sidebands associated with the faulty gearbox signals can be observed around GMF and its harmonics, as shown in the envelope and FFT frequency spectrum.

Figure 7.

Input ranges for CNN-B; gearbox unhealthy signals in frequency domain (Sharif); green windows contain GMF and its second and third harmonics with their four sidebands; (a) envelope spectrum of without-load signal, (b) envelope spectrum of with-load signal, (c) FFT spectrum of without-load signal, and (d) FFT spectrum of with-load signal, first, second, and third harmonics of GMF with a sideband.

As demonstrated in Figure 7(d), components of the gearbox fault, in loaded condition, are observed at frequencies ( $F$ ) of 243.75 Hz (i.e. GMF 2, see Figure 5) and its second and third harmonics, also at a sideband frequency of around $F\pm 6.25$  Hz. The value of 6.25 Hz is equal to the output shaft speed (see Table 3) that indicates a fault in the output gear. Similar analysis can be provided on the without-load condition, where the output shaft speed is 7.05 Hz. To sum up, based on the vibration analysis explained above and the feature extraction method, described in Section Feature extraction, the feature extraction of unhealthy data (with and without load) is performed in such a way that the input of CNN-B, with envelope or FFT analysis, includes GMF and its second and third harmonics with adjacent sidebands of each. As the experimental frequency spectrums contain multiple excitations in the GMF harmonics’ range (Figure 7), due to noise and other nonidealities in the system, the feature extraction module is utilized in CNN-B to deal with these extra excitations.

Fault detection by the diagnosis models

For the fault detection purpose, two commonly used inputs, FFT and envelope signals, are used to train the FNN and CNN models, as well as the data-driven module (CNN-A) of the proposed physics-guided framework.

Data preparation

The datasets are normalized and also randomly divided into training and testing sets before being fed into the model. The frequency range for FFT is from 5 to 4000 Hz, and for the envelope, from 2.5 to 4000 Hz. The length of the vector for FFT and envelope is 3197, and randomly-selected 20% of the data are considered for testing and the remaining 80% for training. The splitting details are summarized in Figure 8.

Figure 8.

Splitting the datasets into training and testing data (number of samples in the datasets and further details of the measurements are reported in Table 2).

Experimental test rig results (Sharif) and comparative analysis

The fault detection performance of the CNN, FNN, and PGNN models, using the envelope and FFT signals and the frequency-domain features (inputs), are compared in the following. Figure 9 represents the learning curves comparing the training loss to the validation loss, which helps to obtain the appropriate number of epochs to train the model and prevent overfitting.

Figure 9.

Training loss versus validation loss (Sharif); (a) CNN, (b) FNN, and (c) PGNN.

The fault detection accuracy is presented in the form of confusion matrices in Figure 10. The diagonal elements of the matrix represent the correct fault detections, while the elements above and below the diagonal line indicate false fault detections. The results with test data are summarized in Table 4.

Figure 10.

Confusion matrices for fault detection accuracy (Sharif); CNN, FNN, PGNN.

Table 4.

Fault detection accuracy with test data.

Model	Envelope	FFT
PGNN	0.96	0.92
CNN	0.90	0.90
FNN	0.88	0.90

Experimental data acquired from the gearbox test rig (Sharif).

Experimental validation using gearbox vibration condition monitoring benchmarking data

Open-source vibration condition monitoring benchmarking datasets of a planetary gearbox test bench, set up in Center for Asset Integrity Management (CAIM) at University of Pretoria, ⁴³ are used to further investigate the CNN, FNN, and PGNN models developed in this study, and to validate the performance of the proposed physics-guided DL approach in gearbox fault detection. The test bench is shown in Figure 11, and the components numbered in the figure are described as follows ⁴³ :

1. Hydraulic load system (pump) that allows changing the torque applied to the gearbox output.

2. Step-up gearbox that is instrumented for vibration measurements using two 9 mV/g uniaxial accelerometers mounted on one side of the ring gear at the top of the gearbox housing.

3. Step-down gearbox that is utilized to decrease the speed and increase the torque at the input of the speed-up gearbox.

4. 3 kW encoder DC motor that is controlled using a variable-frequency drive, with a maximum speed of 3000 rpm.

Figure 11.

Drivetrain of the experimental validation test bench. ⁴³

The gearbox specifications are listed in Table 5. Datasets are gathered on a single planet gear with an artificially-created missing tooth at varying speeds. The measured data are acquired at a sampling frequency of 38.4 kHz. ⁴³

Table 5.

Specifications of the gearbox system used for validation. ⁴³

Specifications	Value
Gearbox model	Bonfiglioli 300-L 1-5.77-PC-V01 B·E
Number of ring gear teeth	62
Number of sun gear teeth	13
Number of planet gear teeth	24

Healthy and impulsive unhealthy signals and the envelope frequency spectrum are illustrated in Figure 12. Because of the better performance obtained with the envelope input in the case study (compared to FFT, see Table 4), the experimental validation is implemented only with the envelope spectrum. The confusion matrices are shown in Figure 13 for the CNN, FNN, and PGNN models. As demonstrated, there is a decrease in the number of outliers in the physics-guided framework due to fault-related information extracted from the unhealthy envelope signals.

Figure 12.

Time signals: (a) healthy, (b) unhealthy, and (c) unhealthy normalized envelope spectrum. ⁴³

Figure 13.

Confusion matrices for fault detection accuracy validation; CNN, FNN, PGNN; envelope with test data.

Summary, accuracy analysis, and discussion

This study, in addition to implementing the classical CNN and FNN models for gearbox fault detection, introduces a novel physics-guided DL approach, incorporating physical knowledge of gearbox faults, as prior information and constraints, into the learning process to improve detection accuracy. GMF and its harmonics along with their sidebands are defined as input features under with- and without-load conditions, as they contain vital fault-related information. Each of these inputs is processed separately using both FFT and envelope analyses.

Developing a physics-guided framework, two CNN models, A and B, are integrated through a PDD parameter fusion mechanism to combine the weights of both models. Fusion is applied after CNN convergence, and the fusion coefficient $\alpha$ is optimized via outer-loop minimization. When CNN-A makes an incorrect detection, the weight of CNN-B, trained using physical information (features) extracted from the faulty response, is amplified, thereby improving fault detection accuracy and ensuring that the model is physically relevant and effective. For a new gearbox system with the same gear specifications and operating conditions, only CNN-A is updated with new input data, while CNN-B, as a guide, has been trained with the GMF harmonics and their sidebands. This is useful from a computational efficiency perspective, especially when dealing with high sampling frequencies.

The performance of the models is evaluated utilizing experimental measurement data from two different gearbox test rigs, that is, vibration measurements from the test rig at Sharif University of Technology and validation data from a planetary gearbox at University of Pretoria. The results illustrate the effectiveness of the proposed physics-guided framework in providing accurate fault detections under varying and non-varying operating conditions, that is, speeds and loads. CNN and FNN models show good learning capabilities with test data having non-varying operating condition. However, according to Table 6, the CNN model loses accuracy in the experimental validation test with varying speeds and loads, whereas FNN achieves better accuracy. The argument would be that the feedforward networks, contrary to CNN models, are not constrained by local or hierarchical feature extraction and can adapt to nonuniform changes. Nevertheless, the physics-guided approach, offering a promising robust solution, is able to achieve high and stable accuracy in both experimental setup tests. The reason is that CNN-B and PDD parameter fusion mechanism, which adjusts the weights of CNN-A and CNN-B in the combined weight, will enhance the model generalization by integrating physical information of faults, extracted from the gearbox unhealthy response, into the PICCN framework. Hence, the framework compensates for small-sample conditions using a complementary CNN-B and a PDD parameter fusion.

Table 6.

Fault detection accuracy validation; envelope with test data.

Model	Test rig (Sharif)	Experimental benchmark
PGNN	0.96	0.97
CNN	0.90	0.82
FNN	0.88	0.92

Experimental benchmarking data from the validation test bench. ⁴³

As reported in Table 6, the results for the first test show that the PGNN model achieves an accuracy of 0.96, outperforming the classical CNN model with 0.90, and the FNN model with 0.88. Similarly, the results for the validation test demonstrate that the PGNN model achieves an accuracy of 0.97, surpassing the CNN model with 0.82, and the FNN model with 0.92. As mentioned in experimental validation using benchmarking data, the improved performance in the physics-guided framework is also demonstrated in the confusion matrices with a decrease in the number of outliers (Figure 13).

Taking into account the results analyzed above, the proposed physics-guided model shows a higher and more stable accuracy than the classical models across both test rigs under different operating conditions. In contrast, the FNN and CNN models show fluctuating fault detection accuracies, that is, less reliable performance than the PGNN model.

The physics-guided approach proposed in this study is capable of detecting any initial defects and damages in gear teeth if the fault causes excitations at the GMF harmonic components and their sidebands in the frequency spectrum. In addition, from a signal-processing point of view, comparing FFT and envelope signals obtained in the framework are also useful for extracting the fault excitation frequencies out of the row vibration signals, increasing the capability of fault detection in the proposed methodology.

To further contextualize the advantages of the proposed physics-guided framework, Table 7 provides a concise comparison between the framework and previous data-driven, physics-guided, and physics-informed fault-diagnosis approaches.

Table 7.

Conceptual comparison of the proposed physics-guided framework with previous and similar approaches.

Method	Physics integration	Key characteristics	Limitations	Improvements in the proposed framework (PGNN)
Classical CNNs10,12,13	None	Learns features directly from vibration signals.	Sensitive to noise and varying speed/load, lacks physical interpretability.	PGNN corrects misclassifications via physics-guided CNN-B.
Classical FNNs5,6,14	None	Simple fully connected architecture, moderately well generalization.	Limited feature extraction capability, prone to overfitting, no physics awareness.	PGNN stabilizes learning process using GMF harmonics and sidebands.
Physics-guided CNNs (e.g., spectral filtering, envelope sideband isolation)33–36	Feature-level	Uses, e.g., spectral kurtosis, filtered envelopes, or extracted fault-related frequency bands.	Physics only in preprocessing, no influence (guidance) on network weight updates.	PGNN embeds physics at the weight-update level inside training to adaptively combine the learned parameters of CNN-A and CNN-B via PDD weight fusion and optimization of the fusion coefficient.
Classical PINNs 26 , 27	ODE/PDE-based constraints	Integrate governing equations into the loss function.	Require explicit system equations, computationally expensive, not ideal for complex gearbox vibration.	PGNN utilizes measurable spectral physics (GMF and sidebands).
Threshold-based models 29	Fixed physics-based thresholds	Simple predefined physics-based indicators (e.g., GMF amplitude) to guide classification.	Not adaptive, thresholds may fail under varying operating conditions.	CNN-B learns physics-guided patterns adaptively rather than relying on fixed thresholds.
Hybrid residual networks (e.g., PIResNet) 30	Architecture-level	Combines domain conversion, modal features, and bi-channel residual learning.	Higher complexity requiring more data, limited explicit physical correction of learning trajectory.	PGNN’s dual-CNN with adaptive weight fusion offers lower complexity and better generalization with integrated physics under sparse conditions.
Digital-twin-based methods32,37,38	Simulation-driven	Uses dynamics models, finite element methods, and simulation, also may use synthetic data augmentation.	Simulation–reality gap, high computational cost, sensitivity to modeling assumptions.	PGNN relies on measured vibration signals and physically grounded spectral information through PDD parameter fusion mechanism.

As shown in Table 7, the proposed framework uniquely integrates physics-derived spectral information directly into the training process via the PDD parameter fusion, offering robustness under varying operating conditions with lower model complexity than classical physics-informed approaches. The key module of the proposed methodology is adding physical information as prior and constraints, and adaptively combining the learned parameters of CNN-A and CNN-B in equation (8) to integrate physics at the weight-update level in the training process. This integration, that is, PDD weight fusion, and the outer-loop optimization of the fusion coefficient $\alpha$ , in equation (10), develop the methodology beyond feature-level physics-guided frameworks.

The sensitivity of the weight-combination factor $\alpha$ is quantified using a robustness interval defined as the range of $\alpha$ values for which the classification accuracy of the fused model remains within 75% of the optimal value (i.e. 25% deviation from the optimal accuracy). The resulting interval indicates that the framework maintains stable performance for $\alpha \in [0,0.4]$ , meaning that the proposed parameter fusion mechanism is robust over a relatively wide range, that is, 40%, of $\alpha$ . This interval captures both the optimal fusion point and a region of consistent performance, while excluding the degradation region at higher values of $\alpha$ . Values of $\alpha \le 0.05$ correspond to a physics-dominated model, while $\alpha \in [0.1,0.4]$ reflect effective fusion.

Conclusion and future works

Gearboxes are critical components in industrial systems, where early and reliable fault detection has substantial impact on performance, availability, and maintenance cost. The key contribution of this study is the introduction of a physics-and-data-driven deep learning methodology for gearbox fault diagnosis, in which physically meaningful spectral information, that is, GMF harmonics and their sidebands, directly influence the weight-update process of a data-driven CNN through a dedicated PDD parameter fusion mechanism. This integration, distinguishing the proposed framework from existing physics-guided methods, enables the model to incorporate fault-specific physical knowledge at the parameter-update level during training, thereby improving interpretability and robustness compared to, purely data-driven, classical FNN and CNN models (which are also developed and evaluated in the study).

The experimental validation strategy is twofold, evaluating the methodology’s performance by two experimental test rigs: one with the wear-type pitting defect (Section Experimental test rig results (Sharif) and comparative analysis) and another one with a missing tooth fault under different operating conditions (Section Experimental validation using gearbox vibration condition monitoring benchmarking data):

• Sharif test rig: Evaluating the proposed physics-guided framework on incipient-like wear-type pitting defect, where it is expected that the defect in the output gear tooth will impact the measured vibration on the gearbox housing.

• Pretoria test rig: Testing the model’s robustness on a missing tooth defect (different fault condition) under varying operating conditions, that is, variable speeds and loads.

This two-stage validation demonstrates both:

• PGNN’s sensitivity to early surface-level defects (Sharif test rig, fault detection accuracy: 0.96, Table 6), and

• PGNN’s robustness to different fault types and operating conditions (Pretoria test rig, accuracy: 0.97, Table 6).

Furthermore, compared to the classical models (CNN and FNN), in addition to a higher accuracy, the proposed PGNN model demonstrates a more stable accuracy, that is, more reliable and robust fault detection, across both test rigs under different operating conditions (CNN: 0.90 and 0.82, FNN: 0.88 and 0.92, respectively, Table 6).

The experimental results from the two independent gearbox test rigs under constant and varying loads and speeds demonstrate that the proposed physics-guided framework consistently outperforms the classical CNN and FNN models. The higher accuracy and notably more stable fault detections across operating conditions confirm the framework’s enhanced reliability in practical scenarios. Furthermore, because CNN-B encodes physics-driven spectral information and requires only one-time training, the method generalizes effectively and efficiently to new vibration datasets of gearboxes with matching specifications, reducing retraining effort and computational cost and making it well-suited for practical condition-monitoring deployment.

The experimental findings indicate that the proposed physics-guided framework improves upon data-driven approaches by incorporating physics-based principles (see Section Methodologies for gearbox fault detection), resulting in a more robust model with high accuracy and reliability. Figure 14 shows the fault detection range of the framework under different gear faults and operating conditions.

Figure 14.

Fault detection range of the physics-guided framework under different gear faults and operating conditions.

The physics-guided approach not only enhances the interpretability of the model, but also helps in addressing the limitations of classical learning methods by combining the strengths of data-driven techniques with the logic and reasoning of physics. PGNNs, which incorporate physical information and constraints into the learning process, affect the parameters of the DL model, that is, the weights and loss function, and fundamentally change how the model generalizes and adapts under data-scarce conditions. Physical laws guide learning as a prior knowledge, reducing overfitting and improving generalization, and this leads to a model less reliant on large training datasets. By adhering to physical laws, PGNNs require fewer labeled data points and generalize better to out-of-sample domains that are not covered by training data. In addition, physics-related constraints help filter out noise, ensuring a more stable and accurate prediction (i.e. robustness). Furthermore, by transferring across related problems, PGNNs require minimal retraining and few data points. Therefore, in fault diagnosis, PGNNs provide accurate, robust, and generalizable models that effectively perform fault detection even with limited samples, which makes them particularly useful for scenarios where obtaining data is expensive or impractical.

In contrast to simulation-driven digital twin frameworks, the proposed methodology does not rely on computationally demanding models. Instead, it leverages directly observable fault signatures, making it more suitable for real-time industrial deployment where vibration sensors are already widely used. The demonstrated robustness under both constant and varying operating conditions highlights the potential of the physics-guided framework as a practical and reliable tool for making informed and timely decisions in condition-based maintenance (CBM).

Future studies may lead to the development of hybrid physics-guided frameworks that can tackle a wider range and more complex fault detection tasks, with high accuracy and efficiency. Bridging from artificial to fully service-induced initial damage requires more detailed experimental setups (e.g. controlled fatigue testing, metallography). Further development on the initial damage identification and generalization to naturally evolved pitting or crack initiation would strengthen this line of research and can also be considered as future works.