Method for improving precision and convergence of solving diffusion equation based on PINN

Abstract

The invention belongs to the technical field of nuclear reactor physical numerical calculation, and particularly relates to a method for improving precision and convergence of solving a diffusion equation based on a PINN, and the method comprises the following steps: S1, building a neural network algorithm model combining a physical information neural network PINN and source iteration; s2, aiming at a multi-region smooth transition neutron diffusion problem with small cross section difference between materials, optimizing the neural network algorithm model by adopting an optimization strategy of a training level; and S3, aiming at the multi-region neutron diffusion problem of large neutron flux gradient at the boundary due to large cross section difference between materials, improving the neural network algorithm model by adopting a domain-divided neural network architecture. According to the method, the problem that the calculation cost is exponentially increased in a high-dimensional or complex geometric scene due to the fact that a traditional finite difference method and a finite element method depend on grid division is avoided.

Description

Technical Field

[0001] This invention belongs to the field of numerical calculation technology of nuclear reactor physics, and in particular relates to a method for improving the accuracy and convergence of solving diffusion equations based on PINN. Background Technology

[0002] Neutron diffusion theory is one of the most important theories in reactor physics design and safety analysis. Traditional numerical methods, such as the finite difference method and the finite element method, solve the neutron diffusion equation by discretizing the solution domain into a mesh. The computational accuracy of these methods is heavily dependent on the mesh quality, and the number of meshes increases exponentially with spatial dimensionality, leading to high computational costs. With the development of deep learning, the PINN method proposed by Raissi et al. directly searches for approximate solutions to the problem in a continuous function space defined by a deep neural network. This method embeds the physical constraints, such as the equations to be solved and boundary conditions, into the loss function of the neural network, and through optimization training, the network's predicted solution gradually approximates the true solution. As a meshless deep learning method, PINN has significant advantages in applications involving complex geometries and high-dimensional problems.

[0003] Elhareef and Wu et al. integrated the PINN model into the traditional source iteration method, briefly explaining the computational principle of this fusion algorithm and the implementation forms of various physical constraint terms. They applied it to the neutron diffusion fixed-source problem and the eigenvalue problem, and validated it using uniform and non-uniform examples of multiple dimensions. Although this study verified the feasibility of the PINN method for solving the neutron diffusion equation, its solution accuracy, especially in complex multi-region problems, cannot meet the accuracy requirements of reactor physics numerical calculations. On the other hand, Yang et al. proposed a Data-Enabled Physics Informed Neural Network (DEPINN) to improve the model's prediction accuracy and convergence performance by introducing a small amount of high-precision prior data to assist PINN. The successful application of DEPINN relies on high-precision prior data, which is difficult to obtain in many initial design stages or scenarios lacking experimental data. Therefore, there is an urgent need to develop a PINN optimization method that does not rely on prior data and can solve complex multi-region problems with high accuracy and good convergence under a purely physics-driven framework.

[0004] Based on the above analysis, the problems and shortcomings of the existing technology are as follows: (1) Existing traditional numerical methods require the generation of a large number of grids under high-dimensional and complex geometric conditions, resulting in high computational costs.

[0005] (2) The existing PINN method driven by pure physics has insufficient prediction accuracy and poor convergence in solving the problem of neutron diffusion in multiple materials. In particular, when there is a strong gradient change at the partition interface, the network has poor expressive ability.

[0006] (3) Existing DEPINN methods require the support of high-precision prior data, which limits their applicability in the early stages of design or in scenarios with insufficient data. Summary of the Invention

[0007] The purpose of this invention is to address the development of numerical computation based on reactor physics, considering the low prediction accuracy and poor convergence stability of existing single-network PINN methods in the case of cross-sectional differences between materials in multi-zone problems and drastic changes in neutron flux gradients at the boundaries, while the DEPINN method relies on high-precision prior data and has limited applicability in scenarios with insufficient data. This invention provides a method to improve the accuracy and convergence of solving diffusion equations based on PINN.

[0008] In view of this, the present invention provides a method for improving the accuracy and convergence of solving diffusion equations based on PINN, comprising the following steps: S1, Establish a neural network algorithm model that combines the physical information neural network PINN with source iteration; S2, For the problem of neutron diffusion in a multi-regional smooth transition with small cross-sectional differences between materials, an optimization strategy at the training level is adopted to optimize the neural network algorithm model; S3. To address the problem of multi-region neutron diffusion caused by large differences in cross-sections between materials leading to large neutron flux gradients at the boundaries, a domain-specific neural network architecture is adopted to improve the neural network algorithm model.

[0009] Furthermore, in step S1, the specific steps for establishing the neural network algorithm model combining PINN and source iteration include: S101, Construct the PINN neural network structure within the deep learning framework, and set the number of network layers, the number of neurons per layer, and the activation function; S102, a spatial coordinate training set is generated according to the geometric range of the neutron diffusion problem. The steady-state multi-group neutron diffusion equation without external source is taken as the equation to be solved. The automatic differentiation function of the deep learning framework is used to obtain the correlation between the network output flux and the input spatial coordinates. Based on this correlation, the diffusion equation constraint terms are established, and the boundary conditions are transformed into the constraint terms in the loss function. S103. A two-layer iterative structure is established. In the inner iteration, the PINN network is used as the neutron flux solver. In the outer iteration, the fission source term and the effective multiplication factor keff are updated, and the iteration process is terminated according to the preset convergence criterion.

[0010] The above scheme, by constructing a PINN neural network structure within a deep learning framework and setting the number of network layers, neurons, and activation functions, ensures that the neural network possesses the fundamental expressive ability to adapt to the continuously changing characteristics of neutron flux, avoiding solution deviations caused by unreasonable network structure parameters. Utilizing the automatic differentiation function of the deep learning framework to obtain the correlation between the network output flux and the input spatial coordinates, and establishing diffusion equation constraints based on this correlation, transforming boundary conditions into loss function constraints, reduces errors introduced during manual derivation of derivatives, ensuring the accuracy of physical and boundary constraints. Establishing a two-layer iterative structure with PINN as the neutron flux solver in the inner iteration and the fission source term and effective multiplication factor keff in the outer iteration enables coordinated optimization of neutron flux solving and core physical parameter updates, avoiding slow convergence or insufficient accuracy caused by single-level iteration, effectively improving the overall solution efficiency and stability of the model.

[0011] Furthermore, in step S2, the optimization strategy at the training level includes a residual adaptive resampling (RAR) strategy and a weight dynamic adjustment strategy, wherein: The residual adaptive resampling strategy periodically identifies high residual regions in the neural network algorithm model and increases the local sampling point density in these regions to improve the model's expressive ability in key high residual regions. The dynamic weight adjustment strategy dynamically adjusts the weights of each constraint term in the loss function based on the gradient contribution of each constraint term during training, thereby avoiding model convergence difficulties caused by improper weight allocation.

[0012] The above-mentioned design can specifically address the bottleneck in solving the problem of neutron diffusion in multi-regional smooth transitions with small cross-sectional differences between materials. The residual adaptive resampling strategy periodically identifies high residual regions in the neural network algorithm model and increases the local sampling point density in these regions. This can focus on key regions where the model fitting is weak, improve the sampling representativeness of high residual regions, avoid insufficient representation of key regions caused by global uniform sampling, and thus improve the model's fitting accuracy of local physical properties. The weight dynamic adjustment strategy dynamically adjusts the weights of each constraint term according to the gradient contribution of each constraint term in the loss function during training. This can automatically balance the influence of various sub-terms such as diffusion equation constraint terms and boundary condition constraint terms, avoiding training process oscillations and convergence difficulties caused by improper fixed weight allocation. This significantly improves the convergence stability and final prediction accuracy of the model in multi-regional smooth transition scenarios.

[0013] Furthermore, in step S3, the improvement using the domain-specific neural network XPINN architecture specifically includes: S301, based on the geometric structure and material distribution of the multi-region neutron diffusion problem, divides the physical domain into several non-overlapping subdomains; S302, establish independent PINN subnets in each subdomain, set network hyperparameters for each subnet to adapt to the physical characteristics of its corresponding subdomain, and assign training sets to each subnet that match the subdomain range; S303, construct an independent loss function for each PINN subnet, wherein the independent loss function includes the diffusion equation constraint terms and local boundary condition constraint terms of the corresponding subdomain of the subnet; S304, add neutron flux continuity constraint terms and neutron flux density continuity constraint terms at the boundary to the loss function of adjacent PINN subnets; S305: After each iteration of calculation, the prediction results of each subdomain PINN subnet are spliced ​​together to form a global neutron flux distribution, and the fission source term and effective multiplication factor keff are updated based on this global distribution.

[0014] The above scheme is specifically designed for multi-region neutron diffusion problems with large differences in cross-sections between materials and large neutron flux gradients at the boundaries. It effectively overcomes the expressive capacity bottleneck of single-network PINN. Based on the geometric structure and material distribution, the physical domain is divided into several non-overlapping subdomains, and independent PINN subnetworks are established. This decomposes the globally complex strong gradient problem into a smooth physical problem within multiple subdomains, allowing each subnetwork to focus only on fitting the stable physical characteristics of its corresponding subdomain. This significantly simplifies the learning task of the subnetworks and avoids the insufficient expressive capacity of single networks due to the need to simultaneously fit the strong differences in characteristics across multiple regions. By adding neutron flux continuity constraints and neutron flux density continuity constraints at the boundaries to the loss functions of adjacent PINN subnetworks, the scheme can eliminate the abrupt errors in physical quantities that may occur due to independent training of subnetworks at the subdomain boundaries, ensuring that the physical laws at the boundaries conform to the requirements of neutron diffusion theory. After each iteration, the prediction results of each subdomain are spliced ​​together to form a global neutron flux distribution and the fission source term and keff are updated. This achieves consistent integration of global physical quantities while ensuring the solution accuracy of each subdomain, ultimately obtaining a global neutron diffusion solution that meets engineering accuracy requirements. Furthermore, the specific implementation process of the residual adaptive resampling strategy includes: S2011 divides the physical domain of the neutron diffusion problem into several sub-grid regions evenly. S2012, During model training, the residual value of each sub-grid region is evaluated at preset intervals of a number of steps, and the sub-grid region with the highest current residual value is located. S2013, add a preset number of sampling points in the sub-grid region with the highest residual value, and combine the added sampling points with the original training set to form a new training set; S2014. Repeat steps S2012-S2013 until the total number of sampling points in the training set reaches the preset upper limit or the neural network algorithm model converges.

[0015] The above scheme ensures the coverage and uniformity of residual evaluation by uniformly dividing the physical domain of the neutron diffusion problem into several sub-grid regions, avoiding the omission of high residual regions caused by uneven factor grid division. During model training, the residual values ​​of each sub-grid region are evaluated at preset intervals to locate high residual regions. This dynamically tracks the residual change trend during model training, avoiding computational redundancy caused by a large increase in sampling points at once. Sampling points are only selectively added to key regions requiring optimization, improving the fitting accuracy of key regions with lower computational overhead. Repeating the residual evaluation and sampling point supplementation steps until the total number of sampling points reaches a preset upper limit or the model converges allows for a flexible balance between sampling cost and solution accuracy, avoiding resource waste from oversampling or accuracy degradation from undersampling. This ensures the efficiency and reliability of the RAR strategy in practical applications. Furthermore, the specific implementation process of the dynamic weight adjustment strategy includes: S2021, During model training, at preset intervals, the average gradient of each constraint term in the loss function with respect to the neural network parameters is calculated. S2022, calculate the temporary weight factor for each constraint term based on the average gradient of each constraint term; S2023, the final weights of each constraint item are adjusted through a smooth update mechanism, wherein the smooth update mechanism is calculated by fusing historical weights and temporary weight factors at a preset ratio; S2024 limits the maximum value of the final weight of each constraint to avoid extreme imbalance of weights.

[0016] The above scheme calculates the average gradient of each constraint with respect to the neural network parameters at preset intervals during model training. This allows for the determination of the actual impact of each constraint based on the real-time training state of the model, providing an objective and accurate basis for weight adjustment and avoiding bias caused by subjective experience in setting weights. Calculating temporary weight factors based on the average gradient and integrating historical weights with temporary weight factors through a smooth update mechanism avoids sudden weight changes caused by single gradient fluctuations, ensuring the continuity and stability of weight adjustment and reducing oscillations during training. Limiting the maximum value of the final weight of each constraint effectively prevents the model from becoming overly biased towards that constraint and ignoring other physical laws due to excessively high weights, avoiding a decrease in solution accuracy caused by extreme weight imbalance, and ensuring that each constraint works synergistically to improve the stability of model training and the final prediction accuracy.

[0017] Furthermore, the network hyperparameters set for each PINN subnet include the number of hidden layers, the number of neurons per layer, the weight initialization method, and the initial bias value. PINN subnets in different subdomains can be set with different hyperparameters according to the differences in the physical characteristics of the corresponding subdomains.

[0018] The above scheme incorporates the number of hidden layers, the number of neurons per layer, the weight initialization method, and the initial bias value of the subnet into the hyperparameter range, which can optimize the expressive power of the subnet from key dimensions such as network structure and parameter initialization. It allows PINN subnets of different subdomains to set different hyperparameters according to the differences in the physical characteristics of the corresponding subdomains. For subdomains with simple physical characteristics (such as single material and gradual changes in neutron flux), simpler hyperparameters can be set to reduce computational costs. For subdomains with complex physical characteristics (such as multi-material interfaces and large neutron flux gradients), more complex hyperparameters can be set to improve the expressive power of the subnet. This avoids the "one-size-fits-all" hyperparameter setting that leads to underfitting or computational redundancy in some subdomains, ensuring that each subnet is in the optimal solution state, thereby improving the accuracy and solution efficiency of the global neutron diffusion solution.

[0019] Furthermore, the neutron flux continuity constraint term in the loss function of the adjacent PINN subnets is used to ensure that the neutron flux values ​​output by the subnets on both sides of the boundary are consistent, and the neutron flux density continuity constraint term is used to ensure that the neutron flux density values ​​output by the subnets on both sides of the boundary are consistent, so as to eliminate the abrupt error of physical quantities at the subdomain boundary.

[0020] In the above scheme, the neutron flux continuity constraint term ensures that the neutron flux values ​​output by the subnets on both sides of the subdomain boundary are consistent, and the neutron flux density continuity constraint term ensures that the neutron flux density values ​​output by the subnets on both sides of the boundary are consistent. Together, they ensure that the physical quantities at the subdomain boundary meet the basic continuity requirements of neutron diffusion theory. This constraint can effectively eliminate the "jump" error of physical quantities at the boundary that may be caused by the independent training of each subnet, avoid such errors from being propagated to the global level and causing distortion of the global flux distribution, ensure the physical rationality and accuracy of the global neutron flux distribution, and provide a reliable basis for the accurate updating of the subsequent fission source term and effective multiplication factor (keff).

[0021] Furthermore, when constructing the PINN neural network structure, the weights of the fully connected layers are initialized using a uniform Xavier distribution, and the initial value of the bias term is set to a preset small value; the activation function adopts a function type that adapts to the continuous variation characteristics of neutron flux.

[0022] In the above scheme, the weights of the fully connected layers are initialized using a uniform Xavier distribution, which ensures that the input and output variances of each layer of the network are consistent, avoiding gradient vanishing or gradient explosion problems caused by excessively large or small weights in the early stage of training, and accelerating model convergence. The initial value of the bias term is set to a preset small value, which can avoid the network initialization being too homogeneous due to an initial bias of 0, and help the network enter the effective training stage more quickly. The activation function adopts a function type that adapts to the continuous variation characteristics of neutron flux, which can ensure that the neural network accurately fits the continuous distribution law of neutron flux, reduce the fitting bias caused by the mismatch between the activation function and the physical quantity characteristics, and significantly improve the model's prediction accuracy of neutron flux.

[0023] Furthermore, in step S1, the preset convergence criteria include an inner iteration convergence criterion and an outer iteration convergence criterion: The internal iteration convergence criterion is that the loss value of the neural network algorithm model is less than a preset loss threshold; The external iteration convergence criteria include that the relative error between two adjacent iterations of the effective multiplication factor keff is less than a preset keff error threshold, and the ratio of the L2 norm of two adjacent iterations of the fission source term is less than a preset source term error threshold.

[0024] In the above scheme, the inner iteration convergence criterion is set as the loss value of the neural network algorithm model being less than a preset loss threshold. This ensures that the PINN solution for neutron flux in the inner iteration stage meets the accuracy requirements of the current outer iteration step, avoiding the propagation of flux errors caused by insufficient inner iteration to the outer iteration, or the waste of computational resources caused by excessive inner iteration. The outer iteration convergence criterion also limits the relative error between two adjacent iterations of the effective multiplication factor (keff) to be less than a preset keff error threshold, and the L2 norm ratio between two adjacent iterations of the fission source term to be less than a preset source term error threshold. This allows for a comprehensive judgment on whether the global solution has converged from two dimensions: the key physical parameters of the reactor (keff) and the core factors of neutron production (fission source term). This avoids misjudgment of convergence caused by a single index and ensures that the final output keff and the global neutron flux distribution both meet the accuracy requirements of nuclear reactor physics numerical calculation.

[0025] The beneficial effects of this invention are: This invention addresses key shortcomings in existing technologies by constructing a fundamental algorithm model combining the Physical Information Neural Network (PINN) with source iteration, and by employing training-level optimization strategies and domain-specific neural network architecture improvements for two types of multi-region neutron diffusion problems with different cross-sectional differences between materials. 1. Avoids the problem that traditional finite difference methods and finite element methods rely on mesh generation, and the computational cost increases exponentially in high-dimensional or complex geometric scenarios; 2. Overcome the shortcomings of the purely physics-driven PINN method in multi-region neutron diffusion problems, especially when there are strong neutron flux gradient changes at the region boundaries, such as insufficient prediction accuracy and poor convergence; 3. Without relying on the high-precision prior data required by the data-driven physical information neural network DEPINN, it can cover two types of multi-region scenarios under a pure physics-driven framework: "smooth transition of material cross sections" and "large differences in material cross sections and large neutron flux gradient at the boundary". This significantly improves the accuracy and convergence stability of solving the neutron diffusion equation and meets the needs of nuclear reactor physics numerical calculations for solving complex multi-region problems. Attached Figure Description

[0026] Figure 1 This is a flowchart of a method for improving the accuracy and convergence of solving diffusion equations based on PINN, provided by an embodiment of the present invention. Figure 2 This is a schematic diagram of the PINN source iteration algorithm provided in an embodiment of the present invention; Figure 3 This is a sample point map with RAR-encrypted area provided in an embodiment of the present invention; Figure 4 This is a geometric structure diagram of a two-dimensional rectangular geometric multi-group multi-material cross-section smooth transition problem provided in an embodiment of the present invention; Figure 5 This is a heatmap of neutron flux distribution in a two-dimensional rectangular geometric multi-group multi-material cross-section with a smooth transition, provided in an embodiment of the present invention. Figure 6 This is a Keff convergence curve diagram of a two-dimensional rectangular geometric multi-group multi-material cross-section smooth transition problem provided in an embodiment of the present invention; Figure 7 This is a geometric structure diagram of a two-dimensional TWIGL benchmark problem provided in an embodiment of the present invention; Figure 8 This is a diagram of the XPINN subnet structure for the two-dimensional TWIGL benchmark problem provided in this embodiment of the invention; Figure 9 This is a heatmap of neutron flux distribution in a two-dimensional TWIGL benchmark problem provided in an embodiment of the present invention; Detailed Implementation

[0027] The technical solutions of the embodiments of this application will be clearly described below with reference to the accompanying drawings. Obviously, the described embodiments are only some, not all, of the embodiments of this application. All other embodiments obtained by those skilled in the art based on the embodiments of this application are within the scope of protection of this application.

[0028] In the description of this application, it should be noted that the terminology used herein is for the purpose of describing particular embodiments only and is not intended to limit the exemplary embodiments according to this application. For ease of description, the dimensions of the various parts shown in the drawings are not drawn to actual scale. Techniques, methods, and devices known to those skilled in the art may not be discussed in detail, but where appropriate, such techniques, methods, and devices should be considered part of the specification. In all examples shown and discussed herein, any specific values ​​should be interpreted as merely exemplary and not as limitations. Therefore, other examples of exemplary embodiments may have different values. It should be noted that similar reference numerals and letters in the following drawings denote similar items; therefore, once an item is defined in one drawing, it need not be further discussed in subsequent drawings.

[0029] It should be noted that the terms "first," "second," etc., used in the specification and claims of this application are used to distinguish similar objects and not to describe a specific order or sequence. It should be understood that such use of data can be interchanged where appropriate so that embodiments of this application can be implemented in orders other than those illustrated or described herein, and the objects distinguished by "first," "second," etc., are generally of the same class and are not limited in number; for example, a first object can be one or more. Furthermore, in the specification and claims, "and / or" indicates at least one of the connected objects, and the character " / " generally indicates that the preceding and following objects are in an "or" relationship.

[0030] It should be noted that in the description of this application, the directional terms such as "front, back, up, down, left, right", "horizontal, vertical, horizontal" and "top, bottom" indicate the orientation or positional relationship based on the orientation or positional relationship shown in the accompanying drawings. They are only for the convenience of describing this application and simplifying the description. Unless otherwise stated, these directional terms do not indicate or imply that the device or element referred to must have a specific orientation or be constructed and operated in a specific orientation, and therefore should not be construed as a limitation on the scope of protection of this application. The directional terms "inner" and "outer" refer to the inner and outer contours relative to the outline of each component itself.

[0031] It should be noted that, in this application, the terms "comprising," "including," or any other variations thereof are intended to cover non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such a process, method, article, or apparatus. Without further limitations, an element defined by the phrase "comprising one..." does not exclude the presence of other identical elements in the process, method, article, or apparatus that includes that element. Furthermore, it should be noted that the scope of the methods and apparatuses in the embodiments of this application is not limited to performing functions in the order shown or discussed, but may also include performing functions substantially simultaneously or in the reverse order, depending on the functions involved. For example, the described methods may be performed in a different order than described, and various steps may be added, omitted, or combined. Additionally, features described with reference to certain examples may be combined in other examples.

[0032] like Figures 1-9 As shown, the method for improving the accuracy and convergence of solving diffusion equations based on PINN provided by embodiments of the present invention includes the following steps: S101, Establish the PINN source iterative algorithm model; S102, the problem of smooth transition in multiple zones is optimized by introducing RAR and dynamic weight adjustment strategies; S103, the problem of large neutron flux gradient in multiple regions is improved by adopting the XPINN domain-sharing architecture; S104, perform network prediction accuracy and convergence stability analysis.

[0033] Based on the development of numerical computation in reactor physics, and considering the differences in cross-sections between materials in multi-region problems and the drastic changes in neutron flux gradients at the boundaries, existing single-network PINN methods suffer from low prediction accuracy and poor convergence stability. The DEPINN method, on the other hand, relies on high-precision prior data, limiting its applicability in scenarios with insufficient data. This invention develops an optimization method to improve the accuracy and convergence of solving diffusion equations based on PINN, specifically for addressing the eigenvalue problem of neutron diffusion equations under complex multi-region conditions.

[0034] 1) Establish the PINN source iterative algorithm model: In the absence of external sources, the steady-state multi-group neutron diffusion equation can be expressed as: In this invention, the established algorithm model is mainly aimed at solving the two-dimensional steady-state two-group neutron diffusion equation.

[0035] The PINN algorithm, combined with source iteration, updates the source term during outer iterations. and k effIn the inner iteration, a neural network model with two outputs is used to output the neutron flux of the hot group and the fast group, respectively. The algorithm flow is as follows: (1) Set the maximum number of outer iterations and generate training set sampling points. Given initial values ​​of fluxes for two groups. and initial value of effective proliferation factor And calculate the initial source term based on the initial flux. .

[0036] (2) and Substituting into the right side of equation (1), the right side of the equation now becomes a known term. The training set is then input into the neural network, and after processing by the network, the flux prediction value is obtained. At this point, based on the current network output and the known source terms, the equation constraint terms can be obtained, and together with the boundary condition constraint terms, a loss function can be constructed. This loss function has the following form: In this equation, the first term on the right-hand side is the boundary loss, which is calculated as follows: boundary points Network prediction With true boundary value The first term is the mean square error between the two terms; the second term is the loss from the physical equations, which is calculated in... Location points within the solution domain Above, physical residual The mean squared error is calculated. The network parameters are optimized by minimizing this loss function, ensuring that the neutron flux output by the network satisfies the current equation as closely as possible. During the inner iteration process, the convergence criterion is set as follows: ,in Adjustments are made based on the specific calculation example, typically within 10. -3 Up to 10 -5 The values ​​are selected within a certain range to ensure that the model achieves a reasonable balance between accuracy and computational cost.

[0037] (3) Update the fission source term using the output convergent neutron flux density. and effective proliferation factor k eff .

[0038] (4) To determine whether the outer iteration satisfies the convergence condition, the following convergence criteria are usually adopted: in It is an absolute value function; It is an L2 norm; Usually, 10 is taken respectively. -6 and 10 -5If convergence is not achieved, repeat steps (2) and (3) until the criterion is met and the final predicted solution is output.

[0039] 2) Optimization of the multi-zone smooth transition problem by introducing RAR and dynamic weight adjustment strategies: The RAR strategy enhances the model's ability to represent regions by dynamically identifying high residual regions and periodically increasing the sampling point density of these regions during training. The specific process is as follows: (1) Divide the physical domain into uniform parts Individual grid areas.

[0040] (2) During the training process, at each interval The residuals of each sub-region are evaluated step by step to locate the sub-region with the highest current residual value.

[0041] (3) Supplementing within this area (4) Take one sampling point and combine it with the original training set to form a new training set. Repeat this process until the total number of sampling points reaches the preset upper limit or the network converges.

[0042] In the above process, , as well as The parameters can be flexibly set according to the specific calculation example to achieve the optimal effect of the method.

[0043] The dynamic weight adjustment strategy can automatically balance various physical constraints during training, avoiding convergence difficulties caused by improper weight allocation and improving the stability and accuracy of network training. The specific process is as follows: (1) Every 10 steps, calculate the mean gradient of each constraint term with respect to the network parameters. : Where M is the total number of network parameters. Let j be the j-th network parameter.

[0044] (2) For each constraint term, calculate the temporary weighting factor. : (3) Smoothly update the weights of each constraint term: in, The value is set to 0.9 to avoid excessively drastic weight changes. To prevent extreme weight imbalance, a limit is set... The maximum value is 20.

[0045] (4) The final loss function is as follows: in, For the loss of the diffusion equation, This is the boundary condition loss.

[0046] 3) The problem of large neutron flux gradient in multiple regions is improved by adopting the XPINN domain architecture: (1) According to the geometric distribution of multiple material regions and the differences in material cross sections, the global physical domain is divided into several non-overlapping sub-regions. For each sub-region, an independent PINN network model is established.

[0047] (2) Within each subdomain, set the neural network hyperparameters independently and construct a loss function that includes the diffusion equation residual term and the local boundary condition residual term of that subdomain.

[0048] (3) On the common boundary of adjacent subnets, add neutron flux continuity constraints and neutron flux density continuity constraints, in the form of: in, This represents the number of sampling points at the interface. , These are the predicted flux values ​​of the two subnets at the interface, respectively; , These represent the neutron flux densities at the interface of the two subnets, respectively.

[0049] (4) After each round of external iteration, the outputs of each subdomain network are concatenated into a global flux distribution, and the fission source terms are iteratively updated. With k eff .

[0050] 4) Perform network prediction accuracy and convergence stability analysis: The method was used to numerically verify examples of varying complexity, including a two-dimensional rectangular geometric multigroup single-material homogeneous problem, a two-dimensional rectangular geometric multigroup multimaterial cross-section with smooth transition, and a two-dimensional TWIGL benchmark problem. The k-value of the network model was also used. eff The prediction results and the k obtained by the variational block method eff By comparing and analyzing the reference solution, the advantages of the method of the present invention in terms of prediction accuracy and convergence in solving the neutron diffusion equation are verified.

[0051] In summary, the innovative aspects of this invention compared to the prior art are: First, a training optimization method based on RAR and dynamic weight adjustment strategy is introduced into the established PINN source iterative model. Its application in solving the neutron diffusion eigenvalue problem in PINN is innovative. Secondly, the XPINN method is used to improve the network architecture to solve the multi-region strongly non-uniform problem, which effectively breaks through the expressive power bottleneck of the single network model in the problem of drastic changes in neutron flux gradient at the boundary.

[0052] The specific principles and methods are as follows: (1) Introduce the RAR strategy during the training process to periodically identify high residual regions and increase the sampling point density to improve the fitting accuracy of residual cluster regions.

[0053] (2) A dynamic weight adjustment mechanism is proposed during the training process to dynamically update the weights based on the gradient of each physical constraint term.

[0054] (3) To address the difficulty of fitting a single network to the strong gradient interface problem, an improved method for the XPINN network architecture is proposed. The computational domain is divided into multiple subdomains and the subnetworks are trained independently to establish a multi-subnetwork collaborative solution framework.

[0055] To demonstrate the inventiveness and technical value of this invention, the correctness of the PINN-based source iteration algorithm is an important basis. First, an example verification of the PINN-based source iteration algorithm is conducted: Example 1: This example uses a two-dimensional rectangular geometric multigroup single-material domain problem to verify and analyze the PINN source iteration method. The problem is a square domain with a side length of 100 cm, considering two boundary conditions: total internal reflection and vacuum boundary conditions. Specific material parameters are shown in the table below (Material Properties of the Single-Material Calculation Domain). In the calculation, the neural network is set to 10 layers, with 40 neurons per hidden layer. All fully connected layer weights w are initialized using a Xavier uniform distribution, and the bias term b is initialized to 0.01. The learning rate is fixed at 5 × 10⁻⁶. -5 The maximum number of training steps for the outer iteration of the source iterative algorithm is 20,000. The convergence criterion for the inner iteration is σ = 1 × 10⁻⁶. -4 Each training iteration uses all sampling points. The Latin supersampling method samples a total of 15,000 points within the geometric region and 1,500 points along each edge at the boundary.

[0056] The calculation results are shown in the table below (two-dimensional multi-group single-region calculation results table). Under the condition of total reflection boundary, k eff The prediction error is only 16 pcm; under more complex vacuum boundary conditions, the error is 56 pcm.

[0057] Example 2: This example uses a two-dimensional rectangular geometric multi-group multi-material region smooth transition problem to verify the RAR sampling and dynamic weight adjustment optimization method.

[0058] A problem involving a smooth transition in a two-dimensional rectangular geometric multi-group, multi-material region. The geometric region is a square with a side length of 100 cm, and the geometric structure is as follows: Figure 4 The five regions shown in the figure are labeled with their corresponding material numbers. The left and lower boundaries are total internal reflection boundary conditions, while the right and upper boundaries are vacuum boundary conditions. The material parameters are shown in the table below (Material Property Table for Multi-Material Calculation Regions). In the calculation, the neural network is set to 10 layers, with 50 neurons per hidden layer. All other parameters are set the same as in Example 1, with the learning rate fixed at 1×10⁻⁶. -4 The maximum number of training steps in the outer iteration of the source iterative algorithm is 40,000, and the inner iteration exit condition is σ = 1 × 10⁻⁶. -3 Each training iteration uses all sampling points. The RAR sampling grid α = 8, and the number of points n_points = 500 is increased every 1000 steps. The Latin supersampling method samples a total of 15000 points inside the geometric region, and samples 1000 points on each edge at the boundary.

[0059] The calculation results are shown in the table below (two-dimensional multi-group multi-material region calculation results table), and the flux distribution heatmap is shown in [reference needed]. Figure 5 k eff Convergence curves are shown Figure 6 As shown in the table above, for multi-region problems with gently transitioning cross-sections in this type of material, the k-value of the PINN source iterative algorithm without an optimization strategy is significantly lower. eff The prediction error is relatively large, at 989 pcm. After introducing the RAR sampling strategy, k eff The error was reduced to 402 pcm. k was obtained using a dynamic weight adjustment method. eff The error was only 63 pcm. Figure 6 k eff As can be seen from the convergence curve, dynamic weight adjustment improves the oscillation phenomenon during model training and enhances convergence stability.

[0060] Example 3: This example employs the XPINN domain-specific solution algorithm in the PINN source iteration method and verifies it using the TWIGL benchmark problem with large neutron flux gradients at the partitions.

[0061] The geometry of the two-dimensional TWIGL benchmark problem is as follows: Figure 7As shown, the left and lower boundaries are total internal reflection boundary conditions, and the right and upper boundaries are vacuum boundary conditions. The material parameters are shown in the table below (material properties of the computational domain in the TWIGL benchmark problem). During the calculation, the XPINN algorithm is used to divide the computational domain as shown... Figure 8 The diagram shows three subdomains, each modeled by a separate subnetwork. The Net2 subnetwork contains three different material regions, with the labels corresponding to the material numbers in the multi-material computation region material property table. In the computation, Net1 is set to 3 layers, each containing 30 neurons; Net2 is set to 4 layers, each containing 40 neurons; and Net3 is also set to 4 layers, each containing 40 neurons. All other network structure parameters are consistent with those described above. The learning rate for all three subnetworks is fixed at 1×10⁻⁶. -4 The convergence criterion for the inner iteration is that the loss of all three subnetworks is simultaneously less than σ = 1 × 10. -4 Within Net1, 900 points are sampled, and 150 points are sampled along each edge of the boundary. Within Net2, 4000 points are sampled, and 200 points are sampled along each edge of the boundary. Within Net3, 5100 points are sampled, and 150 points are sampled along each edge of the boundary. At the boundary between Net1 and Net2, 200 points are sampled along each edge, and at the boundary between Net2 and Net3, 600 points are sampled along each edge. All sampled points are generated using the Latin hypercube sampling method.

[0062] The calculation results are shown in the table below (Calculation results table of TWIGL benchmark problems under the XPINN framework). For TWIGL benchmark problems with large neutron flux gradients, the optimization method k is not used. eff The prediction error was 1253 pcm, while the weighted dynamic strategy, which performed better in the training and optimization process, had an error of 1060 pcm. However, after adopting the XPINN domain-specific framework, k eff The error is only 25 pcm, which is about 97.6% lower than that of dynamic weight adjustment, and is significantly better than the single-network PINN optimization method.

[0063] Evidence of the effects of the embodiments. The embodiments of the present invention have achieved some positive effects during research and development or use, and indeed have significant advantages compared with existing technologies. The following description combines data, charts, etc., from the experimental process.

[0064] Firstly, regarding Example 1, the experimental process mainly verifies the correctness of the PINN source iteration algorithm. Specifically, regarding k in this example... eff The prediction results show that, in a two-dimensional single-region homogeneous problem using total internal reflection boundary and vacuum boundary conditions, the k-value of this iterative algorithm is [missing information]. effThe feasibility of the established PINN source iterative algorithm was confirmed when the error between the predicted and reference values ​​was less than 100 pcm.

[0065] Secondly, in Example 2, for the smooth transition problem with small cross-sectional differences in multi-material regions, compared to the PINN calculation model without optimization strategy, the RAR strategy yields better results after k... eff The prediction error was smaller at 402 pcm, while the error of the unoptimized model reached 989 pcm, indicating that denser sampling in high residual regions can improve the model's ability to fit local physical properties. Furthermore, after dynamic weight adjustment, k... eff The prediction error is only 63 pcm, far less than the error of the unoptimized model. And regarding k... eff From the perspective of convergence stability comparison, combined with Figure 6 It can be seen that after adopting dynamic weight adjustment, k eff The oscillations during convergence are significantly reduced, and the model converges to the vicinity of the reference value more quickly. Overall, the results indicate that in problems with relatively smooth material cross-section transitions, both RAR sampling and dynamic weight adjustment optimization strategies at the training level can effectively improve the prediction accuracy of the PINN computational model.

[0066] Furthermore, regarding Example 3, a comparison of the calculation results table for the TWIGL benchmark problem under the XPINN framework shows that the k-value of the unoptimized model is... eff The error was 1253 pcm, while the optimization methods at the training level performed poorly, with the error reaching 1060 pcm even after adopting dynamic weight adjustment. The k-value of the single-network PINN model... eff The prediction error is over 1000 pcm, far from meeting the accuracy requirements for reactor physics numerical calculations. However, after adopting the XPINN domain-based framework improvement, k eff The error is only 25 pcm, a reduction of approximately 97.6% compared to dynamic weight adjustment. For multi-region problems with large cross-sectional differences between regions, the single-network PINN struggles to effectively fit such complex situations due to the large neutron flux gradient at the boundaries. However, after domain division, XPINN allows each subnetwork to focus on fitting a function with stable physical properties within its own subdomain, simplifying the learning task and effectively improving prediction accuracy. Overall, for this benchmark case, the model's prediction accuracy is significantly better than the single-network PINN model after framework improvement using XPINN, which to some extent addresses the current limitations of PINN methods in computational accuracy for strongly non-uniform multi-region problems.

[0067] Based on the above cases, this invention establishes the PINN source iterative algorithm and verifies the correctness of its calculated values. As can be seen in Case 2 and Case 3, the optimization method based on this iterative algorithm achieves higher solution accuracy compared to the current purely physics-driven PINN method, providing new ideas and references for the application of the PINN method in multi-region neutron diffusion problems.

[0068] The embodiments of this application have been described above with reference to the accompanying drawings. Unless otherwise specified, the embodiments and features in the embodiments of this application can be combined with each other. This application is not limited to the specific embodiments described above. The specific embodiments described above are merely illustrative and not restrictive. Those skilled in the art can make many other forms under the guidance of this application without departing from the spirit and scope of the claims, and all of these forms are within the protection scope of this application.

Claims

1. A method for improving the accuracy and convergence of solving diffusion equations based on PINN, characterized in that, Includes the following steps: S1, Establish a neural network algorithm model that combines the physical information neural network PINN with source iteration; S2, For the problem of neutron diffusion in a multi-regional smooth transition with small cross-sectional differences between materials, an optimization strategy at the training level is adopted to optimize the neural network algorithm model; S3. To address the problem of multi-region neutron diffusion caused by large differences in cross-sections between materials leading to large neutron flux gradients at the boundaries, a domain-specific neural network architecture is adopted to improve the neural network algorithm model.

2. The method for improving the accuracy and convergence of solving diffusion equations based on PINN according to claim 1, characterized in that, In step S1, the specific steps for establishing the neural network algorithm model combining PINN and source iteration include: S101, Construct the PINN neural network structure within the deep learning framework, and set the number of network layers, the number of neurons per layer, and the activation function; S102, a spatial coordinate training set is generated according to the geometric range of the neutron diffusion problem. The steady-state multi-group neutron diffusion equation without external source is taken as the equation to be solved. The automatic differentiation function of the deep learning framework is used to obtain the correlation between the network output flux and the input spatial coordinates. Based on this correlation, the diffusion equation constraint terms are established, and the boundary conditions are transformed into the constraint terms in the loss function. S103. A two-layer iterative structure is established. In the inner iteration, the PINN network is used as the neutron flux solver. In the outer iteration, the fission source term and the effective multiplication factor keff are updated, and the iteration process is terminated according to the preset convergence criterion.

3. The method for improving the accuracy and convergence of solving diffusion equations based on PINN according to claim 2, characterized in that, In step S2, the optimization strategies at the training level include a residual adaptive resampling (RAR) strategy and a weight dynamic adjustment strategy, wherein: The residual adaptive resampling strategy periodically identifies high residual regions in the neural network algorithm model and increases the local sampling point density in these regions to improve the model's expressive ability in key high residual regions. The dynamic weight adjustment strategy dynamically adjusts the weights of each constraint term in the loss function based on the gradient contribution of each constraint term during training, thereby avoiding model convergence difficulties caused by improper weight allocation.

4. The method for improving the accuracy and convergence of solving diffusion equations based on PINN according to claim 3, characterized in that, In step S3, the improvement using the domain-specific neural network XPINN architecture specifically includes: S301, based on the geometric structure and material distribution of the multi-region neutron diffusion problem, divides the physical domain into several non-overlapping subdomains; S302, establish independent PINN subnets in each subdomain, set network hyperparameters for each subnet to adapt to the physical characteristics of its corresponding subdomain, and assign training sets to each subnet that match the subdomain range; S303, construct an independent loss function for each PINN subnet, wherein the independent loss function includes the diffusion equation constraint terms and local boundary condition constraint terms of the corresponding subdomain of the subnet; S304, add neutron flux continuity constraint terms and neutron flux density continuity constraint terms at the boundary to the loss function of adjacent PINN subnets; S305: After each iteration of calculation, the prediction results of each subdomain PINN subnet are spliced ​​together to form a global neutron flux distribution, and the fission source term and effective multiplication factor keff are updated based on this global distribution.

5. The method for improving the accuracy and convergence of solving diffusion equations based on PINN according to claim 3, characterized in that, The specific implementation process of the residual adaptive resampling strategy includes: S2011 divides the physical domain of the neutron diffusion problem into several sub-grid regions evenly. S2012, During model training, the residual value of each sub-grid region is evaluated at preset intervals of a number of steps, and the sub-grid region with the highest current residual value is located. S2013, add a preset number of sampling points in the sub-grid region with the highest residual value, and combine the added sampling points with the original training set to form a new training set; S2014. Repeat steps S2012-S2013 until the total number of sampling points in the training set reaches the preset upper limit or the neural network algorithm model converges.

6. The method for improving the accuracy and convergence of solving diffusion equations based on PINN according to claim 3, characterized in that, The specific implementation process of the dynamic weight adjustment strategy includes: S2021, During model training, at preset intervals, the average gradient of each constraint term in the loss function with respect to the neural network parameters is calculated. S2022, calculate the temporary weight factor for each constraint term based on the average gradient of each constraint term; S2023, the final weights of each constraint item are adjusted through a smooth update mechanism, wherein the smooth update mechanism is calculated by fusing historical weights and temporary weight factors at a preset ratio; S2024 limits the maximum value of the final weight of each constraint to avoid extreme imbalance of weights.

7. The method for improving the accuracy and convergence of solving diffusion equations based on PINN according to claim 4, characterized in that, The network hyperparameters set for each PINN subnet include the number of hidden layers, the number of neurons per layer, the weight initialization method, and the initial bias value. Furthermore, PINN subnets in different subdomains can be set with different hyperparameters according to the physical characteristics of the corresponding subdomains.

8. The method for improving the accuracy and convergence of solving diffusion equations based on PINN according to claim 4, characterized in that, The neutron flux continuity constraint term in the loss function of the adjacent PINN subnets is used to ensure that the neutron flux values ​​output by the subnets on both sides of the boundary are consistent, and the neutron flux density continuity constraint term is used to ensure that the neutron flux density values ​​output by the subnets on both sides of the boundary are consistent, so as to eliminate the abrupt error of physical quantities at the subdomain boundary.

9. The method for improving the accuracy and convergence of solving diffusion equations based on PINN according to claim 2, characterized in that, When constructing the PINN neural network structure, the weights of the fully connected layers are initialized using a uniform Xavier distribution, and the initial value of the bias term is set to a preset small value; the activation function adopts a function type that adapts to the continuous variation characteristics of neutron flux.

10. A method for improving the accuracy and convergence of solving diffusion equations based on PINN, as described in claim 2, characterized in that, In step S1, the preset convergence criteria include an inner iteration convergence criterion and an outer iteration convergence criterion: The internal iteration convergence criterion is that the loss value of the neural network algorithm model is less than a preset loss threshold; The external iteration convergence criteria include that the relative error between two adjacent iterations of the effective multiplication factor keff is less than a preset keff error threshold, and the ratio of the L2 norm of two adjacent iterations of the fission source term is less than a preset source term error threshold.

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