Introduction
Radon (Rn-222) is a colorless, odorless, inert gas widely found in rocks, soil, and groundwater. It is a radioactive product of the uranium-238 decay series [1-3]. Owing to its chemical inertness, radon can migrate freely through soil pores and enter building environments through cracks or cavities [4, 5]. Prolonged exposure to radon and its progeny through inhalation into the lungs (in which alpha decay occurs) can damage lung tissue and increase the risk of lung cancer [6, 7]. The International Agency for Research on Cancer (IARC) and the World Health Organization (WHO) have both classified radon as the second leading cause of lung cancer after tobacco use [8, 9].
Radon has attracted attention not only owing to its potential health risks but also because of its relevance in fields such as environmental science [10], geological exploration [11], and radioactive contamination assessment [12]. As a natural tracer, radon helps scientists study various environmental processes such as greenhouse gas emissions, atmospheric circulation, groundwater flow, and volcanic activity [13-16]. Accurate measurements of the concentration and exhalation of radon provide crucial data to support the study of these natural phenomena.
The exhalation rate of radon refers to the quantity of radon released per unit area of a given medium per unit time. This metric is critical to assess the hazards associated with radon exposure in various environments where humans are expected to be present. Accurate measurement of the rate of radon exhalation not only helps to identify hazardous areas with high levels of radon but also provides essential data for atmospheric and environmental pollution assessments. Currently, several techniques are commonly used to measure the rate at which radon is exhaled to ensure safety, including the closed-loop method (also known as the accumulation method) [17, 18] as well as an open-loop method [19, 20], activated charcoal adsorption [21, 22] and solid-state nuclear track detectors (SSNTDs) [23, 24]. However, in complex real-world applications, conventional methods still involve some limitations of note in terms of the accuracy of the obtained data and the efficiency of the measurement process. Additionally, the results are often associated with uncertainty due to statistical fluctuations in nuclear measurements and instrument errors. Errors become more pronounced on surfaces with low exhalation rates. This uncertainty can lead to discrepancies between estimated and true radon exhalation rates. Therefore, the development of methods to improve the accuracy with which the rate of radon exhalation can be measured has been recognized as a key issue in research on radon monitoring.
In recent years, with the rapid development of machine learning and deep learning technologies, these techniques have gradually been applied in environmental data analysis. Conventional machine learning methods such as support-vector machines (SVMs) [25], random forest (RF) models [26], and eXtreme gradient boosting (XGBoost) [27], have already shown promising results in anomaly detection and spatial prediction of radon concentration. Using external parameters and manually designed features, machine learning methods can effectively handle the relationships between known variables and thus provide significant support for analyses of the concentration of radon in complex environments.
With the increased scale of data and advancements in computing power, deep learning technologies have attracted increasing attention owing to their powerful capability to automatically extract features and model nonlinear relationships. Huang et al. successfully identified anomalous radon sources and quantified the radon precipitation rate by combining particle swarm optimization (PSO) and a long short-term memory network (LSTM) [28]. McKenzie et al. modeled groundwater tracer radon in coastal areas using a one-dimensional convolutional neural network (CNN), recurrent neural network (RNN) model (1D-CNN-RNN), and fully connected neural network (FCNN) models [29]. Luo et al. proposed a calibration method based on fast local outlier factor (FastLOF) and nonuniform popularity-similarity optimization (NPSO) with backpropagation (BP) (NPSO-BP) neural network models, which significantly improved the accuracy of radon concentration measurements [30]. In addition, a reduced-order model [31] and an intelligent prediction method [32] proposed by He et al. based on deep learning have made significant progress in applications within the nuclear industry. Deep learning has demonstrated great potential for handling complex data and improving the accuracy of radon monitoring. The development of integrated and hybrid tools based on artificial intelligence may offer a new approach for environmental radioactivity modeling. Complex nonlinear relationships can be better captured by combining multiple deep learning models, which can significantly enhance the accuracy and timeliness of the obtained predictions. The development of integrated and hybrid artificial intelligence tools may offer a new approach for environmental radioactivity modeling. By combining multiple deep learning models, complex nonlinear relationships can be captured more effectively, which can significantly enhance prediction accuracy and timeliness.
Temporal variations in radon concentration typically exhibit significant nonlinear characteristics that encompass both local rapid fluctuations (such as statistical noise and instrumental disturbances) and long-term dynamic equilibrium trends (such as the saturation phase of radon exhalation). Conventional methods have both advantages and limitations in modeling such data as time series. Long short-term memory (LSTM) and gated recurrent unit (GRU) models rely on recurrent structures for sequential computation to capture long-term dependencies effectively. However, their sequential nature limits parallelization, which makes them less efficient for large-scale data processing and sensitive to local noise. Pure convolutional neural networks (CNNs) extract features using local receptive fields, which makes them well-suited for short-term pattern recognition. However, their fixed convolutional window constrains their ability to model long-range dependencies. Transformer models excel in global temporal modeling and capture long-range dependencies effectively. However, they may overlook local patterns on short time scales. Consequently, a single deep learning model cannot fully accommodate the complex characteristics of radon concentration data. Thus, leveraging the strengths of different models is crucial to enhance prediction accuracy and stability.
In the field of deep learning, the hybrid models combining CNN structures with transformer models have demonstrated outstanding performance in handling complex tasks, which makes these techniques an effective tool for processing intricate time-series data. For example, Xu et al. achieved efficient real-time semantic segmentation by using a transformer model to extract semantic information during training while using a single-branch CNN for inference [33]. Additionally, Peng et al. proposed a feature-coupling unit that integrates the local feature extraction capability of a CNN with the global representation learning of the transformer to significantly improve image classification and object detection performance [34]. Compared to conventional multilayer perceptrons (MLPs), transformers, and pure CNNs, the CNN-Transformer hybrid model not only retains the ability of CNNs to learn local patterns but also leverages the transformer to capture long-range dependencies for better computational efficiency and parallelization [35, 36].
Based on this background, we propose an optimization method for measured data on radon exhalation rates based on a hybrid model that combines CNN and transformer architectures. The hybrid model combines the advantages of one-dimensional convolutional neural networks (1D-CNNs) in local feature extraction [37, 38] and the relative advantages of transformer models in modeling time-series data [39]. Complex nonlinear relationships can be captured without relying on external parameters by directly modeling variations in time-series data representing the concentration of radon. By learning from a large-scale dataset, the proposed models optimized the measurement results to reduce the uncertainties caused by statistical fluctuations and instrument errors and improve the goodness of fit of the data on exhalation rates.
Materials and methods
Data set acquisition
The core of deep-learning technology lies in training models on large datasets to construct accurate and complex predictive models. However, acquiring data on nuclear radiation is challenging, especially with regard to the collection and compilation of large-scale and diverse datasets that conventional methods often fail to meet. Therefore, we adopted a strategy to generate simulated data. A large-scale dataset containing 6 million samples of data on radon concentrations was constructed to train the model based on a numerical model of radon exhalation on the surface of the medium and the principle of statistical fluctuation. The model was tested and validated using actual measurement data.
The measurement chamber of a radon monitor based on the electrostatic collection method (e.g., RAD7) is conventionally used to measure the rate of radon exhalation on the surface of a medium using the accumulation method. During radon decay, the rate at which alpha particles are emitted by 218Po can be expressed by the following formula [40].
Typically, the calibration factor of a radon monitor (i.e., kp) is obtained through a calibration process in a standard radon chamber. Therefore, the true concentration of radon as a function of time can be expressed by the following formula [41].
Assuming that the period of each measurement cycle is T, the rate at which alpha particles are emitted by 218Po during the n-th measurement cycle can be expressed by the following formula.
Based on the above equations, the theoretical true radon concentration values required to train a learning model during the simulation of the measurement cycles can be obtained using the following formula.
Two kinds of uncertainty caused by the error are primarily considered in simulating the true radon concentration measured by the measuring instrument, including the statistical fluctuation uncertainty and the measurement error of the instrument itself (±1 counting error).
The uncertainty caused by statistical fluctuations can be derived by calculating the square root of the number of measurement events [42]. For radon measurements with a 218Po count recorded during the n-th measurement cycle of N, a statistical uncertainty of one-sigma is given by 
The measurement error of the instrument itself (±1 count error) also introduces some additional uncertainty that results from a mismatch between the precise timing of the gate opening and closing and the alpha particle events on the detector [43]. A slight misalignment between the timing window of the counter and the occurrence of a particle event can cause a count difference between the readings, which can thus affect the accuracy of the resulting measurements. Therefore, the analog count can be further represented by
Based on Eq. (1) to (6), the final simulated radon concentration measured by the instrument, considering the effects of statistical fluctuations and measurement errors, can be calculated using the following equation:
The number of cycles was set to 12, and the values of key parameters, such as the radon exhalation rate and effective decay constant, were randomly combined as outlined in Table 1. Theoretical true radon concentration values were generated based on Eq. (4), and simulated radon concentration values designed to mimic those obtained in real measurements were generated using Eq. (7). A total of 6 million samples of simulated radon concentration data were generated across a wide range of concentrations from zero to several hundred thousand.
| Fixed parameter | Value | Random parameter | Value range |
|---|---|---|---|
| ηPo | 0.3 | λeff | 1×10−3∼5×10−5 |
| VR | 0.007 | V | 0.001~0.01 |
| S | 0.04 | J | 0.005~1 |
| kp | 4.7619 × 103 | T | 240 |
The dataset was divided into training and validation sets with a ratio of 8:2. The training set was used to train the parameters of the model, whereas the validation set was used to evaluate the generalization performance of the model to ensure its predictive capability with input data that were not present in the training dataset. To improve the stability of the learning process and accelerate convergence, the raw radon concentration data were standardized before being input into the model. Standardization was performed by calculating the mean and standard deviation of the sample data and converting each feature into a zero-mean unit-variance form. This process can be expressed as follows.
Model Construction
In the process of measuring radon concentration using the accumulation method, the radon concentration data showed significant nonlinear changes over time, which mainly manifested as a rapid growth stage in the early stage and stable dynamic equilibrium in the later stage. These complex timing changes are subject to a combination of statistical fluctuations and instrumental errors. To meet the requirements of optimizing radon exhalation rate measurement data, we propose a deep learning framework based on a CNN-Transformer hybrid model to optimize radon concentration data and improve the accuracy with which measured radon exhalation rate can be fit to the expected data for a given condition. The model was trained iteratively on a desktop system equipped with an NVIDIA RTX 3060 GPU. The implementation was developed using the Python 3.9 programming language using the PyTorch library, and the source code was managed and developed using PyCharm.
The proposed CNN-Transformer hybrid model adopts a concatenated fusion approach [44], fully combines the advantages of local feature extraction and global time series modeling, and predicts radon concentrations optimally based on an input time series. The network structure of the model is shown in Fig. 1. In this hybrid model, the 1D-CNN extracts local features from the time series through sliding window operations and multilayer convolutional stacking. This is particularly effective in capturing dynamic change patterns during the initial rapid growth phase to provide high-quality input for global modeling. The transformer module uses the multi-head self-attention mechanism to model long- and short-term dependence relationships of the time series from a global perspective. Especially in the dynamic equilibrium stage, the results show that the transformer model was able to capture the global characteristics of the change in the concentration becoming stable accurately, and it further improved the fitting accuracy and modeling capability of existing methods.
One-dimensional convolutional neural network (1D-CNN)
The core task of the 1D-CNN module is to extract key local features from the input radon concentration time series to provide basic support for improving the radon exhalation rate-fitting accuracy. Through the convolution operation of the sliding window, 1D-CNN is able to capture local patterns in the time series, such as the change trend and fluctuation characteristics of the radon concentration over time. Furthermore, the introduction of the ReLU activation function enhances the nonlinear expression capability of the network so that it can learn complex data relations more effectively. This module consists of five convolutional layers and one fully connected layer (FC).
Convolutional layers are the core component of 1D-CNN models, and their main function is to extract local features from input data through a sliding window operation. The convolution operation enables local perception of an input sequence by sliding a window across it. The extent of each local perception is determined by the size of the convolution kernel. The model progressively extracts local dynamic features of the time series, ranging from short-term fluctuations to high-level feature patterns, by stacking multiple convolutional layers. For the input time-series X=[x1, x2, ..., xT] of data on radon concentrations, each layer of the convolution operation can be expressed as:
The output of the convolution operation is processed by the nonlinear activation function ReLU to introduce nonlinear capability, the formula of which is given as follows.
With the progressive processing of multilayer convolution and activation operations, the feature dimension gradually expands, and the network can extract higher-level patterns and structures. After all the convolution operations are completed, the convolution stacked high-dimensional feature An (the output of the last convolution layer) is input to the Fully Connected layer (FC). The fully connected layer integrates these features through linear transformations and maps them to the target space as expressed by the following formula:
Transformer
In this study, the primary role of the transformer module was to perform global temporal modeling on local feature sequences extracted by a one-dimensional convolutional neural network. The core design of the transformer aims to capture the long-range dependencies between different time steps in the input data, thereby generating features with a global representation. This module comprises two key components: an encoder stack based on a self-attention mechanism and a simplified decoder output structure.
The transformer model included a four-layer encoder stack, with each layer comprising a multi-head self-attention mechanism, a feedforward neural network (FFN), and layer normalization.
The multi-head self-attention mechanism updates the feature representations by calculating the correlation between each time step and the other time steps in the feature sequence. The formula for each attention head is given as follows.
After the attention mechanism, the features of each time step pass through a two-layer feedforward neural network (FFN), the nonlinear transformation of which is expressed as follows.
Layer normalization improves the stability of the gradient flow by standardizing the inputs and outputs of each layer. The formula is given as follows.
In contrast to the conventional complex transformer decoder, we designed a simplified decoder structure specifically tailored to the requirements of radon concentration time-series data by removing the conventional autoregressive mechanism. Because the input features have already been fully extracted by the 1D-CNN and transformed into a nonsequential structure, the original sequence-to-sequence structure is simplified to nonsequential sequence mapping. Consequently, the decoder consists of a single linear layer that directly maps the high-dimensional feature output from the encoder to the target space. The formula is given as follows.
Training process and performance evaluation
In this study, the training process of the CNN-Transformer hybrid model used the mean squared error (MSE) as the loss function to measure the difference between the predicted and true values. The formula is given as follows.
During the optimization process, a stochastic gradient descent (SGD) algorithm is used as the base optimization method [45]. The SGD algorithm iteratively computes the gradient of the loss function and updates the parameters of the model to reduce the loss and enhance the learning capability of the model. However, relying solely on SGD may lead to overfitting, especially with complex models. Therefore, we incorporated an L2 regularization (weight decay) term into the loss function to limit the excessive growth of the parameters and control the complexity of the model [46]. The final optimization objective function is expressed as follows.
In each epoch, the update rule for the parameters is given as follows.
We also adopted a cosine-annealing learning rate scheduler with a warm-up strategy to further enhance the stability of the training process and accelerate the convergence [47]. This combined approach optimizes the adjustment of the learning to enable the model to update parameters more efficiently during different stages of training.
During the initial few training epochs, a warm-up strategy was applied that gradually increased the learning rate from a small initial value to the preset initial learning rate. This phase was designed to stabilize the parameter updates and prevent instability caused by an excessively large learning rate. By gradually increasing the learning rate, the model can enter an appropriate parameter space smoothly. The learning rate follows a linear growth pattern expressed by the following formula.
After the warm-up phase, the learning rate decayed periodically according to a cosine function, starting from the initial learning rate η0 and gradually decreasing towards the set minimum learning rate ηmin. This approach enabled the model to fine-tune the parameters more precisely in the later stages of training. The updated formula for the cosine annealing learning rate scheduler is given as follows.
The combination of the warm-up and cosine annealing strategies for learning rate adjustment ensured stable training in the early stages to prevent rapid parameter updates that could lead to poor convergence. In the later stages, the gradual decay of the learning rate enhanced the precision of the optimization, which helped to avoid local minima and ultimately improved the overall effectiveness of the training process and the performance of the model.
After the training process was completed, in addition to the evaluation dependent on the loss function, a series of comprehensive evaluation indicators were introduced to comprehensively measure the performance of the model and ensure its performance in terms of correction effect and generalization capability. These metrics included the mean absolute percentage error (MAPE), root mean square error (RMSE), mean absolute error (MAE), coefficient of determination (R2), and correlation coefficient (r). By evaluating the predictive performance of the model from multiple perspectives, these metrics effectively validate the accuracy and reliability of the model in predicting the concentration of radon to provide important references for further optimization.
Results and discussion
In this study, the CNN-Transformer hybrid model was trained for 1000 epochs, with optimal results achieved at the 990th epoch. The specific performance metrics of the training and validation sets are listed in Table 2. Figure 2 presents the changes in the loss function for the training and validation sets with respect to the number of epochs during the training process. This intuitively illustrates the training process of the model and its convergence characteristics.
| Performance index | Training set | Validation set |
|---|---|---|
| Loss | 0.002096 | 0.002093 |
| RMSE (Bq m-3) | 171.0977 | 171.893 |
| MAPE (%) | 0.0569 | 0.057 |
| MAE (Bq m-3) | 103.1781 | 103.5404 |
| R2 | 0.9979 | 0.9979 |
| r | 0.9996 | 0.9996 |
As indicated by Table 2 and Fig. 2, the performance of the CNN-Transformer hybrid model was highly consistent across both the training and validation sets, with no signs of overfitting. The loss of this model, root mean square error (RMSE), mean absolute percentage error (MAPE), and mean absolute error (MAE) were all relatively low, indicating high prediction accuracy and stability. Additionally, the coefficient of determination (R2) and correlation coefficient (r) were close to the ideal values, further confirming the strong linear correlation between the predicted and actual values. During the training process, the loss value decreases gradually, especially in the early stage, which indicates that the model can quickly capture the main features of the data. In the subsequent fine-tuning phase, the loss reduction becomes more gradual, reflecting the model’s optimization of the finer details. Overall, the gap between the training and validation losses was always small, and the model had a good generalization capability.
To further verify the performance of the CNN-Transformer hybrid model in the optimization and prediction of radon concentration data, this study adopted the above-mentioned empirical formula to generate additional radon concentration data as a test set, which was not included in the training or validation sets. After using the trained CNN-transformer model to predict and optimize the test set data, the performance evaluation results showed that the mean absolute percentage error (MAPE), root mean square error (RMSE), and MAE were 0.320%, 306.49 Bq m-3, and 165.01 Bq m-3, respectively. Compared to the training and validation sets, the error of the test set was improved, which may be related to the small amount of test data and the diversity of the data distribution in different concentration ranges. However, the determination coefficient (R2 = 0.9979) and correlation coefficient (r = 0.9999) of the model were close to the ideal values, indicating that the fitting capability and prediction accuracy of the model on the test data remained high.
To optimize the radon concentration data and improve the radon exhalation rate fitting accuracy, further analysis was conducted on the data from different concentration ranges in the test set. Nonlinear fitting of the simulated instrument measurement data, model-predicted optimized data, and theoretical true data was performed using Origin software, with the fitting performed according to Eq. (2). The coefficients of determination (R2) and other related performance metrics are summarized in Table 3, and Fig. 3 displays the curve plots after fitting, illustrating the variation in the radon concentration over time.
| Experimental class | Simulated instrument measurement data | Hybrid model predicted data | Simulation of theoretical true data | ||||||
|---|---|---|---|---|---|---|---|---|---|
| J ± σ (mBq m-2 s-1) | λeff ± σ (× 10-5) (s-1) | R2 | J ± σ (mBq m-2 s-1) | λeff ± σ (× 10-5) (s-1) | R2 | J ± σ (mBq m-2 s-1) | λeff ± σ (× 10-5) (s-1) | R2 | |
| High | 997.18±30.78 | 7.90±3.13 | 0.998 | 966.88±2.29 | 3.97±0.24 | 0.999 | 964.61±1.63 | 3.38±0.17 | 0.999 |
| Moderate | 79.61±10.5 | 20.46±14.23 | 0.966 | 70.01±1.50 | 5.41±2.14 | 0.999 | 70.04±0.12 | 3.38±0.17 | 0.999 |
| Low | 11.25±3.39 | 57.25±39.11 | 0.820 | 8.61±1.12 | 14.52±13.62 | 0.970 | 5.80±0.01 | 3.38±0.17 | 0.999 |
As indicated by Table 3 and Fig. 3, the CNN-Transformer hybrid model demonstrates strong predictive capabilities across different ranges of radon concentration data. In terms of the fitting effect of the radon exhalation rate, the optimized data of the model prediction are superior to the simulated instrument measurement data; particularly, the optimization effect was more obvious at higher concentrations, and the fitted curve was closer to the theoretical real value. Moreover, the coefficient of determination (R2) for all predicted data exceeded that of the simulated instrument measurement data, with R2 values consistently ranging from 0.97 to 0.99. Thus, the results confirm the high stability and reliability of the proposed model in processing different concentration data.
To further validate the optimization performance of the model in practical applications, this study employed a solid radon source exhalation reference model [48] with a reference radon exhalation rate of 43 ± 0.48 mBq m-2 s-1. The test setup and the connections of the experimental instruments are shown in Fig. 4. The experimental apparatus primarily included a drying tube, filter membrane, brick, glass plate, RAD7 radon detector, a radon accumulation chamber, and a solid radon source. These components were connected and sealed with silicone tubing to ensure that no gas leaked during the experiment.
In practical measurements, a set of measurements is usually taken in ten-minute cycles to minimize the experimental error, rather than the four-minute cycles used to train the model. Therefore, to enhance the practical applicability of the model, radon concentration data were collected over a period of 10 min. Data acquisition was performed using a RAD7 instrument combined with an accumulation method. A total of six sets of repeated measurements were carried out to ensure the reliability of experimental results. Subsequently, the designed CNN-Transformer deep learning hybrid model was applied to optimize and predict the measurement data. A nonlinear fitting analysis of the measurement results was conducted using the Origin software, and the fitting results are listed in Table 4.
| Experimental Number | Instrument measurement data | Hybrid model predicted data | ||||
|---|---|---|---|---|---|---|
| J ± σ (mBq m-2 s-1) | λeff ± σ (× 10-4) (s-1) | R2 | J ± σ (mBq m-2 s-1) | λeff ± σ (× 10-4) (s-1) | R2 | |
| 1 | 45.72 ± 4.13 | 2.18 ± 0.45 | 0.973 | 44.51 ± 1.90 | 2.04 ± 0.21 | 0.994 |
| 2 | 43.95 ± 10.14 | 2.18 ± 1.14 | 0.851 | 43.33 ± 1.97 | 2.12 ± 0.22 | 0.993 |
| 3 | 44.08 ± 7.82 | 1.86 ± 0.84 | 0.911 | 43.03 ± 2.68 | 1.72 ± 0.29 | 0.988 |
| 4 | 57.29 ± 7.91 | 3.38 ± 0.79 | 0.937 | 48.02 ± 2.13 | 2.39 ± 0.22 | 0.993 |
| 5 | 35.76 ± 8.39 | 0.25 ± 0.90 | 0.875 | 45.60 ± 1.61 | 1.38 ± 0.16 | 0.996 |
| 6 | 51.19 ± 10.90 | 2.90 ± 1.15 | 0.872 | 46.22 ± 1.92 | 2.40 ± 0.21 | 0.994 |
As shown in Fig. 5, after optimization and prediction using the CNN-Transformer hybrid model, the fitting results for each experimental group showed a significant improvement. The optimized radon exhalation rate (J) was much closer to the reference value, with a significant reduction in deviation. Furthermore, the optimized fitting coefficients (R2) were generally higher than those for the unoptimized data, ranging from 0.988 to 0.996, which significantly outperformed the fitting results of the instrument measurements (R2, which ranged from 0.851 to 0.973). These results indicate that the performance of the proposed CNN-Transformer hybrid model in terms of optimization exhibited high stability and reliability in practical tests. This effectively reduced the uncertainty in the experimental data and thus improved the accuracy with which the radon exhalation rate was fitted to the data.
Conclusion
In this study, we have proposed a deep learning approach based on a CNN-Transformer hybrid model to optimize radon concentration data and enhance the accuracy of radon exhalation rate fitting. The model demonstrated high stability and good generalization capability during both training and validation, with strong learning capabilities. In the testing phase, the optimized radon exhalation rate closely matched the theoretical values, and the coefficient of determination (R2) remained stable between 0.97 and 0.99, which outperformed the simulated instrument measurement data. The model exhibited high prediction accuracy and stability across a range of different radon concentrations, with the optimization effect becoming more pronounced as the radon concentration increased.
In the practical application test, the solid radon source exhalation reference model, combined with the measurement data collected by the RAD7 radon detector, was used to optimize the prediction. The fitting results from the various experimental groups showed significant improvements, with the optimized radon exhalation rate (J) approaching the reference value (43 ± 0.48 mBq m-2 s-1), and the deviation was notably reduced. Further, the optimized R2 values were stable between 0.988 and 0.996, which was significantly better than the instrumental measurement data (R2 ranging from 0.851 to 0.973). These results indicate that the CNN-Transformer hybrid model offers high stability and reliability in optimizing radon exhalation rate measurement data. The proposed approach reduced the impact of statistical fluctuations and instrument errors and significantly improved the accuracy with which the obtained predictions of radon exhalation rates fit the measured data.
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