Multi-task Deep Learning

Multitask deep learning (MTDL) [34] is a model training paradigm wherein the main idea involves training with multiple tasks simultaneously using shared representations to learn information between related tasks. Shared representations can improve the efficiency of data feature extraction, improve the learning speed of related tasks, and alleviate the weaknesses of deep learning, such as large data scale and time consumption. Zhao et al. [35] introduced a classification method that divides MTDL architectures into four types: cascaded, parallel, interactive, and hybrid.

Figure 1(a) presents a typical cascaded architecture. This architecture does not have a shared task layer and the output of the previous task is fed directly to the subsequent training task. The cascaded approach is primarily suitable for cases wherein the subsequent task relies heavily on the output of the former task. Thus, the results of the former task significantly affect the performance of the subsequent task in the task combination.

Fig. 1Full-screen View

(Color online) Schematic of four multi-task deep learning architectures [35]: (a) cascaded architecture, (b) parallel architecture, (c) interactive architecture, and (d) hybrid architecture

Figure 1(b) presents a typical parallel architecture. In this architecture, a partially shared representation layer exists and different tasks are computed in parallel on specific task networks. This parallel approach may be more suitable for addressing task combinations that are related but exhibit different levels of complexity. Examples include the image segmentation and image generation tasks.

An interactive architecture refers to the presence of one or more connections between task-specific layers, which enables the exploration of deep correlations between tasks, as illustrated in Fig. 1(c). It is more suitable for solving situations wherein there is a significant correlation between multiple tasks, and can enable related tasks to provide useful auxiliary information.

Owing to the diversity of detailed network structures, researchers have defined the fourth category as a hybrid architecture (Fig.1(d)). Typically, in a hybrid architecture, at least two of the following strategies are used: cascaded, parallel, or interactive. The hybrid architecture can fully leverage the advantages of the three architectures and is well suited for complex task combinations.

Core neutron field data exhibit the characteristics of high-dimensional and large-scale data, and there are certain correlations among the multiple tasks involved in core reconstruction. Therefore, this study applied MDTL to the reconstruction of core neutron fields. Through cooperative learning among different tasks, the aim was to further enhance the efficiency of the core neutron field reconstruction using ex-core detection signals.

Pipeline Reconstruction Architecture

The pipeline reconstruction architecture (PRA) proposed by the authors in 2022 [33] is a novel and effective algorithm for studying 3D neutron field [33]. The architecture ranging from “coarse-grained” to “fine-grained” reconstruction comprises three separate tasks. When all the tasks are trained individually, the core neutron field is sequentially reconstructed.

As shown in Fig. 2, the reconstruction architecture is based on a pipeline pattern. The first stage of the architecture is a core changed-region recognition module, designed by convolutional neural networks. It accurately identifies the core-changed region based on ex-core detection signals. In the second stage, an improved variational autoencoder is designed to reconstruct the core relay region. After determining the core-change region, the neutron field of the relay region is predicted by inputting ex-core detection signals. In the third stage of reconstruction, the neutron field in the relay region is converted from two-dimensions (2D) to three-dimensions (3D), and then the distribution of the 3D core neutron field is predicted using the 3D full convolution-based reconstruction module. By sequentially executing these three modules, the pipeline-based architecture enables the reconstruction of the 3D core neutron field.

Fig. 2Full-screen View

(Color online) Reconstruction architecture based on a pipeline pattern. This architecture comprises three main modules: the core changed region recognition, reactor relay reconstruction, and core neutron field reconstruction modules

Notably, the PRA must identify the location of a single-core changed region. However, the location and number of core-changed regions are generally uncertain. Therefore, this architecture has certain limitations.

Image processing of neutron data

The purpose of image processing is to organize the neutron distribution in the reactor based on the arrangement of image pixels to facilitate efficient feature extraction and transformation by the neural network layer. The reasons for this design are as follows/ The core neutron field has strong correlations between adjacent assemblies in 3D space, such as the relationships between pixels in an image. Therefore, each assembly to be reconstructed can be regarded as a pixel, and the number of energy groups corresponds to the number of channels in the image (such as the RGB channels in an image).

Typically, both discrete ex-core detection signals and continuous neutron fields are the objects of image processing. For discrete ex-core detection signals, imaging processing forms a 3D signals tensor according to the spatial position of each detection point. Subsequently the 3D signals are augmented to four-dimensions (4D) by adding the energy dimension to facilitate the extraction and allocation of the ex-core detection signal features through the three-dimensional convolutional network layer. For a continuous core neutron field, image processing discretizes the neutron field from both space and energy to form a 4D tensor (energy, height, weight, and length); where, height is the height after discretization of a neutron field in the core, weight is the discretized width of the neutron field in the core, length is the discretized length of the neutron field in the core, and energy is the number of discretized energy groups.

According to the above design, the sample after image processing includes the following steps. $S=\left\{s\mathrm{,}{(\varphi )}_{\text{coreEdge}}^{E\times H\times W\times L}\mathrm{,}{(\varphi )}_{\text{core}}^{E\times H\times W\times L}\right\}$ (1) where s represents the ex-core detection signals, (ϕ)_coreEdge represents the neutron field of the outermost core assemblies, (ϕ)_core represents the core neutron field, and E× H × W × L is the sample size, which represents the number of reconstruction units. In the study, the number of energy groups was used as the convolutional depth. In addition, the magnitude of the neutron field distribution values calculated by the Monte Carlo method is small, which is not conducive to model training. Therefore, this study adopted normalization processing to limit the magnitude of the neutron field to the range of [0,1].

Global-Local Feature upsampling Module

For a multitask architecture, it is essential to extract the shared features of tasks. By upsampling the features contained in the ex-core detection signals, the features carried by the signals can be evenly distributed to each reconstruction unit. Owing to the large magnitude difference between the values of the core neutron field, it is difficult to assign features to certain reconstruction units, and the model cannot reconstruct units with small numerical values.

Therefore, a global-local feature upsampling module was designed in this study. The architecture diagram of the module is shown in Fig. 4. The module was divided into two parts to upsample the information carried by the ex-core detection signals. The first was the global feature upsampling network, which ensured that the ex-core detection signals covered every reconstruction unit, and the network output was H_Global. The second was a local feature upsampling network, which was primarily used to assign features to energy groups with less neutron distribution. The network output was H_Local. Finally, the sampling results of the two parts were concatenated as inputs for the following two reconstruction tasks.

Fig. 4Full-screen View

(Color online) Global-local feature upsampling module. Both global and local sub-modules upsample ex-core signals. Finally, the outputs of sub-module are concatenated together as inputs for downstream tasks

Dual-task Optimization Function

The hidden layer vector H_GLFS obtained from the global-local feature upsampling module was used to train the reconstruction task of the neutron field in the outermost core assemblies and that of the core neutron field. The formula for the reconstruction of the neutron field in the outermost core assemblies is as follows: $L_{\text{A}}=\frac{1}{N}{\displaystyle \sum _{j=1}^N}{\left(f_{\text{A}}^j\left(H_{\text{GLFS}}\right)-{\varphi }_{\text{A}}^j\right)}^2$ (2) where N = E × H × C_A, E is the number of discretized energy groups, H is the height of the core neutron field, and $C_{\text{A}}={\displaystyle {\sum }_{i=1}^W{n}_i}$, where W is the width of the core neutron field, ni is the number of outermost core assemblies in the i-th line. Further, H_GLFS is the output vector of GLFS module, $f_{\text{A}}^j\left(H_{\text{GLFS}}\right)$ represents the reconstructed data of the j-th outermost core assemblies, and ${\varphi }_{\text{A}}^j$ represents the reference data of the j-th outermost core assemblies obtained through Monte Carlo simulations.

Subsequently, the results of the outermost core assemblies were concatenated to the hidden layer vectors to obtain new hidden layer vectors that were used for core neutron field reconstruction training. This step is similar to a residual network, which not only retains the original sampled features but also incorporates certain learned knowledge features, thereby improving the effectiveness of core neutron field reconstruction. Therefore, the loss function for the second task is as follows: $L_{\text{B}}=\frac{1}{M}{\displaystyle \sum _{j=1}^M}{\left(f_{\text{B}}^j\left(H_{\text{GLFS}}\right)-{\varphi }_{\text{B}}^j\right)}^2$ (3) where M = E × H × C_B, E is the number of discretized energy groups, H is the height of the core neutron field, and $C_{\text{B}}={\displaystyle {\sum }_{i=1}^W{l}_i}$, where W is the width of the core neutron field, li is the number of core assemblies in the i-th line of width. Further, H_GLFS is the output vector of GLFS module, $f_{\text{B}}^j\left(H_{\text{GLFS}}\right)$ represents the reconstructed data of the j-th-core assemblies, and ${\varphi }_{\text{B}}^j$ represents the reference data of the j-th-core assemblies obtained through Monte Carlo simulations.

Finally, the loss function of the entire model training comprised L_A and L_B. The formula is as follows: $L_{\text{DTHNet}}=\lambda *L_{\text{A}}+\beta *L_{\text{B}}$ (4) where λ and β are hyperparameters. They are used to balance the two tasks during the training of the neural network, and their values are in the range [0,1]. Typically, after multiple manual attempts, an appropriate value can be selected to obtain better model learning efficiency.

Benchmark and Dataset

Lead-cooled fast reactors, recommended as fourth-generation nuclear energy systems, have made significant progress in their development. The China Lead-based Experimental Advanced Reactor-I (CLEAR-I) [36] is a lead-cooled fast reactor that operates in both critical and accelerator-driven subcritical modes. The numbers of core assemblies and material summaries are listed in Table 1. As shown in Fig. 5, the CLEAR-I core adopts the commonly used hexagonal prism assemblies of fast reactors. According to the different functions of the assemblies, the core was divided into target, buffer, active, reflecting, and shielding regions from the inside to the outside. The buffer region comprised six assemblies, and the core active region consisted of 94 assemblies, including 86 fuel assemblies (52 of which were replaced with reflector assemblies when the reactor was in subcritical operation) and eight control-rod assemblies.

Fig. 5Full-screen View

(Color online) Radial profile of the CLEAR-I reactor

In this study, the critical mode was selected as the simulated operation mode of the reactor to verify the reconstruction of the neutron field under transient changes in the core. In addition, to reduce the time required for model training, half of the reactor model was selected for verification of the reconstruction method.

According to the characteristics of the neutron distribution in fast reactors, the energy groups for neutron field reconstruction in this study were set as follows: [0, 1.00×10^-2] MeV, [1.00×10^-2, 1.00×10^-1] MeV, [1.00×10^-1, 5.00×10^-1] MeV, [5.00×10^-1, 1.00] MeV, and [1.00, 2.00×10¹] MeV. Figure 6 shows the ex-core detection points and relay region for CLEAR-I model in this study. Herein, 8×8 detection points were set outside the shielding layer; thus, the ex-core detection signals were s^{8 × 8}, and the detection signals were s^1×8×1×8 after image processing. In addition, the core assembly region was divided into two parts according to the material distribution; thus, the height of the core was 2. Based on the radial profile of the reactor (Fig. 7), the core had a length of 11 and a width of 7. Therefore, in this case, the size of the neutron flux distribution for the outermost assemblies (relay region) of the core was 5×2×18 and that after image processing was 5×2×7×11. The dimensions of the neutron field core were 5×2×7×11.

Fig. 6Full-screen View

(Color online) Settings of ex-core detection points and relay region. The relay region is composed of the outermost assemblies of the core

Fig. 7Full-screen View

(Color online) Core assembly cross sections and their serial number(upper layer)

In line with the symmetry of the model, a Monte Carlo fixed-source calculation was conducted in this study, and the response matrix of the ex-core detection points was obtained. In this study, assembly was used as the basic spatial unit for reconstruction. When the neutron distribution of an assembly changes, the neutron fields of the surrounding assemblies may be affected. Thus, 5–6 adjacent assemblies were chosen as changed regions. In this context, this study set 18 regions where the core changes, and the upper and lower layers of the core contained nine changes. Table 2 lists the assembly serial numbers of each region of the upper core. The nine regions of the lower core corresponded to the upper core and were not repeated.

In a previous study [21], the authors assumed that only one region of the core was changed at a given time and that the region must be identified first; therefore, a single change dataset (SCD) was constructed. However, in practice, both the location and the number of the changed region are uncertain. Thus, to systematically verify the performance of the architecture proposed in this study, two types of mixed datasets were constructed based on the 18 regions where the core may change: single-region mixed datasets (SRMDs) and multi-region mixed datasets (MRMDs). For SRMDs, the authors assumed that only one region of the core changed at a given time; however, this region was uncertain. In this study, 17 datasets, $\text{SRMDs}=\left\{D_m|2\le m\le 18\right\}$, were generated, where m indicates that a single change occurred in one of m certain regions. Thus, m can also be considered the location uncertainty of a single-changed region. For example, when m=3, a region change occurs in regions 3, 4, or 5, and the location uncertainty is 3. For MRMDs, we assumed that there were multiple changed regions at a certain time. Seventeen datasets $\text{MRMDs}=\left\{D_n|2\le n\le 18\right\}$ were generated. Further, n represents the number of changed regions in all 18 regions. For example, when n=3, there were 3 certain changed regions in the core, where these 3 regions were randomly selected among the 18 regions. In fact, the magnitude of the core changes in reactor operation was also uncertain; therefore, this study increased the diversity of the neutron field changes; that is, even if any two samples of the training set exhibited a consistently changed region, the magnitude of the change was not consistent. The amplitude of the core-changed region was 0~15%; these datasets are listed in Table 3.

Hyperparameter Settings

Table 4 lists the hyperparameter of DTHNet utilized in the experiment. In the experiment employs a three-dimensional convolutional network layer was employed to reconstruct a four-dimensional core neutron field. Moreover, to preserve the inherent features of the ex-core detection signals, the network architecture in this study did not utilize a pooling layer. Finally, the training parameters of the DTHNet in this study were set to λ=0.5, β=0.1, epochs=1500, batch_size=100.

Evaluation criteria for neutron field reconstruction

The relative deviation is expressed as RD, where the value quantifies the relative difference between the reconstructed and reference data. Within the context of this study, RD served as an assessment metric for appraising the reconstruction effect of each reconstruction unit within the phase–space neutron field. Meanwhile, the closer ARD is to 0, the smaller the overall difference between the reconstructed data and reference data. The mathematical expression for RD is as follows: $RD=\frac{\left|{\varphi }_{\text{reconstructed}}-{\varphi }_{\text{reference}}\right|}{{\varphi }_{\text{reference}}}$ (5) where ϕ_{reconstructed} represents the reconstructed data obtained by using the trained model, and ϕ_reference represents the neutron flux obtained through Monte Carlo simulations, where the number of particles for the Monte Carlo fixed source simulation was 10⁹.

Based on the calculation results of the above relative deviation, the average relative deviation ARD and maximum relative deviation RD_max were used as the evaluation criteria. As there were 460 reconstruction units in this study, taking the average relative deviation as the evaluation criterion will inevitably cover certain of the larger deviations, which will affect the final judgment. Therefore, 10% was selected as the reconstruction deviation threshold, and the reconstruction unit with a relative deviation of more than 10% RD_≥10% between the reconstructed value and the Monte Carlo simulation reference value was considered as an additional evaluation criterion.

Experiments on SCDs

To test the reconstruction effect of DTHNet on SCDs, the core neutron field reconstruction performance was tested on 18 datasets of SCDs, and the reconstruction results were compared with those based on PRA. The statistical results in Table 5 revealed that with the utilization of DTHNet, on average, the RD_max was 10.82%, and the number of RD_≥10% was 2. When employing PRA, the RD_max is 12.39%, and the number of RD_≥10% was 3. Meanwhile, the average relative deviations ARD of the two methods were slightly different, and both remained within 2.5%.

To facilitate a more intuitive comparison of the reconstruction outcomes of the two architectures, a line chart was generated as shown in Fig. 8. The chart illustrates RD_max and RD_≥10% for the 18 datasets of SCDs. Upon examination of the chart, it is evident that the reconstruction RD_max of PRA was inferior to that of DTHNet on 14 datasets, and the reconstruction RD_≥10% of PRA was inferior to that of DTHNet on 11 datasets.

Fig. 8Full-screen View

(Color online) Maximum relative deviation RD_max and the number of RD_≥10% evaluated by PRA and DTHNet on SCDs

Thus, this analysis indicates that the DTHNet was well suited for core neutron field reconstruction in the case of a single changed region, and it offered certain advantages over the PRA architecture.

Experiments on SRMDs and MRMDs

The change in SCDs is relatively simple, and there is only a single changed region in each dataset. To assess the adaptability of DTHNet to complex core changes, we evaluated the core neutron field reconstruction efficacy of this model on both SRMDs and MRMDs.

Table 6 presents the reconstruction effect on SRMDs and MRMDs; compared with the ARD of the reconstructed results on SCDs, the ARD on SRMDs and MRMDs increased; however, it could be maintained within 3%. On average, RD_max of SRMDs was 21.25%, and the number of RD_≥10% was 12. When the location uncertainty m of a single changed region was small (9), the number of RD_≥10% could be maintained to within 10. Meanwhile, on average, RD_max of MRMDs was 22.92%, and the number of RD_≥10% was 23. When the number of changed regions n in a dataset was less than five, RD_≥10% was maintained within 10. The results show that DTHNet can achieve core neutron field reconstruction on some datasets of SRMDs and MRMDs, further proving the applicability of the architecture for neutron field reconstruction under complex core changes.

To more intuitively analyze the reconstruction effect of DTHNet under complex core changes. A line chart is drawn for the data in Table 6, as shown in Fig. 9. It displays the RD_max and maximum number of RD_≥10% values of the reconstruction results for SRMDs and MRMDs, respectively. As depicted in Fig. 9(a), the reconstruction deviation increased with location uncertainty of single changed region. Similarly, when a larger number of regions underwent changes, as shown in Fig. 9(b), the efficacy of the reconstruction diminished. These results indicate that the location uncertainty of a single core-changed region and the number of core-changed regions at a certain time significantly impact the reconstruction effect of DTHNet. Therefore, the continuous improvement of the network architecture of the DTHNet method and further enhancement of its applicability under various complex changes in the core will be the focus of subsequent research.

Fig. 9Full-screen View

(Color online) RD_max for each dataset on the SRMDs and MRMDs. The m value of SRMDs and n value of MRMDs increase linearly with the serial number of dataset

Experiment on noise data

To verify the effect of neutron field reconstruction in the case of signals containing noise, reconstruction experiments were performed wherein the ex-core detection signals were affected by noise. When the location uncertainty m of a single changed region did not exceed nine, and the number of multiple changed regions n did not exceed five, the ex-core detection signals were disturbed by noise to obtain noisy data (SRMDs_noise and MRMDs_noise). Noise generation is based on the inherent error of the detector, which ranges as ± 20%–± 30% in current engineering applications. The process of adding noise is as follows. First, six interference intervals (± 2%, ± 5%, ± 10%, ± 15%, ± 20% and ± 25%) for the ex-core detection are set, and then the SRMDs_noise and MRMDs_noise are generated by adding noise to each detection signal of SRMDs_{(n ≤ 9)} and MRMDs_{(m ≤ 5)}.

The DTHNet method was trained and tested based on the acquired noise data. Table 7 presents the results of the reconstruction using the noise data. Compared with the results presented in Table 6, for SRMDs_noise, when the degree of interference was ± 2%, on average, ARD increased by approximately 0.22%, RD_max increased by approximately 8.33%, and among the 460 reconstructed units, the number of RD_≥10% increased by approximately 6. However, the reconstruction accuracy decreased sharply as the degree of interference increased. For MRMDs_noise, when the degree of interference was ± 2%, among the 460 reconstructed units, although the average number of RD_≥10% increased by approximately six, the average ARD increased by approximately 0.23%. Further, the average RD_max increased by approximately 5.05%. As the degree of interference continued to increase, the reconstructed ARD and RD_max did not change significantly, and only the average number of RD_≥10% increased. Thus, when the ex-core detection signals were noisy, the location uncertainty of the core change had a significant impact on the accuracy of the reconstruction; however, an increase in the number of changed regions had a slight impact on the reconstruction accuracy of the method.

In summary, when the noise interference amplitude is within ± 2%, the DTHNet method can accurately and stably reconstruct most neutron fields. Subsequently, we will continue to optimize the reconstruction method under noise interference to improve the stability of neutron field off situ reconstruction methods.

Reconstruction time

To test the reconstruction speed of the proposed method, the time required for model training and reconstruction (experimental environment: windows11+NVIDIA 2080Ti) was determined. The model was trained for approximately 3333 s on each dataset, and the results of neutron field reconstruction could theoretically be obtained in approximately 3 s. Therefore, it can be considered that this method satisfies the real-time monitoring requirements for the neutron field.

Ablation study

By analyzing the aforementioned experimental effect, it is evident that DTHNet exhibits superiority not only in comparison to PRA under a single-region change, but also in its capacity to achieve precise reconstruction under some complex core changes. To further validate the effectiveness of the method design, ablation experiments were conducted on a global-local feature upsampling module and dual-task training. The results are presented in Table 8.

To assign the features of the ex-core detection signals to each reconstruction unit to improve the overall reconstruction accuracy, DTHNet incorporates a global-local feature upsampling module. To validate the efficacy of this module, we conducted ablation experiments, the results of which are presented in Table 8.

When the global-local feature upsampling module was replaced by a common upsampling module, the average RD_max on SCDs, SRMDs, and MRMDs was approximately 89%, 81%, and 74%, respectively. In addition, the average number of RD_≥10% in SCDs, SRMDs, and MRMDs increased significantly. From this observation, it is evident that global-local feature upsampling captures ex-core detection information and enhances the core reconstruction efficiency.

In DTHNet, the reconstruction task of neutron field in the outermost core assemblies (Task A) was used to assist the core neutron field reconstruction task (Task B). To verify the assistance effect of Task A, core neutron field reconstruction experiments were performed after Task A was removed. Comparing the results of DTHNet and DTHNet (without Task A) in Table 8, it can be observed that ARD increased slightly for SCDs, SRMDs, and MRMDs. In addition, the average RD_max values of the SCDs, SRMDs, and MRMDs were approximately 2.57%, 2.99%, and 2.29%, respectively. The average number of RD_≥10% increased by 2~3 for SCDs and SRMDs but increased significantly for MRMDs. These results show that the reconstruction task of the neutron field in the outermost core assemblies improved the accuracy of core neutron field reconstruction.

Error analysis

To further analyze the details of the model prediction, two cases of core changes were randomly selected, and the ex-core detection signals of the two cases were simulated using the Monte Carlo transport program. The neutron flux distribution of the core assembly calculated by Monte Carlo transport was used as the reference data to evaluate the reconstruction effect. Figure 10 shows the two types of the ex-core detection signals and their changes compared to the initial signals. The upper part of the core assembly changed in case1, and the lower part of the core changed in case2.

Fig. 10Full-screen View

(Color online) Visualization of two ex-core detection signals. The change image indicates that the upper of the core has changed under case1, and the lower of the core has changed under case2

In this section, we analyze the reconstruction effect of the neutron flux distribution of the core assembly and the distribution of the reconstruction deviation. According to the reconstruction evaluation criteria in Sect. 5.1, the relative deviation RD between the reconstructed and reference neutron fluxes was calculated, and the errors were analyzed based on RD. Figures 11 and 12 show RD visualizations of core neutron flux reconstruction when two and four regions of the core changed, respectively.

Fig. 11Full-screen View

(Color online) RD of certain core reconstruction units under case1. The redder the color, the larger the RD; and the bluer the color, the smaller the RD

Fig. 12Full-screen View

(Color online) RD of certain core reconstruction units under case2. The redder the color, the larger the RD; and the bluer the color, the smaller the RD

In Figs. 11 and 12, the bold part of the frame line indicates the changed regions. The assemblies with large errors were mainly concentrated in the core-changed regions and [1.00, 2.00×10¹] MeV. Therefore, it can be concluded that the large error in certain assemblies was related to the spatial position and energy group.

First, the spatial influencing factors were analyzed. Figures 11(f) and 12(f) show that the reconstruction error of the changed regions was generally greater than that of the unchanged regions. The reasons are as follows. This study provided the model with sufficient pairs of ex-core detection signals and in-core neutron field data, and the model was trained to learn some nonlinear relationship between the ex-core detection signals and each unit to be reconstructed. However, in the learning process, if the change in the amplitude of the unit to be reconstructed is uncertain, then it is difficult for the model to learn the exact nonlinear relationship between the ex-core detection signals and the value of the reconstruction unit. According to the dataset settings in Sect. 4.1, the diversity of the neutron field change amplitude was increased to simulate the uncertainty of the core change during the reactor operation. Thus, even if any two samples of the training set consistently changed their regions, the magnitude of the change was inconsistent. Thus, when the changed and unchanged regions were trained synchronously, it was difficult to fit the nonlinear relationship between the changed regions of the core and ex-core detection signals, which is the primary cause of the uneven distribution of the error space.

Next, the energy-influencing factors were analyzed. This study mainly selected the RD of the reconstruction unit in the [0, 1.00×10^-2] MeV, [1.00×10^-2, 1.00×10^-1] MeV and [1.00, 2.00×10^-1] MeV with obvious error distribution for visualization. As evident from the two figures, the RD was within 13%, the number of RD_≥10% did not exceed 3, but the RD distribution of [0, 1.00×10^-2] MeV and [1.00, 2.00×10¹] MeV exceeded that of [1.00×10^-2, 1.00×10^-1] MeV. This is because if the magnitude of the values is different among reconstruction units, neural network models may struggle to learn suitable weights and biases for each reconstruction unit simultaneously, such that the individual reconstruction units cannot obtain the optimal results at the same time. Because this study is focused a fast neutron reactor, the distribution of neutrons among the five energy groups is uneven; therefore, error distribution between different energy groups is inevitable.

In summary, the errors of some reconstruction units may be affected by both spatial and energy factors. The design of the method will be further improved according to the above analysis to improve the accuracy of the reconstruction under complex changes in neutron fields.