Preamble

An Explicit Off-Situ Inversion Method for Neutron Fields in Reactor Core Based on Graph Structure

Pei Cao¹,²,†, Xue-Tao Cao¹, Hui Ding¹, Si-Xuan Zhuang³, Zi-Hui Yang², and Guo-Min Sun²
¹Hefei University, Anhui Hefei 230006, China
²Hefei Institutes of Physical Sciences, Chinese Academy of Sciences, Anhui Province Key Laboratory of Small Reactor and Micro-reactor Technology, Anhui Hefei 230031, China
³Institute of High Energy Physics, Chinese Academy of Sciences, Dongguan 523803, China

Future advanced reactors will operate in harsh core environments. Using off-situ detection signals to invert the neutron distribution within the core represents a critical means for ensuring safe reactor operation. However, recent data-driven off-situ inversion algorithms have not fully accounted for correlations among various regions or assembly segments of the reactor core, leading to limited accuracy when performing high-resolution neutron field inversion. To address this limitation, this study proposes an explicit off-situ inversion model for reactor core neutron fields based on graph structure. By structurally modeling both the attribute information of each discretized assembly segment and the relational information between segments, the model explores interactions and correlations among core assembly segments to improve inversion accuracy for high-dimensional neutron distributions. Validation results for the Chinese Lead-based Research Reactor (CLEAR-I) demonstrate that, compared to previous MCRNet and DETNet frameworks, the maximum relative deviation (RDmax) decreases from 21.8% to 14.01% when a single core region varies, with fewer than 5 inversion units exhibiting relative deviations ≥10%. For multi-region core variations, RDmax decreases from 28.08% to approximately 20%, while the proportion of relative deviations ≥10% (RRD≥10%) drops from 3.14% to 1.77%. These results confirm that explicit structural modeling of the reactor core enhances the accuracy of off-situ inversion calculations.

Keywords: Advanced reactor; graph structure; in-core neutron field; off-situ inversion

Introduction

Neutrons, as the primary agents driving nuclear reactions, exhibit dynamic behavior within a reactor core that is intrinsically influenced by multiple operational and physical factors, including control system interventions, progressive fuel burnup, and accumulation of fission products \cite{1,2}. These influences produce temporal and spatial variations in key neutronic parameters such as neutron distribution and reaction rates. Accurate, real-time monitoring of these parameters is essential for maintaining reactor safety and operational reliability. Among these parameters, neutron flux density serves as a fundamental descriptor of the reaction rate within the core. Its spatial distribution critically determines the precision of power density calculations, thereby underpinning the validation of core physics designs and informing safety analyses, accident diagnostics, and the development of real-time monitoring algorithms \cite{3}. Equally important is the neutron energy spectrum, defined as a probability density function representing the distribution of neutron energies. Measurement fidelity of the spectrum is critical for core reactivity control and safety margin evaluations. In fast neutron reactors, axial gradient analysis based on the fast neutron spectrum (E > 0.1 MeV) enables development of quantitative models linking the fuel breeding ratio to refueling cycle length. Such models provide a rigorous foundation for multi-objective optimization of refueling strategies, balancing breeding performance, operational longevity, and safety constraints \cite{4,5}.

Currently, neutron flux measurement in reactors relies primarily on online core monitoring systems \cite{6-9}. These systems employ miniature fission chamber neutron detectors or self-powered neutron detectors (SPNDs) arranged in core instrument channels to measure neutron flux data, which are then used to calculate safety-related parameters. However, for Generation IV nuclear energy systems, in-core neutron detectors face challenges from extreme environments including high temperature, intense irradiation, corrosion, and strong magnetic fields \cite{10,11}, which severely restrict their reliability and economic viability. For example, megawatt-level space nuclear thermal propulsion systems envisioned for deep space exploration typically feature core heights between 0.5 and 2 meters \cite{12,13}. Adopting traditional in-core detection layouts for such systems would compromise safety, reliability, and propulsion performance. Consequently, efficient inversion research for core neutron parameters based on off-situ (outside-core) detection signals has become urgently necessary.

Numerous studies have investigated inversion methods for reconstructing power (neutron) distributions throughout the reactor core from off-situ detection signals. Traditional approaches rely on mechanistic models describing the relationship between detection signals and core physical fields, optimizing or solving for physical field parameter terms in the governing equations given known detection signals. Examples include harmonic synthesis methods \cite{14}, proper orthogonal decomposition (POD) methods \cite{15}, correlation-solving methods \cite{16}, and principal component analysis methods \cite{17}. However, these methods depend on fixed prior knowledge bases, making it difficult to dynamically adjust for rapid power distribution variations under transient conditions, which seriously impacts real-time monitoring efficiency for future advanced reactors.

With the rapid development of artificial intelligence technology, data-driven intelligent models have emerged as effective solutions for complex engineering problems in reactors, replacing traditional mechanistic models \cite{18-20}. Suitable solutions have been developed for intelligent optimization design of shielding for mobile micro nuclear reactors \cite{21}, power distribution inversion \cite{22-25}, and neutron distribution inversion \cite{26-30}. Analysis reveals that for low-resolution core power or neutron distribution inversion, the accuracy of RBFNN \cite{22,23} and IVAE \cite{27} models can theoretically be maintained between 3% and 5%, approaching practical engineering standards. However, recent research by Cao Pei et al. \cite{27-29} found that in off-situ monitoring, inversion accuracy of data-driven models degrades significantly as the complexity and fineness of core variations increase. For instance, as the resolution of the target core increases, the nonlinearity between neutron detection signals and neutron distribution intensifies, causing inversion accuracy of DCNN-based models \cite{27} to rise to 10%—substantially higher than the 3% to 5% observed in low-resolution scenarios. Additionally, under identical scenarios and hardware configurations, the MCRNet model proposed by Cao Pei et al. \cite{29} achieved 6.94% higher accuracy than DTHNet \cite{28}, though the increased model parameters added approximately one second to inversion calculation time.

These observations indicate that while complex network structures can enhance prediction capability for high-resolution core neutron field inversion, data-driven black-box modeling struggles to learn internal physical laws of the core. Blindly adding network structures does not necessarily improve accuracy and instead reduces prediction real-time performance. Therefore, this study proposes a graph structure-based explicit inversion network framework (GS-EINF) for off-situ inversion of reactor core neutron fields. This model effectively aggregates neighborhood information by representing the reactor core as a graph structure, thereby enhancing inversion accuracy for high-resolution neutron distributions.

Methodology

To achieve high-resolution off-situ inversion of reactor core neutron fields, this paper proposes an explicit off-situ inversion model based on graph structure, designated GS-EINF. Building upon MCRNet \cite{29}, the model first fuses off-situ detection signals and allocates them to each assembly segment of the target reactor core. Subsequently, it constructs the spatial and physical information of each segment using a graph-structured network, decoupling the spatial topology and energy spectrum information of each segment to obtain neutron contribution relationships between segments and characteristic representation forms. Finally, the model transforms neutron distribution based on these feature representations to improve inversion accuracy. Figure 1 [FIGURE:1] illustrates the proposed method schematically.

A. Graph Structure Representation of the Reactor Core

To accurately capture distribution characteristics of physical quantities in the reactor core, this paper divides the core into assembly-level segments and constructs a graph structure model based on physical properties and geometric adjacency relationships. The reactor core is first discretized into segments according to structures such as fuel assemblies, control rod regions, and reflector layers. As shown in Figure 2 [FIGURE:2], each assembly is divided into two segments, with each segment treated as a node in the graph. Each node possesses spatial coordinates for geometric information modeling and contains physical attribute information including neutron flux (or power density) $\phi_i$, material information $m_i$, local temperature $T_i$, and fuel abundance. These physical attributes are encoded as node feature vectors $\mathbf{x}_{\text{phys}}$ as follows:

$$
\mathbf{x}_{\text{phys}} = [\phi_i, m_i, T_i, \ldots] \in \mathbb{R}^d
$$

For edge construction between core segments (graph nodes), the definition of segment adjacency varies depending on assembly geometry. For square assemblies, each segment establishes edge connections with its adjacent segments above, below, left, and right to simulate direct geometric adjacency. For hexagonal assemblies, each segment establishes edge connections with six or eight surrounding adjacent nodes to better simulate layout characteristics of fast reactors or other advanced reactor designs.

Furthermore, due to the non-local effect of neutron diffusion, the neutron flux of each segment is influenced not only by directly adjacent segments but also by segments at greater distances. This phenomenon becomes particularly prominent when core geometry is complex, material distribution is non-uniform, or local power fluctuates sharply. To explicitly model this physical phenomenon in the graph structure, this paper introduces the concept of higher-order adjacency relationships beyond traditional first-order adjacency (directly adjacent nodes) to describe non-local neutron field effects.

As shown in Equation (2), let $N(i)$ represent the set of first-order neighbors of the $i$-th segment—i.e., all segments with direct edge connections to segment $i$:

$$
N_1(i) = {j \mid A_{ij} = 1}
$$

where $A_{ij}$ are elements of the adjacency matrix. To incorporate higher-order topological information in the reactor core structure, this work generalizes adjacency relationships beyond first-order neighbors. The $k$-th order neighborhood of a given segment $i$, denoted as $N_k(i)$, includes all segments reachable from segment $i$ through a path of exactly $k$ edges:

$$
N_k(i) =
\begin{cases}
{j \mid A_{ij} = 1}, & \text{if } k = 1 \
\bigcup_{j \in N_{k-1}(i)} N_1(j), & \text{if } k > 1
\end{cases}
$$

Here, $N_k(i)$ denotes the set of $k$-hop neighbors of segment $i$, defined recursively based on first-order neighborhood relationships. The set includes all segments reachable from segment $i$ via paths of exactly $k$ edges in the adjacency graph. The power operation of the adjacency matrix, $A^k$, encodes the number of distinct $k$-step paths between each pair of segments:

$$
(A^k){ij} = \sum}} A_{il_1}A_{l_1l_2}\ldots A_{l_{k-1}j
$$

where $\gamma$ is the attenuation coefficient for higher-order neighbors ($\gamma < 1$), $\alpha$ is the distance attenuation coefficient, and $d_{uv}$ denotes the Euclidean distance between segments $u$ and $v$ along path $P_k$. Finally, the enhanced adjacency matrix used for graph-based modeling is constructed as a weighted sum of multiple adjacency orders:

$$
\tilde{A} = A^{(1)} + \sum w^{(k)} \odot A^{(k)}
$$

where $A^{(1)}$ represents the first-order adjacency matrix, $A^{(k)}$ represents the $k$-order adjacency matrix, $\odot$ denotes the Hadamard product (element-wise multiplication), and $w^{(k)}$ is the attenuation coefficient matrix for $k$-order edges. By introducing these higher-order adjacency relationships, the model can explicitly capture neutron diffusion effects between reactor core blocks, better fitting the non-locality of neutron diffusion in complex reactor cores and effectively improving the accuracy and physical consistency of neutron field inversion.

B. Network Framework Design Based on Graph Structure

In this manner, higher-order neighbor information can be efficiently extracted from the adjacency matrix without explicitly constructing a new edge set. To ensure physical consistency and prevent excessive information diffusion, an attenuation mechanism is introduced. Specifically, for each $k$-path order adjacency, a decaying weight is applied based on path length and spatial distances between intermediate nodes. The edge weight between segment $i$ and segment $j$ at order $k$ is defined as:

$$
w_{ij}^{(k)} = \gamma^k \cdot \sum_{\text{paths } P_k} \prod_{(u,v) \in P_k} A_{uv} \cdot \exp\left(-\alpha \cdot \sum_{(u,v) \in P_k} d_{uv}\right)
$$

Based on the constructed structured core graph, this paper proposes an off-situ inversion framework integrating graph neural networks and 3D convolutional neural networks. By fusing off-situ detection signals, this approach reconstructs the 3D distribution of the neutron field in the reactor core. The overall framework comprises three stages: upsampling observed signals, diffusing information through the graph structure, and transforming spatial features.

Let the observed signal collected by the off-situ detector be:

$$
\mathbf{s} = [s_1, s_2, \ldots, s_M]^\top
$$

where $M \ll N$ indicates that the number of measurement points is much smaller than the total number of segments $N$ in the core. To achieve distribution reconstruction of the observed signal across the entire reactor core, this paper adopts the upsampling mechanism from convolutional neural networks (Upsampling CNN) to map the sparse signal $\mathbf{s}$ to a feature tensor aligned with the number of segments $N$:

$$
\mathbf{S}{\text{up}} = \text{UpsampleCNN}(\mathbf{s}; \theta)}
$$

where $\theta_{\text{cnn}}$ represents trainable parameters. This upsampling module consists of three-dimensional transposed convolution and convolution layers, with output dimension $N \times d_s$. Subsequently, the upsampled observed features are concatenated with physical attributes of core segments to form the input to the graph neural network:

$$
\mathbf{X}^{(0)} = [\mathbf{X}{\text{phys}}; \mathbf{S}}}] \in \mathbb{R}^{N \times (d + d_s)
$$

Next, multi-layer graph convolutional networks (GCN) are applied to the graph structure to achieve spatial diffusion and fusion of information between segments. The graph convolution operation is defined as:

$$
\mathbf{h}i^{(l+1)} = \sigma\left(\sum\right)} \frac{1}{\sqrt{\hat{d}_i \hat{d}_j}} \mathbf{W}^{(l)} \mathbf{h}_j^{(l)
$$

where $\hat{d}i = \sum_j W$ are learnable parameters of layer $l$, and $\sigma(\cdot)$ is the activation function (ReLU). In actual network implementation, residual connections and dropout mechanisms can be introduced to enhance representational capacity and robustness.}$, $\mathbf{W}^{(l)

To further capture structural continuity of the neutron field in three-dimensional space, the segment features $\mathbf{H}^{(L)} \in \mathbb{R}^{N \times d_h}$ obtained after graph convolution are rearranged into a three-dimensional tensor $\mathbf{H}_{3D} \in \mathbb{R}^{D_x \times D_y \times D_z \times d_h}$, where $D_x, D_y, D_z$ represent the number of discrete grids in the $x$, $y$, and $z$ directions, respectively, satisfying $D_x \cdot D_y \cdot D_z = N$. This tensor serves as input to the three-dimensional convolutional network for voxel-level spatial feature transformation. The 3D convolution module uses fixed stride $s$ and window size $k$ to perform multi-channel volumetric convolution and output the final predicted neutron flux tensor:

$$
\hat{\mathbf{Y}} = \text{Conv3D}(\mathbf{H}{3D}; \theta)
$$

where $\theta_{3D}$ represents parameters of the 3D convolutional layer, and the output tensor size is $D_x \times D_y \times D_z \times d_h$, representing the inverted neutron flux values of the reactor core.

Previous studies \cite{26-30} have shown that outermost core assemblies, due to their proximity to ex-core detectors, exhibit stronger response characteristics in neutron leakage signals and thus possess higher utilization value in neutron field inversion. As "boundary information" of the neutron field distribution, the state of outermost assemblies can reflect variation trends of in-core neutron distribution to a certain extent. Therefore, this paper continues and strengthens this concept by adopting a dual loss function mechanism to supervise prediction of the neutron field in both outermost assemblies and the entire core. The loss function comprises two components: prediction loss for outer assemblies, which enhances response consistency between boundary assembly neutron flux and off-situ detection signals; and prediction loss across the entire core range, reflecting inversion accuracy of the global neutron flux distribution. The joint loss function is expressed as:

$$
\mathcal{L} = \mathcal{L}{\text{core}} + \lambda}} \mathcal{L}_{\text{outer}
$$

$$
\mathcal{L}{\text{core}} = \sum_i - y_i)^2} (\hat{y
$$

$$
\mathcal{L}{\text{outer}} = \sum_j - y_j)^2} (\hat{y
$$

where $N$ is the set of all core segments, $N_b$ is the set of outermost segments, and $\lambda_{\text{outer}}$ is the adjustment coefficient for the auxiliary loss term. This strategy not only enhances the model's perception of boundary region characteristics but also improves accuracy and stability of neutron field inversion for the entire reactor core. The model is trained using the Adam optimizer with an initial learning rate of 0.0015. To prevent overfitting, dropout mechanisms are applied after certain network layers. Additionally, to improve training efficiency, the paper designs mechanisms such as early stopping.

Experiment Settings

A. Reactor Model

The effectiveness of the proposed inversion method is evaluated under transient core conditions at the assembly-energy level, based on simulations of the China Lead-based Research Reactor (CLEAR-I). As shown in Figure 2 [FIGURE:2], 8×8 detection points are located outside the shielding layer. Each core assembly is divided into two segments according to material distribution, resulting in a core height factor of 2. Based on the CLEAR-I profile shown in Figure 2 [FIGURE:2], the reactor core has a length of 11, width of 7, and 5 energy groups. Therefore, the neutron field size in the reactor core is $5 \times 2 \times 7 \times 11$.

B. Datasets

Based on previous research \cite{29,30}, the core of the aforementioned lead-based reactor model is divided into 17 regions, each containing 3 to 6 segments. Two databases were simulated using Monte Carlo transport software. The SRMDs database contains datasets for 17 different single-region core variations. Local disturbances are configured at various spatial positions in the reactor core (such as middle, boundary, or local assemblies) to simulate non-uniform neutron field variations that may occur under local power fluctuations, thereby verifying model inversion performance when variations occur at different core positions. Furthermore, disturbances are configured in multiple spatial regions to simulate complex operating conditions such as multi-point power fluctuations that may occur during actual operation. Consequently, the MRMDs database contains 17 datasets with different numbers of reactor core variations to verify model inversion performance under varying complexities of core disturbances. Dataset details are shown in Table 1 [TABLE:1].

C. Evaluation Criteria

For each inversion unit in the neutron field, this paper adopts relative deviation (RD) to evaluate inversion performance. The closer RD is to 0, the smaller the overall difference between inverted data $\phi_{\text{recon}}$ and reference data $\phi_{\text{ref}}$. The mathematical expression for RD is:

$$
\text{RD} = \frac{|\phi_{\text{recon}} - \phi_{\text{ref}}|}{\phi_{\text{ref}}} \times 100\%
$$

Based on RD calculations, this paper employs average relative deviation (ARD) to measure the overall average deviation level between predicted results of all reconstruction units and reference values. Maximum relative deviation (RDmax) captures extreme cases of local deviations, determining whether the model exhibits serious inaccuracies in certain areas. Finally, the proportion of units with relative deviation ≥10% is statistically analyzed to measure the fraction of regions with large prediction deviations, reflecting uniformity of spatial deviation distribution or the extent of widespread local deviations:

$$
\text{RRD}_{\geq 10\%} = \frac{\text{Count(Units with RD} \geq 10\%)}{\text{Count(Units)}}
$$

In actual reactor operation, observation signals collected by off-situ detectors are inevitably affected by various noise factors such as electronic noise, temperature drift, and detector aging. These uncertainties propagate directly to the inversion model input, affecting stability and reliability of the reconstructed neutron field. To quantitatively evaluate the influence of input noise on inversion results, this paper adopts standard deviation as the primary quantitative index of observation uncertainty, statistically analyzing the fluctuation degree of model prediction results under multiple independent observation noise disturbances.

For selected reactor variation scenarios and set noise levels, the original ex-core detection signal is subjected to $n$ independent random noise perturbations to generate $n$ sets of input data with typical observation uncertainties. The proposed GS-EINF method is used to predict the neutron field for these $n$ data groups, yielding $n$ inversion results. For each inversion unit $i$ in the reactor core, denote the $k$-th predicted value as $\phi_i^{(k)}$. The predicted mean $\mu_i$ and standard deviation $\sigma_i$ for this unit under observation disturbances are defined as:

$$
\mu_i = \frac{1}{n} \sum_{k=1}^n \phi_i^{(k)}
$$

$$
\sigma_i = \sqrt{\frac{1}{n-1} \sum_{k=1}^n \left(\phi_i^{(k)} - \mu_i\right)^2}
$$

Here, $\mu_i$ represents the average inverted neutron flux, while $\sigma_i$ measures sensitivity and stability of neutron flux prediction at this position. Larger standard deviation indicates greater sensitivity to observation deviations, while smaller standard deviation indicates better model robustness in that region. Through the spatial distribution map of $\sigma_i$ across the entire reactor core, the model's response characteristics to observation uncertainty in different regions and under different operating conditions can be systematically revealed.

Experimental Results and Analysis

Figure 2 [FIGURE:2] shows the settings of ex-core detection points and division of reconstructed regions. Assemblies are divided into Group 1, Group 2, and Group 3 according to their proximity to ex-core detectors, where Group 1 corresponds to the outermost layer of the reactor core.

1. Single-Region Variation in the Reactor Core

Based on overall prediction accuracy of inversion results, the average relative deviation (ARD) of GS-EINF remains in the low range of 0.08%–0.59% across all variation scenarios, demonstrating highly consistent neutron field inversion accuracy. In contrast, MCRNet's ARD remains at 0.9%–1.4%, while DETNet results are generally above 1.3%, exceeding 2.0% in some scenarios. This result confirms that GS-EINF achieves superior overall inversion performance under single-region core variations.

Furthermore, as shown in Figure 3 [FIGURE:3], regarding local extreme deviation control, GS-EINF's maximum relative deviation (RDmax) for core neutron field inversion is mostly maintained between 2% and 10%, only slightly higher in a few strong disturbance scenarios. This is significantly lower than MCRNet's 10%–17% range and substantially below DETNet's repeated peaks exceeding 25%. This phenomenon indicates that GS-EINF maintains stability of inversion results in locally abnormal disturbance regions and effectively avoids extreme distortion of neutron field inversion. From the perspective of spatial deviation distribution, in the vast majority of scenarios, the proportion of inversion units with relative deviation ≥10% (RRD≥10%) in GS-EINF is 0, indicating no obvious spatial deviation diffusion phenomenon. MCRNet's RRD≥10% generally ranges from 0.2% to 1.7%, while DETNet can reach 5.4% in many scenarios, indicating its deviations are prone to local aggregation or spatial diffusion. These results demonstrate that GS-EINF exhibits better spatial robustness for reactor core neutron field inversion.

2. Multi-Region Variation in the Reactor Core

When disturbances occur simultaneously in multiple reactor core regions, GS-EINF's ARD mostly remains between 0.2% and 0.9%, only exceeding 1.0% in some high-intensity disturbance scenarios. This significantly outperforms DETNet, which achieves deviation levels of 1.8%–2.4% in many scenarios. MCRNet performance lies at an intermediate level, with average deviations primarily between 1.3% and 1.9%. This result indicates that GS-EINF maintains good overall inversion accuracy even when facing complex core interference.

In terms of local deviation control, observation of RDmax for core inversion in Figure 4 [FIGURE:4] reveals that GS-EINF's RDmax does not exceed 15% in most scenarios, while MCRNet and DETNet inversion RDmax values mostly exceed 20%, reaching extreme deviations of 30.7% and 27.4%, respectively. This demonstrates that GS-EINF effectively alleviates violent local prediction oscillations through graph adjacency modeling, exhibiting high spatial robustness. Regarding deviation distribution breadth, GS-EINF controls RRD≥10% within 0%–1.5% in most scenarios, while MCRNet repeatedly reaches approximately 1%–2%, significantly lower than the diffusion levels of MCRNet and DETNet, which mostly exceed 2%–4%. Especially for inversion of assemblies near the core interior, GS-EINF effectively limits deviation propagation in local regions.

Table 3 [TABLE:3] compares inversion results using three methods when the number of core variation regions is ≤9. GS-EINF maintains an advantage in overall accuracy, with ARD mostly within 0.3%–1.3%, while MCRNet remains within 0.8%–1.4%. Both outperform DETNet, whose ARD ranges between 1.4%–1.8%. Regarding RDmax, GS-EINF is controlled within 15%–19% in most scenarios, while MCRNet reaches 25.5% in some scenarios and DETNet frequently exceeds 20%. From the perspective of spatial deviation distribution, GS-EINF stably controls RRD≥10% within 2%–4% in most scenarios, while MCRNet rises to 5.4% in some scenarios and DETNet frequently exceeds 4.7%. Notably, when the number of core variation regions is ≤9, MCRNet and DETNet exhibit significant loss of control over inversion accuracy, with prediction deviations far exceeding acceptable ranges and even failing to converge or produce valid results under certain conditions. In contrast, GS-EINF consistently provides prediction results, further verifying its good adaptability for neutron field inversion under complex core variations.

3. Observation Uncertainty Analysis

In actual reactor operation, ex-core detector signals are inevitably affected by various noise factors such as electronic noise, temperature drift, and detector aging. These uncertainties propagate directly to the inversion model input, affecting stability and reliability of the reconstructed neutron field. To further evaluate robustness of the proposed GS-EINF method against random disturbances in observed signals, this paper constructs an off-situ inversion experimental framework with observation noise. Specifically, for a randomly selected variation scenario, five independent ±2% Gaussian random noise perturbations are applied to ex-core detection signals, and neutron field inversion is performed using both GS-EINF and the optimal comparison model MCRNet according to the design in Section III.C.

Figure 5 [FIGURE:5] shows the distribution statistics of uncertainty values for each inversion unit. GS-EINF's standard deviation in the vast majority of inversion units is below 1E-03, primarily concentrated within 1E-04 to 1E-03, with well-converged long-tail distribution. In contrast, MCRNet exhibits more unstable units, with predicted standard deviation exceeding 5E-03 in multiple regions and some units reaching or exceeding 0.01. From overall statistical indicators, GS-EINF's average standard deviation is 5.99E-04 and maximum standard deviation is 1.20E-02. MCRNet's average standard deviation is 7.54E-04 and maximum standard deviation is 1.04E-02. Although GS-EINF shows relatively large standard deviation in individual units, its overall fluctuation level is significantly lower than MCRNet, indicating that its inversion results are less sensitive to input interference and possess strong anti-noise capability.

Inversion Results Under Noise Interference

To verify stability and robustness of the GS-EINF method when actual observed signals are disturbed by noise, this paper further introduces ±2% random noise to ex-core detection signals based on single-region and multi-region core variations, simulating uncertainty effects of sensor output during reactor operation. Under this setting, neutron field inversion results of three models—GS-EINF, MCRNet, and DETNet—are compared.

1. Single-Region Variation in the Reactor Core

As shown in Table 2 [TABLE:2], when ex-core detection signals are disturbed, GS-EINF maintains relatively low ARD, mostly within 0.3%–0.6%, while MCRNet's ARD ranges between 1.1%–1.4% and DETNet between 1.2%–1.8%. Furthermore, regarding RDmax, GS-EINF remains below 20% in the vast majority of scenarios, while MCRNet mostly exceeds 25% and DETNet results even exceed 35% in some cases. From the perspective of spatial deviation distribution, GS-EINF's RRD≥10% consistently remains below 1%, while MCRNet reaches 6% and DETNet exceeds 8% in many scenarios, peaking at 8.9%. This indicates that GS-EINF suppresses spatial diffusion of systematic deviations by learning physical continuity and structural consistency characteristics of the inversion field, ultimately achieving precise control over relative deviation of predicted values for each inversion unit.

2. Multi-Region Variation in the Reactor Core

Table 3 [TABLE:3] compares inversion results under noise interference for multi-region core variations. GS-EINF maintains an advantage in overall accuracy, with ARD mostly within 0.3%–1.3%, while MCRNet remains within 0.8%–1.4%. Both outperform DETNet, whose ARD ranges between 1.4%–1.8%. Regarding RDmax, GS-EINF is controlled within 15%–19% in most scenarios, while MCRNet reaches 25.5% in some scenarios and DETNet frequently exceeds 20%. From the perspective of spatial deviation distribution, GS-EINF stably controls RRD≥10% within 2%–4% in most scenarios, while MCRNet rises to 5.4% in some scenarios and DETNet frequently exceeds 4.7%. It is particularly noteworthy that when the number of core variation regions is ≤9, MCRNet and DETNet exhibit significant loss of control over inversion accuracy, with prediction deviations far exceeding acceptable ranges and even failing to converge or output valid results under certain conditions. In contrast, GS-EINF consistently provides prediction results, further verifying its good adaptability for neutron field inversion under complex core variations.

Parameter Sensitivity Analysis

A. Analysis of Adjacency Configuration

To investigate the influence of adjacency order between segments in the graph structure on off-situ inversion performance, this paper conducts comparative analysis of two different graph structure modeling methods while keeping other network layers and training strategies unchanged. The first is a first-order adjacency structure containing only connections of directly adjacent core segments. The second introduces a second-order adjacency structure that includes indirectly adjacent core segments. Inversion results of these two models are compared and analyzed.

As shown in Figures 6 [FIGURE:6] and 7 [FIGURE:7], under the SRMDs dataset, the second-order graph structure is slightly superior to the first-order structure in overall deviation control. Although median deviation and distribution center are slightly higher in some scenarios, it performs better in suppressing extreme deviation tails. According to the violin plot in Figure 6 [FIGURE:6], the RDmax distribution of the second-order structure converges significantly, with probability density in the high deviation region decreasing markedly and RRD≥10% approaching 0. The overall volatility is lower than that of the first-order structure, indicating that in single-region variation scenarios, using a two-stage adjacency structure between reactor core blocks better enhances the model's ability to characterize core disturbances.

In DRMDs multi-region variation scenarios, first-order and second-order adjacency structures exhibit different characteristics in deviation unit distribution control. The RDmax violin plot indicates that the second-order graph structure maintains an advantage in suppressing extreme deviations and can more effectively limit the upper bound of maximum prediction errors. However, the RRD≥10% of the first-order structure is actually more stable. Statistical results show that the average value of the first-order adjacency structure on this indicator is 1.10% with a median of 0.43%, while the second-order adjacency structure averages 1.51% with a median of 1.30%. This indicates that in scenarios with multiple disturbed core regions, although the second-order graph structure has certain advantages in controlling maximum deviation, there may be excessive propagation of secondary neighborhood information when controlling deviation diffusion across the entire reactor.

B. Effect of Graph Convolution Layer Count

To explore the influence of graph convolution layer count on off-situ inversion performance, GS-EINF models containing 1, 2, and 3 graph convolution layers are constructed while maintaining consistent training strategies. Model performance is then quantitatively evaluated on SRMDs and MRMDs.

As observed in violin plots Figures 8 [FIGURE:8] and 9 [FIGURE:9], on the SRMDs dataset, increasing the number of graph convolution layers yields varying degrees of performance improvement. Due to receptive field limitations, GS-EINF with 1 graph convolution layer can only fuse local neighborhood information of nodes. Although RRD≥10% remains at 0 in most scenarios, RDmax still exceeds 20% in some scenarios, indicating insufficient modeling capability for distant disturbance propagation. GS-EINF with 2 graph convolution layers significantly reduces extreme prediction deviations while maintaining the proportion of low-deviation units. The RDmax fluctuation range converges overall, demonstrating superior ability to model local disturbance propagation. However, GS-EINF with 3 graph convolution layers shows no further significant performance improvement on this dataset, with slight deviation rebound in some scenarios, possibly due to overfitting from excessive depth.

Additionally, experimental results on MRMDs reveal that in cases of simultaneous multi-region core variations, the influence of graph convolution layer count on inversion accuracy and stability is more significant. GS-EINF with 1 graph convolution layer shows relatively high RDmax in some scenarios, reaching 20.78%, with RRD≥10% up to 6.74%, indicating limited capability for coupled modeling of non-local disturbances and deviation diffusion phenomena. GS-EINF with 2 graph convolution layers effectively reduces extreme deviations and controls deviation unit diffusion in most scenarios, achieving the most balanced comprehensive performance. GS-EINF with 3 graph convolution layers further suppresses the upper bound of maximum error and achieves better deviation control in multiple scenarios.

In summary, GS-EINF with 2 graph convolution layers achieves an optimal balance between deviation control capability and prediction stability for this task, delivering the best performance across most perturbation scenarios. However, in specific applications, the number of graph convolution layers should be carefully adjusted according to spatial complexity of core variations. For instances involving strong spatial coupling or broad variation ranges, increasing graph convolution layer count can enhance the inversion model's ability to capture long-range dependencies. Conversely, for localized variations or limited data availability, a shallower graph convolution network design is more appropriate to minimize overfitting risk and reduce unnecessary computational overhead. This adaptive strategy ensures a balanced trade-off between predictive accuracy and computational efficiency.

Case Analysis

To analyze the detailed effect of the GS-EINF method in local neutron field inversion, this study randomly selected two core variation conditions and used Monte Carlo transport programs to simulate ex-core detection signals. Subsequently, the core assembly neutron flux distribution obtained through Monte Carlo calculations served as benchmark data to evaluate inversion performance. According to evaluation criteria in Section III.C, RD between inverted neutron flux and reference neutron flux was calculated, and deviation analysis was conducted based on RD. Figures 10 [FIGURE:10] and 11 [FIGURE:11] show RD visualization results of core neutron flux reconstruction using GS-EINF and MCRNet, respectively, where black boxes represent core variation regions. Comparing RD distributions between GS-EINF and MCRNet reveals that GS-EINF significantly reduces relative deviation in neutron field inversion, with uniform and relatively low RD value distribution. This is primarily because, compared to existing data-driven models, the graph structure introduced by GS-EINF can effectively learn relationships between reactor core segments, thereby better capturing impacts from core variations and producing more accurate and stable core inversion results.

Conclusion

This paper proposes GS-EINF, an explicit inversion framework for reactor core neutron fields based on graph structures. By structurally modeling both attribute information of discretized core segments and their interrelationships, GS-EINF captures spatial interactions and correlations across different core regions. This structural representation enhances inversion accuracy of high-resolution neutron field distributions. Comparative experiments demonstrate that under different core variation scenarios and ex-core detection signal noises, GS-EINF consistently outperforms existing algorithms in accuracy and stability. Notably, in environments where out-of-core detection signals contain noise, performance of traditional methods degrades significantly as the number of variation regions increases. In contrast, GS-EINF effectively controls deviation propagation and exhibits strong noise robustness and scalability. Overall, this method provides a reliable foundation for accurate characterization of complex reactor core states. Experimental validation under diverse operating scenarios also offers a reference framework for developing more advanced inversion models tailored to complex core conditions.

Although the GS-EINF method proposed in this paper improves off-situ inversion of reactor core neutron fields to a certain extent, it still fails to achieve ideal performance in cases of extreme core variations and large ex-core detection signal noise. Future work will continue to improve the structured representation method of the reactor core to further address these issues.

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