Method for solving multi-group neutron diffusion characteristic value by adopting decoupling residual error loss function

Abstract

The invention relates to the technical field of neutron diffusion characteristic values, in particular to a method for solving multi-group neutron diffusion characteristic values by adopting a decoupling residual error loss function, which comprises the following steps of: randomly taking points on a calculation area, and dividing the points into a boundary point set, an interface point set and a residual error point set; determining the number of neurons of an output layer according to the number of group neutron fluxes and the number of materials in the calculation area; neurons of the output layer are subjected to weighted summation through an indicative function to obtain a neural network numerical solution; constructing a corresponding loss function; constructing a decoupling loss function containing the characteristic function information of the previous step; and optimizing the neural network by using an optimization algorithm, obtaining a numerical result when the loss function reaches a preset condition or the neural network completes a given number of training times, and according to the method, a neutron diffusion characteristic value equation source iteration solving method is absorbed, the characteristic function information of the previous step is inherited in an iteration form, and a new decoupling residual loss function is constructed. And the solving precision of the multi-group neutron diffusion characteristic value problem is improved.

Description

Technical Field

[0001] The present invention relates to the technical field of neutron diffusion eigenvalues, and in particular to a method for solving multi-group neutron diffusion eigenvalues ​​by using a decoupled residual loss function. Background Art

[0002] Neutron diffusion equations play a fundamental role in nuclear reactor physics, describing the propagation characteristics of neutrons in the reactor according to neutron diffusion theory. These equations can be extended to more complex forms including multi-group diffusion and neutron noise models to adapt to different physical requirements.

[0003] In the field of nuclear energy, the neutron diffusion equation is crucial for fuel management, optimizing fuel rod layout, and controlling reactor power and safety by adjusting neutron flux. Although there are many numerical solutions such as finite difference method, finite element method, nodal method, Monte Carlo method, etc., when solving the multi-group neutron diffusion eigenvalue problem, the Physics Constrained-General Inverse Power Method Neural Network (PC-GIPMNN) based on physical constraints has large errors when solving the multi-group neutron diffusion eigenvalue problem in the absence of data. Summary of the invention

[0004] The purpose of the present invention is to provide a method for solving multi-group neutron diffusion eigenvalues ​​by using a decoupled residual loss function, aiming to solve the problem of large errors when solving multi-group neutron diffusion eigenvalues ​​by using the existing generalized inverse power method neural network based on physical constraints.

[0005] To achieve the above object, the present invention provides a method for solving multi-group neutron diffusion eigenvalues ​​using a decoupled residual loss function, comprising the following steps:

[0006] Randomly select points on the calculation area and divide them into boundary point set, interface point set and residual point set according to the position of the sampling points;

[0007] According to the number of neutron fluxes in the energy group to be determined in the equation and the number of materials in the calculation area, the number of neurons in the output layer of the neural network is determined, an initial guess about the characteristic function is given, and an initial characteristic value is given based on the Rayleigh quotient form;

[0008] The output of the neural network is squared, and the neurons in the output layer are weighted summed through the indicative function to obtain the numerical solution of the neural network;

[0009] The corresponding loss function is constructed based on the interface conditions and boundary conditions of the neutron diffusion eigenvalue equation group;

[0010] Based on the decoupling idea of ​​the control equations of the neutron diffusion eigenvalue equations and the source iteration method, a decoupling loss function containing the information of the eigenfunction of the previous step is constructed, and at the same time, the eigenvalue in the form of Rayleigh quotient is constructed;

[0011] The neural network is optimized using an optimization algorithm based on a weighted sum of a loss function formed by a set of boundary points and a set of interface points and a decoupling loss function, and a numerical result is obtained when the loss function reaches a preset condition or the neural network completes a given number of training times.

[0012] The neural network is a fully connected neural network with added jump connections, and automatic differentiation technology is used in all differential operations of the neural network.

[0013] The output of each output layer neuron is subjected to a square operation to ensure the non-negativity of the neuron output, so as to satisfy the non-negative property of the characteristic function of the neutron diffusion eigenvalue problem in the critical state.

[0014] Wherein, the neutron diffusion characteristic value equation group is as follows:

[0015] And G≥2.

[0016] The given initial guess about the characteristic function is a constant function whose values ​​corresponding to all calculation area points of all characteristic functions are 1.

[0017] The method of the present invention adopts a decoupled residual loss function to solve the characteristic values ​​of multiple groups of neutron diffusion. Points are randomly selected on the calculation area, and are divided into a boundary point set, an interface point set and a residual point set according to the positions of the sampling points; the number of neurons in the output layer of the neural network is determined according to the number of neutron fluxes in the energy group to be solved in the equation and the number of materials in the calculation area, an initial guess about the characteristic function is given, and the initial characteristic value is given based on the Rayleigh quotient form; the output of the neural network is subjected to a square operation, and the neurons in the output layer are weighted summed in the form of an indicator function to obtain a numerical solution of the neural network; a corresponding loss function is constructed based on the interface conditions and boundary conditions of the neutron diffusion characteristic value equation group; a control equation group and a source iteration method based on the neutron diffusion characteristic value equation group are used. The decoupling idea is used to construct a decoupled loss function containing the characteristic function information of the previous step, and at the same time construct the characteristic value in the form of Rayleigh quotient; the weighted sum form of the loss function formed by the boundary point set and the interface point set and the decoupled loss function is used to optimize the neural network using an optimization algorithm. When the loss function reaches the preset condition or the neural network completes a given number of training times, a numerical result is obtained. This method absorbs the iterative solution method of the neutron diffusion characteristic value equation source, inherits the characteristic function information of the previous step in the form of iteration, and constructs a new decoupled residual loss function, which greatly improves the solution accuracy of the multi-group neutron diffusion characteristic value problem; the calculation accuracy is significantly better than the unimproved PC-GIPMNN method for solving the neutron diffusion characteristic value problem. The problem of large errors in solving the multi-group neutron diffusion characteristic value of the existing generalized inverse power method neural network based on physical constraints is solved. BRIEF DESCRIPTION OF THE DRAWINGS

[0018] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings required for use in the embodiments or the description of the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative work.

[0019] Figure 1 It is a schematic diagram of the computational domain of the two-group neutron diffusion eigenvalue equation for the 1-D Swedish Ringhals-4 pressurized water reactor problem.

[0020] Figure 2 This is the distribution diagram of the coefficients of the two-group neutron diffusion eigenvalue equation for the 1-D Swedish Ringhals-4 pressurized water reactor problem.

[0021] Figure 3 It is a schematic diagram of key parameter settings of a method for solving the eigenvalue problem of a multi-group neutron diffusion partial differential equation system using a decoupled residual loss function (DRLF) provided by the present invention.

[0022] Figure 4 The present invention provides a computational flow chart of a method for solving the eigenvalue problem of a multi-group neutron diffusion partial differential equation system using a decoupled residual loss function (DRLF).

[0023] Figure 5 It is a comparison diagram of the calculation process of a method for solving the eigenvalue problem of multi-group neutron diffusion partial differential equations using a decoupled residual loss function (DRLF) provided by the present invention and a traditional source iteration method for the eigenvalue problem of multi-group neutron diffusion.

[0024] Figure 6 is the prediction result of Example 1 of the present invention, the reference solution and the point-by-point error image, (a)φ 1 Prediction results, (b)φ 1 Reference solution, (c)φ 1 Pointwise error, (d)φ 2 Prediction results, (e)φ 2 Reference solution and (f)φ 2 Point-by-point error.

[0025] Figure 7 is the prediction result of Example 2 of the present invention, the reference solution and the point-by-point error image, (a)φ 1 Prediction results, (b)φ 1 Reference solution, (c)φ 1 Pointwise error, (d)φ 2 Prediction results, (e)φ 2 Reference solution and (f)φ 2 Point-by-point error

[0026] Figure 8 is a numerical comparison of the calculation results of Example 1, Example 2 and PC-GIPMNN. σ represents the different displacements used in different examples, E(k eff ) represents the relative error of the effective multiplication coefficient, E(φ g ) R,∞ represents the relative infinite norm error of the g-th energy group, E(φ g ) R,2 represents the relative 2-norm error of the g-th energy group.

[0027] Fig. 9 The present invention provides a flowchart of a method for solving the eigenvalue problem of a multi-group neutron diffusion partial differential equation system by using a decoupled residual loss function (DRLF). DETAILED DESCRIPTION

[0028] Embodiments of the present invention are described in detail below, examples of which are shown in the accompanying drawings, wherein the same or similar reference numerals throughout represent the same or similar elements or elements having the same or similar functions. The embodiments described below with reference to the accompanying drawings are exemplary and are intended to be used to explain the present invention, and should not be construed as limiting the present invention.

[0029] See also Figures 1 to 9 The present invention provides a method for solving multi-group neutron diffusion eigenvalues ​​by using a decoupled residual loss function, comprising the following steps:

[0030] S1 randomly selects points in the calculation area and divides them into boundary point set, interface point set and residual point set according to the position of the sampling points;

[0031] In the embodiment of the present invention, the neutron diffusion characteristic value equation of the dual group and its calculation area are determined, and the boundary points, interface points and residual points are generated randomly and uniformly.

[0032] S2 determines the number of neurons in the output layer of the neural network according to the number of neutron fluxes in the energy group to be determined in the equation and the number of materials in the calculation area, gives an initial guess about the characteristic function, and gives an initial characteristic value based on the Rayleigh quotient form;

[0033] In an embodiment of the present invention, the number of neurons n*P in the output layer of the neural network is determined according to the number n of neutron fluxes in the energy group to be determined in the equation and the number P of materials in the calculation area, an initial guess about the characteristic function is given, and an initial eigenvalue is given in the form of Rayleigh quotient, wherein the initial guess about the characteristic function is a constant function whose values ​​corresponding to all calculation area points of all characteristic functions are 1.

[0034] The output of the S3 neural network is squared, and the neurons in the output layer are weighted summed through the indicative function to obtain the numerical solution of the neural network;

[0035] S4 constructs the corresponding loss function based on the interface conditions and boundary conditions of the neutron diffusion eigenvalue equation group;

[0036] In an embodiment of the present invention,

[0037]

[0038] The corresponding interface conditions are:

[0039]

[0040] Among them, p≠q, the values ​​of p and q represent the areas corresponding to different materials. Represents the interface between different materials. The above equation involves the following parameters: Ω is the calculation area, is the external boundary, It is a symmetric boundary, where the boundary condition corresponding to the external boundary can be determined as a Dirichlet boundary condition or a Robin boundary condition depending on the specific problem; g is the diffusion coefficient of the g-th energy group; Σ a,g is the absorption cross section of the g-th energy group; Σ g'→g is the scattering cross section from the g'th group to the gth group; g is the neutron fission spectrum of the gth energy group; νΣ f,g is the neutron production cross section of the g-th energy group; φ g is the neutron flux of the gth energy group, k eff is the effective multiplication coefficient.

[0041] In this system of partial differential equations, φ g and k eff is unknown, and 1 / k eff is the eigenvalue of the characteristic partial differential equation, φ g Then the corresponding characteristic function becomes. In nuclear reactor physics, it is usually required to find the minimum eigenvalue of the equation (i.e. the maximum value of the effective multiplication coefficient) and the corresponding characteristic function.

[0042] The specific interface loss function and boundary loss function are:

[0043]

[0044] S5 constructs a decoupling loss function containing the characteristic function information of the previous step based on the control equations of the neutron diffusion characteristic value equations and the decoupling idea of ​​the source iteration method, and constructs the characteristic value in the form of Rayleigh quotient;

[0045] In this embodiment of the present invention, the specific decoupled residual loss function is:

[0046]

[0047] The specific Ruili quotient form characteristic value is:

[0048]

[0049] S6 optimizes the neural network using an optimization algorithm based on the weighted sum of the loss function formed by the boundary point set and the interface point set and the decoupling loss function, and obtains a numerical result when the loss function reaches a preset condition or the neural network completes a given number of training times.

[0050] In the embodiment of the present invention, the initial guess of the characteristic function is:

[0051]

[0052] The final loss function form is:

[0053] Loss = α De Loss Decoupling +α Int (Loss Int0 +Loss Int1 )

[0054] +α Bou (Loss Dirichlet +Loss Robin +Loss Neumann ),

[0055] where α De , α Int and α Bou Represents the proportion of different types of losses in the total losses.

[0056] The working principle is: the previous model that uses neural network to solve the neutron diffusion eigenvalue problem without data drive has low accuracy if solving the neutron diffusion eigenvalue problem of multiple groups. In fact, it is because in the process of inverse power method iteration, due to the morbid nature of the right side of the equation group, the explicit previous iteration information of the neutron flux of certain energy groups disappears during the iteration process. The present invention adopts a new design concept, designs a decoupled residual loss function for the neutron flux iteration information of all energy groups, and explicitly passes the iteration information of each energy group into the next iteration process, which greatly reduces the impact of the morbid nature of the right side of the multi-group neutron diffusion problem, improves the accuracy of the model solution, and the calculation accuracy is obviously better than the previous PC-GIPMNN method, which has high practical engineering application value.

[0057] In order to better understand the present technical solution, the following embodiments are provided for further explanation:

[0058] Example 1

[0059] The purpose of this embodiment is to test the effectiveness of the neural network method of solving the multi-group neutron diffusion eigenvalue problem using a decoupled residual loss function (DRLF) on the 1-D Swedish Ringhals-4 pressurized water reactor problem. This embodiment includes:

[0060] S101: Determine the neutron diffusion eigenvalue equation and its calculation area for the dual group

[0061] Specifically, the neutron diffusion characteristic value equation of the double group is as follows:

[0062]

[0063] The calculation area is shown in the figure below. Due to symmetry, we only need to take the positive half axis of x as the entire calculation area, implement the symmetric boundary conditions (Neumann boundary conditions) at the origin, and implement the Robin boundary conditions at the endpoints.

[0064] S102: Determine that the number of neutron fluxes in the energy group to be solved in the equation is 2, determine that the number of materials in the calculation area is 2, and therefore determine that the number of neurons in the output layer of the neural network is 2*2; then give an initial guess about the characteristic function and eigenvalue.

[0065] The specific initial guess is as follows:

[0066]

[0067] S103: The output of the neural network will be squared, and the neurons in the output layer will be weighted and summed through the indicative function to obtain the final neural network numerical solution.

[0068] Specifically:

[0069] in It is an indicative function, which means that in the jth region, the function takes the value of 1, otherwise it takes the value of 0. The mathematical definition is as follows:

[0070]

[0071] Ψ g,j It represents the function in the output layer of the neural network related to the value of the g-th energy group in the j-th region.

[0072] S104: Construct the corresponding loss function based on the interface conditions and boundary conditions of the neutron diffusion eigenvalue equation group; construct the loss function at the interface and boundary according to the initial guess about the eigenfunction obtained in S102, the initial eigenvalue given in the form of Rayleigh quotient and the neural network numerical solution obtained in step S103.

[0073] Specifically, the loss function is as follows:

[0074]

[0075] S105: Based on the control equations of the neutron diffusion eigenvalue equation group and the decoupling idea of ​​the source iteration method, a decoupled residual loss function containing the eigenfunction information of the previous step is constructed, the displacement σ=0 of the eigenvalue is selected, and the eigenvalue in the form of Rayleigh quotient is constructed at the same time.

[0076] Specifically, the decoupled residual loss function is as follows:

[0077]

[0078] The eigenvalue expression is as follows:

[0079]

[0080] S106: Optimize the neural network using an optimization algorithm based on the weighted sum of the loss function formed by the boundary point set and the interface point set and the decoupling loss function, and obtain a numerical result when the loss function reaches a preset condition or the neural network completes a given number of training times.

[0081] Example 2

[0082] The purpose of this embodiment is to test the effectiveness of the neural network method of solving the multi-group neutron diffusion eigenvalue problem using the decoupled residual loss function (DRLF) on the 1-D Swedish Ringhals-4 pressurized water reactor problem. This embodiment includes:

[0083] S201: Determine the neutron diffusion eigenvalue equation and its calculation area for the dual group

[0084] Specifically, the neutron diffusion characteristic value equation of the double group is as follows:

[0085]

[0086] The calculation area is shown in the figure below. Due to symmetry, we only need to take the positive half axis of x as the entire calculation area, implement the symmetric boundary conditions (Neumann boundary conditions) at the origin, and implement the Robin boundary conditions at the endpoints.

[0087] S202: Determine that the number of neutron fluxes in the energy group to be solved in the equation is 2, determine that the number of materials in the calculation area is 2, and therefore determine that the number of neurons in the output layer of the neural network is 2*2; then give an initial guess about the characteristic function and eigenvalue.

[0088] The specific initial guess is as follows:

[0089]

[0090] S203: The output of the neural network will be squared, and the neurons in the output layer will be weighted and summed through the indicative function to obtain the final neural network numerical solution.

[0091] Specifically:

[0092]

[0093] in It means that in the jth region, the function takes the value of 1, otherwise it takes the value of 0. The mathematical definition is as follows:

[0094]

[0095] Ψ g,j It represents the function of the g-th energy group in the j-th region in the output layer of the neural network.

[0096] S204: Constructing corresponding loss functions based on interface conditions and boundary conditions of the neutron diffusion eigenvalue equation group; constructing loss functions at interfaces and boundaries according to the initial guess about the eigenvalue obtained in S202, the initial eigenvalue given in the form of Rayleigh quotient, and the numerical solution of the neural network obtained in step S203.

[0097] Specifically, the loss function is as follows:

[0098]

[0099]

[0100] S205: Based on the control equations of the neutron diffusion eigenvalue equation group and the decoupling idea of ​​the source iteration method, a decoupled residual loss function containing the eigenfunction information of the previous step is constructed, the displacement of the eigenvalue σ=1 is selected, and the eigenvalue in the form of Rayleigh quotient is constructed at the same time.

[0101] Specifically, the decoupled residual loss function is as follows:

[0102]

[0103] The eigenvalue expression is as follows:

[0104]

[0105] S206: Optimize the neural network using an optimization algorithm based on the weighted sum of the loss function formed by the boundary point set and the interface point set and the decoupling loss function, and obtain a numerical result when the loss function reaches a preset condition or the neural network completes a given number of training times.

[0106] What is disclosed above is only a preferred embodiment of the method of solving multi-group neutron diffusion eigenvalues ​​using a decoupled residual loss function of the present invention. Of course, this cannot be used to limit the scope of rights of the present invention. Ordinary technicians in this field can understand that all or part of the processes of the above embodiments and equivalent changes made according to the claims of the present invention still fall within the scope of the invention.

Claims

1. A method for solving multi-group neutron diffusion eigenvalues ​​using a decoupled residual loss function, characterized in that: The following steps are involved: Randomly select points on the calculation area and divide them into boundary point set, interface point set and residual point set according to the position of the sampling points; According to the number of neutron fluxes in the energy group to be determined in the equation and the number of materials in the calculation area, the number of neurons in the output layer of the neural network is determined, an initial guess about the characteristic function is given, and an initial characteristic value is given based on the Rayleigh quotient form; The output of the neural network is squared, and the neurons in the output layer are weighted summed through the indicative function to obtain the numerical solution of the neural network; The corresponding loss function is constructed based on the interface conditions and boundary conditions of the neutron diffusion eigenvalue equation group; Based on the decoupling idea of ​​the control equations of the neutron diffusion eigenvalue equations and the source iteration method, a decoupling loss function containing the information of the eigenfunction of the previous step is constructed, and at the same time, the eigenvalue in the form of Rayleigh quotient is constructed; The neural network is optimized using an optimization algorithm based on a weighted sum of a loss function formed by a set of boundary points and a set of interface points and a decoupling loss function, and a numerical result is obtained when the loss function reaches a preset condition or the neural network completes a given number of training times.

2. The method for solving multi-group neutron diffusion eigenvalues ​​using a decoupled residual loss function as claimed in claim 1, characterized in that: The neural network is a fully connected neural network with added skip connections, and automatic differentiation technology is used in all differentiation operations of the neural network.

3. The method for solving multi-group neutron diffusion eigenvalues ​​using a decoupled residual loss function as claimed in claim 1, characterized in that: The output of each output layer neuron is subjected to a square operation to ensure the non-negativity of the neuron output, so as to satisfy the non-negative property of the characteristic function of the neutron diffusion eigenvalue problem in the critical state.

4. The method for solving multi-group neutron diffusion eigenvalues ​​using a decoupled residual loss function as claimed in claim 1, characterized in that: The neutron diffusion characteristic value equation group is as follows: And G≥2.

5. The method for solving multi-group neutron diffusion eigenvalues ​​using a decoupled residual loss function as claimed in claim 1, characterized in that: The given initial guess about the characteristic function is a constant function whose values ​​corresponding to all calculation area points of all characteristic functions are 1.

Images