A method for predicting neutron flux in a nuclear reactor based on global conservation constraints

Abstract

The application discloses a kind of nuclear reactor neutron flux prediction methods based on global conservation constraint, belong to nuclear reactor physics and machine learning cross field, including: constructing deep operator network model based on global neutron conservation constraint;Collect the solution data of neutron transport equation or neutron diffusion equation under different parameters, construct training data set, and train deep operator network model, obtain optimal model weight;Optimal model weight is used to initialize deep operator network model after being applied to the solution of neutron transport equation or neutron diffusion equation under different parameters, to predict the neutron flux distribution in the corresponding area of nuclear reactor.The application can significantly improve the accuracy and physical consistency of DeepONet network in predicting neutron transport or neutron diffusion solution by introducing a physical constraint term describing global neutron conservation, and can also enhance the model generalization ability and robustness, especially suitable for fast and high-precision calculation requirements in the field of reactor design, core analysis and nuclear safety evaluation.

Description

Technical Field

[0001] This invention belongs to the interdisciplinary field of nuclear reactor physics and machine learning, specifically involving a method for predicting nuclear reactor neutron flux based on global conservation constraints. Background Technology

[0002] In the field of nuclear energy, the neutron transport equation and the neutron diffusion equation are two key mathematical models describing neutron behavior in nuclear reactors, playing a crucial role in various stages such as reactor design, core analysis, and nuclear safety assessment. The neutron transport equation accurately describes the spatial, energy, angular, and temporal distribution of neutrons, comprehensively covering the physical processes of neutron generation, scattering, absorption, and fission, and realistically reflects the migration patterns of neutrons in complex media. It forms the theoretical foundation for determining reactor power distribution, fuel utilization, and safety margin. However, due to its high dimensionality, strong nonlinearity, and multi-scale coupling characteristics, numerical solutions to this equation face significant challenges. In contrast, the neutron diffusion equation is a simplified approximation of neutron transport behavior under certain physical assumptions, assuming isotropic neutron scattering and weak absorption. It boasts higher computational efficiency and is widely used in scenarios such as neutron flux distribution prediction, core criticality analysis, and reactivity assessment. Although its accuracy is not as high as the neutron transport equation, it has significant advantages in providing macroscopic trends and preliminary estimates of flux distribution. Therefore, in-depth research on the neutron transport equation and neutron diffusion equation, along with their numerical solution methods, is of significant theoretical and engineering value for improving reactor operating efficiency, ensuring core safety, and extending fuel lifespan. This direction not only helps to balance the computational requirements of high precision and high efficiency but also enhances the physical consistency and practicality of the models, providing more reliable technical support for reactor system design optimization and safety assessment.

[0003] Neutron flux is a crucial physical quantity describing the behavior of neutrons in a nuclear reactor. It reflects the number of neutrons passing through a unit area per unit space and unit time, and is of great significance for reactor power distribution, fuel utilization efficiency, and safety margin assessment. The distribution of neutron flux is influenced by the properties of nuclear materials, geometry, control rod positions, reflector layers, and the interaction between neutrons and the dielectric, exhibiting high spatial non-uniformity and temporal dynamics. Accurately predicting neutron flux distribution is a key task in reactor design and operation optimization, contributing to improved nuclear fuel utilization, reduced power deviation risks, and enhanced operational safety.

[0004] In neutron flux prediction, the neutron transport equation is the fundamental physical equation describing neutron behavior. Starting from first principles, it characterizes the entire process of neutron migration, scattering, absorption, and fission in the nuclear medium. The neutron transport equation is usually expressed as a partial differential-integral equation in the following form:

[0005]

[0006] Where ψ(r,Ω,E,t) represents the neutron angular flux distribution at position r, direction Ω, energy E, and time t; v(E) is the energy-dependent neutron velocity; ∑ t Let be the total cross section, representing the probability of a neutron being removed per unit distance; ∑ s The scattering cross section is denoted by S, which represents the source term, including fission sources or external source terms. This equation fully describes the physical processes of neutron spatial migration, energy change, and angle transfer, and possesses high accuracy and versatility. However, due to its high-dimensional coupling properties, the computational cost in numerical solutions is extremely high.

[0007] To improve computational efficiency, the neutron diffusion equation is often used as an approximate model of neutron transport behavior in engineering practice. Under assumptions such as isotropic scattering and weakly absorbing media, the neutron diffusion equation simplifies the integral along the angular dimension, significantly reducing computational complexity. Its steady-state form can be written as:

[0008]

[0009] Where φ(r) represents the scalar neutron flux, D(r) is the diffusion coefficient, and ∑ α S(r) represents the absorption cross section, and S(r) represents the source term. This equation expresses the diffusion and loss behavior of neutron flux in space. It has high solution efficiency and is widely used in tasks such as criticality calculation, power distribution estimation, and core layout optimization.

[0010] In the field of neutron flux prediction, traditional numerical methods such as the Monte Carlo method (MC), the finite difference method (FDM), and the finite element method (FEM) are the mainstream solutions. The Monte Carlo method simulates neutron transport processes through random sampling, accurately handling complex geometries and anisotropic scattering. However, its computational efficiency is highly dependent on the sample size, facing the "curse of dimensionality" in real-time prediction or multi-parameter optimization scenarios, requiring hours to days to complete a single high-fidelity simulation. The finite difference method and the finite element method solve the neutron diffusion equations through spatial discretization. Although computationally faster than the Monte Carlo method, they are limited by grid resolution and boundary condition handling capabilities, easily generating numerical oscillations in highly inhomogeneous media such as the interface between fuel assemblies and reflectors. This leads to a significant increase in flux distribution prediction errors, and it is particularly difficult to capture the subtle changes in local hotspot regions.

[0011] In recent years, advancements in deep learning technology have provided new approaches to solving Partial Differential Equations (PDEs). Among these, Deep Operator Networks (DeepONets) are a class of deep models used to learn function-to-function mappings, directly establishing the mapping relationship between the input and solution of a PDE. By inputting initial conditions and boundary conditions into the network as functions, DeepONets can quickly output the solution function for the entire region, thereby achieving efficient prediction of neutron flux in nuclear reactors and demonstrating advantages in high computational speed and inference efficiency. In nuclear reactor neutron flux prediction, this network can input spatially relevant nuclear cross-section parameters, source term distributions, and boundary conditions as functions, quickly outputting the neutron flux field across the entire reactor core. This avoids the dependence of traditional numerical methods on grid discretization and iterative solutions, significantly improving prediction efficiency and accuracy.

[0012] The subsequently proposed Physics-informed DeepONet adds a partial differential equation (PDE) loss term to the loss function of deep operator networks, significantly reducing or even completely eliminating the dependence on labeled training data, thus further expanding the application scenarios of deep operator networks. However, the complexity of the neutron transport and diffusion equations makes directly using PDE residuals as physical constraints a significant challenge. The neutron transport equations involve multidimensional coupling of space, angle, energy, and even time. Directly constructing their PDE residuals is not only computationally expensive but also makes gradient propagation extremely difficult, leading to training instability. Furthermore, calculating PDE residuals requires solving for higher-order derivatives of the neural network predictions, which can easily introduce numerical errors in complex models, reducing the convergence stability of the constraints. Introducing high-dimensional PDE residuals in each training step significantly increases the computational burden, thereby greatly extending the training time, especially when multiple samples and high-fidelity simulation data are involved, making the cost prohibitive.

[0013] While DeepONet demonstrates excellent efficiency and predictive capabilities in solving partial differential equations, existing models still suffer from physical inconsistencies and global conservation biases in predicting neutron flux in nuclear reactors. Their main drawback lies in failing to consider the physical constraints of neutron transport and diffusion processes, which can lead to predicted solutions that may not satisfy the global balance between neutron production, absorption, and loss, thus affecting prediction accuracy and physical consistency. Summary of the Invention

[0014] To address the aforementioned shortcomings in existing technologies, the nuclear reactor neutron flux prediction method based on global conservation constraints provided by this invention solves the problems of insufficient physical consistency, global conservation bias, and limitations in the training process due to the introduction of traditional physical losses in existing deep operator networks for nuclear reactor neutron flux prediction. This method thus meets the high-precision computing requirements in the fields of nuclear reactor design, core analysis, and nuclear safety assessment.

[0015] To achieve the aforementioned objectives, the present invention employs the following technical solution: a method for predicting neutron flux in nuclear reactors based on global conservation constraints, comprising the following steps:

[0016] Construct a deep operator network model based on global neutron conservation constraints;

[0017] Collect solution data of neutron transport equations or neutron diffusion equations under different parameters, and construct a training dataset;

[0018] Using the training dataset, the deep operator network model is trained using a data-driven supervised learning approach to obtain the optimal model weights;

[0019] The deep operator network model is initialized using the optimal model weights. The initialized deep operator network model is then applied to solve the neutron transport equation or neutron diffusion equation under different parameters, thereby predicting the neutron flux distribution in the corresponding region of the nuclear reactor.

[0020] The beneficial effects of this invention are as follows:

[0021] (1) The present invention constructs a deep operator network model for predicting neutron flux in nuclear reactors. By introducing a global neutron conservation constraint physical loss term, the physical consistency of neutron flux prediction in nuclear reactors is ensured, which solves the problem of insufficient physical constraints in the prior art, thereby improving the accuracy and rationality of the solution.

[0022] (2) Compared with the traditional deep operator network method, the method of the present invention significantly improves the training efficiency and can more accurately predict the neutron flux distribution in complex nuclear reactor environments, enhancing the prediction accuracy, robustness and adaptability of the model. It is particularly suitable for high-precision calculation needs such as nuclear reactor design, core analysis and nuclear safety assessment.

[0023] Furthermore, the deep operator network model based on the global neutron conservation constraint is a deep operator network model constructed by introducing a physical constraint term that reflects the global neutron conservation law into the loss function of the deep operator network model.

[0024] The physical constraint of the global neutron conservation law transforms the neutron transport or diffusion process into a data-driven physical consistency optimization process during the training of the deep operator network model.

[0025] Furthermore, the physical constraint of the global neutron conservation law is: for any closed volume, the total neutron production rate within it is equal to the sum of the neutron absorption rate and the neutrons flowing out from the boundary, expressed as:

[0026]

[0027] In the formula, J NN (r) represents the neutron flux density predicted at position r by the deep operator network model, φ NN (r) represents the scalar neutron flux predicted by the neural network at position r, ∑ a (r) represents the neutron absorption cross section at position r, S eff (r) represents the overall neutron production rate at position r. dS represents the boundary of the volume V under consideration, and dS represents the area element on the boundary.

[0028] The beneficial effects of the above-mentioned further solutions are as follows:

[0029] Based on the aforementioned physical law reflecting the global conservation of neutrons, this method introduces new physical constraints to ensure that the model accurately reflects neutron behavior while maintaining overall balance in neutron production, absorption, and transport. This improves the accuracy and physical consistency of the solutions to the neutron transport equation or neutron diffusion equation. While strictly adhering to physical laws, this method avoids excessive reliance on large amounts of high-precision training data, simplifying the computation process. In nuclear reactor design and analysis, this method enables faster and more accurate predictions, exhibiting high computational efficiency and robustness, especially in multi-parameter and complex environments.

[0030] Furthermore, the deep operator network model includes a backbone network, branch networks, and an output layer;

[0031] The backbone network is used to extract the corresponding feature representation based on the location coordinates of the predicted solution;

[0032] The branch network is used to extract a global feature representation of the input source term function based on the function values ​​of the input source term function corresponding to the neutron transport equation or diffusion equation in the partial differential equation at multiple sampling points;

[0033] The output layer is used to perform inner product calculation on the feature vectors output by the backbone network and the branch network to obtain the predicted value at the predicted solution.

[0034] Furthermore, when the deep operator network model solves the neutron transport equation, the inputs to the backbone network include the position coordinates of the predicted solution, the direction of neutron motion, and energy parameters; the inputs to the branch networks include the neutron source term, the scattering cross section, and the discrete values ​​of the absorption cross section function at multiple sampling points; and the output layer outputs the predicted neutron angular flux at the predicted solution.

[0035] When the deep operator network model solves the neutron diffusion equation, the input of the backbone network is the position coordinates of the predicted solution; the input of the branch network includes the values ​​of the neutron source term, diffusion coefficient, and macroscopic absorption cross-section function at multiple sampling points; and the output layer outputs the predicted neutron scalar flux value at the predicted solution.

[0036] The beneficial effects of the above-mentioned further solutions are as follows:

[0037] The aforementioned deep operator network model design effectively combines the local features of the location to be predicted with the global characteristics of the input source term function, significantly improving the model's ability to learn neutron flux distribution under complex spatial variations and multi-source term interference. Through the interaction of features between the backbone network and branch networks, the model can more accurately capture neutron behavior under different material properties, source term distributions, and geometric changes, improving prediction accuracy and generalization ability, and is particularly suitable for scenarios where there is strong non-uniformity in the core region of nuclear reactors.

[0038] Furthermore, the loss function for training the deep operator network model includes a supervised loss term and a physical constraint term for the global neutron conservation bias;

[0039] The loss function Represented as:

[0040]

[0041] in, Indicates the monitoring of loss items, λ represents the physical constraint term, and λ represents the balance coefficient, which is used to adjust the importance of the physical conservation constraint term relative to the supervised loss term.

[0042] Furthermore, the supervision loss term is the mean square error between the output of the solver of the neutron transport equation or the neutron diffusion equation and the true solution, expressed as:

[0043]

[0044] In the formula, Let f represent the supervised loss term, x represent the location coordinates of the predicted solution, N represent the number of training samples, and f i Let represent the value of the input source term function for the i-th sample at the source term point. This represents the operator learned by the deep operator network model for the input source term function f. i The output, u i (x) represents the value of the true solution function corresponding to the i-th input at x.

[0045] Furthermore, the physical constraint term of the global neutron conservation bias is constructed based on the global conservation relation in the global neutron conservation physical law. It is used to measure whether the predicted solution satisfies the global balance requirement of neutron production, absorption, and loss throughout the entire computational domain Ω, and is expressed as follows:

[0046]

[0047] In the formula, Let Ω represent the physical constraint term representing the global neutron conservation bias, Ω represent the computational domain, and S(r) represent the external neutron source term at position r. Σ represents the neutron source term produced by fission or other physical processes. a (r) represents the neutron absorption cross section at position r. f represents the angularly distributed neutron flux density predicted based on the deep operator network. i Let r represent the value of the input source term function for the i-th sample at position r. This represents the operator learned by the deep operator network model for the input source term function f. i The output at position r.

[0048] The beneficial effects of the above-mentioned further solutions are as follows:

[0049] In the further scheme described above, by incorporating both traditional supervision errors and conservation constraints reflecting physical laws into the loss function, the model's inherent adherence to physical consistency can be significantly improved while ensuring data fitting accuracy. Unlike high-dimensional PDE residuals that directly introduce neutron transport or diffusion equations, this invention employs integral constraints based on global conservation laws, avoiding the instability and high computational cost associated with high-order derivative calculations, significantly reducing training difficulty and resource consumption. This hybrid loss design not only improves convergence accuracy, speed, and stability during training but also effectively prevents overfitting. Simultaneously, the global conservation constraints, as physical priors, maintain a conservation balance between neutron production and loss under different material parameters, geometric configurations, and source terms, ensuring that the model can still provide accurate and physically consistent neutron flux predictions when facing diverse reactor operating conditions, enhancing the model's engineering application value and generalizability. This conservation term is not a general physical constraint form but is specifically constructed based on the physical conservation relationships in the neutron flux distribution prediction task, fully integrating the energy migration and absorption characteristics of nuclear reactor systems, reflecting scenario adaptability and problem relevance. Attached Figure Description

[0050] Figure 1 The flowchart of the nuclear reactor neutron flux prediction method based on global conservation constraints provided by the present invention is shown.

[0051] Figure 2A schematic diagram of the deep operator network model structure based on global neutron conservation constraints provided by this invention. Detailed Implementation

[0052] The specific embodiments of the present invention are described below to enable those skilled in the art to understand the present invention. However, it should be understood that the present invention is not limited to the scope of the specific embodiments. For those skilled in the art, various changes are obvious as long as they are within the spirit and scope of the present invention as defined and determined by the appended claims. All inventions utilizing the concept of the present invention are protected.

[0053] This invention provides a method for predicting neutron flux in nuclear reactors based on global conservation constraints. This method predicts neutron flux in nuclear reactors by introducing physical constraints based on the global neutron conservation law into a deep operator network model. The specific implementation method is as follows: Figure 1 As shown, it includes the following steps:

[0054] Construct a deep operator network model based on global neutron conservation constraints;

[0055] Collect solution data of neutron transport equations or neutron diffusion equations under different parameters, and construct a training dataset;

[0056] Using the training dataset, the deep operator network model is trained using a data-driven supervised learning approach to obtain the optimal model weights;

[0057] The deep operator network model is initialized using the optimal model weights. The initialized deep operator network model is then applied to solve the neutron transport equation or neutron diffusion equation under different parameters, thereby predicting the neutron flux distribution in the corresponding region of the nuclear reactor.

[0058] In this embodiment of the invention, the deep operator network model based on the global neutron conservation constraint is a deep operator network model constructed by introducing a physical constraint term that reflects the global neutron conservation law into the loss function of the deep operator network model.

[0059] The physical constraint of the global neutron conservation law transforms the neutron transport or diffusion process into a data-driven physical consistency optimization process during the training of the deep operator network model.

[0060] In this embodiment, the global neutron conservation law is a fundamental principle in nuclear reactor physics, describing the balance between neutron production, absorption, and efflux within the reaction region. Specifically, the physical constraint of the global neutron conservation law is: for any closed volume, the total neutron production rate within it is equal to the sum of the neutron absorption rate and the neutrons effluxing from the boundary, expressed as:

[0061]

[0062] In the formula, J NN (r) represents the neutron flux density predicted at position r by the deep operator network model (where "NN" refers to a neural network), i.e., the number of neutrons passing through a unit area per unit time, φ NN (r) represents the scalar neutron flux predicted by the neural network at position r, i.e., the neutron number density obtained by integrating over all directions; ∑ a (r) represents the neutron absorption cross section at position r, describing the medium's ability to absorb neutrons; S eff (r) represents the overall neutron production rate at position r, including new neutrons produced by fission and other source terms; dS represents the boundary of the volume V under consideration, and dS represents the area element on the boundary, which points outward.

[0063] In this embodiment, the above formula indicates that for the reaction region V, the neutron outflow, absorption and generation within it should be in balance, thereby satisfying the global neutron conservation law.

[0064] Specifically, in this embodiment, based on the aforementioned physical constraints, when predicting neutron flux, it is first necessary to perform accurate geometric modeling and discretization of the reaction region V. Based on a deep operator network model with global conservation constraints, the deep operator network uses relevant parameters of position r (such as material properties, coordinates, etc.) to predict the neutron flux density J. NN (r) and scalar neutron flux φ NN (r) Make predictions. During the prediction process, the multi-layered neurons within the network complete feature extraction and result output through a series of complex linear and nonlinear transformations. To ensure the accuracy of the prediction results, a data-driven supervised learning approach is used to train the model. The training loss function is composed of the supervised loss function and the physical constraint term of the global neutron conservation bias. Neutron absorption cross section ∑ a The determination of (r) depends on the material properties at different locations within the reaction region; neutron production rate S eff (r) requires comprehensive consideration of fission and other source terms. Subsequently, numerical integration methods are used to calculate the integral terms in the global neutron conservation formula. By continuously optimizing the model parameters, the equations are made to fit closely, ensuring that the prediction results strictly follow the global neutron conservation law, thereby achieving high-precision prediction of neutron flux in nuclear reactors.

[0065] In embodiments of the present invention, such as Figure 2 As shown, the deep operator network model includes a backbone network, branch networks, and an output layer;

[0066] The trunk network extracts the feature representation of the solution to be predicted based on its position coordinates (x,t). The branch network extracts the global feature representation of the input source term function based on the function values ​​of the input source term function corresponding to the neutron transport equation or diffusion equation in the partial differential equation at multiple sampling points. The output layer performs an inner product calculation on the feature vectors output by the trunk network and the branch network to obtain the predicted value u(x,t) at the predicted solution (x,t), thus realizing the mapping from the input source term function space to the target solution space.

[0067] Specifically, in Figure 2 In the diagram, u represents the source term function of the branch network input, and t represents the spatial and temporal coordinates of the backbone network input; b1 to b p For the branch network output, t1 to t p The output of the backbone network is used to obtain the predicted output u(x,t) through the inner product operation.

[0068] In this embodiment, the input-output mechanisms of the neutron transport equation and the neutron diffusion equation differ in the deep operator network model based on the global neutron conservation constraint.

[0069] Specifically, when the deep operator network model solves the neutron transport equation, given the need to accurately characterize the complex transport process of neutrons in terms of spatial location, direction of motion, and energy dimension, the backbone network's input includes not only the position coordinates of the predicted solution but also the neutron motion direction (such as polar cosine and azimuth angle) and energy parameters to extract multi-dimensional feature representations. The branch networks, on the other hand, input discrete values ​​of functions such as neutron source terms, scattering cross sections, and absorption cross sections at multiple sampling points to extract global features of the input source term functions. The output layer outputs the predicted neutron angular flux value at the solution to be predicted through inner product operations. This value fully describes the spatial, directional, and energy distribution characteristics of neutrons.

[0070] When the deep operator network model solves the neutron diffusion equation, the neutron diffusion equation is used as an approximate simplified form of the transport equation. The input to the backbone network is only the position coordinates of the predicted solution. If the medium energy dependence is significant, the average energy parameter can be additionally input. The branch networks input the values ​​of functions such as neutron source terms, diffusion coefficients, and macroscopic absorption cross sections at the sampling points to capture the spatial distribution characteristics of the input source term functions. The output layer outputs the predicted value of the neutron scalar flux, which is obtained by integrating over the direction and represents the overall density of neutrons at the corresponding position, ignoring the direction dependence.

[0071] In this embodiment of the invention, the constructed deep operator network model is trained using a data-driven supervised learning method. The loss function for training the deep operator network model includes a supervised loss term and a physical constraint term for global neutron conservation bias.

[0072] Specifically, the loss function is expressed as:

[0073]

[0074] in, Indicates the monitoring of loss items, The physical constraint term represents the global neutron conservation bias, and λ represents the balance coefficient, which is used to adjust the importance of the physical conservation constraint term relative to the supervision loss term. By reasonably selecting the value of λ, the model's adherence to global physical laws can be strengthened while ensuring prediction accuracy.

[0075] In actual training, the supervised loss term primarily drives the network to learn the local relationship between the input source terms and the corresponding solutions, while the physical constraint term guides the network to maintain a conservation balance of neutron production, absorption, and loss at the global scale. This synergistic optimization of both factors results in a deep operator network model that not only possesses high prediction accuracy but also significantly outperforms traditional data-driven methods in terms of physical consistency and interpretability.

[0076] In this embodiment, the supervision loss term is the mean square error between the output of the solver of the neutron transport equation or the neutron diffusion equation and the true solution, which is expressed as:

[0077]

[0078] In the formula, Let f represent the supervised loss term, x represent the location coordinates of the predicted solution, N represent the number of training samples, and f i Let represent the value of the input source term function for the i-th sample at the source term point. u represents the output of the operator learned by the deep operator network model on the input source term function fi, i.e., the predicted value of the solution function at point x. i (x) represents the value of the true solution function corresponding to the i-th input at x.

[0079] In this embodiment, considering that neutron flux prediction is a core task in reactor physics analysis and is essentially constrained by global conservation laws, the physical constraint terms constructed in this embodiment are specifically designed based on the characteristics of this scenario.

[0080] Specifically, the physical constraint term of the global neutron conservation bias is constructed based on the global conservation relation in the global neutron conservation physical law. It is used to measure whether the predicted solution satisfies the global balance requirement of neutron production, absorption, and loss throughout the entire computational domain Ω, and is expressed as follows:

[0081]

[0082] In the formula, Let Ω represent the physical constraint term representing the global neutron conservation bias, Ω represent the computational domain, and S(r) represent the external neutron source term at position r. Σ represents the neutron source term produced by fission or other physical processes, whose value depends on the prediction output of the deep operator network for the input source term function fi. a (r) represents the neutron absorption cross section at position r. f represents the scalar neutron flux obtained by integrating over all angles the angularly distributed neutron flux density predicted by the deep operator network. i Let r represent the value of the input source term function for the i-th sample at position r. This represents the operator learned by the deep operator network model for the input source term function f. i The output at position r.

[0083] In one example of this embodiment, when collecting training data, the data source is the MIT OpenMC dataset. This dataset includes numerical solution data obtained from reactor neutron transport or diffusion simulations using OpenMC. This dataset covers basic information such as material properties, geometric configurations, operating settings, and angular flux detection, reflecting neutron transport or diffusion behavior under different parameters. Using the MIT OpenMC dataset ensures that the training data covers a variety of real-world operating conditions, including different fuel assembly layouts, control rod positions, reflector structures, and operating states, thereby improving the model's adaptability and robustness under diverse reactor designs and operating conditions. Simultaneously, training with high-fidelity simulation data helps improve the model's perception and generalization ability to complex physical processes, reduces reliance on expensive experimental data, and shortens the development and deployment cycle.

[0084] In this embodiment, a deep operator network model based on global conservation constraints is used to solve the neutron transport or diffusion equation. This equation serves as a basic physical model describing the evolution of neutron behavior in a nuclear reactor. It comprehensively reflects various physical processes such as neutron diffusion, absorption, scattering, and fission reactions in space, and can effectively capture the evolution of neutron flux density over time and space.

[0085] In summary, compared with traditional deep operator network methods, this invention fully integrates prior physical knowledge and data-driven strategies. By introducing a physical constraint term representing a global neutron conservation bias during training, it achieves efficient and physically consistent solutions to neutron transport or diffusion equations in nuclear reactors. This conservation constraint originates from fundamental equilibrium laws in neutron physics, representing the global conservation relationship of neutron generation, absorption, and loss within the reaction region in integral form. This avoids the high-dimensional derivative calculations and numerical stability issues required by directly introducing PDE residuals, thus reducing training costs and implementation complexity. Compared to traditional deep networks that rely solely on supervised loss, this method employs a weighted combination of supervised error and global conservation constraints in the loss function. This ensures both the accuracy of local fitting and enhances global physical consistency through the physical conservation term, significantly improving the model's training convergence speed and stability while suppressing overfitting risks. The model structure effectively combines local spatial features with global input source term function features, possessing stronger adaptability and accurately characterizing neutron flux distribution under varying material properties, complex geometric configurations, and multi-source terms. Leveraging the diverse and high-fidelity simulation results from the MIT OpenMC dataset, the model requires no complex preprocessing during training. It can directly use discrete source terms and material property data as input to quickly construct neutron flux prediction mappings, significantly improving training efficiency and inference speed. The method of this invention not only achieves significant improvements in neutron flux prediction accuracy and physical consistency but also demonstrates excellent generalization performance and engineering application potential, providing an efficient, reliable, and scalable numerical simulation solution for nuclear reactor design, core safety analysis, and operational optimization.

[0086] Specific embodiments have been used to illustrate the principles and implementation methods of this invention. The descriptions of the embodiments above are only for the purpose of helping to understand the method and core ideas of this invention. At the same time, for those skilled in the art, there will be changes in the specific implementation methods and application scope based on the ideas of this invention. Therefore, the content of this specification should not be construed as a limitation of this invention.

[0087] Those skilled in the art will recognize that the embodiments described herein are intended to help the reader understand the principles of the invention, and should be understood that the scope of protection of the invention is not limited to such specific statements and embodiments. Those skilled in the art can make various other specific modifications and combinations based on the technical teachings disclosed in this invention without departing from the spirit of the invention, and these modifications and combinations are still within the scope of protection of this invention.

Claims

1. A method for predicting neutron flux in nuclear reactors based on global conservation constraints, characterized in that, Includes the following steps: Construct a deep operator network model based on global neutron conservation constraints; Collect solution data of neutron transport equations or neutron diffusion equations under different parameters, and construct a training dataset; Using the training dataset, the deep operator network model is trained using a data-driven supervised learning approach to obtain the optimal model weights; The deep operator network model is initialized using the optimal model weights. The initialized deep operator network model is then applied to solve the neutron transport equation or neutron diffusion equation under different parameters, thereby predicting the neutron flux distribution in the corresponding region of the nuclear reactor. The deep operator network model based on the global neutron conservation constraint is a deep operator network model constructed by introducing a physical constraint term that reflects the global neutron conservation law into the loss function of the deep operator network model. The physical constraint of the global neutron conservation law transforms the neutron transport or diffusion process into a data-driven physical consistency optimization process during the training of the deep operator network model. The physical constraint of the global neutron conservation law is as follows: For any closed volume, the total neutron production rate within it is equal to the sum of the neutron absorption rate and the neutrons flowing out from the boundary, expressed as: In the formula, This indicates that the deep operator network model is at the location The predicted neutron flux density is obtained at that location. For deep operator network models at location The predicted scalar neutron flux, Indicates the location Neutron absorption cross section at that location This represents the overall neutron production rate at position r. Indicates the volume under consideration The boundary, This represents the area element on the boundary.

2. The method for predicting neutron flux in nuclear reactors based on global conservation constraints according to claim 1, characterized in that, The deep operator network model includes a backbone network, branch networks, and an output layer; The backbone network is used to extract the corresponding feature representation based on the location coordinates of the predicted solution; The branch network is used to extract a global feature representation of the input source term function based on the function values ​​of the input source term function corresponding to the neutron transport equation or diffusion equation in the partial differential equation at multiple sampling points; The output layer is used to perform inner product calculation on the feature vectors output by the backbone network and the branch network to obtain the predicted value at the predicted solution.

3. The method for predicting neutron flux in nuclear reactors based on global conservation constraints according to claim 2, characterized in that, When the deep operator network model solves the neutron transport equation, the inputs to the backbone network include the position coordinates of the predicted solution, the direction of neutron motion, and energy parameters; the inputs to the branch networks include the neutron source term, the scattering cross section, and the discrete values ​​of the absorption cross section function at multiple sampling points; and the output layer outputs the predicted neutron angular flux at the predicted solution. When the deep operator network model solves the neutron diffusion equation, the input of the backbone network is the position coordinates of the predicted solution; the input of the branch network includes the values ​​of the neutron source term, diffusion coefficient, and macroscopic absorption cross-section function at multiple sampling points; and the output layer outputs the predicted neutron scalar flux value at the predicted solution.

4. The method for predicting neutron flux in nuclear reactors based on global conservation constraints according to claim 1, characterized in that, The loss function for training the deep operator network model includes a supervision loss term and a physical constraint term for the global neutron conservation bias; The loss function Represented as: in, Indicates monitoring loss items, Represents physical constraint terms. This represents the balance coefficient, used to adjust the importance of the physical conservation constraint term relative to the monitoring loss term.

5. The method for predicting neutron flux in nuclear reactors based on global conservation constraints according to claim 4, characterized in that, The monitoring loss term is the mean square error between the output of the solver of the neutron transport equation or the neutron diffusion equation and the true solution, and it is expressed as: In the formula, This represents the monitoring loss term, and x represents the location coordinates of the predicted solution. Indicates the number of training samples. Indicates the first The value of the input source term function for each sample at the source term point. This represents the operator pair of input source term functions learned by the deep operator network model. At position x, the output Indicates the first The value of the true solution function at x corresponding to each input.

6. The method for predicting neutron flux in nuclear reactors based on global conservation constraints according to claim 4, characterized in that, The physical constraint term of the global neutron conservation bias is constructed based on the global conservation relations in the physical law of global neutron conservation, and is used to measure the predicted solution across the entire computational domain. Whether the global equilibrium requirement for neutron production, absorption, and loss is satisfied within the space is expressed as follows: In the formula, The physical constraint term representing the global neutron conservation deviation. Indicates the calculation area. This represents the external neutron source term at position r. This represents the neutron source term produced by fission or other physical processes. This represents the neutron absorption cross section at position r. This represents the angular distribution neutron flux density predicted based on the deep operator network. Indicates the first The value of the input source term function at position r for each sample. This represents the operator pair of input source term functions learned by the deep operator network model. The output at position r.

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