Deep learning numerical solution method for anisotropic scattering neutron transport equation

Abstract

The invention particularly relates to a deep learning numerical solution method for an anisotropic scattering neutron transport equation, belongs to the technical field of neutron transport equation solution, and solves the anisotropic scattering neutron transport equation through multi-primitive function variable order iteration deep learning. Comprising the following steps: step 1, establishing an expression of a neutron transport equation according to a geometric form and boundary conditions of a scattering source; step 2, performing differential order change on each integral item in the neutron transport equation into an original function corresponding to an integrand function under an integral number to obtain a differential equation, and establishing a definite solution constraint condition of the original function; and step 3, respectively mapping an original function and an unknown angular flux density function in the neutron transport equation into neural network functions, solving a framework of a differential equation based on deep learning, and carrying out alternate iteration deep learning until a loss function is smaller than a preset value, so as to obtain a numerical solution of the unknown angular flux density function. The invention provides a new technical approach for the numerical solution method of the anisotropic scattering neutron transport equation.

Description

Technical Field

[0001] This invention relates to the field of neutron transport equation solving technology, and in particular to a deep learning numerical solution method for anisotropic scattering neutron transport equations. Background Technology

[0002] The neutron transport equation is a fundamental equation in nuclear reactor physics. The main purpose of solving the neutron transport equation is to determine the spatiotemporal characteristics of neutron flux density in the reactor core, which plays a crucial role in reactor engineering design, development, and operational support. Traditional theoretical methods for solving the neutron transport equation include deterministic methods such as the Discrete Scalar Method (SN) and the Method of Characteristics (MOC), as well as statistical methods such as the Monte Carlo method. These methods each have their own characteristics and complement each other. Although they have been successfully applied extensively, they still have certain shortcomings in terms of accuracy and efficiency. How to further improve and enhance the performance of numerical calculation methods for the neutron transport equation has always been a key research area in the field.

[0003] In recent years, using artificial intelligence deep learning methods to solve complex partial differential equations has become a cutting-edge research hotspot in the field of computational physics. In the field of reactor physics, deep learning-based numerical computation methods have also made a series of advances, which have laid a solid foundation for the deep learning numerical solution of neutron transport equations.

[0004] However, due to the differential-integral nature of the neutron transport equation, which contains multiple integral terms representing neutron scattering and fission sources in terms of energy and angular flux density, directly applying existing deep learning methods to solve the neutron transport equation presents a significant challenge. To address this issue, some researchers, referencing traditional SN methods, discretize the angle variable of the integrand function, construct an approximate discrete sum of definite integral terms using Gaussian quadrature, build a loss function for machine learning, and provide discrete-time and angular flux density calculation results for one-dimensional geometry in the transient case; however, their accuracy is not high. Furthermore, many current theories only address the simplified form of the neutron transport equation when the scattering and fission sources are isotropic, and cannot be applied to the anisotropic neutron scattering sources commonly encountered in practical engineering. Solving the anisotropic scattering neutron transport equation requires further in-depth research. Summary of the Invention

[0005] The purpose of this invention is to provide a deep learning numerical solution method for the anisotropic scattering neutron transport equation. This method involves changing the order of the antiderivatives for different forms of integral terms in the neutron transport equation, transforming the neutron transport equation into a pure differential equation represented by multiple antiderivatives. Based on this, different neural network functions are used to represent these antiderivatives and the unknown angular flux density. Deep learning is performed through an alternating iterative method to gradually approximate the numerical solution of the neutron transport equation, thus providing a new technical approach for the numerical solution of the anisotropic scattering neutron transport equation.

[0006] To achieve the above objectives, the present invention provides the following technical solution:

[0007] A deep learning numerical solution method for the anisotropic scattering neutron transport equation is proposed. This method utilizes multi-prime function variable-order iterative deep learning to solve the anisotropic scattering neutron transport equation, and includes the following steps:

[0008] Step 1: Establish the expression for the neutron transport equation based on the geometry and boundary conditions of the scattering source;

[0009] Step 2: Differentiate each integral term in the neutron transport equation and change its order to the antiderivative corresponding to the integrand under the integral sign to obtain the differential equation, and establish the boundary value constraint conditions of the antiderivative.

[0010] Step 3: Map the original function and the unknown angular flux density function in the neutron transport equation to neural network functions respectively. Based on the framework of solving differential equations using deep learning, perform alternating iterative deep learning until the loss function is less than the preset value, and obtain the numerical solution of the unknown angular flux density function.

[0011] In this invention, in step 2, the Newton-Leibniz formula is used to transform the definite integral term into the subtraction of the constant values ​​of the antiderivative at the upper and lower limits of integration.

[0012] In this invention, in step 2, the Newton-Leibniz formula is used to transform the indefinite integral term into the subtraction of the function values ​​of the original function at the corresponding variables of the upper and lower limits of integration.

[0013] In this invention, in step 2, when the neutron transport equation has multiple integral terms, multiple antiderivatives are used to represent different integral terms, and in step 3, a neural network function is established for each antiderivative.

[0014] In this invention, in step 2, a boundary value constraint is established for the integrand antiderivative, such that the function value of the antiderivative is a constant at the upper or lower limit of integration.

[0015] In this invention, in step 2, the constant value of the integrand antiderivative at the lower or upper limit of integration is set to 0, such that the expression of the antiderivative expressed by the upper and lower limits of integration is equal to that with respect to the definite integral term.

[0016] In this invention, step 3 includes:

[0017] Step 30: Based on the physical information neural network or other neural network models, establish neural network functions for the original function and the angular flux density function, respectively;

[0018] Step 31: Substitute the neural network function as the trial function of the angular flux density function and the original function into the differential equation to obtain the loss function of the control equation. Establish the loss function of the original function transformation equation based on the transformation between the integrand and the corresponding original function. Establish the loss function of the original function boundary constraint based on the boundary constraint of the original function. Establish the boundary condition loss function.

[0019] Step 32: Establish a unified loss function based on the loss function of the governing equation, the loss function of the original function transformation equation, the loss function of the original function boundary condition constraint, and the loss function of the boundary condition.

[0020] Step 33: Based on the unified loss function in step 32, adjust the neural network functions corresponding to the angular flux density function and the original function respectively, and perform deep machine learning alternately. When the unified loss function approaches the minimum value, the neural network approaches the numerical solution of the angular flux density function and the original function.

[0021] In this invention, in step 32, different weights are set for the unified loss function established for the control equation loss function, the original function transformation equation loss function, the original function boundary constraint loss function, and the boundary condition loss function, thereby constructing a unified loss function.

[0022] In this invention, in step 33, deep learning is performed alternately and iteratively on the neural network functions corresponding to the angular flux density function and the original function to gradually approximate the angular flux density function and the original function.

[0023] In this invention, during the deep machine learning process with a unified loss function as the target, the weight values ​​of the control equation loss function, the original function transformation equation loss function, the original function boundary constraint loss function, and the boundary condition loss function are continuously adjusted to reduce the possibility of getting trapped in local optima until the weighted unified loss function approaches its minimum value.

[0024] Beneficial technical effects of the present invention:

[0025] This invention presents a deep learning numerical solution method for the anisotropic scattering neutron transport equation. This method utilizes multi-antiderivative order-variable iterative deep learning to solve the anisotropic scattering neutron transport equation, avoiding systematic errors introduced by converting integral terms into finite sums and addressing deep learning convergence issues. Specifically, by transforming the order of antiderivatives in different forms of integral terms in the neutron transport equation, the differential-integral form of the neutron transport equation is transformed into a pure differential equation represented by multiple antiderivatives, and constraints are established to ensure the existence of definite solutions. Simultaneously, multiple deep neural networks are used to map the unknown angular flux density function and various forms of antiderivatives of the neutron transport equation, and corresponding weighted loss functions are constructed. Then, deep learning methods are used to iteratively machine learn multiple neural network functions, gradually reducing the loss function. When the loss function is reduced to a preset minimum value, the neural network approaches the numerical solution of the angular flux density and antiderivatives. Finally, experimental examples are used to verify the principle, thus exploring a new technical approach for deep learning solutions to the anisotropic scattering neutron transport equation. Attached Figure Description

[0026] Figure 1 A flowchart illustrating an embodiment of a deep learning numerical solution method for the anisotropic scattering neutron transport equation;

[0027] Figure 2(a) is a schematic diagram of the calculated angular flux density φ(x,μ) distribution of the single-group plate geometric anisotropic neutron transport equation;

[0028] Figure 2(b) shows the antiderivative of the geometric anisotropic neutron transport equation for a single-group flat plate. A schematic diagram of the distribution calculation results;

[0029] Figure 2(c) shows the antiderivative of the geometrically anisotropic neutron transport equation for a single-group flat plate. A schematic diagram of the distribution calculation results;

[0030] Figure 2(d) shows the antiderivative of the geometrically anisotropic neutron transport equation for a single-group flat plate. A schematic diagram of the distribution calculation results;

[0031] Figure 2(e) is a scatter plot of the central normalized standard flux density of the geometric anisotropic neutron transport equation for a single-group flat plate. Detailed Implementation

[0032] Unless otherwise defined, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this application pertains; the terminology used herein in the specification is for the purpose of describing particular embodiments only and is not intended to limit the application; the terms “comprising” and “including”, and any variations thereof, in the specification, claims and foregoing description of the drawings are intended to cover non-exclusive inclusion.

[0033] In this document, the term "embodiment" means that a particular feature, structure, or characteristic described in connection with an embodiment may be included in at least one embodiment of this application. The appearance of this phrase in various places throughout the specification does not necessarily refer to the same embodiment, nor is it a separate or alternative embodiment mutually exclusive with other embodiments. It will be explicitly and implicitly understood by those skilled in the art that the embodiments described herein can be combined with other embodiments.

[0034] The technical solution of the present invention will be clearly and completely described below with reference to the accompanying drawings and specific embodiments.

[0035] This invention provides a deep learning numerical solution method for the anisotropic scattering neutron transport equation. The method utilizes multi-prime variable-order iterative deep learning to solve the anisotropic scattering neutron transport equation, comprising the following steps:

[0036] Step 1: Establish the expression for the neutron transport equation based on the geometry and boundary conditions of the scattering source;

[0037] Step 2: Differentiate each integral term in the neutron transport equation and change its order to the antiderivative corresponding to the integrand under the integral sign to obtain the differential equation, and establish the boundary value constraint conditions of the antiderivative.

[0038] Step 3: Map the original function and the unknown angular flux density function in the neutron transport equation to neural network functions respectively. Based on the framework of solving differential equations using deep learning, perform alternating iterative deep learning until the loss function is less than the preset value, and obtain the numerical solution of the unknown angular flux density function.

[0039] In this invention, in step 2, the Newton-Leibniz formula is used to transform the definite integral term into the subtraction of the constant values ​​of the antiderivative at the upper and lower limits of integration.

[0040] In this invention, in step 2, the Newton-Leibniz formula is used to transform the indefinite integral term into the subtraction of the function values ​​of the original function at the corresponding variables of the upper and lower limits of integration.

[0041] In this invention, in step 2, when the neutron transport equation has multiple integral terms, multiple antiderivatives are used to represent different integral terms, and in step 3, a neural network function is established for each antiderivative.

[0042] In this invention, in step 2, a boundary value constraint is established for the integrand antiderivative, such that the function value of the antiderivative is a constant at the upper or lower limit of integration.

[0043] In this invention, in step 2, the constant value of the integrand antiderivative at the lower or upper limit of integration is set to 0, such that the expression of the antiderivative expressed by the upper and lower limits of integration is equal to that with respect to the definite integral term.

[0044] In this invention, step 3 includes:

[0045] Step 30: Based on the physical information neural network or other neural network models, establish neural network functions for the original function and the angular flux density function, respectively;

[0046] Step 31: Substitute the neural network function as the trial function of the angular flux density function and the original function into the differential equation to obtain the loss function of the control equation. Establish the loss function of the original function transformation equation based on the transformation between the integrand and the corresponding original function. Establish the loss function of the original function boundary constraint based on the boundary constraint of the original function. Establish the boundary condition loss function.

[0047] Step 32: Establish a unified loss function based on the loss function of the governing equation, the loss function of the original function transformation equation, the loss function of the original function boundary condition constraint, and the loss function of the boundary condition.

[0048] Step 33: Based on the unified loss function in step 32, adjust the neural network functions corresponding to the angular flux density function and the original function respectively, and perform deep machine learning alternately. When the unified loss function approaches the minimum value, the neural network approaches the numerical solution of the angular flux density function and the original function.

[0049] In this invention, in step 32, different weights are set for the unified loss function established for the control equation loss function, the original function transformation equation loss function, the original function boundary constraint loss function, and the boundary condition loss function, thereby constructing a unified loss function.

[0050] In this invention, in step 33, deep learning is performed alternately and iteratively on the neural network functions corresponding to the angular flux density function and the original function to gradually approximate the angular flux density function and the original function.

[0051] In this invention, during the deep machine learning process with a unified loss function as the target, the weight values ​​of the control equation loss function, the original function transformation equation loss function, the original function boundary constraint loss function, and the boundary condition loss function are continuously adjusted to reduce the possibility of getting trapped in local optima until the weighted unified loss function approaches its minimum value.

[0052] In step 1, the neutron transport equation has several equivalent mathematical forms. The most widely studied and applied form is the differential-integral form of the neutron transport equation, which is expressed as follows:

[0053]

[0054] In the formula, φ(r,Ω,E,t) is the neutron angular flux density at position r, angle Ω, and energy E at time t, Σ t Let v be the total reaction cross section, v be the neutron velocity, and Q be the total reaction cross section. s Q f Q e These are anisotropic scattering sources, fission sources, and external sources, respectively.

[0055] Among them, the anisotropic scattering source Q s for:

[0056] Q s =∫∫Σ s (r,E')f(r,Ω',E'→Ω,E)φ(r,Ω',E',t)dΩ'dE' (2)

[0057] Anisotropic fission source Q f for:

[0058]

[0059] In the formula, f(r,Ω',E'→Ω,E) is the scattering function; Σ s For the scattering cross section, Σ f Let ν be the fission cross section, ν be the number of neutrons produced in each fission, and χ(E) be the energy spectrum distribution of the fission neutrons.

[0060] The common form of the neutron transport equations for discrete energy groups under steady-state conditions, without considering external sources, is as follows:

[0061]

[0062] In the formula, k eff is the effective multiplication coefficient; g is the segmented energy group number.

[0063] Meanwhile, the anisotropic scattering source (2) is usually expanded using Legendre polynomials, and its multigroup form of anisotropic scattering source can be expressed as:

[0064]

[0065] In the formula, the neutron flux density moment is expressed as: It is a spherical harmonic function.

[0066] In practical applications, the specific form of the neutron transport equation is usually given by combining factors such as the geometric form of the equation, the number of discrete energy groups, the order of the anisotropic scattering source, and whether there is an external source.

[0067] The specific form of equation (4) for a multi-group flat plate geometric first-order anisotropic scattering source is as follows:

[0068]

[0069] In the formula, x is the thickness of the plate, and μ is the direction cosine in rectangular coordinates.

[0070] The specific form of equation (4) for a first-order anisotropic scattering source in a single-group spherical geometry is as follows:

[0071]

[0072] In the formula, r is the geometric radius of the sphere.

[0073] For neutron transport equations of different forms such as (4), (6), and (7), by shifting the terms of the equations from one side of the equation to the other, they can be described as differential-integral equations in a unified functional form:

[0074]

[0075] In step 2, the integral term in the neutron transport equation includes the scattering source term; for equation (2), the integral term of the scattering source term can be expressed as:

[0076]

[0077] in:

[0078]

[0079] In the formula, Ω1 and Ω0 are the upper and lower limits of the integral of the angle variable, respectively;

[0080] According to the principles of calculus, F s,Ω (r,Ω,E,Ω′,E′,t) is a set of functions that differ by any real number C. Let F c,s,Ω (r,Ω,E,Ω′,e′,t) is a definite function, and F s,Ω (r,Ω,E,Ω′,E′,t)=F c,s,Ω (r,Ω,E,Ω′,E′,t)+C;

[0081] Let C = -F c,s,Ω If (r,Ω,E,Ω0,E′,t), then there exists a specific antiderivative F. 0,s,Ω (r,Ω,E,Ω0,e′,t)=F c,s,Ω (r,Ω,E,Ω′,E′,t)-F c,s,Ω(r,Ω,E,Ω0,E′,t), and when Ω′=Ω0, F 0,s,Ω (r,Ω,E,Ω0,E′,t)=0;

[0082] When this constraint condition is met, combining with equation (9), we get:

[0083]

[0084] Similarly, by the principle of repeated integration, substituting equation (11) into equation (2), the scattering source term can be expressed as:

[0085]

[0086] in:

[0087]

[0088] In the formula, E1 and E0 are the upper and lower limits of the energy variable integral, respectively;

[0089] Similarly, F s,E Given a set of functions (r, Ω, E, Ω1, E′, t) that differ by a real number C, it is still possible to find a definite function F. 0,s,E (r,Ω,E,Ω1,E′,t), when E′=E0, F 0,s,E When (r,Ω,E,Ω1,E0,t)=0, we have:

[0090]

[0091] In step 2, the integral term in the neutron transport equation includes the fission source term; for the fission source term in equation (3), we have:

[0092]

[0093] in:

[0094]

[0095] As before, there exists a definite function F. 0,f,Ω (r,Ω′,E′,t), when Ω′=Ω0, F 0,f,Ω (r,Ω0,E′,t)=0 serves as a constraint for the definite solution of the antiderivative. That is, when this condition is satisfied, we have:

[0096]

[0097] Similarly, the fission source term can be derived as follows:

[0098]

[0099] and:

[0100]

[0101] As before, a definite function F can still be found. 0,f,E (r,Ω1,E′,t), when E′=E0, F 0,f,E (r,Ω1,E0,t)=0, under this condition:

[0102]

[0103] Therefore, substituting equations (14) and (20) into equation (1) yields:

[0104]

[0105] The boundary condition constraints are:

[0106]

[0107] In step 2, the neutron transport equation is transformed from differential-integral form to pure differential form by transforming the original functions of the scattering and fission terms. The differential form of the neutron transport equation is a multigroup form composed of equations (10), (13), (16), (19), (21) and (22).

[0108] In step 2, for the neutron transport equation (4) in the multi-group form, the neutron flux density moment contained in the scattering source term (5) is transformed by the original function. Since the angle variables Ω and Ω′ are separated, the transformation form is greatly simplified.

[0109] make for If the original function is , then we have:

[0110]

[0111] and:

[0112]

[0113] As before, there exists a specific function. When Ω′=Ω0 At this point, we can conclude:

[0114]

[0115] Substituting equation (25) into equation (5), we can obtain the scattering source term in the multigroup form:

[0116]

[0117] Similarly, for the integral term ∫φ contained in the fission source term in equation (4)g′ Let F f,g′ (r,Ω′) is φ g′ The antiderivative of (r,Ω′) is given by:

[0118]

[0119] Under these conditions:

[0120]

[0121] The definite solution constraint is: when Ω′=Ω0, we have F 0,f,g′ (r,Ω0)=0;

[0122] Therefore, substituting equations (26) and (27) into the multigroup neutron transport equation (4) yields:

[0123]

[0124] Its definite solution condition is: when Ω′=Ω0, F 0,f,g′ (r,Ω0)=0;

[0125] Therefore, equations (24), (28) and (29) constitute the multigroup neutron transport equations in fully differential form.

[0126] In step 2, when Ω is a binary variable, the direction cosine μ and the argument are used. Let Ω be the flux density, and the double integral of the flux density with respect to the angle can be expressed as:

[0127]

[0128] Similarly, the integral term can be transformed into an antiderivative step by step using the method of multiple integral antiderivative transformation. Taking the isotropic multi-group fission source term in equation (4) as an example, the multi-group fission integral term in equation (27) can be expressed as follows:

[0129]

[0130] and,

[0131] The boundary value constraint is: when μ = μ0, when hour,

[0132] For higher-order scattering multigroup forms with two-dimensional angle variables, the antiderivative transformation under the multigroup form can be referenced, and the higher-order scattering antiderivative is defined as follows: The flux density moment can then be expressed as:

[0133]

[0134] At this point, the boundary value constraints are:

[0135] and:

[0136]

[0137] Other forms of two-dimensional angular variable neutron transport equations can be deduced by analogy.

[0138] In step 2, by transforming the antiderivatives of the integral terms, the neutron transport equation, which has a calculus form, is transformed into a pure differential equation composed of multiple antiderivative terms, the standard form of which is as follows:

[0139]

[0140] At this point, there exist original function transformation equations (24) and (28), and at the same time, there is a set of corresponding boundary value constraints;

[0141] For equation (34) of a practical problem, if multiple integral terms have the same form, they can be combined and simplified:

[0142] use [F 0,f,g′ [r,Ω1] respectively represent F 0,f,g′ (r,Ω1) does not include the constant term; for example, in equation (25), if the neutron angular flux density moment Included There is a constant term. If the constant is moved outside the integral sign and then transformed, then (24) can be transformed into... At the same time, using Denotes a set of non-repeating elements, i.e., if [F 0,f,g′ Since terms of the same form exist in [r,Ω1], they should be combined. Therefore, equation (34) can be written in simplified form:

[0143]

[0144] In step 3, equations (21), (29), and (34) of the neutron transport equations after the original function transformation are pure differential equations. The subsequent solution is based on various fundamental theories and methods for solving differential equations using deep learning. A crucial step is to use a neural network function as a trial function, substituting it into the neutron transport control equations, the original function transformation equations, the boundary constraints, and the boundary / initial condition constraints to generate the loss function required for machine learning. For the standard form (34), the fully connected form of the deep neural network represents the function as follows:

[0145] N(x)=f(x,w,b,l,n) (36)

[0146] In the formula, x is the neural network input vector, N(x) is the neural network output, w is the neural network connection weight; b is the neural network bias term, l is the network depth, n is the number of hidden neurons, and f is the activation function.

[0147] Furthermore, neural networks can also have multiple outputs, which can often be used to represent a set of functions with similar forms. For example, for multi-group transport equations, different outputs of the same neural network can be used to represent the angular flux density of different energy groups.

[0148] (a) Loss function of neutron transport control equation

[0149] make Where the neural network vector x r ,x Ω Let r, Ω, and x represent the spatial and angular variables of the neutron transport equation, respectively. r x Ω The dimensions should be adaptively adjusted according to the specific form of the neutron transport equation.

[0150] Substituting these equivalent relations into equation (34), we obtain the transport equation loss function in residual form:

[0151]

[0152] In the formula, i represents the sample points required for machine learning, which can be generated according to a certain probability density distribution, or generated using different density distribution strategies based on the form of the equation.

[0153] It should be noted that when g = g′, N g The input and output representations of (r,Ω′) are given by equation (37) φ g (r,Ω′), N g′ (r,Ω′) represents φ in equation (24). g′ (r,Ω′). Similarly, equation (28) uses N. g (r,Ω′) represents N g′ (r,Ω′).

[0154] (b) Original function transformation equation and loss function

[0155] For the transformation equation of the scattering source integral term in equation (24), let The loss function formed by the residuals of the original function transformation equation is obtained as follows:

[0156]

[0157] Similarly, for the transformation equation of the fission source integral term in equation (28), let The loss function can then be expressed as:

[0158]

[0159] (c) Original function boundary value constraint loss function

[0160] For the boundary value constraint of the original function, it is equivalent to having a corresponding function value given the independent variable, which can be expressed as a data-driven loss function.

[0161] Regarding the boundary value constraint Ω′=Ω0 for the integral term of the scattering source (24), make Then we can obtain the boundary value constraint loss function of the scattering source integral term antiderivative:

[0162]

[0163] In the formula, j represents the machine learning sample point in the domain of the boundary constraint term.

[0164] Similarly, for the boundary value constraint Ω′=Ω0 of the fission source integral term (28), F 0,f,g′ (r,Ω0)=0, let Then we can obtain the boundary value constraint loss function of the fission source integral term antiderivative:

[0165]

[0166] In equations (40) and (41), The input variable x represents the angle of the neural network. Ω j sample points, all taking the value Ω0, corresponding to The sample point values ​​should cover F 0,f,g′ The range of all values ​​of variable r in (r,Ω0).

[0167] (d) Boundary condition loss function

[0168] The neutron transport equation requires additional boundary and initial conditions to have a solution. In this case, existing deep learning methods for solving neutron physics diffusion and transport equations can be referenced. Based on the various boundary condition forms of the equation, additional boundary / initial condition loss functions should be set for the corresponding neural network functions and their corresponding differential forms. If the equation includes a time term, appropriate initial conditions should be set. For multi-group transport equations, the following form exists:

[0169]

[0170] In the formula, Let ω be the angular flux density boundary function of energy group g, k be the machine learning sample point on the boundary domain, and r and Ω take values ​​at the boundary of the equation, the form of which is determined according to the specific form of the neutron transport equation, such as the vacuum boundary and the reflection boundary.

[0171] As an example, the following is an example form of the non-negative boundary loss function:

[0172]

[0173] (e) Weighted loss function

[0174] Using different neural network functions as the objects, weighted loss functions composed of the above loss functions are constructed respectively.

[0175] (1) Angle flux density weighted loss function

[0176] Neural Network for Angular Flux Density of Neutron Transport Equations By weighting the neutron transport equations and their boundary / initial condition loss functions, we obtain the corresponding weighted loss functions. If each neural network's different outputs represent the angular flux density of different energy groups, then the weighted loss functions for each group are:

[0177]

[0178] If multiple outputs of the same neural network function represent the angular flux density of different energy groups, then the overall loss function of this neural network is:

[0179]

[0180] In the formula, P φ P b These are the weights corresponding to flux density and boundary / initial condition loss functions, respectively, which can be determined based on experimental experience.

[0181] (2) Weighted loss function corresponding to the original function of the equation

[0182] Similarly, for different neural network functions Construct a weighted loss function by selecting the corresponding original function transformation equation and determining the solution constraint loss function.

[0183] Neural Network of Integral Term of Scattering Source The weighted loss function is:

[0184]

[0185] In the formula, P F,s P d,s These represent the transformation equation of the original function of the scattering source integral term and the weights of the corresponding deterministic solution loss function.

[0186] Fission source integral term antiderivative neural network The weighted loss function is:

[0187]

[0188] In the formula, P F,f P d,f These represent the transformation equation of the original function of the fission source integral term and the weights of the corresponding deterministic solution loss function.

[0189] If equations (46) and (47) are represented by multiple outputs of the same neural network function, they can be expressed as follows:

[0190]

[0191] In step 5, after transforming the neutron transport equation into a pure differential equation through the antiderivative and defining the loss function of each neural network, we can use automatic differentiation techniques to perform alternating iterative deep learning based on standard deep learning architectures such as PINN or various improved deep learning architectures for differential equations. This will gradually reduce the loss function of all neural networks to a minimum value. At this point, the output of the neural network representing the neutron flux density of the neutron transport equation is the numerical solution of the equation.

[0192] It should be noted that the effective proliferation coefficient k in equation (34) is calculated as follows. eff The traditional source iteration method can be used, with the method presented in this invention serving as the inner iteration process of the source iteration method. Alternatively, methods specifically designed for deep learning techniques, such as indirect methods for efficiently calculating the multiplication coefficient or direct search methods, can be employed. To focus on describing the theory of order change of the original function, this invention uses the traditional source iteration method for k... eff Please provide a solution.

[0193] For the multi-group neutron transport equation, the detailed deep learning process involves alternately adjusting the neural network function based on the determined weighted loss functions (44)-(49). In equation (36), w and b are used for iterative deep machine learning, resulting in various neural networks corresponding to... As the loss functions gradually decrease, when all these loss functions approach their minimum values, It is close to φ g The numerical solution of (r,Ω). In practice, specific machine learning convergence criteria are set based on various loss function values ​​and the number of machine learning iterations. For example, the machine learning process is completed after all loss functions are less than a pre-set minimum value ε; or a maximum number of machine learning iterations N is set, and training stops when this number is exceeded.

[0194] If a neural network has only one output, and a convergence criterion is based on pre-setting a minimum value for the loss function of multiple neural networks, the deep learning flowchart is as follows: Figure 1 As shown, the angular flux density function and the neural network loss functions corresponding to the original functions of the scattering source and fission source terms are respectively (44), (46), and (47).

[0195] The technical solution of the present invention will be clearly and completely described below with reference to the accompanying drawings and specific embodiments.

[0196] Example 1

[0197] This embodiment provides a deep learning numerical solution method for the anisotropic scattering neutron transport equation, applied to the anisotropic transport problem of a critical single-group flat plate geometric scattering source. ， Includes the following steps:

[0198] Step 1: Establish the expression for the neutron transport equation based on the geometry and boundary conditions of the scattering source;

[0199] Step 2: Differentiate each integral term in the neutron transport equation into multiple antiderivatives corresponding to the integrand under the integral sign, and establish the boundary value constraint conditions of the antiderivatives.

[0200] Step 3: Map the original function after differential transformation and the unknown angular flux density function in the neutron transport equation to neural network functions respectively. Based on the framework of solving differential equations using deep learning, perform alternating iterative deep learning until the loss function is less than the preset value, and obtain the numerical solution of the unknown angular flux density function.

[0201] In this embodiment, in step 1, the critical single-group plate is subject to vacuum boundary conditions on both sides, and the cross-section of the plate material is Σ. t =0.050cm -1 , Σ s0 =0.030cm -1 , Σ s1 =0.001705cm -1 ,νΣ f =0.0225cm -1 The critical half-thickness is b = 66.00527544 cm;

[0202] Based on the description of the anisotropic transport problem of a critical single-group plate geometry scattering source, the neutron transport equation under the critical single-group plate geometry is established:

[0203]

[0204] In this embodiment, in step 2, the neutron transport equation after differential order change is in the form of:

[0205]

[0206] The boundary value constraint condition of the original function is: μ0 = -1. and,

[0207] In this embodiment, in step 3, both the 0th-order scattering source and the fission source in formula (51) are isotropic, therefore... Using neural network N φ (x,μ) represents the angular flux density function φ(x,μ), and the 0th-order scattering source neural network. Equal to fission source neural network Simplified into a neural network This indicates that a first-order scattering source neural network uses... This indicates that a total of 3 neural networks were formed;

[0208] The machine learning method and parameter selection are as follows: the neural network is configured with 9 layers (l=9), 20 hidden neurons (n=20), tanh as the activation function, and Gaussian random distribution for initial network values. The ADAM machine learning algorithm is used; the machine learning rate is 1×10⁻⁶. -4 Initially, as the number of learning sessions increased, the number gradually decreased to 5×10. -7 Training 1×10 4 The process stops at this point; to accelerate convergence, the eigenvalue constraint φ(0,±1)=0.2 is set; the control equation has 8000 sample points, and the boundary conditions have 100 sample points; the weight coefficients of the loss function are empirical parameters, with P... φ =1 as the baseline, and different P values ​​are periodically changed during the training process. b P F,s P F,f P d,s The value ranges from 50 to 10. Figure 2 shows the calculation results of the geometric anisotropic neutron transport equation for the critical single-group flat plate.

[0209] The embodiments described above are merely illustrative of several implementations of the present invention, and while the descriptions are specific and detailed, they should not be construed as limiting the scope of the present invention. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of the present invention, and these modifications and improvements all fall within the scope of protection of the present invention. Therefore, the scope of protection of this patent should be determined by the appended claims.

Claims

1. A deep learning numerical solution method for anisotropic scattering neutron transport equations, characterized in that, Solving the anisotropic scattering neutron transport equations using multiprimal function variable-order iterative deep learning includes the following steps: Step 1: Establish the expression for the neutron transport equation based on the geometry and boundary conditions of the scattering source; Step 2: Differentiate each integral term in the neutron transport equation and change its order to the antiderivative corresponding to the integrand under the integral sign to obtain the differential equation, and establish the boundary value constraint conditions of the antiderivative. Step 3: Map the original function and the unknown angular flux density function in the neutron transport equation to neural network functions respectively. Based on the framework of solving differential equations using deep learning, perform alternating iterative deep learning until the loss function is less than the preset value, and obtain the numerical solution of the unknown angular flux density function.

2. The deep learning numerical solution method for the anisotropic scattering neutron transport equation according to claim 1, characterized in that, In step 2, the Newton-Leibniz formula is used to transform the definite integral term into the subtraction of the constant values ​​of the antiderivative at the upper and lower limits of integration.

3. The deep learning numerical solution method for the anisotropic scattering neutron transport equation according to claim 1, characterized in that, In step 2, the Newton-Leibniz formula is used to transform the indefinite integral term into the subtraction of the function values ​​of the original function at the corresponding variables of the upper and lower limits of integration.

4. The deep learning numerical solution method for the anisotropic scattering neutron transport equation according to claim 1, characterized in that, In step 2, when the neutron transport equation has multiple integral terms, multiple antiderivatives are used to represent different integral terms, and in step 3, a neural network function is established for each antiderivative.

5. The deep learning numerical solution method for the anisotropic scattering neutron transport equation according to claim 1, characterized in that, In step 2, boundary value constraints are established for the integrand antiderivative, such that the function value of the antiderivative is constant at the upper or lower limit of integration.

6. The deep learning numerical solution method for the anisotropic scattering neutron transport equation according to claim 5, characterized in that, In step 2, the constant value of the integrand antiderivative at the lower or upper limit of integration is set to 0, such that the expression of the antiderivative expressed by the upper and lower limits of integration is equal to that with respect to the definite integral term.

7. The deep learning numerical solution method for the anisotropic scattering neutron transport equation according to claim 1, characterized in that, Step 3 includes: Step 30: Based on the physical information neural network or other neural network models, establish neural network functions for the original function and the angular flux density function, respectively; Step 31: Substitute the neural network function as the trial function of the angular flux density function and the original function into the differential equation to obtain the loss function of the control equation. Establish the loss function of the original function transformation equation based on the transformation between the integrand and the corresponding original function. Establish the loss function of the original function boundary constraint based on the boundary constraint of the original function. Establish the boundary condition loss function. Step 32: Establish a unified loss function based on the loss function of the governing equation, the loss function of the original function transformation equation, the loss function of the original function boundary condition constraint, and the loss function of the boundary condition. Step 33: Based on the unified loss function in step 32, adjust the neural network functions corresponding to the angular flux density function and the original function respectively, and perform deep machine learning alternately. When the unified loss function approaches the minimum value, the neural network approaches the numerical solution of the angular flux density function and the original function.

8. The deep learning numerical solution method for the anisotropic scattering neutron transport equation according to claim 7, characterized in that, In step 32, different weights are set for the unified loss function established for the control equation loss function, the original function transformation equation loss function, the original function boundary constraint loss function, and the boundary condition loss function, thus constructing a unified loss function.

9. The deep learning numerical solution method for the anisotropic scattering neutron transport equation according to claim 7, characterized in that, In step 33, deep learning is performed alternately and iteratively on the neural network functions corresponding to the angular flux density function and the original function to gradually approximate the angular flux density function and the original function.

10. The deep learning numerical solution method for the anisotropic scattering neutron transport equation according to claim 7, characterized in that, In the process of deep machine learning with a unified loss function as the target, the weight values ​​of the control equation loss function, the original function transformation equation loss function, the original function boundary constraint loss function, and the boundary condition loss function are continuously adjusted to reduce the possibility of getting trapped in local optima until the weighted unified loss function approaches the minimum value.

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