Method, product and system for solving steady-state neutron diffusion problems in heterogeneous reactors

Abstract

The present invention belongs to the technical field of intersection of machine learning and nuclear reactor physics, and discloses a method, product and system for solving the steady-state neutron diffusion problem of heterogeneous reactors, which is particularly suitable for reactor core models with complex multi-region and multi-material arrangements. The method constructs an equivalent form of a first-order derivative equation group containing flow continuity conditions by performing derivative reduction processing on the steady-state multi-group neutron diffusion equation, and uses a neural network to jointly predict the neutron flux density and the neutron current density, and can automatically achieve the continuity of the neutron flux density and the neutron current density at the interface between different material regions. Compared with traditional methods, the present invention can avoid complex sampling operations on the material interface and simplify the modeling process; and reduce the design of the flow continuity constraint loss function at the material interface, effectively improving the prediction accuracy and training efficiency, and has good versatility and robustness.

Description

Technical Field

[0001] The present invention belongs to the technical field of the intersection of machine learning and nuclear reactor physics, and relates to a method, product and system for solving the steady-state neutron diffusion problem in heterogeneous reactors. More specifically, it relates to a method and system for constructing a universal flow continuous neural network model for solving the steady-state neutron diffusion problem in heterogeneous reactors, which is particularly suitable for simulating nuclear reactor core models with complex multi-region and multi-material arrangements. Background Art

[0002] The neutron diffusion equation is one of the core models for nuclear reactor core physics analysis and engineering design. The high-precision solution of this equation has a significant impact on many key links, including core physics analysis, structural optimization, operation control, and safety assessment.

[0003] In recent years, Physics-Informed Neural Networks (PINNs) have attracted widespread attention in the field of numerical solution of partial differential equations due to their good geometric adaptability and versatility. By embedding physical models into the neural network training process and combining them with physical constraints or data-driven mechanisms, the PINNs method provides a new solution for meshless modeling and high-dimensional computation of complex physical problems. However, when the traditional PINNs method is applied to the neutron diffusion problem in actual reactors, especially when faced with nuclear reactor core models with complex multi-region and multi-material arrangements, the accuracy and stability of the solution are significantly limited due to the large differences in physical properties between regions and the complex processing of interfaces between multiple materials.

[0004] In existing research, traditional PINNs methods, when directly applied to neutron diffusion problems in nuclear reactors, often fail to fully consider the continuity characteristics of the neutron flux between different material regions, limiting their accuracy and adaptability in calculating complex multi-region, multi-material core models. To address this issue, some research has introduced improved methods based on the concept of domain decomposition, such as conservative physics-informed neural networks (cPINNs). This method divides the solution domain into multiple material sub-regions, constructs a sub-network model for each sub-region, arranges additional sampling points near material interfaces, and enforces neutron flux continuity constraints, thereby improving the overall solution accuracy.

[0005] While cPINNs enhance the expressive power of traditional PINNs for multi-material problems to a certain extent, their implementation requires specialized processing and additional sampling points at each material interface, as well as the introduction of multiple loss terms and coordination mechanisms, significantly increasing the complexity of model construction and training. In situations where subregion decomposition accuracy is insufficient or where drastic physical changes occur near material interfaces, this method suffers from slow training convergence, poor stability, and large error fluctuations. This is particularly true for nuclear reactor core models with complex multi-region and multi-material configurations. The additional sampling points, loss terms, and coordination mechanisms at numerous material interfaces significantly increase the method's implementation complexity and waste computational resources, making it difficult to generalize and efficiently solve real-world nuclear reactor core models. Summary of the Invention

[0006] In response to the above-mentioned defects or improvement needs of the existing technology, the present invention provides a method, product and system for solving the steady-state neutron diffusion problem in heterogeneous reactors, which aims to solve the problems of slow convergence, poor stability and large error fluctuations in the existing technology.

[0007] To achieve the above objectives, according to one aspect of the present invention, a method for constructing a general flow continuum neural network model for solving the steady-state neutron diffusion problem in heterogeneous reactors is provided, comprising the following steps:

[0008] S1: Perform derivative reduction on the steady-state multi-group neutron diffusion equation and construct an equivalent form of the first-order derivative equation system including the neutron flux continuity condition;

[0009] Determine the basic architecture of the neural network based on the spatial dimensions and number of energy groups of the reactor core problem to be simulated;

[0010] Based on the geometric structure, material properties and boundary conditions of the calculation area, a database of characteristic sampling points in the multi-material area is established;

[0011] Fix the fission source distribution and effective proliferation coefficient, and start the iterative cycle training process;

[0012] S2: Under the current fixed fission source distribution and effective multiplication coefficient, based on the derivative reduction equations constructed in step S1 and the characteristic sampling point database, construct the sub-loss function for each material region respectively, and adaptively assign the loss weights of each region and boundary condition according to the characteristic information of the sampling points to form the total loss function;

[0013] S3: Based on the constructed total loss function, the neural network is trained to optimize the neutron flux and neutron flux density distribution simultaneously, and it is determined whether the set upper limit of the number of inner iterations is reached. If not, the process returns to step S2; if the upper limit of the number of inner iterations is reached, the process goes to step S4;

[0014] S4: Determine whether the current loss function meets the convergence condition. If not, update the fission source distribution and the effective proliferation coefficient, and return to step S2; if satisfied, save the neural network model.

[0015] Furthermore, the physical model of the steady-state multi-group neutron diffusion problem satisfies the following set of partial differential equations:

[0016] The physical model of the steady-state multi-group neutron diffusion problem satisfies the following set of partial differential equations:

[0017] ;

[0018] The fission source term Expressed as:

[0019] ;

[0020] in, is the gradient operator, represents the neutron energy group number, r is the spatial position vector, Representative Neutron flux density distribution function of energy group, Representative Energy group neutron diffusion coefficient, Representative Energy group macroscopic removal cross section, Representatives from Can group to the first The macroscopic scattering cross section of the energy group, Representatives in the The energy spectrum distribution share of the energy group, represents the average number of neutrons produced per fission, Representative The macroscopic fission cross section of the energy group, represents the effective multiplication coefficient;

[0021] The equivalent form of the first-order derivative equations of S1 containing the neutron flux continuity condition is solved by reducing the derivatives of the original equations through the following steps:

[0022] Introducing the neutron flux density vector , No. Neutron flux density vector of energy group Expressed as:

[0023] ;

[0024] The original neutron diffusion equation is transformed into a set of first-order partial differential equations containing the neutron flow continuity condition:

[0025] ;

[0026] The neural network is modeled and solved based on the above derivative reduction form, by using the Laplace operator The operation is converted into a two-step gradient operator Operation, reducing the computational complexity of neural networks in the process of automatic differentiation.

[0027] Furthermore, the basic architecture of the neural network constructed in S1, whose output is used to predict neutron flux density and neutron current density, has a fixed input and output structure, where the input is geometric coordinate data points and the output is neutron flux density and neutron current density; for the core problem in d-dimensional space and G energy group, the dimension of the neural network input layer is d, and the dimension of the neural network output layer is G (d+1);

[0028] The neural network jointly predicts the neutron flux density and neutron current density based on its continuity characteristics, and At , the neutron flux continuity condition is automatically satisfied:

[0029] ;

[0030] in, Material interface The unit normal vector of 、 Respectively Energy Group The diffusion coefficient of the region, Respectively Energy Group The neutron flux density of the region, 、 For the Energy Group The neutron flux density in the region, subscript Indicates adjacent areas of two materials.

[0031] Furthermore, the structure of the flow continuous neural network includes using a single network to predict all output results, or outputting the neutron flux density and the neutron current density respectively through multiple sub-networks.

[0032] Furthermore, the characteristic sampling point database of the multi-material region described in S1 is established based on the geometric information, material properties and boundary conditions of the reactor core problem to be simulated;

[0033] The sampling points include sampling points within each material region combined with multiple groups of cross-sectional parameters, as well as boundary sampling points corresponding to boundary conditions, but do not include sampling points on material interfaces; sampling points with material information characteristics are stored in the form of a high-dimensional array.

[0034] Furthermore, the training process of the neural network uses some or all of the sampling points in the feature sampling point database, and the sampling point data are normalized or standardized before training;

[0035] The selection of the sampling points includes random sampling, systematic sampling or stratified sampling strategies to obtain representative multi-dimensional feature data; during the training process, small batch training is implemented by setting the batch size to reduce memory usage and improve training efficiency.

[0036] Furthermore, the loss function in S2 is constructed under the condition that the current fission source term and the effective proliferation coefficient are fixed. Based on the derivative reduction equation group, characteristic sampling point database and neural network structure constructed in step S1, the residual of the partial differential equation at each sampling point is obtained by automatic differentiation, and the sub-loss function corresponding to each material region and boundary type is constructed respectively, which is used as the optimization target in the neural network training process, specifically including:

[0037] Residual loss term of multi-group neutron diffusion equation based on derivative reduction form P1 , according to the number of sampling points in each material area Relative to the total number of sampling points N P Weighted by the proportion:

[0038] ;

[0039] Based on the residual term loss of the flow continuity equation P2 , weighted summation based on regional sampling points:

[0040] ;

[0041] Based on the boundary condition loss term loss B , different values ​​of the parameter group α, β, γ correspond to different boundary types, and the expression is:

[0042] ;

[0043] where N B Represents the number of sampling points at the boundary, M represents the total number of material types, is the coordinate of the pth sampling point in each material area, is the coordinate of the qth sampling point at the boundary, k is the number of current iterations, and different values ​​of α, β, and γ represent different types of boundary conditions;

[0044] The three types of sub-loss functions are combined to form the total loss function loss T :

[0045]

[0046] Each loss term already includes an adaptive weighting mechanism based on the proportion of regional sampling points or the corresponding regional volume information as the minimization optimization goal during the neural network training process.

[0047] Furthermore, steps S2 to S4 constitute a complete iterative process of machine learning training of the neutron diffusion equation, specifically including:

[0048] Steps S2 to S4 constitute a complete iterative process of machine learning training of the neutron diffusion equation, specifically including:

[0049] In the kth outer iteration, based on the The neutron flux density obtained by the iteration , update the fission source term distribution:

[0050] ;

[0051] Based on the updated neutron flux and neutron current density results, calculate the effective multiplication coefficient:

[0052] ;

[0053] in represents the inner product operation;

[0054] The inner iteration termination condition is: the current number of inner iterations ,in is the maximum number of inner iterations set;

[0055] The termination condition of the outer iteration is: the total loss function ,in is the preset convergence threshold.

[0056] According to another aspect of the present invention, a general flow continuum neural network model product for solving the steady-state neutron diffusion problem of a heterogeneous reactor is provided, comprising a neural network model obtained according to the general flow continuum neural network model construction method as described in any of the preceding items.

[0057] According to another aspect of the present invention, a system for solving the steady-state neutron diffusion problem of a heterogeneous reactor is provided. The system includes a memory and a processor. The memory stores a computer program and the universal flow continuous neural network model product. When the processor executes the computer program, the fixed fission source distribution and the effective proliferation coefficient are input into the universal flow continuous neural network model product, thereby outputting a solution to the steady-state neutron diffusion problem of the heterogeneous reactor.

[0058] In general, the above technical solutions conceived by the present invention can achieve the following beneficial effects compared with the prior art:

[0059] 1. This method uses a reduced-order derivative representation of the neutron diffusion equation to transform the original second-order partial differential problem into multiple first-order forms for solution. Compared to the original second-order partial derivative solution, the first-order partial derivative solution has higher computational efficiency and numerical stability.

[0060] 2. The present invention jointly predicts neutron flux density and neutron current density through a neural network. Utilizing the continuity characteristics of the neural network, the present invention achieves continuous expression of the neutron current at the interface across material regions. Therefore, there is no need to impose constraints or additional sampling points at the material interface, significantly simplifying the modeling process. Furthermore, the present invention has good geometric adaptability and is suitable for core simulations of complex structures.

[0061] 3. This invention effectively coordinates the optimization direction of multi-region loss functions by constructing a unified feature sampling point database and incorporating an adaptive weight allocation strategy based on the number of sampling points. Furthermore, it employs a mini-batch training strategy to reduce resource consumption during neural network training, improve training efficiency, and enhance practicality and scalability for large-scale, multi-operation core problems. BRIEF DESCRIPTION OF THE DRAWINGS

[0062] Figure 1 A schematic flow chart of the construction of a general flow continuum neural network model for solving the steady-state neutron diffusion problem in heterogeneous reactors provided in a preferred embodiment of the present invention.

[0063] Figure 2 This is a schematic diagram of the two-material region model of Example 1, which is used to verify the model's solving ability under different diffusion coefficient ratios.

[0064] Figure 3 3 is a comparison diagram of the normalized neutron flux density at the symmetric center dotted line between the method of the present invention and the traditional neural network method in Example 1.

[0065] Figure 4 This is a schematic diagram of the 1 / 4 BIBLIS reference core structure used in Example 2, which has the characteristics of multi-material heterogeneous distribution and component-level full core model.

[0066] Figure 5 This is a comparison chart of the normalized power of the BIBLIS core using different methods in Example 2.

[0067] Figure 6 This is a schematic diagram of the pin-by-pin C5G2 fine core structure used in Example 3, which has the characteristics of multi-material heterogeneous distribution and gate-level refinement.

[0068] Figure 7 This is a comparison diagram of the pin-by-pin level normalized power of different methods in Example 3. DETAILED DESCRIPTION

[0069] In order to make the objectives, technical solutions and advantages of the present invention more clearly understood, the present invention is further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only intended to illustrate the present invention and are not intended to limit the present invention. In addition, the technical features involved in the various embodiments of the present invention described below may be combined with each other as long as they do not conflict with each other.

[0070] The existing neural network methods have problems such as insufficient solution accuracy, complex interface processing, and low training efficiency when dealing with nuclear reactor core models with complex multi-region and multi-material arrangements. Figure 1 As shown, a preferred embodiment of the present invention provides a method for constructing a general flow continuous neural network model for solving the steady-state neutron diffusion problem in a heterogeneous reactor, comprising the following steps:

[0071] S1: Perform derivative reduction on the steady-state multi-group neutron diffusion equation and construct an equivalent form of the first-order derivative equation system including the neutron flux continuity condition;

[0072] Determine the basic architecture of the neural network based on the spatial dimensions and number of energy groups of the reactor core problem to be simulated;

[0073] Based on the geometric structure, material properties and boundary conditions of the calculation area, a database of characteristic sampling points in the multi-material area is established;

[0074] Fix the fission source distribution and effective proliferation coefficient, and start the iterative cycle training process;

[0075] S2: Under the current fixed fission source distribution and effective multiplication coefficient, based on the derivative reduction equations constructed in step S1 and the characteristic sampling point database, construct the sub-loss function for each material region respectively, and adaptively assign the loss weights of each region and boundary condition according to the characteristic information of the sampling points to form the total loss function;

[0076] S3: Based on the constructed total loss function, the neural network is trained to optimize the neutron flux density distribution, and it is determined whether the set upper limit of the number of inner iterations is reached. If not, the process returns to step S2;

[0077] S4: Determine whether the current loss function meets the convergence conditions. If not, update the fission source distribution and the effective proliferation coefficient, and return to step S2. If it does, save the neural network model, end the training, and output the calculation results of the steady-state neutron diffusion problem.

[0078] The method of this embodiment is mainly used to simulate the steady-state neutron diffusion problem in a reactor core with a highly heterogeneous material distribution. It has the ability to automatically constrain cross-region continuity and is particularly suitable for core model problems with complex multi-region and multi-material arrangements. The specific steps and implementation process are as follows:

[0079] S1 Construct the derivative-reduced equivalent form of the steady-state neutron diffusion equation: The steady-state multi-group neutron diffusion problem can be expressed as follows:

[0080] ;

[0081] The fission source term expression is:

[0082] ;

[0083] in, is the gradient operator, represents the neutron energy group number, r is the spatial position vector, Representative Neutron flux density distribution function of energy group, Representative Energy group neutron diffusion coefficient, Representative Energy group macroscopic removal cross section, Representatives from Can group to the first The macroscopic scattering cross section of the energy group, Representatives in the The energy spectrum distribution share of the energy group, represents the average number of neutrons produced per fission, Representative The macroscopic fission cross section of the energy group, represents the effective multiplication coefficient.

[0084] In order to introduce the concept of flow continuity and reduce the complexity of neural network in automatic differentiation, the neutron flow density vector is introduced. , perform derivative reduction strategy, Neutron flux density vector of energy group Expressed as:

[0085] ;

[0086] Substituting it into the original diffusion equation, we can obtain that the original neutron diffusion equation can be transformed into a set of first-order partial differential equations containing the neutron flow continuity condition:

[0087] ;

[0088] This first-order derivative equivalent avoids the direct calculation of the Laplace operator , reducing the computational complexity of the neural network in the automatic differentiation process; explicitly introducing the neutron flux density so that the network can jointly predict 、 , which is conducive to automatically satisfying flow continuity at the material interface and provides a clear structure for the subsequent construction of the physical residual loss function.

[0089] S1 builds the basic architecture of the neural network: In this embodiment, according to the spatial dimensions of the reactor core to be simulated And the neutron energy group number G, to build a neural network structure with the ability to express the flow continuously. The network is based on the geometric coordinate data points As input, the output includes each energy group The corresponding neutron flux density and neutron flux density .

[0090] Based on the equivalent form of the derivative-reduced neutron diffusion equation in S1, the neural network jointly predicts the neutron flux density and the neutron current density, which can automatically meet the requirements of different material interfaces without forcibly adding interface constraints. Neutron flux continuity condition at:

[0091] ;

[0092] in, Material interface The unit normal vector of 、 Respectively Energy Group The diffusion coefficient of the region, Respectively Energy Group The neutron flux density of the region, 、 For the Energy Group The neutron flux density in the region, subscript Indicates adjacent areas of two materials.

[0093] For a problem with G energy groups and d-dimensional space, the output dimension of the neural network is G (d+1), that is, each energy group corresponds to one neutron flux density scalar and d neutron flux density components.

[0094] The neural network can adopt: a single neural network architecture, which uniformly outputs all physical quantities, has the advantages of high parameter efficiency and simple training, and is suitable for medium-sized problems; a multi-subnetwork architecture, which outputs different energy groups or physical components respectively, has stronger network expression capabilities, and is suitable for multi-task or parallel training frameworks.

[0095] The main body of the network can adopt a multi-layer perceptron (MLP) structure, and the activation function is preferably a function with second-order differentiability such as Tanh and Sigmoid. The number of layers and width of the network can be empirically set according to the spatial dimension of the problem, the size of the energy group, the complexity of the problem and the computing resource conditions, and can also be automatically adjusted through hyperparameter optimization methods. The optimization methods include grid search, Bayesian optimization or other structural search strategies based on model performance feedback to improve training efficiency and prediction accuracy while ensuring convergence.

[0096] S1 Establish a feature sampling point database: Based on the geometric structure, material properties, and boundary condition information of the reactor core problem to be simulated, a multi-region feature sampling point database is constructed and uniformly organized and structurally coded. Specifically, it includes:

[0097] Sampling points inside the region: Inside the core, in each material region, a certain number of sampling points are arranged according to the volume or area of ​​the region, and each sampling point contains its spatial coordinates And the corresponding multi-group cross-sectional parameter information, such as the diffusion coefficient , Macroscopic removal section , scattering cross section , macroscopic fission cross section and fission energy spectrum distribution wait;

[0098] Boundary sampling points: Place boundary sampling points in regions of the geometric model with clear physical boundary conditions (such as zero flux boundaries, vacuum boundaries, reflection boundaries, or other types). Each boundary point records its spatial position, boundary normal vector, and its boundary type, which are used to construct loss function terms related to the imposed boundary conditions.

[0099] Sampling point organization structure: All sampling points and their material information are structured and stored in the form of high-dimensional arrays or tensors, which can be extracted, normalized, or batched as needed during neural network training.

[0100] It is particularly noted that the characteristic sampling point database does not need to include sampling points on the material interface. This is because the method of the present invention naturally expresses the flow continuity on the material interface through network structure design and derivative reduction physical model. Therefore, there is no need to set additional interface sampling points, which simplifies the sampling and modeling process.

[0101] During the neural network training phase, the use of the sampling point database includes:

[0102] All data points are normalized or standardized before neural network training to improve training stability;

[0103] The training samples can be constructed by selecting some or all sampling points from the database through random sampling, systematic sampling, stratified sampling and other strategies;

[0104] By setting the batch size to implement mini-batch training, we can meet the needs of small-batch training, reduce memory usage and improve training efficiency.

[0105] Initialize the fission source distribution and effective proliferation coefficient to provide the initial physical state for neural network training, and enter the iterative cycle and neural network training process.

[0106] Set the initial outer iteration counter , initial fission source term distribution , which can be taken as the weighted sum of the neutron flux density at each spatial point, the initial effective multiplication coefficient , generally takes a value of 1.0 or other empirical values.

[0107] After the above initial conditions are set, the neural network will be distributed with the initial fission source term Initial effective proliferation coefficient The conditions are known and the first round of iterative training process begins, that is, the inner iteration phase is started.

[0108] S2 constructs the total loss function: During the off-round iterative training process, the fission source term distribution Effective proliferation coefficient Under fixed conditions, the total loss function is constructed and used as the objective function to execute the neural network training process.

[0109] Based on the derivative reduction equations and characteristic sampling point database constructed in step S1, the neutron flux density output by the neural network constructed in step S1 is called and neutron flux density , through automatic differentiation, the sub-loss functions of each material region are constructed separately, including:

[0110] Residual loss term of neutron diffusion equation based on derivative reduction form P1 , according to the number of sampling points in each material area Relative to the total number of sampling points N P Weighted by the proportion:

[0111] ;

[0112] Based on the residual term loss of the flow continuity equation P2 , weighted summation based on regional sampling points:

[0113] ;

[0114] Based on the boundary condition loss term loss B, different values ​​of the parameter group α, β, γ correspond to different boundary types, and the expression is:

[0115] ;

[0116] where N B Represents the number of sampling points at the boundary, M represents the total number of material types, is the coordinate of the pth sampling point in each material area, is the coordinate of the qth sampling point at the boundary, k is the current iteration number, where α, β, and γ take different values ​​to represent different boundary condition types; the above three types of sub-loss functions are combined to form the total loss function loss T :

[0117] ;

[0118] The weight within each loss term is determined by the number of sampling points in different material regions. , the total number of sampling points N P , number of boundary sampling points N B It is determined to adaptively reflect the volume differences of material regions and the distribution characteristics of boundary information, and to achieve a balance of training errors among multiple regions.

[0119] S3 is the total loss function loss constructed based on S2 T , execute the neural network training process, and control the number of internal iterations and make termination judgments.

[0120] Using the total loss function loss T As a training optimization goal, the neural network is trained and the parameters of the neural network are updated by gradient through the backpropagation algorithm to optimize the neutron flux density distribution under the current physical conditions. and neutron flux density distribution , each training process is regarded as an inner iteration operation.

[0121] It should be noted that during the training process, the fission source term distribution Effective proliferation coefficient It remains fixed as a known input and does not participate in the gradient update during the back propagation of the loss function.

[0122] Set the maximum number of inner iterations , introduce an internal iteration counter , after each training is performed, the counter is incremented .

[0123] The inner iteration termination condition is If the termination condition is met, enter S4, otherwise return to S2 and continue to calculate the total loss function under the current physical state and perform neural network training.

[0124] S4 updates the fission source distribution and effective proliferation coefficient and determines the termination condition of the outer iteration. After completing the current inner iteration process, it is necessary to update the fission source distribution and effective proliferation coefficient and determine whether the outer iteration has converged to decide whether to continue the next round of training cycle. Specifically:

[0125] Neutron flux density based on the current neural network output Update The distribution of fission sources corresponding to the round:

[0126] ;

[0127] Based on neutron flux density and neutron flux density Results update effective multiplication coefficient:

[0128] ;

[0129] in Represents the inner product operation.

[0130] The termination condition of the outer iteration is: the total loss function ,in If the convergence condition is met, the entire training process ends and the final predicted neutron flux density and effective multiplication coefficient are output as the calculation results of the present invention; if not converged, the external iteration counter is updated. , and returns to S2, reconstructs the loss function based on the updated fission source distribution and effective proliferation coefficient, and conducts the next round of training.

[0131] In practical engineering, reactor core problems often present complex heterogeneity across multiple regions and materials. To validate the applicability and advantages of the proposed method for such complex core problems, several typical validation models and benchmark problems were constructed as examples for experimental testing.

[0132] Figure 2 The figure shows a schematic diagram of the geometric model of the two-substance material region problem used in Example 1, which includes two different material regions. = 1, the diffusion coefficient, absorption cross section (for single group problems, the macroscopic removal cross section is equal to the absorption cross section), fission cross section, and scattering cross section are expressed as 、 、 、 To express it, by changing the ratio D of the diffusion coefficients between the two regions a1 and a2a1 / D a2 , typical problems with different physical heterogeneity strengths can be constructed to verify the calculation accuracy and neutron flux continuity expression ability of the method of the present invention under strong heterogeneous structure conditions. This embodiment 1 sets four examples, and the material parameters are set as shown in the following table (for a single group, i.e. Time fission energy spectrum distribution , not listed in the table):

[0133] ;

[0134] The method of the present invention and the traditional neural network method (Elhareef MH, Wu Z. Physics-informed neural network method and application to nuclear reactor calculations: Apilot study[J]. Nuclear Science and Engineering, 2023, 197(4): 601-622. hereinafter referred to as traditional PINNs) on the effective proliferation coefficient The calculation and comparison results are as follows:

[0135] ;

[0136] Figure 3 The comparison curves of normalized neutron flux density obtained by different methods are shown at the dotted line at the center of symmetry. It can be seen that in the case with the same diffusion coefficient ratio (Example 1), the general flow continuous neural network method proposed in this invention and the traditional neural network method show high consistency in the prediction results of neutron flux density distribution and effective multiplication coefficient. In the cases with heterogeneous differences (Examples 2, 3, and 4), the method of this invention can accurately capture the significant changes in neutron flux density gradient at the material interface and show a clear accuracy advantage in predicting the effective multiplication coefficient and neutron flux density distribution, demonstrating the applicability of the method of this invention in dealing with highly heterogeneous diffusion problems.

[0137] Figure 4 The figure shows a schematic diagram of a 1 / 4 BIBLIS benchmark model used in Example 2. The model is a 17×17 two-dimensional two-group pressurized water reactor model composed of 8 different materials arranged in an alternating manner, which can be used to simulate the multi-material arrangement at the component level. Figure 5 The normalized power comparison results of the method of the present invention and the traditional neural network method are shown, and the effective proliferation coefficient prediction values ​​of the two methods are given as follows:

[0138] ;

[0139] It can be seen that the method of the present invention can effectively improve the prediction accuracy of the effective proliferation coefficient and reduce the relative error of the normalized power error in the component area from 10% to below 3%, which can overcome the problem of excessive prediction deviation caused by insufficient expression of flow continuity constraints in existing methods.

[0140] Figure 6 The figure shows a schematic diagram of the pin-by-pin C5G2 core structure at a higher level of sophistication used in Example 3, used to describe the more sophisticated cell-level core physics modeling problem. This example is more complex in terms of spatial and material partitioning, making it essentially impossible to use traditional methods based on forced flow continuity constraints. Figure 7 The comparison results of the normalized power between the method of the present invention and the traditional neural network method, as well as the prediction results of the effective proliferation coefficient by the two methods are shown as follows:

[0141] ;

[0142] It can be seen that the present invention can effectively improve the prediction accuracy of the effective proliferation coefficient and reduce the relative error of the normalized power error in the component area from 26% to below 5%, demonstrating strong robustness and adaptability in high-precision and high-complexity problems.

[0143] In summary, the above three embodiments have verified the applicability of the general flow continuity neural network method proposed in the present invention in dealing with multi-material, multi-region heterogeneous reactor core problems from the perspective of different model complexity. This method is based on the derivative-reduced equivalent form of the neutron diffusion equation, and jointly predicts the neutron flux density and the neutron current density under the framework of a neural network. It automatically expresses the continuity of the neutron flow at the material interface without taking additional points at the material interface and forcing the loss of continuity constraints. This feature makes this method particularly suitable for core structure problems with complex arrangements of multiple materials and multiple regions, such as those in Examples 2 and 3.

[0144] In summary, this method performs derivative reduction on the steady-state multi-group neutron diffusion equation to construct an equivalent system of first-order derivative equations that includes flow continuity conditions. This method then utilizes a neural network to jointly predict the neutron flux density and neutron current density, automatically achieving continuity of these two parameters at the interfaces between different material regions. Compared to traditional methods, this method avoids complex sampling operations at material interfaces, simplifies the modeling process, and reduces the design of loss functions for flow continuity constraints at material interfaces, effectively improving prediction accuracy and training efficiency, while demonstrating excellent versatility and robustness.

[0145] Compared with existing methods such as cPINNs that require explicit construction of interface loss terms, the neural network framework structure adopted by the present invention is simpler, the training process is more stable, and it has higher versatility. It exhibits stronger versatility and robustness in strong heterogeneity and high complexity scenarios, and is suitable for efficient numerical simulation of complex nuclear reactor core physics problems.

[0146] It will be easily understood by those skilled in the art that the above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included in the scope of protection of the present invention.

Claims

1. A method for constructing a general flow continuous neural network model for solving the steady-state neutron diffusion problem in heterogeneous reactors, characterized in that: The steps include: S1: Perform derivative reduction on the steady-state multi-group neutron diffusion equation and construct an equivalent form of the first-order derivative equation system including the neutron flux continuity condition; The basic architecture of the neural network is determined based on the spatial dimension and number of energy groups of the reactor core problem to be simulated. The basic architecture of the neural network constructed in S1 has an output used to predict neutron flux density and neutron current density, and has a fixed input and output structure, where the input is geometric coordinate data points and the output is neutron flux density and neutron current density. For a core problem with d-dimensional space and G energy groups, the dimension of the neural network input layer is d, and the dimension of the neural network output layer is G (d+1). The neural network jointly predicts the neutron flux density and neutron current density based on its continuity characteristics, and At , the neutron flux continuity condition is automatically satisfied: in, Material interface The unit normal vector of 、 Respectively Energy Group The diffusion coefficient of the region, Respectively Energy Group The neutron flux density of the region, 、 For the Energy Group The neutron flux density in the region, subscript Indicates adjacent areas of two materials; Based on the geometry, material properties, and boundary conditions of the computational domain, a database of characteristic sampling points for multi-material regions is established. These sampling points include sampling points within each material region that combine multiple groups of cross-sectional parameters, as well as boundary sampling points corresponding to the boundary conditions, excluding sampling points on material interfaces. The sampling points with material information characteristics are stored in a high-dimensional array. Fix the fission source distribution and effective proliferation coefficient, and start the iterative cycle training process; S2: Under the current fixed fission source distribution and effective multiplication coefficient, based on the derivative reduction equations constructed in step S1 and the characteristic sampling point database, construct the sub-loss function for each material region respectively, and adaptively assign the loss weights of each region and boundary condition according to the characteristic information of the sampling points to form the total loss function; S3: Based on the constructed total loss function, the neural network is trained to optimize the neutron flux and neutron flux density distribution simultaneously, and it is determined whether the set upper limit of the number of inner iterations is reached. If not, the process returns to step S2; if the upper limit of the number of inner iterations is reached, the process goes to step S4; S4: Determine whether the current loss function meets the convergence condition. If not, update the fission source distribution and the effective proliferation coefficient, and return to step S2; if satisfied, save the neural network model.

2. A method for constructing a general flow continuous neural network model for solving the steady-state neutron diffusion problem in heterogeneous reactors according to claim 1, characterized in that: The physical model of the steady-state multi-group neutron diffusion problem satisfies the following set of partial differential equations: The fission source term Expressed as: in, is the gradient operator, represents the neutron energy group number, r is the spatial position vector, Representative Neutron flux density distribution function of energy group, Representative Energy group neutron diffusion coefficient, Representative Energy group macroscopic removal cross section, Representatives from Can group to the first The macroscopic scattering cross section of the energy group, Representatives in the The energy spectrum distribution share of the energy group, represents the average number of neutrons produced per fission, Representative The macroscopic fission cross section of the energy group, represents the effective multiplication coefficient; The equivalent form of the first-order derivative equations of S1 containing the neutron flux continuity condition is solved by reducing the derivatives of the original equations through the following steps: Introducing the neutron flux density vector , No. Neutron flux density vector of energy group Expressed as: The original neutron diffusion equation is transformed into a set of first-order partial differential equations containing the neutron flow continuity condition: The neural network is modeled and solved based on the above derivative reduction form, by converting the Laplace operator The operation is converted into a two-step gradient operator Operation, reducing the computational complexity of neural networks in the process of automatic differentiation.

3. A method for constructing a general flow continuous neural network model for solving the steady-state neutron diffusion problem in heterogeneous reactors according to claim 1, characterized in that: The structure of the flow continuous neural network includes using a single network to predict all output results, or outputting the neutron flux density and the neutron current density respectively through multiple sub-networks.

4. A method for constructing a general flow continuous neural network model for solving the steady-state neutron diffusion problem in heterogeneous reactors according to claim 1, characterized in that: The training process of the neural network uses some or all of the sampling points in the feature sampling point database, and normalizes or standardizes the sampling point data before training; The selection of the sampling points includes random sampling, systematic sampling or stratified sampling strategies to obtain representative multi-dimensional feature data; During the training process, small batch training is implemented by setting the batch size to reduce memory usage and improve training efficiency.

5. The method for constructing a general flow continuous neural network model for solving the steady-state neutron diffusion problem in heterogeneous reactors according to claim 2, characterized in that: The loss function in S2 is constructed under the condition that the current fission source term and the effective multiplication coefficient are fixed. Based on the derivative reduction equation group, characteristic sampling point database and neural network structure constructed in step S1, the residual of the partial differential equation at each sampling point is obtained by automatic differentiation, and the sub-loss function corresponding to each material region and boundary type is constructed respectively. It is used as the optimization target in the neural network training process, specifically including: Residual loss term of multi-group neutron diffusion equation based on derivative reduction form P1 , according to the number of sampling points in each material area Relative to the total number of sampling points N P Weighted by the proportion: Based on the residual term loss of the flow continuity equation P2 , weighted summation based on regional sampling points: Based on the boundary condition loss term loss B , different values ​​of the parameter group α, β, γ correspond to different boundary types, and the expression is: where N B Represents the number of sampling points at the boundary, M represents the total number of material types, For each material area p The coordinates of the sampling points, For the border q The coordinates of the sampling points, k is the current iteration number, where α 、 β 、 γ Different values ​​represent different types of boundary conditions; The three types of sub-loss functions are combined to form the total loss function loss T : Each loss term already includes an adaptive weighting mechanism based on the proportion of regional sampling points or the corresponding regional volume information as the minimization optimization goal during the neural network training process.

6. A method for constructing a general flow continuous neural network model for solving the steady-state neutron diffusion problem in heterogeneous reactors according to claim 5, characterized in that: Steps S2 to S4 constitute a complete iterative process of machine learning training of the neutron diffusion equation, specifically including: In the k In the outer iteration, based on the k- Neutron flux density obtained after 1 iteration , update the fission source term distribution: Based on the updated neutron flux and neutron current density results, calculate the effective multiplication coefficient: in represents the inner product operation; The inner iteration termination condition is: the current number of inner iterations ,in is the maximum number of inner iterations set; The termination condition of the outer iteration is: the total loss function ,in is the preset convergence threshold.

7. A general flow continuous neural network model product for solving the steady-state neutron diffusion problem in heterogeneous reactors, characterized by: Including a neural network model obtained by the general flow continuous neural network model construction method according to any one of claims 1-6.

8. A system for solving the steady-state neutron diffusion problem in a heterogeneous reactor, characterized in that: The system includes a memory and a processor, the memory stores a computer program and the universal flow continuous neural network model product according to claim 7, and when the processor executes the computer program, the fixed fission source distribution and the effective proliferation coefficient are input into the universal flow continuous neural network model product, thereby outputting the solution to the steady-state neutron diffusion problem of the heterogeneous reactor.

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