## 1 Introduction
They constructed an instance graph to extract the structural feature relationships of the samples and measure the cross-domain structural differences using the maximum mean square error. Since the representations of different GCN nodes become increasingly similar with depth, it becomes challenging to capture subtle differences between nodes in graph data under variable working conditions, thereby affecting diagnostic accuracy and stability. [PERSON] et al. (2023) [12] studied a fault diagnosis method for rotating equipment under variable operating conditions that integrates GNN with one-shot learning. They used a CNN for feature extraction and utilized GNN for one-shot learning to achieve fault diagnosis. [PERSON] et al. (2024) [13] proposed a semi-supervised meta-path space extended graph neural network (ME-GNN) to transform the nearest neighbor relationships among vibration data, fault information, and variable speed information into a graph; facilitate message passing and aggregation across heterogeneous data types; and investigate the fault diagnosis of rotating equipment under time-varying speed conditions. Based on existing research, the current GNN method still has problems that need to be solved, such as unbalanced information transmission, insufficient global and local feature extraction capabilities, and limited cross-condition generalization capabilities. To address the above problems, this paper proposes a Chebyshev graph convolutional neural network modeling method for rotating equipment fault diagnosis under variable working conditions. Its application performance is verified through a series of variable working condition fault diagnosis experiments. The main contributions of this paper are as follows:  
1. To address the problem of unbalanced information transmission faced by feature extraction under variable working conditions, a symmetry processing mechanism is introduced in the ChebyNet framework to ensure that the information exchange between nodes in the graph is conducted with full consideration of the symmetry in their mutual relationships, thus achieving a fair transmission and effective integration of information. 2. To tackle the challenge of interwoven global and local features, as well as the coexistence of complex relationships in variable operating conditions data, The ChebyNet, renowned for its proficiency in capturing global structural features, is integrated with a GAT, which effectively characterizes local dependencies among nodes. This integration yields a model adept at comprehensively mining and analyzing both global and local features, as well as the intricate relationships within data. 3. To address the problem of insufficient generalization across different working conditions, a self-supervised learning mechanism incorporating a node reconstruction task designed as an auxiliary training component is proposed. This allows the model to deeply explore and map the intrinsic feature structure of the original data during the reconstruction process, forming an efficient and collaborative multi-task learning framework alongside the primary node classification task. As a result, the model can more accurately capture the complex and dynamic features of variable working conditions data, fundamentally enhancing its generalization ability across different conditions. The remainder of the paper is organized in the following way: Section 2 elaborates on the architecture of the proposed ChebyNet modeling approach for rotating equipment fault diagnosis under variable operating conditions; Section 3 introduces the experimental setup and provides an analysis of the experimental results; and Section 4 concludes the paper.