## 1 Introduction
[PERSON] [16] proposed using the average instantaneous frequency as the basis for selecting the number of decomposition modes \(K\) in the VMD algorithm. He utilized the unsupervised learning algorithm, self-organizing map (SOM), to classify gear faults, achieving effective gear fault diagnosis. [PERSON] [17] optimized the decomposition layers \(K\) of VMD by combining the artificial fish swarm algorithm with envelope entropy. [PERSON] [18] employed a swarm algorithm using kurtosis as a criterion for parameter optimization. [PERSON] [19] used the particle swarm optimization algorithm to optimize the decomposition layers of VMD. These studies indicate that most scholars focus on the impact of the number of VMD decomposition layers on the decomposition performance, while often neglecting the coupled effect of decomposition layers \(K\) and penalty parameters \(\omega\)[20; 21]. To address this issue, this paper proposes an improved VMD method based on PSO, which comprehensively analyzes the coupled effects of the decomposition layers and penalty factors on the decomposition performance. PSO is an intelligent search method that simulates the collaborative behavior of a group of organisms. It iteratively adjusts computing parameters by searching for local and global optimal solutions to approach the global optimal solution. However, the selection of various hyperparameters in particle swarm will greatly affect the optimization rate of the algorithm, and inappropriate hyperparameters will also cause the algorithm to fall into the local optimal solution prematurely. By incorporating the adaptive inertia weight strategy to accelerate the convergence speed and using the compression factor method to solve the problem of getting stuck in local optima, the optimized particle swarm algorithm is employed to search adaptively for the optimal \(K\) and penalty parameters \(\alpha\). The optimized results are combined with feature extraction on the original signal, and finally, a probabilistic neural network is used for fault mode recognition, achieving high-precision fault diagnosis. The remainder of this paper is structured as follows. To fully leverage the performance advantages of VMD and its suitability for local damage fault feature extraction, as well as to address the difficulty in selecting key parameters in VMD, Section 2 combines VMD with an improved particle swarm optimization algorithm to achieve the optimal combination of VMD key parameters. With this optimal parameter combination, VMD is utilized to decompose vibration signals. In Section 3, effective modal components are extracted based on the correlation coefficient threshold. To mitigate the loss of signal features due to the VMD strong penalty on signal discontinuities, multi-domain feature parameters are selected using the Laplacian score method, collectively constructing a fault-sensitive feature vector. Based on this, a probabilistic neural network model is built to achieve fault diagnosis of rotating machinery. In Section 4, the signal processing and fault diagnosis algorithm proposed in the previous sections are validated using public data of gears and rolling bearings, demonstrating that IPSO-VMD-PNN possesses strong feature extraction capabilities and high fault diagnosis accuracy. In summary, the main contributions of this paper are as follows:  
(1) A vibration analysis model for key components of rotating machinery is established by analyzing failure mechanisms and signal characteristics. VMD is proposed for fault feature extraction and its effectiveness is verified through simulated fault signals. (2) The PSO is improved using adaptive inertia weights and compression factor methods. Optimal parameters \(K\) and penalty parameters \(\alpha\) for VMD are determined. VMD decomposes the vibration signal, and effective components are identified using correlation coefficient thresholding. Multi-domain features are selected using the Laplace score method, constructing a fault-sensitive feature vector.