Principle of Source-Term Inversion

If the source-term geometry is known at the time of inversion, then it can be discretized into several sub-geometries comprising a set $G=\left[g_1,g_2,\dots ,g_n\right]$. Each subgeometry comprises a corresponding activity distribution, the set of which is $S=\left[S_1,S_2,\dots ,S_n\right]$, and the dose rate in space is expressed as follows: $D=F\left(S,G,P\right),$ where F. represents an operator for calculating the dose, $P=\left[P_1,P_2,\dots ,P_m\right]\in R^m$. represents a matrix coaining t positions of $m$. sampling points in space, and $D=\left[D_1,D_2,\dots ,D_m\right]$. is the dose vector corresponding to $P$. Subsequently, the process of inverting the source-term activity distribution can be expressed as follows: $S=I\left(P,G,D\right)+ϵ,$ where $I$. represents the inversion operator and $ϵ\in R^n$. represents the inversion error. The equation above is a general expression for the main source-term inversion method. The general expression of a neural network with input $X$. and output $O$. is $O=f\left(X\right).$

In summary, the general form of the neural network used to reconstruct the source term is as follows: $S=f\left(P,G,D\right)+ϵ.$

For regression problems, the most typically used loss function is the mean squared error (MSE) function,ch is expressed as follows: $\text{MSE}=\frac{1}{n}{\displaystyle \sum _{i=1}^n{S}_i-O_i{}^2}.$

Thus, the method of applying a neural network for source-term distribution inversion can be described as follows: A predictive model is constructed using a neural network dedicated to minimizing the loss function; subsequently, a model with the smallest possible error is determined by training the pameters of the network model.

Method and Process

In this study, deep neural networks and Monte Carlo methods were used to calculate examples with different source-term parameters to form a dataset for model training and to realize source-term iersion. For a series of detectors at different positions in space, the two-dimensional distribution of the source term must be calculated based on the measured parameters of the detectors. Conceptually, this process comprises two aspects:

(1) Extraction of measurement parameter information for the sampling points. The two-dimensional distribution information of the source term is encoded using the relative magnitudes of the measured parameters at the sampling points. Therefore, the measured parameters of each sampling point must be considered as independent variables and information from the relative sizes between the sampling points must be extracted.

(2) Reconstruction of the source-term distribution. The two-dimensional distribution of the source term is reconstructed based on the information extracted in the previous step. Multiple solution strategies are available for calculating the distribution of the measured parameters based on any sampling point; therefore, this problem is pathological and is an indeterminate inverse problem that has no unique solution and typically requires constraining the solution space using the corresponding samples.

In this study, a neural network model dedicated to the inversion of complex source-term distributions based on the idea above is proposed. The radiation parameters measured at n sampling points in the computing space are used to form a $1\times n$ one-dimensional array as the input parameters for the neural network, and the sampling points are independent of each other. Similarly, two-dimensional images of the source-term distribution are used as the output parameters of the neural network. In this study, the SDICNN is classified into two components: a fully connected network and a CNN. The extraction of the measurement parameter information of the sampling points is mainly completed in the fully connected network. The low-resolution source-term distribution parameters are obtained from individual sampling points in the radiation field via the fully connected neural network. The one-dimensional feature representation output by the fully connected neural network is converted into a two-dimensional feature representation to obtain a low-resolution source term distribution image, which typically exhibits 1/2 the resolution of the source term. The CNN mainly uses a structure similar to that of the Super Resolution CNN (SRCNN) to complete the fine reconstruction of the source-term distribution to obtain a high-resolution image from a low-resolution image. The structure of the SDICNN is shown in Figure 1.

Fig. 1Full-screen View

(Color online) Diagram of the SDICNN model

In summary, the inversion process for the source-term distribution shown in Figure 2 is as follows:

Fig. 2Full-screen View

Flowchart for fixed complex source-term inversion

(1) The number and location of sampling points in space, as characteristic variables of the neural network input, are determined based on the geometry of the space; simultaneously, the resolution of the complex source-term distribution is determined and its two-dimensional image is used as the target signal for the output of the network.

(2) A certain amount of data is required to train, test, and verify the neural network. Therefore, different source-term distributions are constructed, and the parameters of the sampling points under the source-term distribution condition are used as datasets and calculated via Monte Carlo simulation. In addition, the input and output of the network are standardized to accelerate the convergence of the neural network.

(3) Based on the input and output parameters of the neural network combined with the complexity of the actual radiation device space, hyperparameter settings such as the number of hidden layers and the number of nodes of this neural network are determined.

(4) A preprocessed dataset is used to train the neural network. First, the linear layer is trained, and the parameters of the previous linear layer are fixed; subsequently, the convolution and deconvolution layers are trained.

(5) Test samples are used to test the source-term inversion of the deep neural network model based on a fixed complex source-term inversion.

(6) The fixed complex source term is inverted using the deep neural network model obtained after testing.

Calculation Example

Many nuclear power plants are beginning to use the dry storage of spent fuel, which requires the construction of storage system facilities for the off-stack storage of spent fuel. Based on a dry spent fuel storage system as a benchmark, we constructed a spent fuel transfer container with a simple geometry to test the inversion method described above.

The calculated area in this example was a 10 m × 10 m × 10 m. cube-shaped room. The surrounding walls were 0.20-m-thick concrete walls. The entire room was partitioned into two areas by a 0.20 m wall. The spent fuel transfer container was placed upright inside the room. The dimensions of the container were as follows: outer diameter, 2.34 m; inner diameter, 2.20 m; height, 5.00 m; a bottom surface, 0.20 m. The container cover, which was made of stainless steel, featured a diameter of 2.40 m and a height of 0.20 m. A cylindrical beam with a diameter of 1.00 m and a height of 5.80 m was placed above the inner room to increase the complexity of the geometry. The positions and coordinates of the main geometric components in this study are listed in Table 2, and a geometric diagram is shown in Fig. 3a.

Fig. 3Full-screen View

(Color online) Simple example of geometric diagram (a); complex surface diagram of the cross-section (b); and circumferential angles (c)

Dataset

In this study, the source term refers to a neutron source and the energy spectrum is the watt fission spectrum. The outside of the container is regarded as the surface source term, which is abstracted as a two-dimensional array. The two dimensions are the circumferential discrete angle of the source term and the axial discrete mesh of the source term. The complex surface source term in the example is discretized into a series of subsurface sources, and the particles are emitted uniformly outward in the region with a certain intensity, based on the assumption that the energy spectrum remains unchanged. In the arithmetic example described herein, the entire cylindrical container is sampled at 360 °and segmented into 30 angles. Each angle is segmented into 20 sampling regions. Therefore, the number of parameters of the source term to be inverted is 30 × 20 = 600, as shown in Figs. 3b and 3c.

In this study, the abovementioned two-dimensional arrays represent the intensity of each source-term discrete mesh. The intensity distribution of the source term in the example assumes the form of a two-dimensional trigonometric function. The parameters of the function are randomly sampled, including the number of levels, the amplitude, the frequency, and the phase of the trigonometric function. The circumferential and axial coordinates of the source term are substituted into the sampled two-dimensional distribution function to generate a set of source-term parameters. To enable the intensity distribution of the generated source term change continuously in both the circumferential and axial directions, we use binary functions to represent the intensity of the source term in each mesh and trigonometric functions with multiple random parameters to represent the variation in the intensity of the source term, as shown in the formula below: $F\left(x,y\right)={\displaystyle \sum _{\text{order}}\begin{array}{l}[{\text{amp}}_1\cdot \sin \left({\text{freq}}_1\cdot x+{\phi }_1\right)+{\text{amp}}_2\cdot \cos \left({\text{freq}}_2\cdot x+{\phi }_2\right)\\ +{\text{amp}}_3\cdot \sin \left({\text{freq}}_3\cdot y+{\phi }_3\right)+{\text{amp}}_4\cdot \cos \left({\text{freq}}_4\cdot y+{\phi }_4\right)],\end{array}}$

where $F\left(x,y\right)$. represents the source-term intensity of meshes $x$ and $y$. in the circumferential and axial directions, respectively; and order, amp, freq, and φ are the parameters of random sampling. Figure 4 shows examples of the two-dimensional distributions of different source terms in the dataset.

Fig. 4Full-screen View

(Color online) Two-dimensional distributions of source terms of different generated datasets

Monte Carlo Calculation

After a series of source-term distributions are generated using the method above, the Monte Carlo method is used to calculate the example. The Monte Carlo method can be used not only to flexibly model the geometry of source terms, but also to accurately describe and analyze their distribution, thus rendering it suitable for rapid calculations and generating datasets required for training [27]. To improve the computational efficiency of Monte Carlo simulations, Dr. Pan [28] from Shanghai Jiao Tong University proposed a single-step Monte Carlo criticality algorithm, which has been used to generate transplutonium isotopes [29]. Similar methods include the DeGVR [30] and PDMC methods [31].

In this study, a Monte Carlo program known as MCShield was used for calculations. MCShield is a Monte Carlo program developed by the Radiation Protection and Environmental Protection Laboratory of Tsinghua University for shielding calculations. The custom source term function in MCShield was used in this study to sample the source terms. After normalizing the activity of the source particles at different locations, energies, and angles, the distribution of particles at different locations, energies, and angles was obtained. Information such as the location, energy, and angle of individual source particles was obtained via sampling.

Subsequently, two cases of the same source term without and with a shield were computed separately, and 10000 sets of arithmetic cases were computed to create a dataset for validating the neural network. In this study, the MESH method was used to calculate the neutron flux over the entire calculation area. Each MESH measured 160 mm ×160 mm ×160 mm. In total, 64 × 64 × 64 MESHs were configured. After the calculation was completed, the MESH parameters at the sampling points were extracted to obtain the measured values of the sampling points. In the case with shielding, the number of simulated particles was 1E7, and the average statistical error was approximately 0.07. In the case without shielding, the number of simulated particles was 1E6, and the average statistical error was approximately 0.10. Figure 5 shows the calculation results obtained with and without shielding under the same source-term distribution.

Fig. 5Full-screen View

(Color online) Calculation results for radiation field with shielding

Inversion calculation results

When training neural networks, normalization preprocessing is necessary to transform different feature ranges to achieve a mean of 0 and a variance of 1. The source-term parameters and flux values are transformed from being several orders of magnitude apart to the same order of magnitude. Subsequently, different features in the dataset are standardized, which allows the network model to learn the fitting relationship between the two more effectively and accelerate convergence. During the training of the neural network in this study, the linear layer was trained first. Subsequently, the parameters of the previous linear layer were fixed, and the convolution and deconvolution layers were trained. During this training process, the dropout technique was used to randomly turn off a certain proportion of neurons to avoid overfitting the model. The dataset used for the training contained 10000 examples, 150 training epochs were performed, and the MSE was selected as the loss function. In this example, different numbers of sampling points were selected on the wall around the spent fuel transfer container to simulate the actual detector arrangement. Owing to the different numbers of sampling points, the training times differed slightly. The time for one training epoch was 20–50 s, and the total training time was 1–2 h.

A total of 96 sampling points were selected around the side of the container as the input to each neural network, where the circumferential direction was segmented into 30 angles, and the axial direction was segmented into 20 regions. The fully connected neural network in the SDICNN contained three hidden layers, and the CNN in the SDICNN model performed two convolution and one upsampling operation, followed by two additional convolutional operations. Figure 7a shows the inversion results of the SDICNN model for source terms outside the dataset.

Fig. 7Full-screen View

(Color online) Comparison of predicted source distribution (left) output by SDICNN model (a) and fully connected neural network (b), and actual source distribution (right)

Comparison of Neural Network Inversion Results

In this study, an experiment was performed to compare between a fully connected neural network model and the SDICNN model to verify the effectiveness of the neural network method for source-term inversion and to analyze the performance of the SDICNN model by comparing it with that of a conventional fully connected neural network model. To be consistent with the SDICNN model, a typical fully connected neural network containing three hidden layers was used, whose structure is the same as that of the fully connected network of the SDICNN model, as shown in Fig. 6.

Fig. 6Full-screen View

(Color online) Schematic diagram of conventional fully connected neural network

For comparison purposes, all results are presented herein based on the same validation case. Different methods (such as the fully connected neural network and SDICNN model) and parameters (such as different numbers of sampling points) were used for source inversion, and the results are presented. Figures 7 and 8 show the inversion results of the fully connected neural network and SDICNN model for source terms outside the same dataset and the relative deviations from the actual source terms. The average absolute error of the source-term inversion results of the fully connected neural network was 87.41%, whereas that of the SDICNN model was 22.16%.

Fig. 8Full-screen View

(Color online) Relative errors of (a) fully connected neural network and (b) SDICNN model

The prediction results of the fully connected neural network and SDICNN model indicate that the fully connected network could not learn the two-dimensional features of the source-term distribution. This is mainly because a fully connected neural network regards the source terms as separate discrete points, whereas in reality, the source-term distribution is a continuous two-dimensional distribution with different continuous features in both the circumferential and axial directions. By contrast, the SDICNN model can extract features of the source-term distribution more effectively and obtain better inversion results.

Figure 9 shows a comparison of the MSEs induced in the validation set during the training processes of the two networks. The fully connected neural network indicated a lower convergence speed and a plateau phase during the source-term inversion, whereas the SDICNN model exhibited a higher convergence speed and achieved better training results for the validation set.

Fig. 9Full-screen View

(Color online) MSEs obtained on validation set by two neural network methods

Figure 10 shows the flux values calculated for 96 sampling points based on the source-term prediction results of the fully connected network and SDICNN model, as well as the relative deviations from the flux values under the conditions of the actual source terms. The average relative deviation under the predicted source term conditions of the fully connected network was 16.29%, whereas the average relative deviation under the predicted source-term conditions of the SDICNN model was 14.92%. The SDICNN model performed slightly better than the fully connected network in predicting the source-term results.

Fig. 10Full-screen View

(Color online) (a) Flux results and (b) relative flux deviations obtained for 96 sampling points by two neural networks

These results indicate that the fully connected neural network is unsuitable for this application. Compared with the conventional fully connected neural network, the SDICNN model achieves better source prediction performance on the same dataset and is more suitable for predicting two-dimensional complex source distributions.

Shielding Effect

In the calculation space, a significant amount of shielding effects (e.g., beams and walls in the calculation example presented herein) are typically observed, which can reduce the dose level and significantly increase the complexity of the calculation space as scattering terms are added. Therefore, the effect of shielding in the radiation field on the source-term inversion effect must be investigated. This section is based on the SDICNN model, where 30 sampling points at fixed positions on the wall are selected as network inputs. Based on two cases, i.e., with and without shielding (in the Monte Carlo simulation, the shielding material is set as air), the model was trained and tested. Figures 11 and 12 show the source-term inversion results obtained for the cases with and without shielding, and the relative deviations from the actual source-term distribution. The results of the two schemes show minimal difference between the inversion results obtained for the source terms with and without shielding.

Fig. 11Full-screen View

(Color online) Comparison of predicted (left) and actual (right) distributions for source terms (a) with and (b) without shielding

Fig. 12Full-screen View

(Color online) Relative error (a) with and (b) without shielding

In the cases without and with shielding, the average absolute errors of the source-term inversion were 78.82%, and 64.73%, respectively. Based on the relative deviation results, the extreme errors of the inversion results obtained with and without shielding were relatively low and relatively high, respectively. In general, the inversion results obtained with shielding were slightly better than those obtained without shielding.

Figure 13 shows a comparison of the MSE values obtained on the verification set when network training was performed under two conditions: with and without shielding. As shown, on the same dataset, the network converged faster without shielding and achieved better training results on the validation set.

Fig. 13Full-screen View

(Color online) MSEs obtained on validation set under two shielding conditions

The inversion results obtained for the source terms with and without shielding exhibited mutual advantages and disadvantages. This is primarily because of the fewer number of scattering terms under the condition without shielding, and that the training process of the network model is relatively simple, thus resulting in a faster convergence process. However, without shielding, the same sampling point position is not shielded from the source, and the radiation parameters at the sampling point position may not accurately reflect the source-term distribution.

Number of Sampling Points

To investigate the effect of the number of sampling points on the source-term inversion results based on the SDICNN model, we selected 10 and 40 sampling points at fixed positions on the wall as network inputs, conducted network model training, and then conducted tests. Figures 14 and 15 show the source-term inversion results obtained based on 10 and 40 sampling points and the relative deviations from the actual values. The average absolute errors of the source-item inversion results based on 10 and 40 sampling points were 66.44% and 39.35%, respectively. By combining the source-term inversion results of the 30 and 96 sampling points (as shown in Figs. 11 and 7, respectively), we clearly observed that the source-term inversion results improved gradually as the number of sampling points increased. A greater number of sampling points corresponded to more source-term information obtained and thus better source-term inversion results.

Fig. 14Full-screen View

(Color online) Predicted source distribution (left) with (a) 10 and (b) 40 sampling points and actual source distribution (right)

Fig. 15Full-screen View

(Color online) Relative deviation with (a) 10 and (b) 40 sampling points

Figure 16 shows a comparison of the MSE values induced on the verification set when the network was trained based on 10, 30, 40, and 96 sampling points. As shown, on the same dataset, an increase in the number of sampling points resulted in a higher network convergence speed, and the training results obtained on the verification set were better.

Fig. 16Full-screen View

(Color online) MSEs obtained on validation set with different numbers of sampling points

Figure 17 shows the flux values calculated at the sampling points based on the prediction results of networks trained on data from 10 and 40 sampling points and the relative deviations from the flux values under the actual source-term conditions. The average relative deviations of the neural network based on 10 and 40 sampling points were 25.27% and 25.64%, respectively; as mentioned above, the average relative deviation of the neural network based on 96 sampling points was 14.92%. In general, the prediction results became increasingly accurate as the number of sampling points increased.

Fig. 17Full-screen View

(Color online) Flux results and relative deviations obtained with (a) 10 and (b) 40 sampling points

Discussion

The effectiveness of the SDICNN model for the source-term distribution inversion problem was verified comparing it with the fully connected neural network. In addition, the effects of factors such as shielding and the number of sampling points on the source-term inversion effect were verified and tested. The findings obtained were as follows: (1) The effect of shielding on source-term inversion results must be considered comprehensively. Shielding introduces complexity into the radiation field, which is not conducive to the neural network training. However, without shielding, the distances between the sampling points and source term must increase accordingly, and the resolution ability of the radiation parameters at the sampling point positions for the source-term distribution is weakened. Therefore, in practical applications, the effects of shielding and sampling point positions on inversion must be considered comprehensively. (2) The test results show that more sampling points in the space yields a better source-term inversion effect. Therefore, in actual application scenarios, the number of sampling points should be increased as much as possible to achieve better results. Based on the verification results above, the proposed SDICNN model offers two advantages:

(1) In the radiation protection scenarios, the radiation parameter measurement values at each sampling point are the result of the comprehensive effect of various positions in the fixed complex source, and they exhibit strong nonlinear characteristics. The proposed network model is based on a deep neural network and is trained on a dataset composed of different source scenarios without considering specific nonlinear processes and complex geometric structures. It can perform complex source inversions rapidly based on the radiation parameter values measured at the sampling points.

(2) The proposed neural network is classified into a fully connected neural network and a CNN. The fully connected neural network obtains a higher-dimensional feature representation of the input, whereas a CNN performs convolution and deconvolution operations to extract more spatial information and high-level features. This network design structure combines the advantages of both types of neural networks, thus effectively solving the source-inversion problem with a certain degree of flexibility.