Nuclear reactor neutron flux prediction method based on global conservation constraint

Abstract

The invention discloses a nuclear reactor neutron flux prediction method based on global conservation constraint, and belongs to the nuclear reactor physics and machine learning crossing field, and the method comprises the following steps: constructing a depth operator network model based on global neutron conservation constraint; collecting solution data of a neutron transport equation or a neutron diffusion equation under different parameters, constructing a training data set, and training the depth operator network model to obtain an optimal model weight; and initializing the depth operator network model by using the optimal model weight, applying the depth operator network model to a neutron transport equation or a neutron diffusion equation under different parameters for solving, and predicting neutron flux distribution in a corresponding region of the nuclear reactor. According to the method, the physical constraint term for describing the global conservation of neutrons is introduced, so that the accuracy and the physical consistency of the DeepONet network for predicting the neutron transport or neutron diffusion solution can be remarkably improved, the generalization ability and the robustness of the model can be enhanced, and the method is particularly suitable for meeting the rapid and high-precision calculation requirements in the fields of reactor design, reactor core analysis and nuclear safety evaluation.

Description

Technical Field

[0001] The present invention belongs to the interdisciplinary field of nuclear reactor physics and machine learning, and specifically relates to a nuclear reactor neutron flux prediction method based on global conservation constraints. Background Art

[0002] In the nuclear energy field, the neutron transport equation and the neutron diffusion equation are two key mathematical models describing neutron behavior in nuclear reactors. They play an important role in multiple aspects, including reactor design, core analysis, and nuclear safety assessment. The neutron transport equation accurately describes the distribution of neutrons in space, energy, angle, and time, comprehensively covering the physical processes of neutron production, scattering, absorption, and fission. It truly reflects the migration of neutrons in complex media and serves as the theoretical foundation for determining reactor power distribution, fuel utilization, and safety margins. However, due to its high dimensionality, strong nonlinearity, and multiscale coupling, the equation faces significant challenges in numerical solution. In comparison, the neutron diffusion equation is a simplified approximation of neutron transport behavior under certain physical assumptions. It assumes isotropic neutron scattering and weak absorption, and is computationally efficient. It is widely used in scenarios such as neutron flux distribution prediction, core criticality analysis, and reactivity assessment. Although its accuracy is not as good as the neutron transport equation, it has significant advantages in providing macroscopic trends and preliminary estimates of flux distribution. Therefore, in-depth research into the neutron transport and diffusion equations and their numerical solutions has important theoretical and engineering implications for improving reactor efficiency, ensuring core safety, and extending fuel life. This research not only helps balance high-precision and high-efficiency computational requirements, but also enhances the physical consistency and practicality of models, providing more reliable technical support for reactor system design optimization and safety assessment.

[0003] Neutron flux is a key physical quantity describing neutron behavior in nuclear reactors. It reflects the number of neutrons passing through a unit area per unit space and per unit time, and is crucial for assessing reactor power distribution, fuel utilization efficiency, and safety margins. Neutron flux distribution is influenced by a combination of nuclear material properties, geometry, control rod position, reflector layers, and neutron-medium interactions, resulting in high spatial heterogeneity and temporal dynamics. Accurately predicting neutron flux distribution is a key task in reactor design and operational optimization, helping to improve nuclear fuel utilization, reduce the risk of power deviation, and enhance operational safety.

[0004] In neutron flux prediction, the neutron transport equation is the fundamental physical equation that describes neutron behavior. Starting from first principles, it describes the entire process of neutron migration, scattering, absorption, and fission in nuclear media. The neutron transport equation is usually expressed as a partial differential integral equation of 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 is the total cross section, which indicates the probability of a neutron being removed per unit distance; ∑ s is the scattering cross section; S is the source term, including fission sources or external sources. This equation fully describes the physical processes of neutron spatial migration, energy changes, and angular shifts, and is highly accurate and universal. However, due to its high-dimensional coupling nature, its numerical solution is extremely computationally expensive.

[0007] To improve computational efficiency, the neutron diffusion equation is often used in engineering practice as an approximate model for neutron transport behavior. Under the assumptions of isotropic scattering and weakly absorbing media, the neutron diffusion equation simplifies 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, ∑ α (r) is the absorption cross section, and S(r) is the source term. This equation describes the spatial diffusion and loss behavior of the neutron flux. 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 mainstream solutions. The Monte Carlo method simulates the neutron transport process through random sampling and can accurately handle complex geometric structures and anisotropic scattering. However, its computational efficiency is highly dependent on the sample size. In real-time prediction or multi-parameter optimization scenarios, it faces the "curse of dimensionality" problem, and it takes hours to days to complete a high-fidelity simulation. The finite difference method and the finite element method solve the neutron diffusion equation through spatial discretization. Although the computational speed is faster than the Monte Carlo method, they are limited by the grid resolution and boundary condition processing capabilities. Numerical oscillations are easily generated in strongly inhomogeneous media such as the interface between the fuel assembly and the reflector layer, resulting in a significant increase in the flux distribution prediction error. In particular, it is difficult to capture the subtle changes in local hot spots.

[0011] In recent years, advances in deep learning technology have provided new ideas for solving PDEs. Among them, Deep Operator Network (DeepONet) is a type of deep model used to learn function-to-function mapping, which can directly establish the mapping relationship between the input and solution of partial differential equations. By inputting initial conditions, boundary conditions, etc. into the network in the form of functions, DeepONet can quickly output the solution function of the entire region, thereby achieving efficient prediction of the neutron flux of nuclear reactors, showing the advantages of high computing speed and inference efficiency. In the prediction of neutron flux in nuclear reactors, the network can input spatially related nuclear cross-section parameters, source term distribution and boundary conditions in the form of functions, and quickly output the neutron flux field within the entire core, avoiding the reliance of traditional numerical methods on grid discretization and iterative solutions, and significantly improving prediction efficiency and accuracy.

[0012] The subsequently proposed physics-informed deep operator network (Physics-informed DeepONet) incorporates a partial differential equation loss term into the deep operator network's loss function, significantly reducing or even completely eliminating the reliance on labeled training data and further expanding the application scenarios of deep operator networks. However, the complexity of the neutron transport and diffusion equations makes the direct use of PDE residuals as physical constraints fraught with challenges. The neutron transport equation involves multidimensional couplings in space, angle, energy, and even time. Directly constructing its PDE residuals is not only computationally expensive but also extremely difficult to propagate gradients, leading to training instability. Furthermore, calculating the PDE residuals requires solving high-order derivatives of the neural network's predictions, which can easily introduce numerical errors in complex models and reduce the convergence stability of the constraints. Introducing high-dimensional PDE residuals in each training step significantly increases the computational burden, significantly extending training time. This is especially cost-prohibitive when multiple samples and high-fidelity simulation data are involved.

[0013] While DeepONet demonstrates excellent efficiency and predictive power in solving partial differential equations, existing models still suffer from physical inconsistencies and global conservation biases in nuclear reactor neutron flux prediction. Their primary drawback is their failure to account for the physical constraints of neutron transport and diffusion. This can lead to predictions that fail to satisfy the global balance between neutron production, absorption, and loss, compromising both accuracy and physical consistency. Summary of the Invention

[0014] In response to the above-mentioned deficiencies in the prior art, the nuclear reactor neutron flux prediction method based on global conservation constraints provided by the present invention solves the problems of insufficient physical consistency, global conservation deviation, and limited introduction of traditional physical loss training process in existing deep operator networks in nuclear reactor neutron flux prediction, thereby meeting the high-precision computing requirements in the fields of nuclear reactor design, core analysis, and nuclear safety assessment.

[0015] In order to achieve the above-mentioned object of the invention, the technical solution adopted by the present invention is: a method for predicting the neutron flux of a nuclear reactor 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 equation or neutron diffusion equation under different parameters to construct a training data set;

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

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

[0020] The beneficial effects of the present invention are:

[0021] (1) The present invention constructs a deep operator network model for realizing nuclear reactor neutron flux prediction. By introducing a global neutron conservation constraint physical loss term, the physical consistency of nuclear reactor neutron flux prediction is ensured, solving 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 a complex nuclear reactor environment, thereby enhancing the prediction accuracy, robustness and adaptability of the model. It is particularly suitable for high-precision computing requirements 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] Among them, the physical constraints of the global neutron conservation law of the reaction enable the neutron transport process or diffusion process to be transformed into a data-driven physical consistency optimization process during the training process of the deep operator network model.

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

[0026]

[0027] Where, J NN (r) represents the neutron flux density predicted by the deep operator network model at position r, φ NN (r) is 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, 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 further scheme are:

[0029] Based on the aforementioned physical laws that reflect global neutron conservation, new physical constraints are introduced to ensure that the model accurately reflects neutron behavior while maintaining an overall balance between neutron production, absorption, and transport. This improves the accuracy and physical consistency of the solutions to the neutron transport or diffusion equations. While strictly adhering to the laws of physics, this method avoids over-reliance on large amounts of high-precision training data and simplifies the computational process. In nuclear reactor design and analysis, this approach enables faster and more accurate predictions, demonstrating high computational efficiency and robustness, particularly in applications with multiple parameters and complex environments.

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

[0031] The backbone network is used to extract the corresponding feature representation according to the position coordinates of the predicted solution;

[0032] The branch network is used to extract a global feature representation representing the input source term function according to the function value of the input source term function corresponding to the neutron transport equation or the 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 input of the backbone network includes the position coordinates, neutron motion direction and energy parameters of the predicted solution; the input of the branch network includes the discrete values ​​of the neutron source term, scattering cross section and absorption cross section function at multiple sampling points; the output layer outputs the predicted value of the 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 value of the neutron scalar flux at the predicted solution.

[0036] The beneficial effects of the above further scheme are:

[0037] The design of this deep operator network model effectively combines the local characteristics of the predicted location with the global properties of the input source term function, significantly improving the model's ability to learn neutron flux distributions under complex spatial variations and multi-source interference. Through the interaction of backbone and branch network features, the model can more accurately capture neutron behavior under varying material properties, source term distributions, and geometric structures, improving prediction accuracy and generalization. This is particularly applicable to scenarios with strong non-uniformities in the nuclear reactor core.

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

[0039] The loss function Expressed as:

[0040]

[0041] in, represents the supervision loss term, 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 supervision 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, which is expressed as:

[0043]

[0044] Where, represents the supervised loss term, x represents the location coordinate of the predicted solution, N represents the number of training samples, and f i Represents the value of the input source function of the i-th sample at the source point, Represents the operator input source function f obtained by learning the deep operator network model 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 deviation is constructed based on the global conservation relationship in the global neutron conservation physical law, which is used to measure whether the predicted solution meets the global balance requirement of neutron production, absorption and loss in the entire calculation area Ω, and is expressed as:

[0046]

[0047] Where, represents the physical constraint term of the global neutron conservation deviation, Ω represents the calculation area, S(r) represents the external neutron source term at position r, represents the neutron source term produced by fission or other physical processes, Σ a (r) represents the neutron absorption cross section at position r, represents the angular distribution neutron flux density predicted by the deep operator network, f i Represents the value of the input source function of the i-th sample at position r, Represents the operator input source function f obtained by learning the deep operator network model i Output at position r.

[0048] The beneficial effects of the above further scheme are:

[0049] In this further approach, 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 directly introducing high-dimensional PDE residuals from neutron transport or diffusion equations, the present invention employs integral constraints based on global conservation laws, avoiding the instability and high computational overhead associated with high-order derivative calculations and 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. Furthermore, the global conservation constraint, acting as a physical prior, maintains a conservation balance between neutron production and loss under varying material parameters, geometric configurations, and source terms, ensuring that the model can still provide accurate and physically consistent neutron flux predictions for diverse reactor operating conditions, enhancing the model's engineering application value and generalizability. This conservation term is not a general physical constraint, but is specifically constructed based on the physical conservation relationships used in neutron flux distribution prediction tasks. It fully incorporates the energy transfer and absorption characteristics of nuclear reactor systems, demonstrating scenario adaptability and problem relevance. BRIEF DESCRIPTION OF THE DRAWINGS

[0050] Figure 1 This is a flow chart of the nuclear reactor neutron flux prediction method based on global conservation constraints provided by the present invention.

[0051] Figure 2Schematic diagram of the structure of the deep operator network model based on global neutron conservation constraints provided by the present invention. DETAILED DESCRIPTION

[0052] The specific embodiments of the present invention are described below to facilitate understanding of the present invention by those skilled in the art. However, it should be clear that the present invention is not limited to the scope of the specific embodiments. For those skilled in the art, as long as various changes are within the spirit and scope of the present invention as defined and determined by the appended claims, these changes are obvious, and all inventions and creations utilizing the concepts of the present invention are protected.

[0053] The embodiment of the present invention provides a method for predicting the neutron flux of a nuclear reactor based on global conservation constraints. The method predicts the neutron flux in a nuclear reactor by introducing a physical constraint term based on the global neutron conservation law into a deep operator network model. The specific implementation method is as follows: Figure 1 As shown, the following steps are included:

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

[0055] Collect solution data of neutron transport equation or neutron diffusion equation under different parameters to construct a training data set;

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

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

[0058] In an embodiment of the present 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] Among them, the physical constraints of the global neutron conservation law of the reaction enable the neutron transport process or diffusion process to be transformed into a data-driven physical consistency optimization process during the training process of the deep operator network model.

[0060] In this embodiment, the global neutron conservation law is a law in nuclear reactor physics that describes the balance between neutron production, absorption, and outflow 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 the volume is equal to the sum of the neutron absorption rate and the neutron outflow from the boundary, which is expressed as:

[0061]

[0062] Where, J NN (r) represents the neutron flux density predicted by the deep operator network model (where “NN” refers to neural network) at position r, that is, the number of neutrons passing through a unit area per unit time, and φ NN (r) is the scalar neutron flux predicted by the neural network at position r, that is, the neutron number density obtained by integrating in all directions; ∑ a (r) represents the neutron absorption cross section at position r, which describes 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 sources; Represents the boundary of the volume V under consideration, dS represents the area element on the boundary, and its direction points outward.

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

[0064] Specifically, in this embodiment, based on the above physical constraints, when realizing the neutron flux prediction, it is necessary to first perform accurate geometric modeling and discretization processing on the reaction area V. Based on the deep operator network model with global conservation constraints, the deep operator network uses the relevant parameters of the position r (such as material properties, coordinates, etc.) to predict the neutron flux density J. NN (r) and the scalar neutron flux φ NN (r) is used for prediction. During the prediction process, the multi-layer neurons in 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, the model is trained using a data-driven supervised learning method. The training loss function is composed of the supervisory loss function and the physical constraint term of the global neutron conservation deviation. Neutron absorption cross section ∑ a The determination of (r) depends on the material properties at different locations within the reaction area; the 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 term in the global neutron conservation equation. By continuously optimizing model parameters, the equation is highly consistent, ensuring that the predicted results strictly adhere to the global neutron conservation law, thereby achieving high-precision predictions of the neutron flux in nuclear reactors.

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

[0066] Among them, the trunk network (Trunk Net) is used to extract the corresponding feature representation according to the position coordinates (x, t) of the solution to be predicted; the branch network (Branch Net) is used to extract the global feature representation of the input source term function according to 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 calculate the inner product of 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), thereby realizing the mapping from the input source term function space to the target solution space.

[0067] Specifically, in Figure 2 In the example, u represents the source function of the branch network input, t represents the space and time coordinates of the backbone network input; b1 to b p is the branch network output, t1 to t p As the output of the backbone network, the predicted output u(x,t) is finally obtained through the inner product operation.

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

[0069] Specifically, when the deep operator network model solves the neutron transport equation, given that it needs to accurately characterize the complex transport process of the neutron's spatial position, movement direction and energy dimension, the input of the backbone network, in addition to the position coordinates of the predicted solution, also includes the neutron movement direction (such as polar angle cosine, azimuth angle) and energy parameters to extract multi-dimensional feature representation; the branch network inputs the discrete values ​​of functions such as neutron source terms, scattering cross sections, absorption cross sections at multiple sampling points to extract the global characteristics of the input source term function; the output layer outputs the predicted value of the neutron angular flux at the solution to be predicted through inner product operation, which fully describes the distribution characteristics of neutrons in space, direction and energy.

[0070] When the deep operator network model solves the neutron diffusion equation, the neutron diffusion equation is an approximate simplified form of the transport equation. The input of 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 network inputs the values ​​of functions such as the neutron source term, diffusion coefficient, and macroscopic absorption cross-section at the sampling points to capture the spatial distribution characteristics of the input source term function; the output layer outputs the predicted value of the neutron scalar flux, which is obtained by integrating the directions and represents the overall density of neutrons at the corresponding position, ignoring the directional dependence.

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

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

[0073]

[0074] in, represents the supervision loss term, represents the physical constraint term of the global neutron conservation deviation, 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, it is possible to ensure the prediction accuracy while strengthening the model's compliance with global physical laws.

[0075] During actual training, the supervised loss term primarily drives the network to learn the local relationship between the input source term and the corresponding solution, while the physical constraint term guides the network to maintain a conservation balance between neutron production, absorption, and loss on a global scale. The coordinated optimization of these two factors results in a trained deep operator network model that not only achieves high prediction accuracy but also significantly outperforms traditional data-driven approaches 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] Where, represents the supervised loss term, x represents the location coordinate of the predicted solution, N represents the number of training samples, and f i Represents the value of the input source function of the i-th sample at the source point, Represents the output of the operator learned by the deep operator network model for the input source function fi, that is, the value of the predicted solution function at point x, u 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 items constructed in this embodiment are specially designed based on the characteristics of this scenario.

[0080] Specifically, the physical constraint term of the global neutron conservation deviation is constructed based on the global conservation relationship in the global neutron conservation physics law. It is used to measure whether the predicted solution meets the global balance requirements of neutron production, absorption, and loss in the entire calculation area Ω. It is expressed as:

[0081]

[0082] Where, represents the physical constraint term of the global neutron conservation deviation, Ω represents the calculation area, S(r) represents the external neutron source term at position r, Represents the neutron source term produced by fission or other physical processes, whose value depends on the predicted output of the deep operator network for the input source term function fi, Σ a (r) represents the neutron absorption cross section at position r, represents the angular distribution neutron flux density predicted by the deep operator network, and the scalar neutron flux obtained by integrating over all angles, f i Represents the value of the input source function of the i-th sample at position r, Represents the operator input source function f obtained by learning the deep operator network model i Output at position r.

[0083] In one example of this embodiment, training data is collected from the MIT OpenMC dataset, including numerical solution data obtained from reactor neutron transport or diffusion simulations using OpenMC. This dataset covers basic information on material properties, geometric configurations, operating settings, and angular flux detection, and is used to reflect neutron transport or diffusion behavior under different parameters. Using the MIT OpenMC dataset ensures that the training data covers a variety of actual operating conditions, including different fuel assembly layouts, control rod positions, reflector layer structures, and operating states, thereby improving the model's adaptability and robustness under diverse reactor designs and operating conditions. Furthermore, using high-fidelity simulation data for training helps improve the model's perception and generalization capabilities for 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, as a basic physical model describing the evolution of neutron behavior in a nuclear reactor, comprehensively reflects various physical processes such as neutron diffusion, absorption, scattering, and fission reaction in space, and can effectively capture the evolution law of neutron flux density with time and space.

[0085] In summary, compared with traditional deep operator network methods, the present invention fully integrates physical prior knowledge and data-driven strategies. By introducing the physical constraint term of global neutron conservation deviation during the training process, it achieves efficient and physically consistent solutions to the neutron transport or diffusion equations in nuclear reactors. This conservation constraint originates from the basic equilibrium law in neutron physics and reflects the global conservation relationship of neutron production, absorption and loss in the reaction region in an integral form, avoiding the high-dimensional derivative calculation and numerical stability issues required for directly introducing PDE residuals, reducing training costs and implementation complexity. Compared with traditional deep networks that rely solely on supervised loss, this method uses a weighted combination of supervised error and global conservation constraints in the loss function, which not only ensures the local fitting accuracy of the solution, but also improves the global physical consistency of the solution through physical conservation terms, significantly improving the training convergence speed and stability of the model and suppressing the risk of overfitting. The model structure effectively combines local spatial characteristics with global input source function characteristics, has stronger adaptability, and can accurately characterize the neutron flux distribution under material property changes, complex geometric configurations and multiple source terms. Leveraging the diverse, high-fidelity simulation results from the MIT OpenMC dataset, the model eliminates the need for complex preprocessing during training. Instead, it can directly use discrete source terms and material property data as input to rapidly construct a neutron flux prediction map, significantly improving both training efficiency and inference speed. The method not only significantly improves 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 are used in the present invention to illustrate the principles and implementation methods of the present invention. The description of the above embodiments is only used to help understand the method of the present invention and its core ideas. At the same time, for those skilled in the art, according to the ideas of the present invention, there may be changes in the specific implementation methods and application scopes. In summary, the contents of this specification should not be understood as limiting the present invention.

[0087] Those skilled in the art will appreciate that the embodiments described herein are intended to help readers understand the principles of the present invention, and it should be understood that the scope of protection of the present invention is not limited to such specific descriptions and embodiments. Those skilled in the art can make various other specific variations and combinations based on the technical teachings disclosed in the present invention without departing from the essence of the present invention, and such variations and combinations are still within the scope of protection of the present invention.

Claims

1. A method for predicting neutron flux in a nuclear reactor based on global conservation constraints, characterized in that: The following steps are involved: Construct a deep operator network model based on global neutron conservation constraints; Collect solution data of neutron transport equation or neutron diffusion equation under different parameters to construct a training data set; Using a training data set, the deep operator network model is trained using a data-driven supervised learning method to obtain the optimal model weights; The deep operator network model is initialized using the optimal model weights, and the initialized deep operator network model is applied to solve the neutron transport equation or neutron diffusion equation under different parameters, thereby predicting the neutron flux distribution in the corresponding area of ​​the nuclear reactor.

2. The method for predicting nuclear reactor neutron flux based on global conservation constraints according to claim 1, characterized in that: 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; Among them, the physical constraints of the global neutron conservation law of the reaction enable the neutron transport process or diffusion process to be transformed into a data-driven physical consistency optimization process during the training process of the deep operator network model.

3. The method for predicting nuclear reactor neutron flux based on global conservation constraints according to claim 2, characterized in that: The physical constraint of the global neutron conservation law for the reaction is that for any closed volume, the total neutron production rate within it is equal to the sum of the neutron absorption rate and the neutron outflow from the boundary, which can be expressed as: Where, J NN (r) represents the neutron flux density predicted by the deep operator network model at position r, φ NN (r) is 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, represents the boundary of the volume V under consideration, and dS represents the area element on the boundary.

4. The method for predicting nuclear reactor neutron flux based on global conservation constraints according to claim 1, characterized in that: The deep operator network model includes a backbone network, a branch network and an output layer; The backbone network is used to extract the corresponding feature representation according to the position coordinates of the predicted solution; The branch network is used to extract a global feature representation representing the input source term function according to the function value of the input source term function corresponding to the neutron transport equation or the 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.

5. The method for predicting nuclear reactor neutron flux based on global conservation constraints according to claim 4, characterized in that: When the deep operator network model solves the neutron transport equation, the input of the backbone network includes the position coordinates of the predicted solution, the neutron motion direction and energy parameters; the input of the branch network includes the discrete values ​​of the neutron source term, the scattering cross section and the absorption cross section function at multiple sampling points; the output layer outputs the predicted value of the 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 value of the neutron scalar flux at the predicted solution.

6. The method for predicting nuclear reactor neutron flux 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 of a global neutron conservation deviation; The loss function Expressed as: in, represents the supervision loss term, 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 supervision loss term.

7. The method for predicting nuclear reactor neutron flux based on global conservation constraints according to claim 6, characterized in that: 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: Where, represents the supervised loss term, x represents the location coordinate of the predicted solution, N represents the number of training samples, and f i Represents the value of the input source function of the i-th sample at the source point, Represents the operator input source function f obtained by learning the deep operator network model i The output, u i (x) represents the value of the true solution function corresponding to the i-th input at x.

8. The method for predicting nuclear reactor neutron flux based on global conservation constraints according to claim 6, characterized in that: The physical constraint term of the global neutron conservation deviation is constructed based on the global conservation relationship in the global neutron conservation physical law, and is used to measure whether the predicted solution meets the global balance requirement of neutron production, absorption and loss in the entire calculation area Ω. It is expressed as: Where, represents the physical constraint term of the global neutron conservation deviation, Ω represents the calculation area, S(r) represents the external neutron source term at position r, represents the neutron source term produced by fission or other physical processes, Σ a (r) represents the neutron absorption cross section at position r, represents the angular distribution neutron flux density predicted by the deep operator network, f i Represents the value of the input source function of the i-th sample at position r, Represents the operator input source function f obtained by learning the deep operator network model i Output at position r.

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