A mutual information-based similarity evaluation method
By employing a similarity evaluation method based on mutual information, and utilizing multi-group covariance matrices and kernel data sampling, the mutual information between nuclear reaction cross sections and responses is calculated. This overcomes the limitations of traditional similarity evaluation methods, enabling efficient and accurate similarity assessment of nuclear reactor systems and reducing model uncertainty.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- SOUTHEAST UNIV
- Filing Date
- 2025-07-31
- Publication Date
- 2026-07-21
AI Technical Summary
In nuclear reactor design, existing similarity evaluation methods rely on expert experience, which cannot be applied on a large scale. Furthermore, they are difficult to accurately assess the similarity between systems in nonlinear systems, making it difficult to meet the requirements for the quantity and quality of experiments in model validation, thus affecting the uncertainty and economy of the model.
A similarity evaluation method based on mutual information is adopted. Nuclear data is sampled through a multi-group covariance matrix, and multiple neutronics calculations are performed using a perturbed nuclear cross-section database to calculate mutual information to evaluate system similarity, including normalized mutual information between nuclear reaction cross-sections and responses, thus quantifying the similarity between systems.
It improves the applicability and accuracy of similarity evaluation, enabling accurate assessment of system similarity in linear and nonlinear systems, reducing model uncertainty, and improving the accuracy and efficiency of model validation.
Smart Images
Figure CN120995118B_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the field of nuclear reactor core physics technology, and in particular to a similarity evaluation method based on mutual information. Background Technology
[0002] Due to the unique nature of nuclear reactors, the process from design to engineering implementation is extremely lengthy, requiring complex and rigorous verification and validation. The high safety requirements of reactors necessitate a significant safety margin in the design phase to prevent accidents. This approach, to some extent, sacrifices the reactor's economic performance and can lead to resource waste in larger-scale applications. This places higher demands on the accuracy of the model; only by minimizing uncertainty while ensuring the model outputs correct results can a balance between economy and safety be achieved. Model validation primarily involves comparing the model's output with actual measurement results to assess the correctness and reliability of the designed model. It reduces model uncertainty to a greater extent by fusing measurement and simulation data, which places higher demands on the selected benchmark experiments. Ideally, a real-world experiment completely identical to the design model would be established, using measured data for direct comparison. However, considering cost and safety, this approach is not feasible. An alternative is to utilize experiments in an existing database with high similarity to the design model instead of a scaled-down model for model validation. Clearly, under this approach, the similarity between the benchmark experiment and the design scheme becomes the most important factor in determining the model's validation effect, which is the significance of conducting system similarity assessment.
[0003] Traditional methods use physical parameters such as geometric dimensions, fuel type, and water-to-uranium ratio as standards for similarity assessment. However, this method relies too heavily on expert experience and cannot be widely applied. Researchers then proposed linear correlation coefficients E and C based on sensitivity and uncertainty analysis. k As a similarity evaluation index, C k The coefficient remains the primary standard for reactor similarity assessment. Nuclear reaction cross section and effective multiplication factor k eff A good linear relationship between them makes C k In the main focus k eff It exhibits high accuracy in calculating numerical criticalities. However, k eff C is not the only response quantity worth noting in a reactor system; in system similarity determination based on some nonlinear responses, C... k It is difficult to meet the needs of practical applications. The application of C in nonlinear systems... k Nonlinear similarities between systems are often overlooked, which to some extent leads to a waste of valuable information. Related research indicates that only C... kExperiments with a similarity score higher than 0.8 are considered, and only those with a score higher than 0.9 can be considered ideal reference benchmarks. Furthermore, there are requirements regarding the number of benchmark experiments. In practical engineering, especially for reactor types still in the research stage and whose technologies are not yet mature, such requirements are often unattainable. In fact, our focus is not on whether the similarity between the two meets a predetermined standard, but rather on whether the benchmark experiments make a positive contribution to reducing application uncertainty—that is, whether they can provide effective information. The reason for this is... k The requirement for a specific numerical value is due to the potential negative impact of experiments with low similarity during kernel data adjustment. These experiments often amplify uncertainty, necessitating experiment screening. In information theory, information entropy is often used to measure the magnitude of uncertainty in a system; a larger entropy value corresponds to greater uncertainty. Therefore, entropy reduction can be used to measure the degree of uncertainty reduction brought about by benchmark experiments, known as information gain. Mathematically, information gain and mutual information are consistent and can be understood from the perspective of information entropy. For two random variables X and Y, let H(X) and H(Y) represent their information entropies, respectively. Mutual information and information gain are essentially cross-entropy, which can be understood as the amount of shared information between two variables. The portion excluding cross-entropy is defined as conditional entropy. Once one random variable is defined, the shared information in another random variable is also defined, reducing the uncertainty of the other random variable, i.e., resulting in entropy reduction. This reduced value is mutual information, while the remaining portion is conditional entropy, which can be understood as the portion of another random variable that remains uncertain after defining one. This portion represents the amount of information that is not shared between the two variables. Information gain possesses the desirable mathematical property of nonnegativity, meaning that even with high uncertainty in benchmark experiments, there's no need to worry about negative impacts, as the system's information entropy will remain constant even in the worst-case scenario. Therefore, the quantity and quality of benchmark experiments are no longer a challenge in this similarity assessment system, as this method extracts usable information from each benchmark experiment to reduce uncertainty in the final application. Summary of the Invention
[0004] The purpose of this invention is to address the problems existing in the background technology by proposing a similarity evaluation method based on mutual information.
[0005] The technical solution of the present invention, in its first aspect, provides a similarity evaluation method based on mutual information, comprising the following steps:
[0006] Step 1: Kernel data sampling based on multi-group covariance matrix;
[0007] Step 2: Perform multiple neutronics calculations using the perturbed nuclear cross-section database as input to obtain multiple types of responses;
[0008] Step 3: Calculate mutual information separately to obtain a comprehensive similarity assessment.
[0009] Preferably, in step one, kernel data random sampling is performed based on a multi-group covariance matrix. For a multi-group covariance matrix Σ, this method performs random sampling based on a normal distribution. We consider a multivariate normal distribution N(0,Σ) based on this covariance matrix. To obtain the perturbation factor required for kernel cross-section sampling from this covariance matrix, we first need to construct a series of independent and identically distributed standard normal distribution sample vectors x. i = [x1,x2,…x m Each sample vector contains m elements, where m is the number of energy groups in the corresponding covariance matrix, and i is the number of samples.
[0010] Based on this, our goal is to transform i independent standard normal distribution sample vectors into multidimensional normal distribution vectors that follow N(0,Σ). In matrix form, this means finding a linear transformation Y = LX such that Y ~ N(0,Σ). By definition, the expectation E(Y) of Y is expressed as...
[0011] E(Y)=E(LX)=LE(X)=0 (1)
[0012] Clearly, the sample mean of Y remains unchanged after the linear transformation. Next, we examine the sample covariance matrix as follows:
[0013]
[0014] At this point, it is only necessary to ensure that the target covariance matrix Σ = LL. T This ensures that the obtained samples follow the target multivariate normal distribution. This can be achieved by performing Cholesky decomposition on the original matrix to obtain two matrices that are transposes of each other; in fact, L is a lower triangular matrix. This decomposition requires the covariance matrix to be positive definite, but in reality, the covariance matrix does not always meet this requirement, meaning it may have negative eigenvalues. In this case, a usable method is to set the negative eigenvalues to 0 to obtain an approximate matrix of the original matrix and then perform relevant operations.
[0015] Preferably, after obtaining the lower triangular matrix L in step one, the cross-sectional perturbation coefficient matrix Y can be derived. For ease of calculation, Δ can be set to Y + I, where I is the identity matrix. Δ can then be directly used for cross-sectional sampling, adhering to the basic principle of synchronous sampling and considering the mutual influence between different cross-sections. This is significant in comprehensively considering the contribution of the kernel data cross-section to the uncertainty of the response.
[0016] Preferably, in step two, models for benchmark experiments and engineering applications are established respectively, and the perturbed nuclear cross-section database obtained in step one is used as input to repeatedly perform neutronics calculations using relevant simulation software to obtain results. The number of simulations is determined according to the required accuracy.
[0017] Preferably, in step three,
[0018] The mutual information between two discrete random variables is calculated as follows:
[0019]
[0020] Where p(x,y) represents the joint probability distribution function between random variables X and Y, while p(x) and p(y) represent the marginal probability distribution functions of the two random variables, respectively. For continuous random variables, the form is integral:
[0021]
[0022] Here, p(x,y) is the joint probability density function between random variables X and Y, while p(x) and p(y) are the marginal probability density functions of X and Y, respectively. In practical applications, the probability density function of random variables is difficult to determine, and it is generally calculated in discrete form. Numerically, nonparametric estimation methods such as kernel density estimation can also be used to obtain an approximate probability density function for calculation.
[0023] Preferably, in step three, the normalized mutual information magnitudes between the nuclear reaction cross section and the response, and between similar responses of the two systems, are calculated respectively. The normalized mutual information NMI between the nuclear data cross section and different responses is written as:
[0024]
[0025] Among them, R p This indicates the output response, where p represents its type, and Let represent the reaction cross section, n, r, and g represent the nuclide type, reaction type, and number of specific energy groups, respectively, and A represent the normalization factor. For data with different structural characteristics, to assess the similarity between two systems, the mutual information of each cross section for responses of the same type can be quantified. The mutual information value indicates the amount of shared information between two random variables, and also indicates the degree to which the uncertainty of one random variable is reduced for the other under certain conditions. The nuclear cross section is the input to the reactor system; high consistency of mutual information across cross sections indicates a high degree of similarity between the systems.
[0026] In step three, the similarity between systems is quantified by calculating the distance between the mutual information vectors of two specific cross-sections and the responses. For multiple specific response types, we select multiple important response cross-sections, such as heavy metal nuclides that have a significant impact during reactor operation, such as uranium and plutonium, and obtain the normalized mutual information vector as follows:
[0027]
[0028] Here, m represents different systems. The similarity coefficient can be expressed as:
[0029]
[0030] 's' represents the cosine of the angle between two mutual information vectors. Since the normalized mutual information value is between [0,1], the value of 's' is also limited to [0,1]. 's=1' represents that the angle between the two vectors is 0, i.e., they are perfectly correlated; while 's=0' represents that the two vectors are perpendicular, i.e., they are completely uncorrelated. Furthermore, since normalized mutual information has a certain degree of comparability, we can determine the degree of influence of the normalized mutual information of different cross-sections on the uncertainty of the result response based on the magnitude of the normalized mutual information. This provides an important reference for screening the main influencing factors of the response quantities of interest and for subsequent model optimization based on these factors. This method eliminates the reliance on sensitivity analysis; in fact, most neutron science simulation software does not have a sufficiently robust sensitivity analysis function to support such cross-section screening. The method proposed in this invention provides an optional solution for subsequent related research.
[0031] In step three, the normalized mutual information of the same response of two systems can also be used as a similarity evaluation criterion. It characterizes the correlation between the changes in the responses of the two systems under the same perturbation. Considering multiple different types of response quantities, a comprehensive assessment of the similarity between the two systems can be made. On the other hand, in information theory, information entropy is used to measure the magnitude of system uncertainty. From the perspective of information entropy, mutual information is essentially the cross-entropy between two random variables. A larger cross-entropy indicates that they share more information and that their respective conditional entropies are lower. The entropy of a random variable can be seen as the sum of cross-entropy and conditional entropy. From this perspective, mutual information represents the extent to which the uncertainty of one random variable is reduced after the other is defined, because this indicates a greater amount of shared information. Therefore, this can be used to test whether the uncertainty brought by different benchmark experiments to applications is consistent with their similarity coefficients and to make certain adjustments.
[0032] A second aspect of the present invention provides a computer device including a memory and a processor, the memory storing a computer program, the processor executing the computer program to implement the steps of the above-described similarity evaluation method based on mutual information.
[0033] Compared with the prior art, the present invention has the following beneficial technical effects:
[0034] Compared to traditional linear coefficient evaluation methods, the method proposed in this invention is based on mutual information and is not limited to specific types of systems. It has high applicability and evaluation accuracy for both linear and nonlinear systems. Furthermore, since it does not rely on the original linear assumption, the similarity coefficients based on mutual information can consider various types of response quantities. After normalization, the normalized mutual information has a certain degree of comparability, allowing various types of mutual information coefficients to be incorporated into the same evaluation system, thus possessing greater universality and versatility. Attached Figure Description
[0035] Figure 1 This is a flowchart illustrating the implementation of a similarity evaluation method based on mutual information. Detailed Implementation
[0036] Example 1
[0037] like Figure 1 As shown, this embodiment provides a similarity evaluation method based on mutual information, including the following specific steps:
[0038] S1. Kernel data sampling based on multi-group covariance matrix;
[0039] S2. Multiple neutronics calculations are performed using the perturbed nuclear cross-section database as input to obtain multiple types of responses;
[0040] S3. Calculate mutual information separately to obtain a comprehensive similarity assessment.
[0041] The following example illustrates this process. First, benchmark experiments and engineering applications are established within the same codebase and subjected to the same perturbations. The nuclear cross-section database version used here is ENDF / B-VIII.0. Since ENDF / B-VIII.0 uses nuclides as the basic storage unit, U-235 is used as an example to illustrate the specific process. The perturbation coefficients are obtained using Cholesky decomposition of the covariance matrix between different reaction cross-sections. The covariance matrix is expressed in multi-group form; the perturbation method applies perturbations of different magnitudes within different energy group intervals so that the overall sample vector still conforms to the original covariance matrix. Then, a random sampling method is used to obtain 500 independent and identically distributed sample vectors from the standard normal distribution. The number of samples is determined based on the required accuracy and the number of cross-sections considered. The original covariance matrix is then subjected to Cholesky decomposition to obtain a lower triangular matrix L, finally yielding the perturbation coefficient matrix. In this matrix, each column corresponds to a sample, and each row represents the amount of perturbation applied to a specific energy region, thus deriving a library of 500 perturbation cross-sections.
[0042] After obtaining a sufficient database of perturbation cross sections, these need to be used as input to run neutronics calculation code. Modeling is performed for both benchmark experiments and engineering applications, and multiple simulations are run, with only the input nuclear cross section data changed in each run to ensure the same perturbation is applied. Three benchmark experiments from the International Critical Safety Test Evaluation Project (ICSBEP) are selected as examples to demonstrate the superiority of the method proposed in this patent. The basic parameters of the three selected benchmark experiments are shown in the table below.
[0043] Table 1. Benchmark Experiment Parameters
[0044]
[0045] After the simulation was completed, the maximum mutual information coefficient and linear correlation coefficient were calculated using the effective proliferation factors of the three benchmark experiments as random variables. The results are shown in the table below.
[0046] Table 2. Maximum mutual information coefficient and linear correlation coefficient of the benchmark experiment.
[0047]
[0048] The mutual information coefficient and linear correlation coefficient between systems A and B are both above 0.9, indicating a high degree of similarity between them. Under both evaluation criteria, this similarity meets the standard for model validation. However, the linear correlation coefficient and maximum mutual information coefficient between systems A and B and C show a significant decrease, especially the linear correlation coefficient, which is close to 0. Therefore, it can be considered that there is no linear correlation between A, B, and C. In traditional similarity evaluation systems, system C would be completely excluded from model validation options for systems A and B. However, based on the maximum mutual information coefficient, although the information contained in C is insufficient, it can still provide some reference. By collecting information from multiple different systems, the uncertainty of the target system can be reduced to a greater extent, thereby improving model performance. This also demonstrates the superiority of this method compared to traditional similarity evaluation systems.
[0049] In this embodiment, nuclear data sampling and neutronics calculations are performed based on a multi-group covariance matrix. The similarity is determined by comprehensively evaluating the mutual information coefficients between different cross-sections of the nuclear data and the output response, as well as the similar responses of the two reactor systems. The method proposed in this embodiment can use any output response of the nuclear reactor as the basis for similarity judgment to evaluate the similarity between the application design system and the benchmark experimental system. Compared to traditional methods that use linear correlation coefficients as the evaluation standard, this method does not rely on linear assumptions and extends the available response quantities from limited to the effective multiplication factor in traditional methods to all response quantities. This is crucial for conducting design, verification, and validation work for different types of nuclear reactors. This method effectively solves the problem of insufficient experimental quantity that meets traditional similarity evaluation standards, which is common in the validation of new reactor models. It is beneficial to further improve the accuracy of model validation work and accelerate the engineering application process of designed reactor types.
[0050] Other embodiments of the invention will readily occur to those skilled in the art upon consideration of the specification and practice of the embodiments disclosed herein. This invention is intended to cover any variations, uses, or adaptations of the invention that follow the general principles of the invention and include common knowledge or customary techniques in the art not disclosed herein. It should be understood that the invention is not limited to the precise structures described above and shown in the accompanying drawings, and various modifications and changes can be made without departing from its scope. The scope of the invention is limited only by the appended claims.
[0051] The embodiments of the present invention have been described in detail above with reference to the accompanying drawings. However, the present invention is not limited thereto. Various changes can be made within the scope of knowledge possessed by those skilled in the art without departing from the spirit of the present invention.
Claims
1. A similarity evaluation method based on mutual information, characterized in that, The specific steps include the following: S1. Kernel data sampling based on multi-group covariance matrix; also includes the following steps: For a multi-group covariance matrix Based on multivariate normal distribution Perform random sampling; First, construct a series of independent, identically distributed, standard normal distribution sample vectors. Each sample vector contains One element, This represents the number of energy groups corresponding to the covariance matrix. That is the sample size; Will The transformation of a set of independent standard normal distribution sample vectors into a vector that follows the... The multidimensional normal distribution vector; converting it into matrix form is the way to find the linear transformation. Make ; According to the definition, Expectations Represented as, ； The covariance matrix of the sample under consideration is: ； Let the target covariance matrix The obtained samples follow the target multivariate normal distribution, where It represents the sample matrix composed of standard normal distribution vectors. This represents the transformed sample matrix; By performing Cholesky decomposition on the original matrix, two matrices that are transposes of each other are obtained. In the above formula... It is a lower triangular matrix, which is the linear transformation required; The mutual information between two discrete random variables is calculated as follows: in, Represents random variables and The joint probability distribution function between them, and and Let these represent the marginal probability distribution functions of the two random variables; for continuous random variables, they are in integral form: ； in The current variable is a random variable. and The joint probability density function between them, and and They are and The marginal probability density function; S2. Multiple neutronics calculations are performed using the perturbed nuclear cross-section database as input to obtain multiple types of responses; S3. Calculate mutual information separately to obtain a comprehensive similarity assessment; Calculate the normalized mutual information between the nuclear reaction cross section and the response, as well as between similar responses in the two systems; For the normalized mutual information (NMI) between kernel data sections and different responses, written as in, Indicates the output response. Indicates its type, and Indicates the reaction cross section. These represent the types of nuclides, types of reactions, and energy group numbers, respectively. Indicates the normalization factor; The similarity between systems is quantified by calculating the cosine similarity between the mutual information vectors of two cross-sections and the response. For multiple response types, multiple response cross-sections are selected, and the normalized mutual information vector is represented as follows: ； in, They represent different systems, and the similarity coefficient is expressed as: ； This represents the cosine of the angle between two mutual information vectors. Since the normalized mutual information takes on values in... between, The value of is also limited to between, This represents two vectors with an angle of 0, meaning they are perfectly correlated; while This represents two vectors that are perpendicular, meaning they are completely unrelated; in the formula... , Don't refer to two systems. This represents the index of an element in the normalized vector.
2. The similarity evaluation method based on mutual information according to claim 1, characterized in that, The lower triangular matrix was obtained. Then, the cross-sectional disturbance coefficient matrix was calculated. ; make , It is the identity matrix; at this time It can be used directly for cross-sectional sampling.
3. The similarity evaluation method based on mutual information according to claim 1, characterized in that, The obtained perturbed nuclear cross-section database is used as input to perform multiple neutronics calculations using particle transport simulation software to obtain results. The appropriate number of simulations is determined based on the required accuracy.
4. The similarity evaluation method based on mutual information according to claim 1, characterized in that, The magnitude of the impact of the same type of disturbance on the two systems can be determined by the angle between the two mutual information vectors.
5. The similarity evaluation method based on mutual information according to claim 1, characterized in that, Using the normalized mutual information of the same response of two systems as a similarity evaluation criterion, this method characterizes the degree of correlation of the changes in the responses of two systems under the same perturbation. Based on considering multiple different types of response quantities, a comprehensive evaluation of the similarity between the two systems is made.
6. A computer device comprising a memory and a processor, wherein the memory stores a computer program, characterized in that: When the processor executes the computer program, it implements the steps of the similarity evaluation method based on mutual information as described in any one of claims 1-5.