Unsupervised physics-informed domain-adversarial graph convolutional network for bearing fault diagnosis under varying operating conditions

Abstract

In real-world industrial settings, bearing operation is influenced by varying operating conditions, resulting in vibration responses that exhibit significant multi-scale feature coupling characteristics. Moreover, reliable data for characterizing these responses remain scarce. Traditional methods lack the ability to adaptively integrate multi-scale information during feature modeling and fail to effectively incorporate physical constraints related to failure mechanisms, resulting in unstable feature representations and diagnostic results that lack physical consistency. To address these challenges, an unsupervised physics-informed domain-adversarial graph convolutional network is proposed for bearing fault diagnosis under varying operating conditions. Within this framework, a Kernel Selective Fusion Attention mechanism is introduced to selectively fuse the extracted multi-scale convolutional responses, thereby enhancing the stability and discriminability of graph features. Furthermore, a physical prior distribution is constructed based on the envelope power spectrum and fault characteristic frequencies, and the prior knowledge of fault mechanisms is incorporated into the model optimization process through a sample-level gating strategy and a delayed-start linear ramp-up weighting. The experimental results demonstrate that the proposed method outperforms the comparison methods in terms of fault diagnosis accuracy in the target domain, cross-domain feature alignment capability, and compound-fault recognition performance, thereby validating its effectiveness in unsupervised domain adaptation for diagnosis under varying operating conditions.

Keywords

domain-adversarial learning fault diagnosis graph convolutional networks physical consistency constraint information fusion

1. Introduction

Rolling bearings are critical supporting components in rotating machinery. The health condition of rolling bearings directly affects equipment safety, reliability, and maintenance costs (Lei et al., 2020). In recent years, with the advancement of deep learning and transfer learning, intelligent fault diagnosis methods based on vibration signals have made significant progress in the identification of rolling bearing conditions (Tang et al., 2024; Tong et al., 2018; Zhang et al., 2023). However, in real-world industrial scenarios, constantly changing rotational speeds, loads, and environmental conditions can cause significant shifts in signal distributions, making it difficult for models trained under fixed operating conditions to generalize to new conditions (Lu et al., 2021; Wu et al., 2022; Zhong et al., 2024). Therefore, reliable cross-condition bearing fault diagnosis with few or no labels in the target domain remains a key challenge in intelligent fault diagnosis (X Li et al., 2020).

Changes in operating conditions can cause systematic shifts in the statistical distributions of vibration signals, the scales of time-frequency features, and the similarity relationships among samples (Xiao et al., 2025; ZHANG et al., 2023). Specifically, changes in load and rotational speed cause variations in the amplitude and modulation structure of the impact responses, which in turn result in shifts in the energy distributions of the frequency spectrum and envelope spectrum (Klausen et al., 2020; Mauricio et al., 2020; Randall and Antoni, 2011). Meanwhile, the coordinated changes in distribution and scale also alter the clustering patterns of the same fault class in the feature space, manifesting as stretching, compression, or local misalignment. As a result, the structural relationships among samples become unstable, thereby weakening similarity-based relational modeling and cross-domain alignment (Li et al., 2021a; Xiao et al., 2025). Consequently, conventional diagnostic models relying on fixed operating conditions or fully labeled data from the target domain are prone to distorted decision boundaries and degraded generalization during cross-domain transfer (Zhang et al., 2018; Zhang et al., 2023).

For bearing fault diagnosis, existing studies have extensively explored the end-to-end modeling capability of deep networks. Convolutional neural networks (CNNs) can directly learn discriminative features from raw vibration sequences, thereby reducing the reliance on manual feature engineering (Guo et al., 2016; Ince et al., 2016). Methods based on time-frequency representations further map non-stationary signals into the time-frequency domain and perform representation learning to capture local patterns (Liu et al., 2016). Although these methods can achieve satisfactory performance under fixed operating conditions or mild distribution shifts, they generally assume similar training and test distributions and therefore suffer from performance degradation under significant condition changes.

To reduce cross-domain distribution discrepancies, unsupervised domain adaptation (UDA) has emerged as an important solution for diagnosis with unlabeled data from the target domain. Existing UDA methods mainly include explicit alignment based on statistical distances and implicit alignment based on adversarial learning. Representative methods in the former category, such as domain adaptation networks and joint adaptation networks, typically align the feature distributions of the source and target domains using metrics such as maximum mean discrepancy, and further extend the alignment to multi-layer representations or joint distributions to enhance transfer performance (Long et al., 2017, 2019). Representative methods in the latter category, such as domain-adversarial neural networks and conditional domain adversarial networks, learn domain-invariant features through domain discriminators and gradient reversal mechanisms, while incorporating conditional adversarial strategies to alleviate class mismatch in multimodal transfer (Ganin et al., 2016; Long et al., 2018). However, these methods primarily focus on class-level and domain-level distribution alignment while neglecting the effective modeling of structural relationships among samples, which may lead to local mismatch and unstable decision boundaries under complex operating conditions.

Graph-based UDA methods have been introduced to exploit inter-sample relationships in cross-condition fault diagnosis. Li et al. proposed a domain-adversarial graph convolutional network that integrates convolutional representation learning, graph construction, graph convolutional propagation, and domain-adversarial learning into a unified framework, thereby improving unsupervised transfer diagnosis performance (Li et al., 2021a). Subsequent studies further explored structured subdomain alignment, meta-learning-based graph adaptation, and multi-source graph feature alignment to improve cross-domain diagnostic generalization (Ghorvei et al., 2023; Hu et al., 2024; Kavianpour et al., 2022; Liu et al., 2024). Although these methods explicitly exploit structural information, two challenges remain under varying operating conditions with severe distribution shifts. First, model optimization is still mainly driven by source domain supervision and data-driven alignment constraints, while explicit physical consistency constraints related to bearing fault mechanisms are insufficiently considered, limiting diagnostic reliability and interpretability. Second, graph construction relies heavily on convolutional node features, making similarity measurement and adjacency construction vulnerable to feature instability.

From the perspective of physical interpretability, bearing fault characteristic frequencies and their energy responses in the envelope spectrum provide mechanism-related prior evidence for diagnosis (Randall and Antoni, 2011; Shen et al., 2021). Existing studies have shown that integrating physically relevant representations, such as the envelope spectrum and wavelet responses, with deep networks can enhance the model’s sensitivity to fault mechanisms, thereby improving diagnostic interpretability and generalization performance (T Li et al., 2020). For example, Lu et al. introduced a physics-informed feature weighting strategy to assign higher weights to spectral features close to bearing fault characteristic frequencies, thereby improving sensitivity to fault-related components under speed variations (Lu et al., 2023). In addition, Ni et al. developed a physics-informed residual network by embedding a modal-property-dominant layer into the diagnostic model to enhance the physical consistency of learned representations (Ni et al., 2023). However, in scenarios where the target domain is unlabeled, stable and controllable integration of mechanism-related information remains challenging, especially because noisy physical priors may induce negative transfer when they are optimized jointly with domain-adversarial learning and structural alignment.

To address the above challenges in unsupervised transfer diagnosis under varying operating conditions, an unsupervised physics-informed domain-adversarial graph convolutional network with multi-scale selective feature enhancement and physical prior consistency constraint, termed PI-KSFA-DAGCN, is proposed. Specifically, a Kernel Selective Fusion Attention (KSFA) mechanism is introduced between convolutional feature extraction and the graph generation layer. By selectively fusing multi-scale candidate responses, it enhances the stability and discriminability of the features used for graph construction, thereby stabilizing similarity measurement and adjacency relationships among samples and improving the robustness of graph construction and relational modeling. Meanwhile, by incorporating the mechanisms reflected by bearing fault characteristic frequencies, a sample-level physical prior distribution is constructed from fault characteristic frequencies and the energy responses within their harmonic neighborhoods in the envelope power spectrum. With a gating strategy and a delayed-start linear ramp-up weighting strategy, the physical prior consistency constraint incorporates mechanism knowledge into classifier optimization, thereby alleviating semantic mismatch in unlabeled target domains and enhancing the physical interpretability of the decision process. Extensive cross-condition transfer experiments conducted on the HUST bearing dataset demonstrate that the proposed method achieves competitive cross-domain diagnostic performance.

The main contributions are summarized as follows.

(1) An unsupervised physics-informed domain-adversarial graph convolutional network with multi-scale selective feature enhancement and physical prior consistency constraint is proposed and successfully applied to cross-domain bearing fault diagnosis.

(2) A sample-level physical prior consistency constraint is proposed, where fault characteristic frequencies and the energy responses within their harmonic neighborhoods are mapped to a class-level prior distribution. Through a sample-level gating strategy and a delayed-start linear ramp-up weighting, mechanism knowledge is robustly incorporated into model optimization in coordination with domain-adversarial training.

(3) A Kernel Selective Fusion Attention mechanism is designed to suppress feature drift through multi-scale selective fusion and improve the stability and discriminability of features used for graph construction, thereby enhancing the robustness of graph generation and structural modeling.

The remainder is organized as follows. Section two reviews the related work. Section three presents the overall framework of PI-KSFA-DAGCN and the design of its key modules. Section four provides the experimental settings, comparative results, ablation analyses, computational cost analysis, and further discussion on practical applicability. Section five concludes the study and outlines possible directions for future research.

2. Related work

2.1. Unsupervised domain adaptation

To address the degradation of cross-domain diagnostic performance caused by distribution shifts under varying operating conditions, unsupervised domain adaptation (UDA) aimed to learn transferable discriminative representations under the setting where labeled data from the source domain and unlabeled data from the target domain were available (Ganin et al., 2016; Long et al., 2017, 2018, 2019; Sun and Saenko, 2016). Existing methods could be generally categorized into adversarial approaches, discrepancy-based approaches, and methods that incorporated conditional constraints, self-training, and entropy regularization. Representative adversarial approaches included DANN, which learned domain-invariant features through a domain discriminator and gradient reversal to reduce domain discrepancy (Ganin et al., 2016). CDAN further introduced classification prediction information for conditional alignment, thereby alleviating class confusion caused by feature-level adversarial learning and improving multimodal transfer capability (Long et al., 2018). Discrepancy-based approaches achieved alignment by explicitly matching statistical distributions. DAN aligned marginal distributions using maximum mean discrepancy (Long et al., 2019). JAN aligned multi-layer joint distributions to enhance semantic transfer (Long et al., 2017). Deep CORAL aligned second-order statistics to mitigate domain shift (Sun and Saenko, 2016). Overall, these methods mainly focused on the optimization of global distributions and prediction confidence, but remained limited in their ability to stably characterize the variations in inter-sample similarity relationships and intra-class structural patterns under varying operating conditions. Consequently, their transfer performance remained constrained when structural misalignment was pronounced. Figure 1 further illustrates the three types of key information involved in UDA, namely class-discriminative information, domain-alignment information, and structural information among samples. Existing UDA methods mainly concentrated on the first two types of information, whereas explicit modeling of structural relationships among samples remained relatively insufficient, which in turn provided the motivation for the subsequent development of graph-based domain adaptation methods.

Figure 1.

Three types of transferable information in UDA.

2.2. Graph convolutional networks

To characterize structural information among samples under varying operating conditions, graph convolutional networks (GCNs) (Kipf and Welling, 2016; Wu et al., 2021) unified convolutional feature learning, graph construction, and graph convolutional propagation within a single modeling framework. By explicitly modeling topological relationships among samples while facilitating feature distribution alignment, these methods improved the robustness of cross-condition transfer diagnosis (Wang et al., 2021; Yu et al., 2023). Such frameworks were typically organized into three stages: feature extraction, relational graph construction, and structural propagation (Wen et al., 2018). Compared with conventional UDA methods, GCNs were able to further exploit structural information among samples and, to some extent, alleviate the degradation in diagnostic performance caused by insufficient structural modeling. Although GCN-based domain adaptation methods provided a new modeling perspective for cross-domain fault diagnosis, they still exhibited notable limitations in unsupervised scenarios. On the one hand, model optimization mainly relied on data-driven alignment constraints and lacked explicit modeling and prior guidance with respect to fault-related physical mechanisms, thereby making the target domain more susceptible to class-level semantic shift, local alignment distortion, and unstable decision boundaries (X Li et al., 2020; Jiang et al., 2025). On the other hand, graph structure construction depended heavily on node representations generated by convolutional networks. Under varying operating conditions, such features were relatively sensitive to scale perturbations and local pattern shifts, which reduced the reliability of similarity measurement and destabilized adjacency relationships, thereby weakening graph convolutional propagation and structural alignment.

3. Methodology

3.1. Overview

The overall framework of the model is shown in Figure 2. The proposed model is built upon a domain-adversarial graph convolutional network. To address the issues of scale fluctuations in convolutional features and local pattern drift caused by varying operating conditions, KSFA is introduced between the final convolutional feature maps of the CNN backbone and the fully connected embedding layer. This mechanism extracts candidate representations through multi-scale deep convolutional branches and achieves adaptive fusion by combining temporal and channel selection, thereby obtaining more robust node embeddings and improving the stability of inter-sample similarity measurement and adjacency relationship learning. Furthermore, a sample-level physical prior distribution is constructed from the fault characteristic frequencies in the envelope power spectrum and the energy responses within their harmonic neighborhoods, and a physical prior consistency constraint is imposed between this distribution and the model prediction in the class-probability space. With a sample-level gating strategy and delayed-start linear ramp-up weighting, the physical prior consistency constraint further injects fault mechanism prior knowledge into classifier optimization in a robust manner. Ultimately, the model is trained end to end under the joint supervision of the source domain classification loss, domain alignment loss, structural alignment loss, and physical prior consistency loss, thereby collaboratively enhancing cross-condition diagnostic discrimination, structural modeling stability, and physical interpretability.

Figure 2.

Overall framework of PI-KSFA-DAGCN.

The labeled dataset in the source domain is defined as $D_{s} = {(x_{i}^{s}, y_{i}^{s})}_{i = 1}^{n_{s}}$ , where $n_{s}$ is the number of source domain samples, $x_{i}^{s} \in X_{s}$ is the $i$ -th sample from the source domain, and $y_{i}^{s} \in Y_{s}$ is its corresponding label. The unlabeled dataset in the target domain is defined as $D_{t} = {x_{j}^{t}}_{j = 1}^{n_{t}}$ , where $n_{t}$ represents the number of samples from the target domain and $x_{j}^{t} \in X_{t}$ is the $j$ -th sample from the target domain. In UDA, it is generally assumed that the source and target domains share an identical label space, that is, $Y_{s} = Y_{t}$ , whereas their input spaces are identical or isomorphic but their data distributions differ due to domain shift induced by varying operating conditions. The objective is to construct a deep network $f : X_{t} \to Y_{t}$ that leverages labeled knowledge from the source domain to accurately identify unlabeled samples in the target domain.

3.2. Kernel selective fusion attention

In the constructed domain-adversarial graph convolutional network, the features extracted by the convolutional backbone are used for subsequent graph generation and structural propagation. When variations in operating conditions induce feature-scale fluctuations or local pattern drift, inter-sample similarity estimation can be disturbed, thereby leading to unstable graph construction and weakened graph convolutional propagation and subsequent cross-domain alignment. To address this issue, inspired by the selective fusion strategy in (Xu et al., 2025), Kernel Selective Fusion Attention (KSFA), tailored for one-dimensional temporal feature enhancement, is introduced between the final convolutional feature maps of the CNN backbone and the fully connected embedding layer to selectively enhance multi-scale convolutional responses, as illustrated in Figure 3. Given an input sample $X_{input}$ , let $X$ denote the final feature maps extracted by the convolutional backbone. KSFA then performs multi-scale modeling and selective fusion on $X$ to obtain the enhanced feature $X^{'}$ . The enhanced feature is further mapped to the node representation $X_{g}$ through the subsequent embedding layer. Finally, $X_{g}$ is fed into the graph generation layer (GGL) for subsequent graph generation and structural propagation.

Figure 3.

The proposed Kernel Selective Fusion Attention (KSFA). DwConv represents depthwise convolution, Avg and Max represent channel-wise average pooling and max pooling, respectively, and GAP means temporal global average pooling.

Let the input feature maps be $X \in R^{B \times C \times L}$ , where $B$ , $C$ , and $L$ denote the batch size, number of channels, and temporal length, respectively. The feature maps $X$ produced by the last convolutional block of the CNN backbone are fed into the KSFA module before the subsequent embedding mapping layer. To capture multi-scale local patterns, KSFA employs two depthwise convolution branches. The first branch uses a depthwise convolution with kernel size 3, whereas the second branch adopts a cascaded structure consisting of a depthwise convolution with kernel size three and dilation rate one followed by a depthwise convolution with kernel size five and dilation rate 2. Then, two $1 \times 1$ convolution layers $F_{1 \times 1} (\cdot)$ are used to integrate channel information and project the branch features into consistent channel dimensions. The resulting branch features are given by

U_{1} = F_{1 \times 1}^{1} (F_{k = 3, d = 1}^{d w c} (X)), U_{2} = F_{1 \times 1}^{2} (F_{k = 5, d = 2}^{d w c} (F_{k = 3, d = 1}^{d w c} (X)))

(1)

Where

U_{1}, U_{2} \in R^{B \times \frac{C}{2} \times L}

,and

F_{k = 3, d = 1}^{d w c}

and

F_{k = 5, d = 2}^{d w c}

represent depthwise convolution operations with kernel size three and dilation rate one and with kernel size five and dilation rate 2, respectively. The fused feature maps are then obtained by concatenating the outputs of the two branches along the channel dimension

U = [U_{1}; U_{2}] \in R^{B \times C \times L}

(2)

Where

U \in R^{B \times C \times L}

is the fused feature representation obtained by concatenating the outputs of the two multi-scale branches along the channel dimension. To obtain temporal selection information, average pooling and max pooling are first performed on

U

along the channel dimension, yielding

{\bar{U}}_{a v g} (b, 1, l) = \frac{1}{C} \sum_{c = 1}^{C} U (b, c, l), {\bar{U}}_{\max} (b, 1, l) = \max_{1 \leq c \leq C} U (b, c, l)

(3)

Where

{\bar{U}}_{a v g}

and

{\bar{U}}_{\max}

denote the temporal descriptors obtained by average pooling and max pooling of the fused feature

U

along the channel dimension, respectively. The two temporal descriptors are then concatenated as

T = [{\bar{U}}_{a v g}; {\bar{U}}_{\max}] \in R^{B \times 2 \times L}

. Subsequently, a

1 \times 1

convolution layer

F_{1 \times 1} (\cdot)

is employed to generate

O

temporal attention maps, with

O

= 2, expressed as

\bar{S} = F_{1 \times 1}^{2 \to O} (T) \in R^{B \times O \times L}, (O = 2)

(4)

The corresponding branch-wise temporal selection masks are then obtained through sigmoid activation, $S_{i} = σ ({\bar{S}}_{i}), i = 1, 2$ , where $S_{i} \in R^{B \times 1 \times L}$ . To obtain channel selection information, global average pooling is performed on the fused feature $U$ along the temporal dimension, yielding the channel descriptor

z (b, c, 1) = \frac{1}{L} \sum_{l = 1}^{L} U (b, c, l), z \in R^{B \times C \times 1}

(5)

To enhance the modeling of channel dependencies while controlling parameter complexity, a bottleneck channel selection structure is implemented using two fully connected layers. Specifically, the channel descriptor is first compressed into a compact representation and then projected back to the original channel space to generate the channel response

\hat{z} = F_{F C 1} (z) \in R^{B \times r \times 1}, Z = F_{F C 2} (\hat{z}) \in R^{B \times C \times 1}

(6)

Where

\hat{z} \in R^{B \times \tilde{C} \times 1}

is the intermediate compact representation, and

\tilde{C}

denotes the reduced channel dimension.

Z \in R^{B \times C \times 1}

is the channel response mapped back to the original channel space.

F_{F C 1} (\cdot)

and

F_{F C 2} (\cdot)

denote the two fully connected layers, respectively. Subsequently,

Z

is reshaped into

\tilde{Z} \in R^{B \times 2 \times \frac{C}{2} \times 1}

, where the original channel dimension is divided into two groups of branch responses of length

\frac{C}{2}

. The softmax function is then applied along the branch dimension to obtain the channel selection weights for the two branches

C_{1}^{[j]} = \frac{e^{{\tilde{Z}}_{1}^{[j]}}}{e^{{\tilde{Z}}_{1}^{[j]}} + e^{{\tilde{Z}}_{2}^{[j]}}}, C_{2}^{[j]} = \frac{e^{{\tilde{Z}}_{2}^{[j]}}}{e^{{\tilde{Z}}_{1}^{[j]}} + e^{{\tilde{Z}}_{2}^{[j]}}}, j = 1, 2, \dots, \frac{C}{2}

(7)

Where

C_{i}^{[j]}

denotes the channel selection weight at the

j

-th channel position in

C_{i} \in R^{B \times \frac{C}{2} \times 1}

for

i = 1, 2

, and

{\tilde{Z}}_{1}^{[j]}

and

{\tilde{Z}}_{2}^{[j]}

denote the response values at the

j

-th position of the reshaped feature

\tilde{Z}

for the two branches, respectively. Subsequently, the channel selection weight

C_{i}

and the temporal selection mask

S_{i}

are broadcast along the corresponding dimensions and multiplied elementwise to obtain the joint weights

W_{i} = C_{i} ⊙ S_{i}, i = 1, 2

(8)

Where

W_{i} \in R^{B \times \frac{C}{2} \times L}

. On this basis, the branch features

U_{i}

are weighted and summed, and the attention feature is obtained through

F_{1 \times 1} (\cdot)

A_{a t t} = F_{1 \times 1} (\sum_{i = 1}^{2} (W_{i} ⊙ U_{i})) \in R^{B \times C \times L}

(9)

The attention gating mask is then obtained by applying the sigmoid function

M = σ (A_{att}) \in R^{B \times C \times L}

(10)

Finally, KSFA performs elementwise gated recalibration of the input feature maps to obtain the enhanced feature maps

X^{'} = X ⊙ M

(11)

After flattening and fully connected projection of the output feature maps $X^{'} \in R^{B \times C \times L}$ , the sample-level embedding $X_{g} \in R^{B \times m}$ is obtained. Enhanced by KSFA, $X_{g}$ integrates multi-scale temporal patterns and branch selection information, thereby improving the stability of node representations in the subsequent stages of instance graph generation and structural propagation.

3.3. Graph generation and structural propagation

Taking the sample-level embedding $X_{g}$ produced by KSFA as input, the graph generation layer (GGL) projects the node features through a multilayer perceptron and constructs an inter-sample similarity matrix based on the inner products of the projected features. The weighted adjacency matrix used for graph convolutional propagation is then obtained through normalization and row-wise Top-k sparsification

{\begin{cases} {\hat{X}}_{g} = M L P (X_{g}) \\ A = normalize ({\hat{X}}_{g} {\hat{X}}_{g}^{T}) \\ \bar{A} = Top - k (A) \end{cases}

(12)

Where

Top - k (\cdot)

retains, for each row, the adjacency entries with relatively large similarity values, thereby reducing computational cost and improving the controllability of graph construction.

normalize (\cdot)

represents the normalization function. Accordingly, the instance graph can be formulated as

G = (V, \bar{A}, X_{g})

, where

V

represents the node set formed by the samples in a mini-batch,

\bar{A}

is the weighted adjacency matrix, and

X_{g}

is the corresponding node feature matrix. The graph generation process is illustrated in Figure 4.

Figure 4.

Process of the graph generation layer (GGL).

On the constructed instance graph, structural propagation is performed using the multi-receptive-field graph convolutional network (MRF-GCN) (Li et al., 2021b), so that neighborhood dependencies at different scales can be modeled in parallel and the structural information among samples can be embedded into node representations. The multi-receptive-field graph convolution operation is defined as

H = [\sum_{k_{1} = 0}^{K_{g_{1}} - 1} θ_{k_{1}} Λ^{k_{1}} X_{g}, \sum_{k_{2} = 0}^{K_{g_{2}} - 1} θ_{k_{2}} Λ^{k_{2}} X_{g}, \dots, \sum_{k_{v} = 0}^{K_{g_{v}} - 1} θ_{k_{v}} Λ^{k_{v}} X_{g}]

(13)

where

Λ = diag ({λ_{i}}_{i = 1}^{N})

is the diagonal matrix of Laplacian eigenvalues,

K_{g_{i}}

controls the neighborhood aggregation range,

θ_{k_{1}}, θ_{k_{2}}, \dots, θ_{k_{v}}

are learnable parameters, and

v

is the number of receptive fields. Furthermore, two stacked MRF-GCN layers are applied to the sparse instance graph to obtain structurally enhanced features

H^{0} = M R F C o n v (\bar{A} X_{g} W^{0})

(14)

H^{1} = M R F C o n v (\bar{A} H^{0} W^{1})

(15)

Where

H^{0}

and

H^{1}

are the outputs of the first and second MRF-GCN layers, respectively, while

W^{0}

and

W^{1}

are trainable weight matrices. The resulting structurally enhanced representations are subsequently used for classification, domain-adversarial alignment, and optimization under the physical prior consistency constraint.

3.4. Physical prior consistency constraint

The gradient updates of DAGCN are primarily governed by data-driven mechanisms, including source domain supervision as well as distributional and structural alignment, while explicit physical consistency constraints associated with bearing fault mechanisms remain insufficiently incorporated. To address this issue, a physical prior consistency constraint based on sample-level physical priors is introduced. Specifically, by mining the energy responses of fault characteristic frequencies and their harmonic neighborhoods in the envelope power spectrum, sample-level physical prior evidence is constructed and further mapped into a class-level physical prior distribution to regularize classifier predictions. Subsequently, in the class-probability space of the classifier output, the physical prior distribution is aligned with the model-predicted distribution through the proposed physical prior consistency constraint. Meanwhile, a sample-level gating strategy and delayed-start linear ramp-up weighting are combined to inject fault-mechanism knowledge into the classifier training process in a stable and controllable manner.

For each unlabeled sample from the target domain, the raw vibration signal is first band-pass filtered within a preset resonance band to enhance components related to impulsive modulation. The Hilbert transform is then applied to extract the envelope signal, and the one-sided envelope power spectrum is obtained through the fast Fourier transform. The envelope spectrum is given by

E [k] = FFT {e [n]} [k]

(16)

Where

e [n]

is the envelope signal obtained from the sample vibration signal through the Hilbert transform, and

E [k]

is its frequency-domain representation after the fast Fourier transform. Accordingly, the one-sided envelope power spectrum

P [k]

and the actual frequency

f_{k}

corresponding to the

k

-th spectral bin are expressed as

P [k] = {| E [k] |}^{2}

(17)

f_{k} = \frac{k}{N} f_{s}, k = 0, \dots, ⌊ \frac{N}{2} ⌋

(18)

Where

f_{s}

is the sampling frequency,

N

is the number of FFT points,

k = 0, \dots, ⌊ \frac{N}{2} ⌋

represents the frequency bins in the one-sided spectrum, and

⌊ \cdot ⌋

is the floor operation. On this basis, the energy distribution over frequency bands associated with different fault mechanisms is characterized in the frequency domain. Considering that fault characteristic frequencies vary with the shaft rotational frequency under different bearing types and operating conditions, a dynamic modeling strategy is adopted. Let

f_{r}

denote the shaft rotational frequency corresponding to the sample.

In general, let $K$ denote the number of fundamental fault mechanisms, and let $t = 1, 2, \dots, K$ be the index of the fundamental fault mechanism. The characteristic frequency associated with the $t$ -th fundamental fault mechanism is denoted as $f_{t}$ . Different fault classes can be represented by either an individual fundamental mechanism or a combination of multiple fundamental mechanisms. In the bearing fault diagnosis task considered in this study, three fundamental fault mechanisms are involved, namely inner race fault, outer race fault, and ball fault. Therefore, $K = 3$ , and the corresponding characteristic frequencies are instantiated as ${f_{t}}_{t = 1}^{K} = {f_{BPFI}, f_{BPFO}, f_{BSF}}$ . Under the deep-groove ball bearing condition with contact angle $ϕ \approx 0$ , the inner race fault characteristic frequency, outer race fault characteristic frequency, and ball fault characteristic frequency are formulated as follows

f_{BPFI} = \frac{n}{2} f_{r} (1 + \frac{d_{a}}{D_{p}})

(19)

f_{BPFO} = \frac{n}{2} f_{r} (1 - \frac{d_{a}}{D_{p}})

(20)

f_{BSF} = \frac{D_{p}}{2 d_{a}} f_{r} (1 - {(\frac{d_{a}}{D_{p}})}^{2})

(21)

Where

n

is the number of rolling elements,

d_{a}

is the diameter of the rolling element, and

D_{p}

is the pitch diameter. Based on dynamic characteristic frequencies, the positions of mechanism-related frequency bands are determined online for target samples across different bearing types and operating conditions, thereby avoiding the prior mismatch issue caused by fixed frequency settings. To characterize the fault characteristic frequencies and their harmonic components in the envelope spectrum, for each fundamental fault mechanism indexed by

t = 1, 2, \dots, K

and harmonic order

h \in H = {1, \dots, H_{\max}}

, the center frequency of the

h

-th harmonic is defined as

f_{t, h} = h f_{t}, f_{t, h} \leq \frac{f_{s}}{2}

(22)

A narrow-band analysis window is further constructed using the proportional bandwidth $ρ \in (0, 1)$ . For the $t$ -th fundamental fault mechanism and its $h$ -th harmonic, the window is centered at $f_{t, h}$ , and its lower and upper bounds are given by

l_{t, h} = f_{t, h} (1 - ρ), u_{t, h} = f_{t, h} (1 + ρ), 0 \leq l_{t, h}, u_{t, h} \leq \frac{f_{s}}{2}

(23)

A schematic illustration of the neighborhood windows around the fault characteristic frequencies and their harmonics is shown in Figure 5. Let the hard window be defined as

W_{t, h} (k) = 1 (l_{t, h} \leq f_{k} \leq u_{t, h})

(24)

Where

W_{t, h} (k)

represents the hard window corresponding to the

h

-th harmonic neighborhood of the

t

-th fundamental mechanism, while

l_{t, h}

and

u_{t, h}

correspond to the lower and upper bounds of that harmonic neighborhood window, respectively. Within the harmonic neighborhoods associated with each fundamental mechanism, the envelope power spectrum energy is accumulated to obtain the K mechanism-level physical evidence scores

s_{t} (x) = \sum_{h \in H} \sum_{k} P [k] W_{t, h} (k), t = 1, \dots, K

(25)

Where

s_{t} (x)

represents the physical evidence intensity of sample

x

over the harmonic neighborhoods corresponding to the

t

-th fundamental mechanism. Accordingly, for a given sample

x

, the mechanism-level physical evidence vector is given by

s (x) = [s_{1} (x), s_{2} (x), \dots, s_{K} (x)]

(26)

Figure 5.

Illustration of fault characteristic frequencies and their harmonic neighborhoods in the envelope power spectrum. The dashed bands denote the narrowband windows centered at the harmonic frequencies of BPFI, BPFO, and BSF.

To map the mechanism-level physical evidence into class-level physical priors, an indicator matrix $R \in {0, 1}^{Q \times K}$ is introduced, in which $Q$ is the number of fault classes and $R_{c, t} = 1$ indicates that class $c$ involves the $t$ -th fundamental fault mechanism. Based on this mapping, the raw class-level score of class $c$ is defined as

r_{c} (x) = \prod_{t = 1}^{K} s_{t} {(x)}^{R_{c, t}}

(27)

This definition accommodates both single-fault and compound-fault classes in a unified form. When a class involves only one mechanism, its score reduces to the evidence associated with that individual mechanism. When a class is composed of multiple fundamental mechanisms, its score is determined by the product of the corresponding mechanism evidence, thereby emphasizing the joint response when multiple mechanisms are simultaneously significant. In the bearing fault diagnosis task considered here, the set of fundamental mechanisms consists of inner race fault, outer race fault, and ball fault. Therefore, $K = 3$ , and the corresponding mechanism evidence vector is specifically written as

s (x) = [s_{I} (x), s_{O} (x), s_{B} (x)]

(28)

Where

s_{I} (x)

s_{O} (x)

, and

s_{B} (x)

represent the energy response intensities of sample

x

within the frequency bands associated with inner race fault, outer race fault, and ball fault, respectively. For the six fault classes

{I, O, B, I O, I B, O B}

considered in this study, the class-to-mechanism mapping matrix is instantiated as

R = [\begin{array}{l} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 1 \end{array}]

(29)

Where the rows correspond to the classes

I, O, B, I O, I B

, and

O B

, respectively, while the columns correspond to the fundamental mechanisms

I, O

, and

B

. The raw scores are then normalized, and a temperature parameter

τ

is introduced to control the smoothness of the prior distribution, thereby yielding the class-level physical prior distribution

p_{phys} (c | x) = \frac{{(r_{c} (x) + ε_{1})}^{\frac{1}{τ}}}{\sum_{j = 1}^{Q} {(r_{j} (x) + ε_{1})}^{\frac{1}{τ}}}, ε_{1} > 0, τ > 0

(30)

Where

ε_{1}

is a numerical stabilizer. When

τ

is small, the physical prior becomes sharper and places greater emphasis on the dominant mechanism class. When

τ

is large, the distribution becomes smoother, which helps mitigate the adverse effect of noisy physical evidence on training. To maintain consistency between model predictions and the physical prior, the unnormalized score vector output by the classifier for sample

x

is defined as

z (x) = C_{θ_{C}} (q (x)) = C_{θ_{C}} (f_{θ_{F}} (x)) \in R^{Q}

(31)

Where

z (x) = [z_{1} (x), \dots, z_{Q} (x)]

represents the logits of all classes,

q (x) = f_{θ_{F}} (x)

denotes the structure-enhanced feature representation of sample

x

obtained through convolutional feature extraction, KSFA-based feature enhancement, graph construction, and MRF-GCN-based structural propagation, and

C_{θ_{C}} (\cdot)

denotes the label classifier parameterized by

θ_{C}

. Accordingly, the class-probability distribution predicted by the model is given by

p_{θ} (c ∣ x) = softmax {(z (x))}_{c} = \frac{\exp (z_{c} (x))}{\sum_{j = 1}^{Q} \exp (z_{j} (x))}

(32)

Considering that not all samples from the target domain provide sufficiently reliable physical evidence, a gating mechanism is further introduced so that the physical prior consistency constraint is imposed only on samples with sufficiently reliable physical evidence. Let $w (x) \in {0, 1}$ denote the sample-level gating weight. Specifically, $w (x)$ is determined by three practical criteria: confidence gating, energy gating, and frequency-validity gating.

First, confidence gating is used to determine whether the constructed physical prior distribution has a sufficiently dominant class response. The confidence score is defined as

g_{conf} (x) = \max_{c} p_{phys} (c ∣ x)

(33)

The confidence-gating indicator is given by

m_{conf} (x) = 1 (g_{conf} (x) \geq δ_{conf})

(34)

Where

δ_{conf}

is the confidence threshold.

Second, energy gating is used to check whether the harmonic neighborhoods contain sufficient mechanism-related spectral energy. To avoid repeated counting caused by overlapping harmonic neighborhoods, a union window over all mechanism-related harmonic bands is first defined, and the mechanism-related energy is computed as

W_{unio n} (k) = 1 (\sum_{t = 1}^{K} \sum_{h \in H} W_{t, h} (k) > 0), E_{mech} (x) = \sum_{k} P [k] W_{union} (k)

(35)

Where

W_{union} (k)

indicates whether the

k

-th frequency bin falls into at least one valid harmonic neighborhood, and

E_{mech} (x)

represents the non-repeated mechanism-related energy within the union of all harmonic neighborhoods.

The total envelope-spectrum energy within the analyzed frequency band is given by

E_{band} (x) = \sum_{k} P [k] 1 (f_{ana}^{low} \leq f_{k} \leq f_{ana}^{high})

(36)

Where

[f_{ana}^{low}, f_{ana}^{high}]

denotes the envelope-spectrum analysis band,

f_{ana}^{low}

and

f_{ana}^{high}

are its lower and upper bounds, respectively, and the summation is performed over the one-sided frequency bins defined in equation (18). The energy ratio is calculated as

g_{eng} (x) = \frac{E_{mech} (x)}{E_{band} (x) + ε_{2}}

(37)

Where

ε_{2}

is a small numerical stabilizer used to avoid division by zero. The energy-gating indicator is defined as

m_{eng} (x) = 1 (g_{eng} (x) \geq δ_{eng})

(38)

Where

δ_{eng}

is the energy-ratio threshold. In implementation, this criterion prevents samples with extremely weak mechanism-related harmonic energy from dominating the physical consistency constraint when the energy threshold is enabled.

Third, frequency-validity gating is used to ensure that the fault characteristic frequencies used for physical prior construction are physically valid. Specifically, $m_{meta} (x) \in {0, 1}$ is introduced to indicate whether the bearing type and rotational-speed information required for calculating the fault characteristic frequencies are available and valid. The frequency-validity indicator is defined as

m_{freq} (x) = m_{meta} (x) \cdot 1 (\sum_{t = 1}^{K} \sum_{h \in H} 1 (0 < f_{t, h} \leq \frac{f_{s}}{2}) > 0)

(39)

Where

m_{meta} (x) = 1

means that the bearing type and rotational speed required for calculating the fault characteristic frequencies of sample

x

are available and valid; otherwise,

m_{meta} (x) = 0

. This condition ensures that at least one valid harmonic center falls within the Nyquist range for physical prior construction, thereby avoiding invalid physical priors caused by missing rotational-speed information, unavailable bearing parameters, or out-of-range harmonic frequencies.

The final sample-level gating weight is obtained by combining the three indicators

w (x) = m_{conf} (x) \cdot m_{eng} (x) \cdot m_{freq} (x)

(40)

Thus,

w (x) = 1

only when the target sample simultaneously satisfies the confidence, energy, and frequency-validity criteria; otherwise,

w (x) = 0

. Based on this design, the physical prior consistency loss is defined as

L_{phys} = \frac{E_{x} [w (x) D_{KL} (p_{phys} (\cdot | x) ‖ p_{θ} (\cdot | x))]}{E_{x} [w (x)]}

(41)

Where

E_{x} [\cdot]

is the expectation over samples from the target domain,

w (x)

is the gating weight of sample

x

, and

D_{KL} (\cdot ‖ \cdot)

represents the Kullback–Leibler divergence. The term

p_{phys} (\cdot | x)

denotes the class distribution constructed from the physical prior, and

p_{θ} (\cdot | x)

is the predicted class distribution of the model for sample

x

. If the denominator becomes zero,

L_{phys}

is set to zero. This normalized form avoids instability in the loss scale caused by fluctuations in the number of valid samples and ensures that the physical constraint acts only on samples from the target domain with sufficiently reliable physical evidence. To further improve training stability, a delayed-start linear ramp-up weighting strategy is designed to regulate the contribution of the physical prior consistency loss. Let the current training epoch be

e

. The physical prior consistency constraint weight is then defined as

λ_{phys} (e) = {\begin{cases} 0, & e < e_{start}, \\ λ_{\max} \frac{e - e_{start}}{T_{ramp}}, & e_{start} \leq e < e_{start} + T_{ramp} \\ λ_{\max}, & e \geq e_{start} + T_{ramp} \end{cases}

(42)

Where

λ_{\max}

is the maximum weight,

e_{start}

is the epoch at which the physical constraint begins to take effect, and

T_{ramp}

is the duration of linear ramp-up. In implementation, when

T_{ramp} \leq 0

, the ramp-up stage is skipped and

λ_{phys} = λ_{\max}

is directly applied once

e \geq e_{start}

. As indicated by equation (42), the model first establishes fundamental discriminative capability and cross-domain alignment capability during the early stage of training, and then progressively introduces mechanism-based constraints in the middle and later stages, thereby preventing the physical prior from imposing excessively strong interference on feature learning at the initial stage.

The hyperparameters associated with the physical prior were selected according to bearing signal-processing knowledge and the sampling setting of the HUST bearing dataset, and were kept fixed for all transfer tasks. The delayed-start ramp-up strategy was configured with $e_{start} = 120$ , $T_{ramp} = 80$ , and $λ_{\max} = 0.005$ , meaning that the physical prior consistency constraint was activated at epoch 120, increased linearly from epoch 120 to epoch 200, and reached its maximum weight at epoch 200. The sampling frequency was set to $f_{s} = 51.2$ kHz. For envelope analysis, the raw vibration signal was first band-pass filtered within the resonance band of 500–5000 Hz, and the envelope-spectrum analysis band was set to 0–2000 Hz to cover the main fault characteristic frequencies and their harmonic components. The first five harmonic orders were considered, that is, $H = {1, 2, 3, 4, 5}$ . The proportional bandwidth of each harmonic neighborhood was set to $ρ = 0.08$ , providing a narrow but tolerant neighborhood around each dynamically calculated harmonic center. The temperature parameter used to normalize the class-level physical prior distribution was set to $τ = 1.5$ . For the gating strategy, the confidence threshold was set to $δ_{conf} = 0.9$ , and the energy-ratio threshold was set to $δ_{eng} = 0.8$ to exclude samples with weak mechanism-related harmonic responses from the physical consistency loss. These parameters were fixed across all transfer tasks and were not tuned on the target test data.

Overall, the proposed physical prior consistency constraint based on sample-level physical priors does not directly impose a fixed frequency template on the model. Instead, through sample-level mechanism evidence extraction, class-level prior mapping, gated consistency regularization, and progressive weight scheduling, bearing fault mechanism knowledge is incorporated into the unsupervised transfer diagnosis process in a data-dependent and sample-adaptive manner. This design not only preserves the modeling capability of data-driven methods for complex operating condition distributions but also provides additional mechanism-based constraints for unlabeled samples from the target domain, thereby enhancing decision stability and physical plausibility in diagnosis under varying operating conditions.

3.5. Overall loss and training strategy

Following DAGCN, the source domain classification loss, domain alignment loss, and structural alignment loss are denoted by $L_{C}$ , $L_{DA}$ , and $L_{MMD}$ , respectively. Let $q_{i}^{s} = f_{θ_{F}} (x_{i}^{s})$ and $q_{j}^{t} = f_{θ_{F}} (x_{j}^{t})$ denote the structure-enhanced feature representations of the source domain sample $x_{i}^{s}$ and the target domain sample $x_{j}^{t}$ , respectively, as defined in equation (31). The source domain classification loss is defined as

L_{C} (D_{s}) = E_{(x_{i}^{s}, y_{i}^{s}) \sim D_{s}} [l_{CE} (C_{θ_{C}} (q_{i}^{s}), y_{i}^{s})]

(43)

The domain alignment loss is defined as

L_{DA} (D_{s}, D_{t}) = - E_{x_{i}^{s} \sim D_{s}} [\log (1 - D_{θ_{D}} (q_{i}^{s}))] - E_{x_{j}^{t} \sim D_{t}} [\log (D_{θ_{D}} (q_{j}^{t}))]

(44)

The structural alignment loss is defined as

L_{MMD} (D_{s}, D_{t}) = {‖ E_{x_{i}^{s} \sim D_{s}} [ϕ (q_{i}^{s})] - E_{x_{j}^{t} \sim D_{t}} [ϕ (q_{j}^{t})] ‖}_{Ω}^{2}

(45)

Where

D_{θ_{D}} (\cdot)

is the domain discriminator that outputs the probability that

q

comes from the target domain,

ϕ (\cdot)

is the nonlinear mapping function,

l_{CE} (\cdot, \cdot)

means the cross-entropy loss and

Ω

is the reproducing kernel Hilbert space. By further incorporating the physical prior consistency loss

L_{phys}

, the overall objective function is obtained as

L_{total} = L_{C} + γ L_{DA} + κ L_{MMD} + λ_{phys} (e) L_{phys}

(46)

Where

γ

and

κ

are the balancing coefficients, and

λ_{phys} (e)

is the dynamic weight assigned to the physical prior consistency loss, which varies dynamically with the training iteration. Under the gradient reversal implementation, the gradient of

L_{DA}

is reversed before being propagated to the feature extractor. Therefore, the discriminator minimizes

L_{DA}

, whereas the feature extractor is optimized in the opposite direction with respect to

L_{DA}

. Let

θ_{F}

θ_{C}

, and

θ_{D}

denote the parameters of the feature extractor, classifier, and domain discriminator, respectively, and let

η

be the learning rate. Since the physical prior consistency loss

L_{phys}

is defined in the class-probability space at the output of the classifier, it acts most directly on the classifier parameters

θ_{C}

. Meanwhile, the gradient of this loss is also backpropagated from

z (x)

through

q (x)

to the feature extractor by the chain rule, thereby jointly updating the feature extractor parameters

θ_{F}

. Under domain-adversarial training implemented with gradient reversal, the parameter updates can be expressed as

θ_{F} = θ_{F} - η \frac{\partial (L_{C} + κ L_{MMD} + λ_{phys} (e) L_{phys} - γ L_{DA})}{\partial θ_{F}}

(47)

θ_{C} = θ_{C} - η \frac{\partial (L_{C} + λ_{phys} (e) L_{phys})}{\partial θ_{C}}

(48)

θ_{D} = θ_{D} - η \frac{\partial L_{DA}}{\partial θ_{D}}

(49)

By optimizing the above overall objective function, the network parameters are jointly updated, enabling the model to learn feature representations with discriminability, domain invariance, and structural robustness, thereby improving the diagnostic performance on unlabeled samples from the target domain. The overall training procedure of PI-KSFA-DAGCN is summarized in Algorithm 1.

4. Case experiment

4.1. Dataset description and preprocessing

The proposed method is validated on the HUST bearing dataset, that is, A Practical Dataset for Ball Bearing Fault Diagnosis (Thuan and Hong, 2023), whose experimental platform and fault-type illustrations are shown in Figure 6. This dataset contains six fault classes composed of inner race fault, outer race fault, ball fault, and their pairwise combinations. Three load conditions, namely 0 W, 200 W, and 400 W, are considered to characterize the bearing vibration behavior under different operating conditions. The raw vibration signals are sampled at 51.2 kHz, and the duration of each raw vibration signal is 10 s. The raw vibration signals are first normalized using the Z-score standardization method. The signals are then segmented by a sliding window with both window length and step size set to 1024, such that no overlap exists between adjacent samples. Within each domain, a unified upper limit is imposed on the number of samples in each fault class, with at most 1000 segments retained per class. Subsequently, stratified sampling is performed according to fault class within each domain, and the samples are divided into training and test sets at a ratio of 8:2, yielding 800 training samples and 200 test samples per class. The three operating conditions are indexed by 0, 1, and 2, corresponding to 0 W, 200 W, and 400 W, respectively. Accordingly, six cross-condition transfer tasks are constructed. For brevity, $H_{s t}$ denotes the transfer task from source domain $s$ to target domain $t$ , where the model is trained on samples from domain $s$ and tested on samples from domain $t$ . It should be noted that, during domain adaptation training, labeled samples from the source domain together with a small subset of unlabeled samples from the target domain are fed into PI-KSFA-DAGCN, whose network architecture is shown in Table 1. The samples from the source domain are used for supervised classification learning. The small subset of samples from the target domain is used without class labels and participates only in feature alignment through domain-adversarial learning and the associated constraints. The remaining unlabeled samples from the target domain are reserved for model evaluation.

Figure 6.

Experimental platform of the HUST bearing dataset and the six fault types considered in this study, including inner race fault, outer race fault, ball fault, and their pairwise compound faults.

Table 1.

Detailed structure and computational complexity of PI-KSFA-DAGCN.

Component	Layer	Filter	Output size	TC (FLOPs)	SC (bytes)
Feature extractor	Input	/	64 × 1024	0	0
	1d conv layer_1	16 × 15 × 1	64 × 16 × 1010	242400	16400
	1d conv layer_2	32 × 3 × 1	64 × 32 × 1008	1548288	33792
	Pooling_2	2 × 1 max-pool	64 × 32 × 504	0	0
	1d conv layer_3	64 × 3 × 1	64 × 64 × 502	3084288	38272
	1d conv layer_4	128 × 3 × 1	64 × 128 × 500	12288000	88576
	Pooling_4	4 × 1 adaptive max-pool	64 × 128 × 4	0	0
	KSFA_DwConv_1	128 × 3 × 1	64 × 128 × 4	1536	896
	KSFA_DwConv_2	128 × 5 × 1, d = 2	64 × 128 × 4	2560	1152
	KSFA_Conv1 × 1_1	128 × 64	64 × 64 × 4	32768	8448
	KSFA_Conv1 × 1_2	128 × 64	64 × 64 × 4	32768	8448
	KSFA_Spatial squeeze	2 × 7 × 1	64 × 2 × 4	112	36
	KSFA_FC_1	128 × 32	64 × 32	4096	4096
	KSFA_FC_2	32 × 128	64 × 128	4096	4096
	KSFA_Fusion conv	64 × 128	64 × 128 × 4	32768	8704
	Linear layer_1	512 × 256	64 × 256	131072	131072
	GGL	256 × 10	64 × 64	2560	2560
	MRF_Conv_1	256 × 1200	64 × 1200	15654912	245760
	MRF_Conv_2	1200 × 300	64 × 300	290880000	4550400
	Linear layer_2	300 × 256	64 × 256	76800	76800
Classifier	Bottleneck	256 × 256	64 × 256	65536	65536
Classifier	Linear layer_3	256 × 6	64 × 6	1536	1536
Discriminator	Linear layer_4	256 × 1024	64 × 1024	262144	262144
	Linear layer_5	1024 × 1024	64 × 1024	1048576	1048576
	Linear layer_6	1024 × 1	64 × 1	1024	1024

4.2. Training visualization

Each model is trained for 300 epochs, with the first 50 epochs used for pretraining, during which no samples from the target domain are involved. The initial learning rate is set to 0.001 and is decayed by a factor of 0.1 at the 150th and 250th epochs, respectively. The two balancing coefficients, $γ$ and $κ$ , are computed as $[1 - \exp (- μ β)] / [1 + \exp (- μ β)]$ , where $μ$ is set to 10. Before the transfer learning strategy is activated, $β = 0$ . After activation, $β = 1$ . The loss curves of the model on the transfer task $H_{20}$ , which exhibits relatively large cross-domain discrepancy, are shown in Figure 7. It can be observed that the variation trends of the individual loss terms are consistent with the designed stage-wise optimization strategy. The classification loss decreases rapidly in the early training stage, indicating that the model is able to establish source domain discriminative capability effectively. After domain alignment and structural alignment are introduced at the 50th epoch, the corresponding losses gradually stabilize after a brief fluctuation, suggesting that the model begins to learn cross-domain consistent distributional and structural representations. After the physical prior consistency constraint is introduced at the 120th epoch, the physical loss first rises and then gradually decreases before eventually reaching a stable state, indicating that the physical prior is progressively incorporated into the optimization process during the later training stage. Meanwhile, the total loss remains bounded throughout all stages without divergence, demonstrating favorable training stability and convergence of the proposed model on transfer tasks with high representational difficulty.

Figure 7.

Training Loss Curves on transfer task $H_{20}$ .

4.3. Fault diagnosis analysis

To intuitively evaluate the feature representation capability of the proposed method, t-SNE is employed to visualize the high-dimensional features extracted from the final layer of the feature extractor after dimensionality reduction. Taking the transfer task $H_{21}$ as an example, the feature distributions learned by the backbone DAGCN and PI-KSFA-DAGCN are compared, and the visualization results are presented in Figures 8(a) and (b), respectively. It can be observed that the features learned by PI-KSFA-DAGCN exhibit clearer distribution boundaries and higher inter-class separability among different fault classes. Source domain and target domain samples exhibit sufficient alignment and overlap within the same fault class, while the class clusters become more compact. Notably, for the sixth fault class, namely OB, corresponding to the combination of outer race fault and ball fault, both models exhibit two relatively independent subclusters. However, each subcluster contains sufficient overlap between samples from the source domain and samples from the target domain. This phenomenon is more likely to reflect the intra-class multimodal distribution characteristics of the same fault under different operating conditions, rather than insufficient cross-domain alignment. Overall, these results demonstrate that the proposed method achieves more stable cross-domain feature alignment while preserving intra-class structural information.

Figure 8.

t-SNE visualization of the feature distributions learned by DAGCN and PI-KSFA-DAGCN on transfer task $H_{21}$ .

To further analyze the diagnostic performance across fault classes, the confusion matrices of the backbone DAGCN and PI-KSFA-DAGCN on the target domain are presented in Figure 9. The results show that, compared with the baseline model, PI-KSFA-DAGCN improves the recognition accuracy for four fault classes, namely B, corresponding to ball fault; IO, corresponding to the combination of inner race fault and outer race fault; IB, corresponding to the combination of inner race fault and ball fault; and OB, corresponding to the combination of outer race fault and ball fault, by 1%, 1%, 7%, and 26%, respectively. These results further demonstrate that the proposed method provides superior discriminative capability in complex fault patterns, particularly for the identification of compound faults.

Figure 9.

Confusion matrices of DAGCN and PI-KSFA-DAGCN on the target domain.

4.4. Comparison with state-of-the-art methods

To evaluate the effectiveness of PI-KSFA-DAGCN, seven advanced unsupervised domain adaptation methods are selected for comparison, including DANN (Ganin et al., 2016), DAN (Long et al., 2019), JAN (Long et al., 2017), CDAN, CDAN-E (Long et al., 2018), CORAL (Sun and Saenko, 2016), and DAGCN (Li et al., 2021a). Here, CDAN-E refers to CDAN with entropy conditioning. Table 2 and Figure 10 present the diagnostic accuracies on the target domain over six cross-condition transfer tasks. It can be observed that PI-KSFA-DAGCN obtains the highest accuracy on all transfer tasks, with an average accuracy of 93.75%, exceeding those of DANN, DAN, JAN, CDAN, CDAN-E, CORAL, and DAGCN by 10.64%, 13.63%, 17.98%, 12.24%, 11.19%, 5.15%, and 3.69%, respectively. Compared with the strongest baseline DAGCN, the average gain is 3.69%, which indicates a moderate rather than dramatic improvement. Therefore, the value of the proposed method should be understood not only from the numerical accuracy gain, but also from its consistent improvements across different transfer tasks and its incorporation of physically interpretable constraints into graph-based domain adaptation. This moderate and consistent improvement can be attributed to two aspects. First, compared with distribution-matching methods such as DAN, JAN, and CORAL, PI-KSFA-DAGCN explicitly models the structural relationships among samples during cross-domain alignment and enhances the stability and discriminability of graph features through the KSFA module, thereby alleviating the adverse effect of local structural mismatch caused by operating condition variations on transfer performance. Second, compared with adversarial methods such as DANN, CDAN, and CDAN-E, the proposed method further introduces physical prior consistency constraints based on fault characteristic frequencies and the energy responses within their harmonic neighborhoods, thereby providing additional mechanism-guided information to the classifier in scenarios where the target domain is unlabeled. As a result, the proposed method can improve cross-domain alignment and decision-boundary stability with improved physical interpretability.

Table 2.

Comparison of different methods on the HUST bearing dataset.

Task	DANN	DAN	JAN	CDAN	CDAN-E	CORAL	DAGCN	PI-KSFA-DAGCN
$H_{01}$	84.15%	79.25%	76.67%	80.42%	83.94%	87.58%	92.83%	95.64%
$H_{02}$	77.86%	77.44%	66.52%	78.12%	78.82%	85.08%	89.08%	93.21%
$H_{10}$	83.66%	80.99%	78.78%	81.13%	81.45%	87.75%	87.50%	91.64%
$H_{12}$	88.58%	86.90%	79.72%	87.13%	84.52%	98.08%	97.33%	98.36%
$H_{20}$	76.19%	71.83%	71.41%	75.22%	78.35%	77.75%	80.42%	84.64%
$H_{21}$	88.22%	84.30%	81.53%	87.02%	88.29%	95.33%	93.17%	99.00%
Average	83.11%	80.12%	75.77%	81.51%	82.56%	88.60%	90.06%	93.75%

Figure 10.

Comparison of different methods on six cross-condition transfer tasks of the HUST bearing dataset.

From the results of individual transfer tasks, PI-KSFA-DAGCN improves upon the baseline DAGCN by 2.81%, 4.13%, 4.14%, 1.03%, 4.22%, and 5.83% on the six transfer tasks, respectively, indicating that the proposed improvement is not only effective on average but also exhibits favorable stability and consistency across different transfer directions. In particular, more pronounced gains are observed on tasks such as $H_{20}$ and $H_{21}$ , suggesting that the proposed method can improve feature discriminability and domain invariance under relatively large operating condition discrepancies. These results also provide useful insight into the applicable scope of the proposed method. Although PI-KSFA-DAGCN achieves consistent improvements across all transfer tasks, the magnitude of improvement varies among different transfer directions. For relatively easier tasks where the baseline model already achieves high diagnostic accuracy, the remaining room for further improvement becomes limited. In contrast, more evident gains are observed under transfer tasks with larger operating-condition discrepancies, indicating that the proposed KSFA module and physical prior consistency constraint are particularly beneficial when feature drift, structural mismatch, and class-level semantic ambiguity are more pronounced. Therefore, the effectiveness of the proposed method should be interpreted not only from the absolute accuracy gain, but also from its consistent cross-task improvement, enhanced physical interpretability, and improved robustness under more challenging transfer scenarios.

4.5. Ablation study

To verify the independent contributions of the KSFA module and the physical prior consistency constraint, as well as their complementarity, ablation experiments are conducted on six cross-condition transfer tasks by comparing DAGCN, KSFA-DAGCN, PI-DAGCN, and the complete PI-KSFA-DAGCN. In this comparison, DAGCN is used as the backbone-only baseline, while KSFA-DAGCN and PI-DAGCN are used to evaluate the independent effects of the KSFA module and the physical prior consistency constraint, respectively. The results are presented in Table 3 and Figure 11. It can be observed that PI-KSFA-DAGCN achieves the best performance on all transfer tasks, with an average accuracy of 93.75%, exceeding those of DAGCN, KSFA-DAGCN and PI-DAGCN by 3.69%, 1.12% and 1.43%, respectively. These gains are moderate in magnitude, but they are consistently observed across the six transfer tasks. Compared with the backbone DAGCN, adding the KSFA module improves the average accuracy from 90.06% to 92.63%, indicating that KSFA contributes to more discriminative graph feature construction by enhancing multi-scale fault-related feature responses. Adding the physical prior consistency constraint improves the average accuracy to 92.32%, suggesting that mechanism-guided regularization helps improve the generalization ability on unlabeled samples from the target domain. The two components exhibit clear complementarity, and their joint use further improves the generalization capability of the model for bearing fault diagnosis under varying operating conditions.

Table 3.

Ablation results on the HUST bearing dataset.

Task	DAGCN	KSFA-DAGCN	PI-DAGCN	PI-KSFA-DAGCN
$H_{01}$	92.83%	94.83%	94.79%	95.64%
$H_{02}$	89.08%	91.92%	92.21%	93.21%
$H_{10}$	87.50%	89.00%	89.36%	91.64%
$H_{12}$	97.33%	98.33%	98.29%	98.36%
$H_{20}$	80.42%	82.92%	84.14%	84.64%
$H_{21}$	93.17%	98.75%	95.14%	99.00%
Average	90.06%	92.63%	92.32%	93.75%

Figure 11.

Ablation results on six cross-condition transfer tasks of the HUST bearing dataset.

4.6. Computational cost analysis

To further evaluate the practical applicability of the proposed method, the model complexity and computational overhead of different variants are analyzed. As shown in Table 4, DAGCN and PI-DAGCN contain the same number of trainable parameters, that is, 1.5846 M, because the physical prior constraint is used only as a training-stage regularization term rather than an additional inference branch. KSFA-DAGCN and PI-KSFA-DAGCN contain 1.6190 M trainable parameters, indicating that the KSFA module introduces only 0.0344 M additional parameters, corresponding to a 2.17% increase over DAGCN.

Table 4.

Computational cost comparison of different model variants.

Method	Trainable parameters	Increase vs. DAGCN	Avg. Training time per epoch (s)
DAGCN	1.5846 M	-	5.65
PI-DAGCN	1.5846 M	0/0.00%	7.18
KSFA-DAGCN	1.6190 M	+0.0344 M/+2.17%	7.12
PI-KSFA-DAGCN	1.6190 M	+0.0344 M/+2.17%	7.20

The average training time per epoch is calculated over epochs 120-300 on six cross-condition transfer tasks, during which the physical prior constraint is activated for PI-DAGCN and PI-KSFA-DAGCN. Under the same hardware and training settings, DAGCN, PI-DAGCN, KSFA-DAGCN, and PI-KSFA-DAGCN require 5.65 s, 7.18 s, 7.12 s, and 7.20 s per epoch on average, respectively. These results show that the proposed modules introduce only moderate training-time overhead.

Specifically, the KSFA module consists mainly of depthwise convolution, 1 × 1 convolution, and lightweight channel-temporal selection operations, resulting in limited additional parameter and computational costs. The graph construction process is based on mini-batch sample similarity calculation and Top-k sparsification, so its computational scale is controlled by the mini-batch size. The physical prior calculation, including band-pass filtering, Hilbert transform, FFT, and harmonic-neighborhood energy accumulation, is only used during training to construct the physical consistency constraint. Therefore, the proposed method improves cross-condition diagnostic performance and physical interpretability with limited additional overhead. The physical prior constraint does not introduce any additional inference branch, and the only increase in inference-stage model size comes from the lightweight KSFA module, which adds merely 0.0344 M trainable parameters.

5. Conclusion

To address distribution shift, graph construction instability, and insufficient physical constraints in bearing fault diagnosis under varying operating conditions, an unsupervised physics-informed domain-adversarial graph convolutional network with multi-scale selective feature enhancement and a physical prior consistency constraint is proposed. Built upon the DAGCN backbone, the proposed method incorporates Kernel Selective Fusion Attention to enhance the stability and discriminability of graph features. Meanwhile, a physical prior is constructed from fault characteristic frequencies in the envelope spectrum and the energy within their harmonic neighborhoods, and mechanism knowledge is integrated into the model optimization process through a sample-level gating strategy and a physical prior consistency constraint with delayed-start and linearly ramped weighting. Experimental results demonstrate that the proposed method achieves consistent improvements across multiple cross-condition transfer tasks. It not only improves the fault recognition accuracy on the target domain but also enhances inter-class separability in the feature space and cross-domain alignment capability, indicating that the collaborative modeling of physical prior and domain adaptation can effectively improve intelligent diagnostic performance under complex operating conditions. Future work will focus on few-shot target domain adaptation, learnable physical prior modeling, and online cross-device deployment.

Footnotes

Acknowledgement

The authors gratefully acknowledge the financial supported by The National Key Research and Development Program of China (2024YFD2100203) and Natural Science Foundation of Guangdong Province (2023A1515011723).

ORCID iD

Jichao Zhuang

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Key Research and Development Program of China (2024YFD2100203) and Natural Science Foundation of Guangdong Province (2023A1515011723).

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

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