# Federated Few-Shot Learning-Based Machinery Fault Diagnosis in the Industrial Internet of Things
## 1 Introduction  
Mechanical fault diagnosis is of paramount importance in modern industrial contexts as it significantly contributes to enhancing equipment reliability, operational safety, and cost reduction. The emergence of the Industrial Internet of Things (IIoT), big data, and AI has propelled the field of fault diagnosis, leading to notable advancements [1; 2]. These advancements have demonstrated their effectiveness in utilizing condition-monitoring data. As a result, extensive research efforts have been conducted to employ conventional machine-learning techniques [3], such as Support Vector Machines (SVM) [4], k-Nearest Neighbors (KNN) [5], decision trees [6], etc. However, the diagnosing accuracy of these methods often relies on the expertise of specialists who manually select relevant features. In light of the above limitations, deep-learning (DL) techniques have emerged as a prominent approach, offering the potential for end-to-end learning and the elimination of manual feature selection [7]. However, industrial equipment generally operates under normal conditions, resulting in a limited availability of fault samples. Unfortunately, the conventional deep-learning approach requires a substantial amount of labeled data to achieve a significant level of accuracy. Consequently, numerous recent studies have concentrated on few-shot fault diagnosis [8; 9]. Furthermore, due to the complex and variable working conditions of the equipment, the results of fault diagnosis vary across specific industrial scenarios, which divides fault samples into different domains based on their feature distribution and category space. To address the challenges posed by limited data availability and domain diversity, few-shot cross-domain fault diagnosis emerges as a compelling approach, often referred to as few-shot transfer learning [10; 11]. This technique leverages knowledge from the source domain to the target domain by training anetwork using data from the source domain on one operational condition and subsequently fine-tuning it with the data from the target domain on different working conditions. This approach has exhibited significant potential to greatly enhance the efficacy of fault diagnosis within the target domain. In recent years, a few scholars have embarked on the investigation of the few-shot cross-domain fault-diagnosis approach. For instance, [PERSON] et al. [12] introduced a novel cross-domain fault-diagnosis method for bearings, which relies on an enhanced semi-supervised meta-learning technique. [PERSON] et al. [11] presented a few-shot transfer learning method that leverages meta-learning for the diagnosis of few-shot samples under varying conditions. [PERSON] et al. [10] systematically summarized the application of deep meta-learning in the context of few-shot cross-domain fault diagnosis. However, the existing research on few-shot cross-domain fault diagnosis has primarily focused on transferring knowledge from a single source domain to the target domain. Nevertheless, several crucial issues remain unresolved: (1) Practical scenarios often still involve a shortage of all types of fault samples within a single source domain, making the utilization of data from multiple source domains a potentially effective solution. (2) Conducting few-shot machinery fault diagnosis across multiple source domains introduces several new challenges. First, the distinct source domains can be considered to be independent factories or data holders, who are reluctant to disclose or share their private data due to concerns related to industry competition, user privacy, and data security [13; 14]. Second, fault samples acquired from multiple source domains may exhibit diverse distributions due to disparate working conditions or industrial scenarios [15], therefore further complicating the complexities associated with few-shot fault diagnosis.