3.1. Problem Formulation

We formulate the problem of cross-domain few-shot industrial defect detection as follows. Let \({C} = \{C_1, C_2, \dots, C_m\}\) be the universal set of all industrial product categories, where each category \(C_i\) is treated as a distinct domain with a unique feature distribution. To facilitate meta-learning, we partition \({C}\) into two disjoint subsets: a meta-training set \({C}_{\text{train}} = \{C_1, C_2, \dots, C_k\}\) and a meta-testing set \({C}_{\text{test}} = \{C_{k+1}, \dots, C_m\}\), where \(k < m\) and their intersection is the null set (\({C}_{\text{train}} \cap {C}_{\text{test}} = \emptyset\)). The learning process is structured around a series of episodic tasks, \(\{T_j\}\). Each task \(T_j\) is designed as an \(n\)-way, \(k\)-shot classification problem and comprises a support set \(S_j\) and a query set \(Q_j\). The support set \(S_j\) consists of \(n \times k\) labeled examples, with \(k\) examples for each of the \(n\) sampled classes. The query set \(Q_j\) contains unlabeled examples from the same \(n\) classes, for which the model must generate predictions. The objective is to train a model \(M\) on tasks sampled from \({C}_{\text{train}}\) such that it learns a transferable meta-knowledge. Subsequently, when presented with a new task from an unseen domain in \({C}_{\text{test}}\), the model must rapidly adapt using only the support set \(S_j\) and demonstrate strong generalization performance on the corresponding query set \(Q_j\).

3.2. Model Architecture

3.2.1. Feature Extractor with Attention Mechanism

Given that industrial defects typically manifest as subtle variations in texture or shape, the selection of a powerful feature extractor, denoted as \(f_{\varphi}\), is crucial. We selected ResNet-18 as our backbone network because it offers an effective balance between representational capacity and computational efficiency, thereby mitigating the risk of overfitting common in few-shot scenarios with deeper networks. To enhance the model’s sensitivity to discriminative features, we integrated a SE module within each residual block of the ResNet-18 architecture. As depicted in Fig. 1, the SE module first applies global average pooling to each channel to produce a channel-wise statistic. This is followed by two fully connected (FC) layers that learn non-linear inter-channel dependencies, generating a set of attention weights for each channel. These weights are then used to rescale the original feature maps, adaptively recalibrating channel-wise feature responses to emphasize informative features and suppress less useful ones. The integration of residual connections and the SE module not only helps mitigate the vanishing or exploding gradient problem but also enables dynamic feature recalibration, contributing to more stable model training.

3.2.3. Meta-Learning Algorithm of R\(^2\)-Net

The novelty of our approach lies in the synergistic fusion of Reptile, an efficient first-order meta-optimization algorithm, with a modified Relation Network tailored for capturing fine-grained inter-sample relationships. This hybrid architecture is designed to learn a meta-initialization that exhibits strong cross-domain generalization, thereby enabling rapid and precise inference on few-shot tasks from unseen domains. The model is parameterized by \(\theta = \{\phi, \varphi\}\), where \(\phi\) and \(\varphi\) denote the parameters of the feature extractor and the Relation Network, respectively. Our meta-learning algorithm optimizes the global parameters \(\theta\) through a bi-level loop structure. Algorithm 1 provides a detailed description of the specific learning process for R\(^2\)-Net. As detailed in Algorithm 1, the R\(^2\)-Net training procedure is structured into two nested phases: an inner loop for task-specific adaptation and an outer loop for meta-optimization. The primary objective of the inner loop is to adapt the model to a specific few-shot task, \(T_i\), sampled from the source domains. To this end, temporary model parameters, \(\theta'\), are fine-tuned using the support set \(S_i\). This adaptation is achieved by computing the task-specific loss \(\mathcal{L}_{T_i}(f_{\theta'})\) on the support set and subsequently updating \(\theta'\) via standard mini-batch stochastic gradient descent (SGD). The update rule is defined as follows:

\begin{equation} \theta_i' \leftarrow \theta_i' - \alpha \nabla_{\theta_i'} \mathcal{L}_{T_i}, \label{eq:UpdateU0020taskU0020parameters} \end{equation}

where \(\alpha\) is the inner-loop learning rate. After \(k\) optimization steps, we obtain a version of the parameters, \(\theta'\), that is fully adapted to the specific task \(T_i\). After the inner loop has found the parameters \(\theta_i'\) that are fully adapted to each specific task, the Reptile algorithm in the outer loop performs a meta-update to enhance the generalization ability of the meta-parameters \(\theta\). The meta-update rule is as follows:

\begin{equation} \theta \leftarrow \theta + \varepsilon \frac{1}{B} \displaystyle\sum\limits_{i=1}^{B}\left(\theta_i' - \theta\right), \label{eq:UpdateU0020meta-parameters} \end{equation}

where \(\varepsilon\) is the outer-loop meta-learning rate, and \(B\) is the number of tasks in a batch.

The learning process can be interpreted as seeking a meta-initialization within the parameter space that is not specialized for any single task, but is instead positioned to be rapidly adaptable across a distribution of tasks. Through iterative training on diverse tasks, the model gradually learns a transferable, task-agnostic meta-knowledge. As a result, when confronted with a new few-shot task from an unseen domain, it can leverage this meta-learned starting point to converge to a high-performance solution with only a few steps of inner-loop adaptation.

As illustrated in Fig. 2, the efficacy of the R\(^2\)-Net framework stems from the synergy between its two core components: the Reptile meta-optimizer and the Relation Network. Reptile is responsible for learning a robust meta-initialization for the entire model—encompassing both the feature extractor and the Relation Network—that exhibits strong cross-domain potential. Building upon this advantageous starting point, our modified Relation Network then performs precise, task-specific reasoning by leveraging its powerful, fine-grained metric based on instance-level comparisons. This symbiotic relationship enables R\(^2\)-Net to achieve both high training stability and superior classification performance on the challenging task of cross-domain industrial defect detection.

4.1. Datasets

To systematically evaluate the performance of our proposed R\(^2\)-Net framework, we conducted experiments on the publicly available MVTec and NEU-CLS benchmark datasets. The MVTec dataset is particularly well-suited for our study, as it comprises 15 distinct industrial product categories, each featuring multiple fine-grained, real-world defect types and their corresponding defect-free counterparts. This diversity enables a rigorous assessment of both fine-grained recognition and cross-domain generalization capabilities. For our experimental setup, we selected 13 categories with multiple defect types and partitioned them into 6 distinct folds. Each fold was split into a meta-training set (\({C}_{\text{train}}\)) and a meta-testing set (\({C}_{\text{test}}\)), ensuring that the product categories in \({C}_{\text{test}}\) were entirely unseen during the meta-training phase. To determine the optimal hyperparameter configuration, we employed a 5-fold cross-validation strategy within the meta-training set of each fold. Fig. 3 visualizes sample images of both defect-free and various defective products from the dataset. The NEU-CLS dataset is a collection of surface defects specific to hot-rolled steel strips, encompassing six typical defect types: rolled-in scale, patches, crazing, pitted surface, inclusion, and scratches. This dataset presents a distinct set of challenges, characterized by significant intra-class morphological variations and high inter-class visual similarities, which provides a rigorous test for fine-grained classification. We include the NEU-CLS dataset in our evaluation to assess the generalizability of our method to a different industrial material and defect taxonomy. Since all defects in this dataset belong to the same product category, it serves as a benchmark for intra-domain few-shot learning. Consequently, for each task, we directly sample \(n\) distinct defect types to construct the classification problem without the need for domain-level partitioning.

4.2. Experimental Setup

The R\(^2\)-Net framework was implemented in PyTorch and trained on a single graphics processing unit (GPU). Additional hardware and software specifications are detailed in Table 1. Our training process adhered to the standard episodic paradigm for few-shot learning. Each task was formulated as an \(n\)-way, \(k\)-shot problem, where we sampled \(n \in \{2,3,4,5\}\) and \(k \in \{1,2,3,4,5\}\). The query set for each task contained 10 samples per class. All input images were resized to \(128 \times 128\) pixels. We employed the SGD optimizer for both the inner and outer loops, with learning rates (\(\alpha\)) of 0.01 and meta-learning rates (\(\varepsilon\)) of 0.001, respectively. The meta-learning rate was selected based on empirical tuning; we observed that higher rates led to training instability, while significantly lower rates resulted in slow convergence. The inner-loop adaptation consisted of seven gradient update steps. To mitigate overfitting, a dropout layer with a rate of 0.5 was applied within the relation module. The model was trained for 100 epochs with a meta-batch size of 16 tasks. During the meta-testing phase, we assessed the model’s generalization performance on three mutually exclusive product domains randomly sampled from \({C}_{\text{test}}\). For each unseen domain, the final performance was reported as the average accuracy over 50 randomly generated evaluation tasks. Each task required the model to classify \(n\) distinct defect types from the same product category.

4.3. Result Analysis

To comprehensively evaluate the performance of R\(^2\)-Net, we benchmark it against seven representative few-shot learning methods, which are described below. (1) Training from scratch: As a fundamental baseline, a standard CNN is trained from scratch using only the \(k\) samples from the support set of a given task. The model starts with randomly initialized weights and is immediately evaluated on the query set. (2) ResNet-12 (fine-tuning): This baseline represents a simple transfer learning approach. A ResNet-12 backbone is first pre-trained on all classes in the meta-training set. During evaluation, the feature extractor is frozen, and a new linear classifier is trained using only the \(k\) support samples. (3) Prototypical Networks (ProtoNet): A classic metric-learning method. ProtoNet computes a class prototype by calculating the mean embedding (centroid) of the support samples for each class. Classification is then performed by assigning a query sample to the class of its nearest prototype in the embedding space based on a distance metric like Euclidean distance. (4) Relation Network (RN): An advancement over methods with fixed distance metrics. RN employs a learnable deep neural network as a non-linear comparator to compute a relation score between a query sample and support samples, allowing it to learn more complex similarity measures. (5) MAML: A seminal optimization-based meta-learning algorithm. MAML aims to find a set of initial model parameters that are highly sensitive to new tasks, such that only a few steps of gradient descent are needed to achieve high performance on a new task. (6) Reptile: An efficient, first-order approximation of MAML. Reptile also learns a meta-initialization but updates the meta-parameters by linearly moving them toward the task-adapted parameters, thereby circumventing the costly second-order derivatives required by MAML. (7) ALFA: An advanced MAML variant that shifts the optimization focus from the initial parameters to the adaptation process itself. It employs a meta-network to dynamically generate task-adaptive hyperparameters (e.g., learning rates) for the inner-loop updates, enabling more effective and efficient adaptation.

To ensure a fair comparison across tasks with a varying number of classes (\(n\)) and to account for the effect of random guessing, we report all results using a normalized accuracy metric. This metric is defined as:

On the MVTec dataset, our evaluation followed a rigorous 6-fold cross-validation protocol, with the final results reported as the average normalized accuracy across the six independent runs. As shown in Table 2, our R\(^2\)-Net model achieves a mean accuracy of 0.5192, outperforming all other compared methods. Notably, it surpasses the next-best baseline, ALFA (0.4840), by a significant margin of 3.5%, demonstrating a substantial advance in overall performance. Furthermore, R\(^2\)-Net not only outperforms advanced optimization-based methods like ALFA but also shows a considerable lead over the classic MAML (0.3569). This suggests that our meta-optimization strategy using Reptile is more effective in practice for finding a generalizable initialization than more complex second-order methods. When compared to metric-based approaches, our model’s performance is also markedly superior to that of Prototypical Networks (0.3910) and a standalone Relation Network (0.1611). This highlights that the potential of metric learning in complex domains is significantly amplified when guided by a meta-optimizer like Reptile. The performance of R\(^2\)-Net (0.5192) is not merely an additive combination of its components; it substantially exceeds the sum of the individual performances of Reptile (0.2565) and the Relation Network (0.1611). This result strongly attests to the synergistic nature of our framework: Reptile provides an advantageous and readily adaptable meta-initialization, which the Relation Network then leverages to realize its full relational reasoning capabilities.

To validate the robustness of our framework beyond the multi-domain setting of MVTec, we conducted further evaluations on the NEU-CLS dataset, which focuses on intra-domain defect classification. This dataset, characterized by a relatively uniform background, presents a distinct challenge due to the subtle and often ambiguous boundaries between defect classes. As reported in Table 3, R\(^2\)-Net again demonstrated superior performance. Specifically, it achieved an accuracy of 0.8233 in the demanding 1-shot setting, which increased to 0.8250 in the 5-shot setting. In the 1-shot scenario, our model surpasses the strong ALFA baseline (0.8000) and outperforms MAML (0.6416) by a substantial margin of nearly 18 percentage points. This superior performance extends to comparisons against its core components, significantly exceeding the results of a standalone Relation Network (0.6361) and Reptile (0.6028). The consistent, top-tier performance across two industrial datasets with disparate visual characteristics and challenges strongly validates the generality and robustness of the R\(^2\)-Net framework. This demonstrates its ability to learn effective, transferable meta-knowledge applicable to both complex cross-domain scenarios and fine-grained, intra-domain classification tasks.

4.4. Ablation Experiments

To systematically investigate the contribution of each core component within the R\(^2\)-Net framework and to validate our choice of backbone network, we conducted a comprehensive set of ablation studies. These studies were performed on the MVTec AD dataset, adhering to the same 3-way, 5-shot protocol as our main experiments. The performance for each configuration is reported as the average normalized accuracy from the 6-fold cross-validation. Specifically, we analyzed the impact of the three primary components of our framework: the Reptile meta-optimizer, the Relation Network metric learner, and the SE attention module. The results of this analysis are presented in Table 4.

Our analysis begins with a baseline ResNet-12 backbone. A simple transfer learning strategy (fine-tuning) yields a normalized accuracy of 0.3036. Interestingly, when the core meta-learning components are applied individually, their performance falls short of this baseline; a standalone Reptile optimizer achieves 0.2565, and a standalone Relation Network scores 0.1611. This finding underscores that these components must be integrated correctly to unlock their potential. The synergy between our chosen components becomes evident when we combine the Reptile meta-optimizer with the Relation Network. As shown in Table 4, this combination boosts the performance to 0.3639, surpassing the fine-tuning baseline. This significant improvement, which is greater than the sum of the individual parts, strongly validates the central hypothesis of our framework; the synergistic fusion of a meta-optimizer and a metric learner is crucial. We further investigated the contribution of the SE module. Integrating the SE module into the Reptile+RN architecture on the ResNet-12 backbone results in a substantial performance leap to 0.4968. This effect is consistent on our final ResNet-18 backbone, where the full R\(^2\)-Net model achieves 0.5048, while removing the SE module causes the performance to drop to 0.4875. This clearly demonstrates that the SE module’s channel attention mechanism is integral to the model’s success, effectively enhancing its ability to capture fine-grained, discriminative features in industrial defects.

Having established the synergy of the core components, we next investigate the impact of the backbone architecture on performance. An ideal backbone for few-shot learning must strike a delicate trade-off between powerful representational capacity and efficient task adaptability. To this end, we evaluated the R\(^2\)-Net framework with four ResNet backbones of varying depths: ResNet-12, ResNet-18, ResNet-50, and ResNet-101. The results, presented in Table 5, reveal a clear trend. Performance improves as the network depth increases from ResNet-12 (0.4824) to ResNet-18, which achieves the peak normalized accuracy of 0.5192. However, further increasing the depth leads to a significant and sustained decline in performance, with ResNet-50 and ResNet-101 scoring 0.4435 and 0.4171, respectively. This outcome provides strong evidence that, within our few-shot learning framework, simply increasing network depth beyond a certain threshold is detrimental to generalization performance.

The performance degradation of deeper networks like ResNet-50 and ResNet-101 can be attributed to their pre-training on large-scale datasets such as ImageNet. This process entrenches highly abstract and specialized features in the deeper layers that are specific to the source domain (i.e., natural images). In a few-shot cross-domain scenario, adapting this large and relatively rigid parameter space using only a handful of samples presents two significant challenges. First, the optimization becomes exceedingly difficult; the Reptile algorithm may fail to find an effective optimization trajectory within the limited inner-loop steps. Second, the risk of overfitting to the source domain increases, as the model’s high capacity gives it a propensity to rely on its memorized ImageNet features rather than learning to generalize from the few new samples, leading to negative transfer. In contrast, ResNet-18 achieves the optimal trade-off. It is sufficiently deep to offer superior representational capacity compared to ResNet-12, providing the foundation for its performance gains. Simultaneously, it is not so deep as to exhibit the significant parameter inertia seen in ResNet-50 and ResNet-101. Consequently, its parameter space retains sufficient plasticity, making it highly suitable for rapid and effective adaptation. The choice of backbone depth, therefore, directly governs the efficacy of our framework’s synergistic design. Our experimental results confirm that ResNet-18 is the optimal choice for the “Reptile+Relation Network+SE” combination, as it provides powerful feature representation while retaining the necessary flexibility for effective meta-learning.

4.5. Model Behavior Analysis

Having established the overall superiority of the R\(^2\)-Net framework, we now further investigate its behavioral properties under varying conditions of task difficulty (\(n\)-way) and sample availability (\(k\)-shot). The following experiments are designed to probe the model’s performance boundaries, providing both qualitative and quantitative insights into the effectiveness of our design philosophy.

1. (1)

   Impact of task difficulty (\(n\)-way): As illustrated in Fig. 4, we evaluated the model on the MVTec dataset with a fixed 5-shot setting while increasing the number of classes from \(n=2\) to \(n=5\). The results show a graceful degradation in performance across all categories as \(n\) increases, with no sudden collapses. This indicates the inherent stability of our framework. We also observe performance variations linked to class characteristics; categories with distinct visual features, such as “Capsule” and “Cable,” consistently maintain high accuracy with a flatter decline. This suggests that performance is correlated with the inherent class separability of the task, offering a valuable reference for assessing the model’s applicability to specific industrial products.

   

2. (2)

   Impact of sample availability (\(k\)-shot): A key finding, made apparent by the clear performance gap between categories in Fig. 5, is the model’s differing performance on shape-based versus texture-based defects. The results indicate a clear difference in the model’s ability to handle different types of visual features. The model demonstrates significantly higher proficiency on shape-based tasks compared to texture-based ones. For categories like “Cable” and “Hazelnut,” the model achieves high levels of accuracy, whereas for texture-rich categories such as “Grid” and “Carpet,” the overall accuracy is noticeably lower. This observation is a significant contribution, as it reveals the performance characteristics of our meta-learning framework on tasks with different visual properties. It not only explains performance variations but also informs future methodological refinements (e.g., designing specialized modules for texture analysis).