## 1 Introduction
Hence, the need to identify fault categories across different working conditions remains a critical aspect of ongoing research in few-shot learning within fault diagnosis. Meta-learning, as an adaptable approach to the few-shot problem, emphasizes the acquisition of learning capabilities rather than the learning process itself [26]. This strategy, pivotal in practical industrial applications, necessitates only minimal adjustments to accommodate new tasks [27]. Contrary to direct learning of a predictive mathematical model, meta-learning endeavors to understand the process of learning a generalized model [17]. Viewing feature extraction as direct data learning, the meta-learner gains insight by evaluating this process, enabling task completion with minimal samples. Among the various optimization-based meta-learning methods, model-agnostic meta-learning (MAML) seeks an initial parameter set sensitive to new tasks [28; 29]. This approach allows for an enhanced performance following a gradient update on a small dataset of new tasks, demonstrating its potential in rapidly adapting to novel challenges. In industrial systems, fault diagnosis relies on complex time-series signals gathered from equipment operating under varying conditions. The scarcity of fault samples and the diversity of working conditions pose challenges in discerning intricate fault information. To address this, the article introduces a novel gradient-oriented prioritization meta-learning (GOPML) approach. This method is designed to enhance the adaptability of meta-learning in recognizing few-shot fault conditions across diverse operational scenarios. GOPML optimizes knowledge adaptability, crucial for accurately identifying faults in data-rich yet sample-scarce environments. The principal contributions of this work are encapsulated as follows:  
1. We introduce a novel gradient-oriented prioritization meta-learning (GOPML) approach that demonstrably mitigates overfitting in few-shot learning scenarios. By refining sensitive initialization parameters and harnessing the robust adaptability of knowledge, GOPML gains a distinct edge in tackling few-shot problems under varying operational conditions. 2. This study advocates a strategic task sequencing strategy inherent to GOPML, bolstering the stability of the diagnostic performance amidst fluctuating operational conditions. The strategy draws inspiration from curriculum learning, methodically arranging tasks from the simplest to the most complex, thus endorsing a more stable, stepwise learning process. This structured learning trajectory is conducive to the development of a generalized knowledge representation, enhancing convergence efficiency with a reduced number of samples. 3. The robustness of the GOPML algorithm is rigorously validated through extensive experiments on the Tennessee Eastman Process (TEP) and Skoltech Anomaly Benchmark (SKAB) datasets. The findings confirm GOPML's adeptness in adapting to new categories in few-shot scenarios and its consistent diagnostic accuracy across varied working conditions. Comparative analysis further underscores the superiority of GOPML over contemporary state-of-the-art methods. The article is organized to present a coherent exploration of the GOPML methodology in the following section, then empirical case studies are discussed to substantiate the method's efficacy, culminating in a synthesis of insights in the concluding section.