Feature selection technique

Feature selection is a methodological approach for extracting the most relevant subset of attributes from a comprehensive feature space based on predefined optimization criteria and statistical evaluation metrics. This process aims to retain only the most relevant features from a dataset, thereby reducing the scale of data processing by eliminating redundant and irrelevant variables.

Residual convolutional neural networks (RCNNs)

CNNs represent a class of deep-learning architectures that employ multiple filters to hierarchically process input samples through convolution and pooling operations across successive layers [27]. Building on this concept, RCNNs introduce skip connections to effectively address the model degradation problem [35, 36]. Instead of relying solely on deeper network structures to approximate complex hidden nonlinearities, RCNNs facilitate the learning of residual representations, which simplifies the optimization process. The architectural configuration of RCNNs is illustrated in Fig. 1.

Fig. 1Full-screen View

(Color online) RCNN architecture

The transformation from input to output can be expressed as follows [36]:(4)where represents the composite mapping learned by the network layers, corresponds to the residual function, and x denotes the input features. The residual function is often parameterized as to explicitly indicate its dependence on both the input and the learnable weights. Accordingly, the nth residual block can be formulated as follows [36]:(5)(6)In this formulation, and represent the input and output feature maps of the nth residual module in the network architecture, respectively. The identity mapping is captured by , whereas denotes the ReLU activation function. According to Eq. (6), the feature propagation from the nth to the Nth layer can be expressed as [36]:(7)In deep network architectures, the backpropagated gradient can be expressed as [36]:(8)Here, denotes the gradient propagated through the nth layer. The identity term 1 in this equation enables direct gradient propagation without attenuation. While the residual component must pass through weighted layers, potentially modifying the gradient, the constant identity term ensures stable gradient flow. Even when the residual gradient approaches zero, the presence of this identity term prevents the gradients from vanishing.

Anti-noise module

In this study, an anti-noise (AN) architecture that combines a soft threshold function with an attention mechanism integrated into an RCNN is proposed. The resulting framework, termed AM-RCNN, enhances noise robustness in MSR accident diagnosis.

The soft thresholding (ST) operator is a widely adopted denoising technique in signal processing applications. Its fundamental principle involves threshold contraction: when the input exceeds a predefined threshold value, the output approaches this boundary; otherwise, it is set to zero. This mechanism effectively suppresses noise interference while preserving signal characteristics. The mathematical representation is as follows [44]:(9)In conventional implementations, the threshold parameter τ typically requires manual configuration by system engineers prior to network deployment. In this study, the threshold is computed using an attention mechanism [45] that can be trained within the network and updated adaptively. The architectural configuration of the AN module is illustrated in Fig. 2.

Fig. 2Full-screen View

(Color online) The architectural configuration of the anti-noise module (in the red box)

The global average pooling (GAP) operation in this submodule computes channel-wise mean values while preserving essential feature information. Parallel to this path, a compact network processes the GAP outputs through two sequential fully connected layers, culminating in a sigmoid activation function that generates scaling factors within the [0, 1] range. This auxiliary network incorporates batch normalization and PReLU layers for feature normalization and nonlinear activation, respectively. The resulting adaptive threshold τ is obtained by multiplying the GAP-derived averages by the scaling coefficients produced by the compact network.

Bayesian theory-based optimization

Hyperparameters play a crucial role in machine learning algorithms because they directly influence the training process and significantly affect model performance. Tuning these hyperparameters often requires professional knowledge and expert experience. Consequently, hyperparameter tuning represents a challenging optimization problem with an implicit objective function. BO is a powerful approach for addressing such optimization problems [46]. Figure 3 presents the workflow of BO.

Fig. 3Full-screen View

(Color online) BO workflow

The BO process begins by defining the hyperparameter space, which encompasses the feasible value ranges for each tunable parameter. The optimization procedure then begins by sampling an initial set of configurations for model training and validation. Following performance evaluation, an acquisition strategy determines the next parameter combination for evaluation, guided by probabilistic surrogate models.

This process iterates through successive cycles, refining the parameter selection based on the performance observed in previous iterations. The iterative procedure continues until a predefined termination condition is satisfied. Once this condition is met, the optimization process concludes by identifying the hyperparameter configuration that yields the best model performance.

To provide a comprehensive and intuitive visualization of the computational procedure, a flowchart summarizing the integrated methodology is presented in Fig. 4. The flowchart illustrates the step-by-step procedure of the proposed methodology. The process begins with feature selection using Pearson correlation analysis and random forest importance ranking, followed by the construction and training of the AM-RCNN model. BO is then applied to fine-tune the model’s hyperparameters, ultimately yielding an optimized diagnostic model for MSR accident scenarios.

Fig. 4Full-screen View

(Color online) Flowchart of the proposed methodology

A designed MSR description

In 2011, the Chinese Academy of Sciences (CAS) initiated a strategic research program focusing on advanced nuclear fission energy systems, in which MSR technology represented a key development pathway. As part of this initiative, the TMSR program [5], a pioneering effort in MSR technology development, was conducted by the Shanghai Institute of Applied Physics (SINAP), CAS. The MSR investigated in this study corresponds to the design version proposed in the TMSR program [47-49]. Figure 5 illustrates the conceptual design of the reactor, and Table 1 summarizes its key technical specifications [50].

Fig. 5Full-screen View

(Color online) The conceptual design of the reactor

The reactor adopts a dual-loop molten salt configuration. The primary loop, designated as the fuel–salt system, comprises a core vessel, primary heat exchanger, circulation pump, and connecting piping. The fuel salt circulates upward from the bottom plenum through the core and then flows into the shell side of the heat exchanger via the circulation pump. Thermal energy is subsequently transferred to the secondary loop, after which the cooled fuel salt completes the cycle by returning to the bottom plenum.

The secondary loop, referred to as the coolant salt system, contains a circulation pump, air-cooled heat exchanger, and associated piping. The coolant salt flows through the tube side of the primary heat exchanger, absorbs thermal energy from the primary loop, and releases heat to the atmosphere through the air-cooled exchanger. Furthermore, the MSR design incorporates a passive reactor vessel air cooling system (RVACS) for effective decay heat removal.

Operation scenario data collection

Given the limited availability of operational data for MSRs, accident diagnosis datasets were generated using the RELAP5-TMSR simulation code. This specialized tool, developed at SINAP [51, 52], extends RELAP5/MOD4.0 with enhanced capabilities for liquid-fueled reactor analysis. The accuracy of the code has been verified through experiments conducted on a high-temperature molten-salt natural circulation test bench [53, 54], and it is currently applied in the design and analysis of MSR systems. Figure 6 illustrates the nodal configuration, where the primary circuit components are labeled 0XX, while 1XX and 2XX denote the secondary loop and the air-cooling system, respectively. The RVACS is modeled using nodes 301–306.

Fig. 6Full-screen View

(Color online) MSR nodal configuration diagram

In this study, eight typical operational scenarios are considered: one steady-state condition and seven transient events. By training the model using data that include both steady-state and transient conditions, the model’s generalization capability is improved, enabling comprehensive accident detection across a wide range of operating states. Table 2 lists these operational conditions.

Given the varying transient characteristics at different power levels, each scenario was generated at reactor power levels ranging from 10% to 120% of full power (FP) in increments of 10% FP. The resulting dataset comprised more than 550,000 data samples, with each sample consisting of 21 monitoring parameters aligned with the architecture of the instrumentation and control system. The data-partitioning scheme allocated 70% of the dataset for model training, while equal proportions of 15% were reserved for validation and testing.

Additionally, to prevent the diagnostic system from ignoring variables with small absolute values caused by differences in measurement scales, the Z-score normalization method was adopted [55].

Feature selection

The 21 monitored parameters are listed in Table 3.

Assuming that all monitoring parameters are used as input features would impose a substantial computational burden, thereby compromising the efficiency and performance of the model. Therefore, feature selection was performed to reduce redundant variables among these parameters.

Reactor monitoring systems typically generate multiple interdependent parameters in which strong correlations may exist (e.g., perfectly correlated variables contain redundant information and can be represented by a single parameter). Therefore, the Pearson correlation metric was used as the primary statistical tool to quantify the linear dependence among system variables. The Pearson correlation coefficients between variables are shown in Fig. 7, where features 0–20 correspond to the parameters listed in Table 3.

Fig. 7Full-screen View

(Color online) The Pearson correlation analysis of the monitoring parameters in data set

A Pearson correlation coefficient with an absolute value exceeding 0.8 is commonly considered to indicate a strong correlation between two parameters [56]. However, this threshold should be evaluated within the specific context of the application. In practical scenarios, when two parameters exhibit strong correlation, preference is typically given to retaining the parameter that is easier to measure and has higher measurement accuracy.

Following this principle, a parameter-screening process was conducted to evaluate parameter significance across three correlation thresholds: , , and perfect linear correlation (). The results indicate that selecting a threshold greater than 0.9 for parameter retention yields optimal model performance. Accordingly, the input feature subset filtered using the Pearson correlation coefficient comprises seven parameters: reactor power; fuel circulating pump speed (FCPS); coolant circulating pump speed (CCPS); inlet, middle, and outlet temperatures of the reactor core (CIT, MCT, and COT); and the inlet temperature of the primary heat exchanger tube side (TITHE).

Uncorrelated features in machine learning models may act as noise sources, potentially reducing model accuracy by introducing classification bias. The random forest variable importance method was therefore applied to identify the relevance between variables and classes. The evaluation and ranking of feature importance are shown in Fig. 8. To ensure that the selected features effectively support the model’s diagnostic capability, scenarios with relative importance thresholds of 0.05 and 0.10 were examined during the parameter-screening process. Rigorous testing indicated that the model achieved the best performance with a threshold of 0.05. Consequently, all seven parameters were retained in the final feature set to preserve diagnostic accuracy and ensure robust system performance.

Fig. 8Full-screen View

(Color online) The evaluation and ranking of feature importance

Model structure and hyperparameter setting

The structure of the AM-RCNN model for MSR accident diagnosis, which integrates advanced deep-learning methodologies, is shown in Fig. 9, and Table 4 provides details of the AM-RCNN architecture.

Fig. 9Full-screen View

(Color online) Architectural comparison of CNN, RCNN, and the proposed AM-RCNN

The AM-RCNN adopts a four-convolutional-layer architecture with batch normalization layers and activation functions incorporated throughout the network to enhance performance. It integrates several key components, including a residual mapping module for feature transformation, an anti-noise (AN) processing unit for signal enhancement, and a GAP layer for feature aggregation.

The subsequent layers include a tensor-flattening operation for dimensionality reduction, a dropout regularization layer to mitigate overfitting [57], and a fully connected transformation layer for feature classification. The network employs a softmax activation function for final classification decisions, while utilizing parametric ReLUs (PReLUs) throughout the model to improve feature representation learning [58]. The shortcut connection in the residual architecture from Layers 3 to 14 is automatically adapted to either identity mapping or a convolutional layer to maintain dimensional consistency with the main network structure. The AN processing module spanning network layers 9–14 is described in detail in Sect. 2.3 and illustrated in Fig. 2.

Evaluation index

Model performance was evaluated using confusion matrices [59], which provide a detailed analysis of classification performance. As illustrated in Fig. 10, the matrix structure for multiclass problems assigns distinct labels to each category, with diagonal elements indicating correct predictions. The row sums correspond to the number of samples belonging to each actual class.

Fig. 10Full-screen View

(Color online) The classification performance matrix for multi-class prediction analysis

In addition, the evaluation incorporated accuracy and the F1-score metric. Classification accuracy is a fundamental performance indicator that quantifies the ratio of correctly predicted instances to the total number of samples. The F1-score represents the harmonic mean of precision and recall and is particularly sensitive to the lower value between these two measures. Higher F1-scores indicate improved model performance, especially in handling imbalanced classification tasks. Detailed definitions of precision, recall, F1-score, and diagnostic accuracy are provided in Eqs. (10)–(13) [60].(10)(11)(12)(13)where TP (true positive) and TN (true negative) represent the numbers of correctly classified positive and negative instances, respectively, while FP (false positive) and FN (false negative) represent the numbers of incorrectly classified negative and positive instances.

Model performance and comparison

The case study primarily evaluated the proposed AM-RCNN against conventional CNN and RCNN architectures, focusing on comparative training and testing performance within similar network frameworks.

Figure 9 illustrates the architectural differences among the baseline CNN, standard RCNN, and the proposed AM-RCNN approach. To ensure consistency, the number of convolutional layers in both the RCNN and CNN models was fixed at four, matching that of the AM-RCNN. In addition, the other layers in the main structures of the RCNN and CNN were identical to those of the AM-RCNN. The shortcut structure in the RCNN was configured as identity mapping.

The training procedure employed cross-entropy as the optimization objective function. Model parameter optimization was performed using stochastic gradient descent (SGD) with momentum acceleration and L2 regularization. The convolutional and fully connected layers were initialized using Kaiming and Xavier initialization methods [61, 62], respectively. Based on prior training experience, the model hyperparameters were configured as follows: a learning rate of 0.005 for gradient updates, a dropout probability of 0.5 for regularization, a mini-batch size of 128 samples per iteration, and 200 training epochs.

To prevent overfitting and determine the optimal model configuration, an early stopping strategy was adopted. The validation loss function served as the primary convergence criterion, triggering early termination when no improvement was observed over ten consecutive training epochs. In such cases, the model parameters corresponding to the minimum validation loss, typically occurring ten epochs prior to termination, were retained as the optimal configuration.

Figure 11 presents the epoch-wise progression of training loss and validation accuracy across the different models, revealing several key observations.

Fig. 11Full-screen View

(Color online) Training loss and validation accuracy trends across model architectures versus training epochs

Figure 12 and Table 5 present the classification results for all evaluated models on the test dataset, including the confusion matrices and performance metrics (accuracy and F1-score). This analysis yielded several significant findings.

Fig. 12Full-screen View

(Color online) Test data classification performance comparison using confusion matrices

Robustness test

Liquid-fueled MSR operational environments inherently contain multiple noise sources that affect detector readings, necessitating a rigorous evaluation of the robustness of the diagnostic model.

Real-world operational conditions typically exhibit composite noise characteristics arising from multiple sources. For analytical purposes, the mixed noise was approximated as Gaussian white noise. The noise impact was quantified using the signal-to-noise ratio (SNR), which evaluates the relative strength between the target signal and background interference. The SNR is mathematically defined as follows [63]:(14)where represents the spectral energy of the target signal component and denotes the corresponding noise energy within the measurement bandwidth.

Monitoring signals in NPPs generally maintain relatively low noise levels, with vibration measurements typically exceeding 30 dB SNR [64]. This threshold was therefore used as the baseline for assessing model robustness, and the corresponding test results are presented in Fig. 13 and Table 6. The results indicate that all three models performed well in noisy environments, achieving diagnostic accuracy exceeding 90%. In particular, the accuracy and F1-score of the proposed model remained close to 97%.

Fig. 13Full-screen View

(Color online) Test data classification performance under noisy conditions comparison using confusion matrices

As shown in Fig. 14, the performance of all three models decreased in noisy environments, with the conventional CNN model experiencing the most significant decline. Performance evaluation under noisy conditions revealed reductions of 2.4% in classification accuracy and 2.6% in the F1 metric for the conventional CNN architecture. Similarly, the standard RCNN implementation exhibited an equivalent decrease of 2.4% in both performance metrics. In contrast, the proposed AM-RCNN framework demonstrated superior noise resilience, with only a 0.4% reduction in both metrics. This result highlights the effectiveness of the integrated anti-noise processing module in maintaining model stability.

Fig. 14Full-screen View

(Color online) Comparison of different model results between noiseless data and noisy data

Model optimization

To further enhance the performance of the AM-RCNN model, hyperparameter optimization was conducted using the BO method. In training deep neural networks, the learning rate, batch size, and dropout rate are critical hyperparameters that significantly influence learning efficiency, convergence speed, and model generalization capability. Consequently, these hyperparameters were selected as the primary optimization variables in this study.

The hyperparameter search space was defined as follows: the learning rate ranged from 1×10^-4 to 1×10^-1; the batch size was randomly selected from the discrete set {32, 64, 128, 256}; and the dropout rate ranged from 0.1 to 0.6.

Figure 15 presents the results of the BO for diagnostic performance. The optimized configuration achieved 98.3% accuracy, representing a 1% improvement over the baseline model, with the following optimal parameters: a learning rate of 0.0003, a batch size of 256, and a dropout rate of 0.28.

Fig. 15Full-screen View

(Color online) Hyperparameters optimization results of proposed AM-RCNN model