Uniform system hyperfine group resonance self-screen deep learning calculation method

Abstract

The invention particularly relates to a homogeneous system hyperfine group resonance self-shielding deep learning calculation method, and belongs to the technical field of nuclear reactor core calculation, and the method comprises the following steps: respectively mapping a neutron moderation equation in a complete differential form and a primitive function transformation equation into neural network functions; constructing a neutron moderation equation loss function, a primitive function transformation equation damage function, a scattering source primitive function definite solution constraint loss function, an energy initial condition loss function and a flux density non-negative boundary condition loss function, and weighting to form a machine learning loss function; according to a machine learning loss function, alternately adjusting neural network function parameters to carry out iterative deep machine learning; a whole training set is divided into small training sets, training is carried out in sequence from high to low according to an energy segment, the training set of a previous energy group is a boundary condition of a next energy group, and segmented solution of a neutron moderation equation is achieved. According to the invention, a new way for solving the neutron moderation equation in the uniform system is realized.

Description

Technical Field

[0001] This invention relates to the field of nuclear reactor core computing technology, and in particular to a deep learning computing method for ultrafine group resonance self-screen in a uniform system. Background Technology

[0002] Resonant self-shield calculation is a key area in reactor physics analysis and core design, providing an important effective cross section for neutron transport calculations. Solving the neutron moderation equation is the core step in resonant self-shield calculation. Currently, commonly used resonance calculation methods include equivalent theory and subgroup methods, which have high computational efficiency and are widely used. However, because equivalent theory uses a series of simplified models for specific geometries, it lacks the ability to handle complex geometries; and subgroup methods often encounter numerical instability problems when generating subgroup parameters.

[0003] Meanwhile, among traditional resonant self-shield calculation methods, the hyperfine group method is recognized as a high-precision method. It solves the moderating equations in extremely detailed energy group structures to obtain an accurate modulated energy spectrum that approximates continuous energy, and has unique advantages in energy spectrum adaptability and handling of resonant interference effects. In particular, the technique of combining the hyperfine group method with the method of characteristics (MOC) developed in recent years can handle complex geometric effects on an almost continuous energy structure, making this method often the calculation benchmark for other multi-group methods. However, current hyperfine group methods still have difficulties such as low computational efficiency and limitations in handling geometric forms. They are mainly used for generating resonant integrals, benchmark calculations of multi-group resonant methods under specific conditions, and cross-section correction in local regions, and have not yet been directly applied to large-scale reactor core engineering design.

[0004] Currently, with the rapid development of artificial intelligence technology, deep learning-based numerical methods for solving differential equations have made significant progress, achieving remarkable results in solving neutron diffusion and transport equations in reactor physics. These methods also demonstrate a series of potential advantages in areas such as the integrity of the governing equation model, the discretization of meshless equations, and the continuity of computational results. This provides a new technical approach for hyperfine group resonance self-screen computation. Preliminary research has been conducted on using deep learning methods for resonance self-screen computation. The main technical approach involves forming a machine learning proxy model for resonance self-screen computation through a data-driven method, thereby accelerating computation. However, this approach requires prior access to computational results from traditional methods as machine learning samples for deep learning, which has certain limitations.

[0005] To address this, this invention employs deep learning computational methods to directly solve the sub-moderation equation in an infinitely uniform system. It proposes a variable-order iterative deep learning method for combining functions, which multiplies the moderated energy spectrum by the total cross section to form a combined function. This reduces the impact of the high-frequency oscillation characteristics of the moderated energy spectrum on the numerical approximation performance of the neural network function, thus establishing a novel method for numerically solving the moderated equation of a uniform system. Summary of the Invention

[0006] The purpose of this invention is to provide a deep learning-based computational method for hyperfine group resonance self-screen in uniform systems. This method combines the PINN method driven by a physical model with the hyperfine group method for traditional resonance self-screen computation. It employs variable-order iterative deep learning with a combined function, multiplying the moderated energy spectrum by the total cross section to form a combined function. This reduces the impact of the high-frequency oscillation characteristics of the moderated energy spectrum on the numerical approximation performance of the neural network function, enabling the solution of the neutron moderated equation in uniform systems. This allows for the computation of the moderated energy spectrum under resonant interference conditions, providing a new technical approach for reactor resonance self-screen computation.

[0007] To achieve the above objectives, the present invention provides the following technical solution:

[0008] A deep learning computation method for hyperfine group resonance self-screen in a uniform system includes the following steps:

[0009] Step 1: Upgrade the scattering source integral term in the neutron moderation equation to a fully differential form to obtain the fully differential form of the neutron moderation equation; map the neutron flux and the upgraded scattering source integral term in the fully differential form of the neutron moderation equation to neural network functions respectively.

[0010] Step 2: Substitute the neural network function as a trial function into the neutron slowing equation and the original function transformation equation in the complete differential form to obtain the loss function of the neutron slowing equation and the damage function of the original function transformation equation; construct the boundary value constraint loss function of the scattering source original function, the energy initial condition loss function, and the flux density non-negative boundary condition loss function.

[0011] Step 3: Weight the loss functions of the neutron slowing equation, the transformation equation, the boundary value constraint, the initial energy condition, and the flux density non-negative boundary condition to form a machine learning loss function.

[0012] Step 4: Iterate deep machine learning by alternating the parameters of the neural network function according to the machine learning loss function;

[0013] Step 5: Based on Step 4, the entire training set is divided into smaller training sets, and training is performed sequentially from high to low energy levels. The training set of the previous energy group serves as the boundary condition for the next energy group, thus achieving a piecewise solution to the neutron moderation equation.

[0014] As one possible approach, in step one, each nuclide corresponds to two neural network functions: one is a neural network function for the neutron moderation equation in fully differential form, and the other is a neural network function for the nuclide antiderivative transformation equation; for each additional nuclide, there will be an additional neural network function for the antiderivative transformation equation.

[0015] As one feasible approach, the homogeneous system is an infinitely homogeneous system. In step one, the neutron moderation equation is a combined function of the neutron reaction cross section and the neutron flux density:

[0016]

[0017] The scattering source integral term in the neutron moderation equation is elevated to a fully differential form using the Newton-Leibniz formula, yielding the fully differential form of the neutron moderation equation:

[0018]

[0019] Where, ∑ t denoted as neutron reaction cross-section function, φ as neutron flux density function, and k as the numbering of multiple nuclides; N k Let σ be the nucleon density of nuclide k. s,k For the microelastic scattering cross section of nuclide k, σ s,k For the scattering cross section, a k α is the value of nuclide k; u is the logarithmic energy decrease, and u′ is the integrand logarithmic energy decrease.

[0020] As one possible approach, in step one, the neural network function is a fully connected deep neural network function:

[0021] N(x)=f(x,w,b,l,n) (3)

[0022] Where N(x) is the output vector of the neural network; x is the input vector of the neural network, including the neutron slowing equation, the antiderivative transformation equation, the boundary value constraint of the antiderivative of the scattering source, the initial energy condition, and the non-negative flux density boundary condition; w is the connection weight of the neural network; b is the bias term of the neural network; l is the depth of the neural network; n is the number of hidden neurons in the neural network; and f is the activation function.

[0023] As one feasible approach, the constraint equations for solving the neutron moderation equations are as follows:

[0024]

[0025] As one possible approach, the loss function of the neutron moderation equation is:

[0026]

[0027] Where i represents the sample points required for machine learning, generated according to a certain probability density distribution method.

[0028] As one possible approach, the loss function of the original function transformation equation is:

[0029]

[0030] Among them, the range of values ​​for machine learning samples in equation (12) is the same as that in equation (11).

[0031] As one feasible approach, the boundary value constraint loss function of the scattering source primitive function is:

[0032]

[0033] As one feasible approach, the energy initial condition loss function is:

[0034]

[0035] Where j represents machine learning sample points between 0 < u < u0.

[0036] As one possible approach, the flux density nonnegative boundary condition loss function is:

[0037]

[0038] In the formula, abs() is the absolute value function.

[0039] As one possible approach, this involves the neural network N corresponding to the combination function M(u). M (x u We select the slowing equation and the corresponding boundary condition loss function for weighting to obtain the weighted single loss function:

[0040]

[0041] In the formula, These are the weights corresponding to the loss function of the neutron slowing equation, the loss function of the initial energy condition, and the loss function of the nonnegative boundary condition, respectively, and their values ​​are determined based on experimental and empirical coefficients.

[0042] As one feasible approach, different neural network functions corresponding to the original function equation of the scattering source are used. By selecting the corresponding original function transformation equation, determining the solution constraints, initial energy conditions, and nonnegative boundary conditions, and constructing the weighted loss function, we obtain:

[0043]

[0044] In the formula, These are the weights corresponding to the loss function of the original function transformation equation for the integral term corresponding to nuclide k, the loss function of the constraint of the deterministic solution, the loss function of the initial energy condition, and the loss function of the nonnegative boundary condition.

[0045] As one possible approach, in equation (19), if multiple outputs of the same neural network are used to represent F corresponding to different nuclides... k,0 If (u), then the overall loss function is expressed as:

[0046]

[0047] As one possible approach, in step one, equation (4) is applied with respect to... 1 The integral term of H is processed, and the logarithmic energy drop u0 of the upper limit of the energy at the highest point of the solution domain is used as the dividing point. 1 Decomposing the integral term of H, we get:

[0048]

[0049] Simplifying the first term by removing the constant term inside the integral sign, we finally only have e remaining below the integral sign. u′ This term can be directly integrated to obtain an analytical solution; from equation (21), the loss function of the neutron moderation equation can be obtained as:

[0050]

[0051] The remaining loss functions remain unchanged.

[0052] As one feasible approach, in actual training, it is assumed that the logarithmic energy reduction range of the m-th group is u. m-1 :u m B m-1 For u m-2 :u m-1 A constant within a range; represented by N φ,m and N k,0,m Let the neural network function represent the m-th group, then the boundary condition loss function is:

[0053]

[0054] In the formula, j represents the sample points required for machine learning.

[0055] As one possible approach, the boundary condition loss functions (24) and (25) are replaced with the initial energy conditions and the original function boundary value constraint functions and weighted to obtain the result for... Deep learning loss function Loss φ-alland targeting Loss function:

[0056] Loss φ-all =P φ Loss φ +P b,φ Loss b,φ +P φ-abs Loss φ-abs (26)

[0057] Loss p-all-k =P p Loss p +P b,F,k Loss b,F,k +P p-abs-k Loss p-abs-k (27)

[0058] In the formula, P φ ,P b,φ ,P φ-abs ,P p ,P b,F,k ,P p-abs-k These are the weights of the corresponding loss function, where Loss p-all-k This assumes the existence of multiple antiderivatives, that is, the number of nuclides is greater than or equal to 2.

[0059] Beneficial technical effects of the present invention:

[0060] This invention presents a deep learning-based computational method for hyperfine group resonance self-screen in uniform systems. It combines the PINN method driven by a physical model with the traditional hyperfine group method for resonance self-screen computation. A variable-order iterative deep learning method using a combinatorial function is employed. The moderated energy spectrum is multiplied by the total cross-section to form a combinatorial function, reducing the impact of high-frequency oscillations in the moderated energy spectrum on the numerical approximation performance of the neural network function. This enables the solution of the neutron moderating equation in uniform systems, thus achieving the computation of the moderated energy spectrum under resonant interference conditions. Numerical calculations using multiple uniform systems verify the correctness of the proposed method's principles and computational process, demonstrating its ability to solve multi-nucleus resonant interference problems. The computational results show a continuous distribution of the moderated energy spectrum with energy, proving the potential advantage of the proposed method in resolving local characteristics of the moderated energy spectrum and exploring a new approach to solving the neutron moderating equation. Attached Figure Description

[0061] Figure 1 A schematic diagram of the resonance peak characteristics of the 235U moderation energy spectrum, cross section, and combination function;

[0062] Figure 2 A flowchart illustrating an embodiment of a deep learning computation method for hyperfine group resonance self-screen in a uniform system;

[0063] Figure 3 for 238 U and 16 O is a schematic diagram of the relative deviation distribution of the total cross section of a uniform system;

[0064] Figure 4 for 238 U and 16 A schematic diagram comparing the continuous energy spectrum of a homogeneous system constructed from O;

[0065] Figure 5 This is a schematic diagram showing the trend of the combination function M(u) as a function of logarithmic energy.

[0066] Figure 6 for 238 U and 16 The antiderivative F corresponding to O k,0 A schematic diagram of the calculation results of (u);

[0067] Figure 7 for 238 U and 1 H is a schematic diagram of the relative error distribution of the total cross section of a uniform system.

[0068] Figure 8 for 238 U and 1 A comparative diagram of the energy spectra of groups 15-18 of the H-structure homogeneous system;

[0069] Figure 9 for 238 U and 1 H corresponds to the antiderivative F k,0 (u) Schematic diagram of calculation results;

[0070] Figure 10(a) shows 235 A comparison and a schematic diagram of the deviation distribution of the U-microscopic resonant scattering cross sections;

[0071] Figure 10(b) shows 238 A comparison and a schematic diagram of the deviation distribution of the U-microscopic resonant scattering cross sections;

[0072] Figure 10(c) is 235 U、 238 U and 16 O is a schematic diagram of the relative deviation distribution of the total cross section of a uniform system;

[0073] Figure 11 for 235 U、 238 U and 16 The antiderivative F corresponding to O k,0 (u) Schematic diagram of calculation results;

[0074] Figure 12(a) shows 239 A comparison of the microscopic resonant scattering cross sections of Pu and a schematic diagram of the relative deviation distribution;

[0075] Figure 12(b) shows 90 A comparison of Zr microscopic resonant scattering cross sections and a schematic diagram of the relative deviation distribution;

[0076] Figure 12(c) is 238 A schematic diagram showing the comparison and relative deviation distribution of the microscopic resonant scattering cross sections of U;

[0077] Figure 13 A schematic diagram showing the calculation results of the integral antiderivative Fk,0(u) corresponding to 238U, 16O, 90Zr and 239Pu. Detailed Implementation

[0078] 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 belongs; the terminology used herein in the specification of the application is for the purpose of describing particular embodiments only and is not intended to be limiting of the application.

[0079] 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.

[0080] The terms “include,” “comprising,” or any other variation thereof are intended to cover non-exclusive inclusion, which includes not only the elements listed but also other elements not expressly listed.

[0081] The technical solution of the present invention will be clearly and completely described below with reference to the accompanying drawings and specific embodiments.

[0082] This embodiment provides a deep learning computation method for hyperfine group resonance self-screen in a uniform system, including the following steps:

[0083] Step 1: Upgrade the scattering source integral term in the neutron moderation equation to a fully differential form to obtain the fully differential form of the neutron moderation equation; map the neutron flux and the upgraded scattering source integral term in the fully differential form of the neutron moderation equation to neural network functions respectively.

[0084] Step 2: Substitute the neural network function as a trial function into the neutron slowing equation and the original function transformation equation in the complete differential form to obtain the loss function of the neutron slowing equation and the damage function of the original function transformation equation; construct the boundary value constraint loss function of the scattering source original function, the energy initial condition loss function, and the flux density non-negative boundary condition loss function.

[0085] Step 3: Weight the loss functions of the neutron slowing equation, the transformation equation, the boundary value constraint loss function of the scattering source original function, the initial energy condition loss function, and the flux density non-negative boundary condition loss function to form a unified machine learning loss function.

[0086] Step 4: Iterate deep machine learning by alternating the parameters of the neural network function according to the machine learning loss function;

[0087] Step 5: Based on step 4, the entire training set is divided into smaller training sets with appropriate data sizes, and the sets are trained sequentially from high to low energy levels. The training set of the previous energy group serves as the boundary condition for the next energy group, thus enabling the piecewise solution of the neutron slowing equation.

[0088] In this embodiment, as one possible approach, in step one, one nuclide corresponds to two neural network functions, one of which is a neural network function for the neutron moderation equation in fully differential form, and the other is a neural network function for the nuclide antiderivative transformation equation; for each additional nuclide, there will be one more neural network function for the antiderivative transformation equation.

[0089] In this embodiment, as one possible approach, the homogeneous system is an infinitely homogeneous system. In step one, the neutron moderation equation is a combined function of the neutron reaction cross section and the neutron flux density:

[0090]

[0091] Where, ∑ t denoted as neutron reaction cross-section function, φ as neutron flux density function, and k as the numbering of multiple nuclides; N k Let σ be the nucleon density of nuclide k. s,k For the microelastic scattering cross section of nuclide k, σ s,k Let u be the scattering cross section, u be the logarithmic energy drop, u′ be the integrand's logarithmic energy drop, and a be the scattering cross section. k Let α be the value of nuclide k; the expression for the logarithmic energy drop is as follows:

[0092]

[0093] The key neutronographic characteristic of resonant self-screening calculations lies in the dramatic fluctuations in the neutron reaction cross-section within the resonance energy range, especially for larger atomic-weight nuclides commonly found in reactors. Near certain energy points, a peak-like shape can be observed in the neutron reaction cross-section, with fluctuations spanning several orders of magnitude. Due to the presence of the resonance peak in the neutron reaction cross-section, the probability of neutrons reacting near this energy point is very high, while the corresponding neutron flux often exhibits a trough, resulting in a very low neutron reaction rate at this point—a phenomenon known as resonant self-screening. Figure 1 The changes in the total transport cross section and the moderation energy spectrum of a uniform system with energy are given, and the resonance peaks of the total cross section and the corresponding neutron moderation energy spectrum can be observed.

[0094] from Figure 1 It can be seen that the size of the neutron reaction cross section of a resonant nuclide is inversely proportional to the neutron flux density. That is, when the neutron reaction cross section is large, the neutron flux density is usually small, while where there is no resonance peak in the corresponding neutron reaction cross section, the neutron flux density is relatively large. For the neutron slowing equation in an infinitely homogeneous system, multiplying the neutron reaction cross section function and the neutron flux density function yields a combined function M(u) that includes both the neutron reaction cross section and the neutron flux density:

[0095] M(u)=Σ t (u)φ(u) (2)

[0096] The degree of fluctuation in the amplitude of the combination function M(u) relative to the neutron reaction cross-section function Σ t (u) and neutron flux density function φ t (u) The fluctuation of amplitude will be greatly reduced, thus reducing the adverse effects of slowed-down high-frequency energy spectrum features during deep learning solution.

[0097] In this embodiment, as one possible approach, in step one, the scattering source integral term in the neutron moderation equation is upgraded to a fully differential form using the Newton-Leibniz formula, resulting in the fully differential form of the neutron moderation equation. Specifically, this includes the following steps:

[0098] The scattering source integral term in the neutron moderation equation is:

[0099]

[0100] The neutron flux density function φ(u′) and nucleon density N in equation (4) k , scattering cross section σ s,k If (u′) is continuous, then the integrand under the integral sign is also continuous, and there exists an antiderivative F of integration with respect to u′. k (u′), whose derivative with respect to u′ is:

[0101]

[0102] Alternatively, from the Newton-Leibniz formula, we have:

[0103]

[0104] In equation (6), the original function F k (u) is not deterministic; it consists of a family of functions with equal derivatives, and the original function F needs to be determined. k (u) The specific form of equation (6) is required to obtain a solution, and F is given. k (u) A concise and deterministic constraint condition:

[0105] u=0,F k,0 (0)=0 (7)

[0106] Therefore, substituting equations (6) and (7) into equation (1) yields the neutron moderation equation in fully differential form:

[0107]

[0108] In this embodiment, as one possible implementation method, in step one, the neural network function is a fully connected deep neural network function:

[0109] N(x)=f(x,w,b,l,n) (3)

[0110] Where N(x) is the output vector of the neural network; x is the input vector of the neural network, including the neutron slowing equation, the antiderivative transformation equation, the boundary value constraint of the antiderivative of the scattering source, the initial energy condition, and the non-negative flux density boundary condition; w is the connection weight of the neural network; b is the bias term of the neural network; l is the depth of the neural network; n is the number of hidden neurons in the neural network; and f is the activation function.

[0111] The neutron moderation equation is a simplified neutron transport equation. Referring to methods for solving neutron transport equations, its boundaries typically include total reflection boundaries, vacuum boundaries, zero boundaries, and non-negative boundaries, and are usually treated as conventional equations. It should be noted that since homogeneous systems lack geometric and angular variables, the corresponding neutron moderation equation does not have geometric or angular boundaries. However, because the neutron moderation equation is a time-delay equation, the moderation energy spectrum at the highest energy level in its solution domain affects the solution. Therefore, initial conditions should be set for the highest energy point to facilitate the solution of the neutron moderation equation.

[0112] In this embodiment, as one possible approach, referring to the traditional hyperfine group calculation method, the energy spectrum from the logarithmic energy drop reference point to the highest energy starting boundary point of the neutron moderation equation is set to a fixed value C0. This means that the moderation energy spectrum in the energy range above the boundary point u0 is set to be equal, and the energy range Σ... t (u), σ s,k (u) The cross-sectional data is also set to the cross-sectional data at u0, that is:

[0113] φ(u)=C0,∑ t (u)=∑ t (u0),σ s,k (u)=σ s,k (u0), 0 < u < u0 (9)

[0114] Rearranging equation (9) and substituting it into the combination function, we obtain the set of constraint equations for solving the neutron moderation equation:

[0115]

[0116] In this embodiment, as one possible approach, the loss function of the neutron slowing equation is obtained through the following steps:

[0117] Let N M (x u )=M(u), Where, x u The input variable for the neural network represents the logarithmic energy drop of the neutron moderation equation;

[0118] N M (x u )=M(u), Substituting the first term into equation (10), we obtain the loss function of the neutron slowing equation in residual form:

[0119]

[0120] Where i represents the sample points required for machine learning, which can be generated according to a certain probability density distribution method.

[0121] In this embodiment, as one possible approach, the loss function of the original function transformation equation is obtained through the following steps:

[0122] For the second term of equation (10), let N Fk,0 (x u ) = F k,0 (u), thus obtaining the loss function of the original function transformation equation:

[0123]

[0124] Among them, the range of values ​​for machine learning samples in equation (12) is the same as that in equation (11).

[0125] In this embodiment, as one possible approach, the boundary value constraint loss function of the scattering source primitive function is obtained through the following steps:

[0126] For the third term of equation (10), let The bounded loss function of the scattering source primitive function is obtained:

[0127]

[0128] The neutron moderation equation addresses the energy spectrum solution of the resonance energy group. Due to the characteristics of the time-delay equation, initial conditions above the highest energy in the solution domain must be considered. In this embodiment, as one possible approach, referring to the fourth term of equation (10), the corresponding loss function is:

[0129]

[0130] In the formula, j represents machine learning sample points between 0 < u < u0;

[0131] Correspondingly for the original function F k,0 (u), according to the second term of equation (10), its corresponding loss function is:

[0132]

[0133] In this embodiment, as one possible approach, for the flux density and original function of the moderated energy spectrum to be non-negative, the corresponding boundary condition corresponds to the last term of equation (10), and its loss function is:

[0134]

[0135] In the formula, abs() is the absolute value function.

[0136] In this embodiment, as one possible approach, the neural network N corresponding to the combination function M(u) is... M (x u We select the slowing equation and the corresponding boundary condition loss function for weighting to obtain the weighted single loss function:

[0137]

[0138] In the formula, These are the weights corresponding to the loss function of the neutron slowing equation, the loss function of the initial energy condition, and the loss function of the nonnegative boundary condition, respectively, and their values ​​are generally determined based on experimental and empirical coefficients.

[0139] In this embodiment, as one possible approach, different neural network functions corresponding to the original function equation of the scattering source are used. By selecting the corresponding original function transformation equation, determining the solution constraints, initial energy conditions, and nonnegative boundary conditions, and constructing the weighted loss function, we obtain:

[0140]

[0141] In the formula, These are the weights corresponding to the loss function of the original function transformation equation for the integral term corresponding to nuclide k, the loss function of the constraint of the deterministic solution, the loss function of the initial energy condition, and the loss function of the nonnegative boundary condition.

[0142] In this embodiment, as one possible approach, in equation (19), if multiple outputs of the same neural network represent F corresponding to different nuclides... k,0 If (u), then the overall loss function is expressed as:

[0143]

[0144] for 1 H, since its α is close to 0, leads to equation (4) regarding 1 The lower bound of the integral term of H tends to -∞, and directly solving it using the deep learning method described above will lead to non-convergence of the training results. In this embodiment, as one possible approach, in step one, equation (4) is modified with respect to... 1 The integral term of H is processed to facilitate deep learning training;

[0145] Using the logarithmic energy drop u0 of the upper limit of the energy at the highest point of the solution domain as the dividing point 1 Decomposing the integral term of H, we get:

[0146]

[0147] For integral terms below the upper limit of the logarithmic energy integral, since the flux transformation is relatively gentle, the microscopic scattering cross section and flux can be directly solved by taking the value of the upper limit of the highest energy of the solution domain.

[0148] Therefore, the first term can be simplified by removing the constant term inside the integral sign, leaving only e below the integral sign. u′ This term can be directly integrated to obtain an analytical solution; from equation (21), the loss function of the neutron moderation equation can be obtained as:

[0149]

[0150] The remaining loss functions remain unchanged.

[0151] Since full-scale training would require too much data for training at once, resulting in low training accuracy, and the current training equipment performance does not support training large datasets, in this embodiment, in step five, a segmented training method is used to solve the uniform multi-nucleus neutron slowing equation. That is, the entire training set is divided into small training sets with appropriate data volume, and training is performed sequentially from high to low energy level. The training set of the previous energy group serves as the boundary condition for the next energy group, thereby solving the uniform multi-nucleus neutron slowing equation.

[0152] In this embodiment, as one possible approach, in step five, during the deep learning solution process, the solution method for the first group is the same as in step three. Starting from the second group, the initial energy conditions and the original function boundary value constraint functions are changed to boundary condition loss functions determined by the previous group.

[0153] In this embodiment, as one possible approach, during actual training, it is assumed that the logarithmic energy reduction range of the m-th group is u. m-1 :u m B m-1 For u m-2 :u m-1 A constant within a certain range. Using N φ,m and N k,0,m Let the neural network function represent the m-th group, then the boundary condition loss function is:

[0154]

[0155] In the formula, j 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.

[0156] In this embodiment, as one possible approach, the boundary condition loss functions (24) and (25) are replaced with the initial energy conditions and the original function boundary value constraint functions and weighted to obtain the result for... Deep learning loss function Loss φ-all and targeting Loss function:

[0157] Loss φ-all =P φ Loss φ +P b,φ Loss b,φ +P φ-abs Loss φ-abs (26)

[0158] Loss p-all-k =P p Loss p +P b,F,k Lossb,F,k +P p-abs-k Loss p-abs-k (27)

[0159] In the formula, P φ ,P b,φ ,P φ-abs ,P p ,P b,F,k ,P p-abs-k , respectively, are the weights of the corresponding loss function, and are adjustable hyperparameters determined based on experimental experience, where Loss , p-all-k This assumes the existence of multiple antiderivatives, that is, the number of nuclides is greater than or equal to 2.

[0160] In this embodiment, as one possible approach, the weighted loss function is substituted into the neural network for alternating iterative training, which allows for the segmented solution of the neutron slowing equation.

[0161] F k,0 and F k Let F represent different antiderivatives. k,0 This represents a definite solution to the antiderivative, while F... k This represents the uncertain solution of the original function. In the formula, u and u′ both represent logarithmic energy decreases; u′ represents the integral sign of the logarithmic energy decrease within the integrand, while u represents the sign of the logarithmic energy decrease outside the integrand.

[0162] In this embodiment, as one possible approach, two nuclides based on uranium dioxide fuel are used. 238 U and 16 A comparative verification experiment was conducted on a homogeneous system of O. Because in a typical reactor system, the resonance effect is caused by... 238 The study primarily tested the predictive ability of deep learning methods for energy spectrum changes caused by single-resonance nuclides, eliminating interference from problems such as resonance interference. The benchmark for resonance calculations was derived from the UFG-MOC method in the published paper "Solution Method for Slowing Equations of Ultrafine Groups Based on the Method of Characteristics" (Qin Shuai, Zhang Qian, Zhao Qiang, et al. Solution Method for Slowing Equations of Ultrafine Groups Based on the Method of Characteristics [J]. Atomic Energy Science and Technology, 2019, 53(12):8.DOI:10.7538 / yzk.2018.youxian.0823.). The cross section used was the hyperfine group point cross section library based on ENDF / B-Ⅶ.0 generated by the NJOY program. The hyperfine group energy group width was 0.000025. The multigroup structure and numbering of the resonance energy group came from the WIMS format 69-group database. Groups 15 to 27 contained a total of 309,200 fine energy groups. For simplicity, groups 15 to 23 were selected as the observation objects. These energy groups contain relatively complex typical spectral oscillations, which are suitable for testing the spectral prediction capabilities of deep learning methods.

[0163] The machine learning method and parameter selection are as follows: a basic fully connected network is used as the neural network, with hidden units n=40, network layers l=9, activation function tanh, and initial values ​​of the network values ​​set to a Gaussian random distribution. The ADAM machine learning algorithm is employed; the machine learning rate starts from 10. -4 Initially, the loss function gradually decreases as the number of training iterations increases, until the loss function stops decreasing.

[0164] The deviation distribution of the total cross section of a uniform system on the energy group is as follows: Figure 3 As shown, groups 15 through 23 are arranged from right to left. It can be seen that... 238 U and 16 The relative deviation of the total cross section of the homogeneous system constructed in the O-structure for groups 15 to 20 is less than 0.4%, achieving high cross section calculation accuracy. However, the relative deviation increases with the increase of the number of energy groups. Figure 4 A comparison of the energy spectra of deep learning methods and hyperfine group methods in groups 15–23 of the resonance energy group is presented, and the overall energy spectrum shapes are in good agreement. Figure 5 The trend of the reaction rate combination function as a function of logarithmic energy is given. Figure 6 Give 238 U in the original function F k,0 (u) The computational results from deep learning show a strong agreement with the trends of traditional hyperfine group methods. The numerical examples above demonstrate that deep learning methods have the ability to accurately capture the physical characteristics of the slowing equations, thus ensuring the accuracy of resonance computation from a mechanistic perspective.

[0165] In this embodiment, as one possible approach, two nuclides are used. 238 U and 1 A comparative verification experiment was conducted on a uniform system of H, with the machine learning method and parameter selection as described above. 238 U and 16 A uniform system of O. Figure 7 Showing 238 U and 1 The calculated total cross-section of the uniform system constructed by H. It can be seen that deep learning methods... 238 U-microscopic scattering cross section, based on the traditional hyperfine group method. 238 The calculation error of the U-group cross section is less than 0.3%, and the calculation results have high accuracy. Figure 8 A comparison of the energy spectra of deep learning methods and hyperfine group methods in the 15th to 18th energy groups of the resonance energy group is presented, and the overall energy spectrum shapes are in good agreement. 238 U and 1 H corresponds to the antiderivative F k,0 (u) The calculation results are as follows Figure 9 As shown, it also conforms well to traditional calculation methods.

[0166] In this embodiment, three nuclides are used as one possible approach. 235 U、 238 U and 16 A uniform system of O was used to test the energy spectrum recovery capability of deep learning methods for real uranium dioxide fuel. The source of the baseline solution remained unchanged, and the machine learning method and parameter selection were the same as described above. 238 U and 16 The uniform system of O is consistent. In this problem, the following is introduced: 235 U、 238 The interference effect of U should be observed in detail. 235 U、 238 The accuracy of the microscopic cross-section. Solving the slowing equation requires only the macroscopic total cross-section of the material and the microscopic scattering cross-section of each nuclide. Figure 10 shows... 235 U and 238 The microscopic resonant scattering cross section of U and 235 U、 238 U and 16 By comparing the total cross-section and deviation distribution of the constructed uniform system, it can be seen that deep learning methods are effective in... 235 U、 238 U-microscopic scattering cross section, based on the traditional hyperfine group method, 15-20 groups 235 The relative deviation in the calculation of the U-group scattering cross section is less than 0.3%. 238 When U is less than 0.7%, the calculation results have high accuracy; as the energy group number increases, 238 The relative deviations of the U-group scattering cross section and the total cross section have increased. 235 U、 238 U and 16 The antiderivative F corresponding to O k,0 (u) The calculation results are as follows Figure 11 As shown, the results agree well with traditional calculation methods. These calculations demonstrate that deep learning methods possess excellent capabilities in handling resonance interference effects.

[0167] In this embodiment, as one possible approach, a four-nucleoside homogeneous system is constructed. 238 U+ 16 O+ 90 Zr+ 239 Pu, this problem considers introducing typical resonance nuclides from MOX fuel. 239 cladding materials in PU and pressurized water reactor fuel cells 90 Zr. Among them 90 Zr is also treated as a resonance nuclide, taking into account the resonance effect of the cladding material. This problem involves more complex multi-nucleus resonance interference effects, which will be the focus of observation. 90 Zr、 239 Pu and238 The calculation results of the micro-section of U are shown in Figure 12. 239 Pu, 90Zr and 238 A comparison of the microscopic resonant scattering cross sections and the distribution of relative deviations shows that... 239 The relative deviation of the Pu cross section is less than 0.1%. 90 The relative deviation of the Zr cross section is less than 0.5%. 238 The relative deviation of the cross section of U is less than 0.7%. 239 Pu、 90 Zr and 238 The antiderivative F corresponding to U k,0 (u) The calculation results are as follows Figure 13 As shown, it is in good agreement with traditional calculation methods.

[0168] The traditional hyperfine group method refers to the published literature "Resonant Self-Shield Calculation in Reactor Physics" (Zhang Qian, Wu Hongchun, Cao Liangzhi. Resonant Self-Shield Calculation in Reactor Physics [M]. Beijing: National Defense Industry Press, 2020.).

[0169] The above calculation results show that deep learning methods have good ability to handle resonance interference effect problems, and still maintain high computational accuracy for cross-sectional calculations of complex nuclide problems.

[0170] This embodiment uses numerical calculations on multiple uniform systems to verify the correctness of the proposed method's principle and calculation process, and demonstrates its ability to solve multi-nucleus resonance interference problems. The verification results show a continuous distribution of the moderation energy spectrum with energy, proving the potential advantage of the proposed method in resolving local characteristics of the moderation energy spectrum, and exploring a new way to solve the neutron moderation equation.

[0171] 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, all of which 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 computation method for hyperfine group resonance self-screen in a uniform system, characterized in that, Includes the following steps: Step 1: Upgrade the scattering source integral term in the neutron moderation equation to a fully differential form to obtain the fully differential form of the neutron moderation equation; map the neutron flux and the upgraded scattering source integral term in the fully differential form of the neutron moderation equation to neural network functions respectively. Step 2: Substitute the neural network function as a trial function into the neutron slowing equation and the original function transformation equation in the complete differential form to obtain the loss function of the neutron slowing equation and the damage function of the original function transformation equation; construct the boundary value constraint loss function of the scattering source original function, the energy initial condition loss function, and the flux density non-negative boundary condition loss function. Step 3: Weight the loss functions of the neutron slowing equation, the transformation equation, the boundary value constraint, the initial energy condition, and the flux density non-negative boundary condition to form a machine learning loss function. Step 4: Iterate deep machine learning by alternating the parameters of the neural network function according to the machine learning loss function; Step 5: Based on Step 4, the entire training set is divided into smaller training sets, and training is performed sequentially from high to low energy levels. The training set of the previous energy group serves as the boundary condition for the next energy group, thus achieving a piecewise solution to the neutron moderation equation.

2. The deep learning computation method for hyperfine group resonance self-screen in a uniform system according to claim 1, characterized in that, In step one, each nuclide corresponds to two neural network functions: one is a neural network function for the neutron moderation equation in fully differential form, and the other is a neural network function for the nuclide antiderivative transformation equation; for each additional nuclide, there will be an additional neural network function for the antiderivative transformation equation.

3. The deep learning computation method for hyperfine group resonance self-screen in a uniform system according to claim 1, characterized in that, The homogeneous system is an infinitely homogeneous system. In step one, the neutron moderation equation is a combined function of the neutron reaction cross section and the neutron flux density: The scattering source integral term in the neutron moderation equation is elevated to a fully differential form using the Newton-Leibniz formula, yielding the fully differential form of the neutron moderation equation: Where, ∑ t denoted as neutron reaction cross-section function, φ as neutron flux density function, and k as the numbering of multiple nuclides; N k Let σ be the nucleon density of nuclide k. s,k For the microelastic scattering cross section of nuclide k, σ s,k For the scattering cross section, a k α is the value of nuclide k; u is the logarithmic energy decrease, and u′ is the integrand logarithmic energy decrease.

4. The deep learning computation method for hyperfine group resonance self-screen in a uniform system according to claim 1, characterized in that, In step one, the neural network function is a fully connected deep neural network function: N(x)=f(x,w,b,l,n) (3) Where N(x) is the output vector of the neural network; x is the input vector of the neural network, including the neutron slowing equation, the antiderivative transformation equation, the boundary value constraint of the antiderivative of the scattering source, the initial energy condition, and the non-negative flux density boundary condition; w is the connection weight of the neural network; b is the bias term of the neural network; l is the depth of the neural network; n is the number of hidden neurons in the neural network; and f is the activation function.

5. The deep learning computation method for hyperfine group resonance self-screen in a uniform system according to claim 1, characterized in that, The constraint equations for solving the neutron moderation equation are:

6. The deep learning computation method for hyperfine group resonance self-screen in a uniform system according to claim 5, characterized in that, The loss function of the neutron moderation equation is: The loss function of the original function transformation equation is: Where i represents the sample points required for machine learning, generated according to a certain probability density distribution method; the boundary value constraint loss function of the scattering source primitive function is: The energy initial condition loss function is: Where j represents machine learning sample points between 0 < u < u0; The flux density non-negative boundary condition loss function is: In the formula, abs() is the absolute value function.

7. The deep learning computation method for hyperfine group resonance self-screen in a uniform system according to claim 1, characterized in that, For the neural network N corresponding to the combination function M(u) M (x u We select the slowing equation and the corresponding boundary condition loss function for weighting to obtain the weighted single loss function: In the formula, These are the weights corresponding to the loss function of the neutron slowing equation, the loss function of the initial energy condition, and the loss function of the nonnegative boundary condition, respectively, and their values ​​are determined based on experimental and empirical coefficients.

8. The deep learning computation method for hyperfine group resonance self-screen in a uniform system according to claim 1, characterized in that, Different neural network functions corresponding to the original function equation of the scattering source By selecting the corresponding original function transformation equation, determining the solution constraints, initial energy conditions, and nonnegative boundary conditions, and constructing the weighted loss function, we obtain: In the formula, These are the weights corresponding to the loss function of the original function transformation equation for the integral term corresponding to nuclide k, the loss function of the constraint of the deterministic solution, the loss function of the initial energy condition, and the loss function of the nonnegative boundary condition.

9. The deep learning computation method for hyperfine group resonance self-screen in a uniform system according to claim 8, characterized in that, In equation (19), if multiple outputs of the same neural network are used to represent F corresponding to different nuclides, k,0 If (u), then the overall loss function is expressed as:

10. The deep learning computation method for hyperfine group resonance self-screen in a uniform system according to claim 1, characterized in that, In step one, equation (4) is related to... 1 The integral term of H is processed, and the logarithmic energy drop u0 of the upper limit of the energy at the highest point of the solution domain is used as the dividing point. 1 Decomposing the integral term of H, we get: Simplifying the first term by removing the constant term inside the integral sign, we finally only have e remaining below the integral sign. u′ This term can be directly integrated to obtain an analytical solution; from equation (21), the loss function of the neutron moderation equation can be obtained as: The remaining loss functions remain unchanged.

11. The deep learning computation method for hyperfine group resonance self-screen in a uniform system according to claim 1, characterized in that, In actual training, assume that the logarithmic energy reduction range of the m-th group is u. m-1 :u m B m-1 For u m-2 :u m-1 A constant within a range; represented by N φ,m and N k,0,m Let the neural network function represent the m-th group, then the boundary condition loss function is: In the formula, j represents the sample points required for machine learning.

12. The deep learning computation method for hyperfine group resonance self-screen in a uniform system according to claim 11, characterized in that, By substituting the boundary condition loss functions (24) and (25) for the initial energy conditions and the original function boundary value constraint functions and weighting them, we obtain the result for... Deep learning loss function Loss φ-all and targeting Loss function: Loss φ-all =P φ Loss φ +P b,φ Loss b,φ +P φ-abs Loss φ-abs (26) Loss p-all-k =P p Loss p +P b,F,k Loss b,F,k +P p-abs-k Loss p-abs-k (27) In the formula, P φ ,P b,φ ,P φ-abs ,P p ,P b,F,k ,P p-abs-k These are the weights of the corresponding loss function. Among them, Loss p-all-k This assumes the existence of multiple antiderivatives, that is, the number of nuclides is greater than or equal to 2.

Images