## 1 Introduction  
As an important part of rotating equipment, rolling bearings have been applied in modern industrial areas such as shipbuilding, vehicles, and aerospace. The failure of rolling bearings can cause massive time and economic losses, especially for large plant equipment. Therefore, the early fault monitoring and diagnosis of rolling bearings are vital for the reliability and safety of industry equipment [1, 2]. However, it is hard to diagnose faults of rolling bearings under complex non-linear and nonstationary vibration signals. It is an outstanding challenge to successfully extract vulnerable fault features under very noisy conditions. Generally, the traditional time-frequency fault diagnosis feature extractors such as the classic Fourier transform (FFT), wavelet transforms (WT), and empirical mode decomposition (EMD) are commonly used to extract features and diagnose faults. However, the vibration signals of faults are nonstationary and non-linear characteristics, so the analysis method of a single time-frequency domain cannot adequately extract the nonstationary features from the fault data of rotating machinery [3]. [PERSON] et al. [4] designed the EMD method as an adaptive analysis method employed extensively for mechanical fault detection and diagnosis. The EMD method can effectively reduce low-frequency interference by highlighting the high-frequency resonant part. However, EMD uses the recursive decomposition method to separate components with similar frequencies, which can easily cause the modal confusion and endpoint effects [5]. The errors in the EMD process are passed in layers, which are prone to false components and endpoint effects that affect the accuracy of fault feature extraction. To overcome these EMD disadvantages, many scholars have tried to adapt the EMD method [6; 7]. These improved EMD methods have some inhibitory effects on the problems of mode mixing but cannot eliminate the effects of mode mixing. In 2014, [PERSON] et al. [8] proposed a variational mode decomposition (VMD) method which has attracted increasing attention in bearing fault diagnosis in industrial engineering. As a non-recursive signal decomposition method, the VMD method, characterized by noise robustness and high accuracy, has an outstanding advantage in conquering the issues of modal confusion and endpoint effects compared with EMD [9]. [PERSON] et al. [10] utilized VMD in the feature extraction of bearing signals, and the experimental results verified that VMD had good results in bearing fault detection. Although the method of the VMD has a better decomposition ability than the EMD, it also has some disadvantages in that the modal decomposition fraction \(K\) and the learning rate \(\alpha\) must be set in advance by the people's prior experience. In addition, the accuracy of decomposition relies heavily on these parameters in the decomposition process of nonstationary signals. An improperly chosen size of the modal number \(K\) can cause over- or under-decomposition phenomena, and the quadratic penalty term \(\alpha\) can affect the bandwidth of the modal components in the VMD solution. With the development of intelligence optimization algorithms such as particle swarm optimization (PSO), genetic, swarm, whale, and grey wolf (GWO), researchers have used these algorithms to optimize and solve the VMD parameters. [PERSON] et al. [11] proposed a fault information guided variational decomposition method (FIVMD) which determines parameters \(\alpha\) and \(K\) based on statistical thresholds of IGGCS\(\backslash\)GGS, IGGCS\(\backslash\)GGS, and RFCA, effectively extracting repeated transients of weak bearings but lacking adaptability. [PERSON] et al. [12] proposed a VMD mode that adaptively obtained the globally optimal mode number \(K\) through a sparrow-inspired algorithm but omitted the impact of penalty factor \(\alpha\) on signal decomposition.