Chromatographic reconstruction method based on Gaussian process

Abstract

The invention discloses a tomography reconstruction method based on a Gaussian process, and relates to the field of data analysis, and the method comprises the steps: dispersing a physical quantity region of plasma into N rectangular grids; establishing a forward modeling calculation model of the multi-dimensional measurement data according to the calculation model of the physical quantity, the to-be-reconstructed physical quantity and a preset data error; establishing a prior probability about to-be-reconstructed physical quantity distribution by using a non-stationary Gaussian process with a hierarchical model; when the data error meets independent normal distribution, establishing a conditional probability about multi-dimensional measurement data distribution by adopting multi-dimensional normal distribution; the condition probability and the prior probability are combined, and the posterior probability of the physical quantity distribution to be reconstructed is determined; according to the method, a covariance function is established, hyper-parameters in the covariance function are optimized, a gradient method is utilized to maximize the probability about the hyper-parameters, the maximum probability value of the hyper-parameters is determined, and under the maximum probability value, the maximum posterior probability estimated value and the confidence interval of the physical quantity to be reconstructed are determined through the posterior probability.

Description

Technical Field

[0001] The present application relates to the field of data analysis of nuclear fusion plasma, and more specifically, to a tomographic reconstruction method based on Gaussian process. Background Art

[0002] In the field of magnetic confinement nuclear fusion, there are hundreds of different diagnostic systems used to measure the spatial distribution of physical quantities such as temperature, density, and radiation of fusion plasma.

[0003] Due to the high temperature characteristics of fusion plasma, the diagnostic system must be installed outside the plasma, which constitutes a typical measurement method of integrating along the observation line of sight. For this multi-channel line integral data, the spatial distribution of the corresponding physical quantity must be obtained through the tomographic reconstruction method. The tomographic reconstruction problem in the field of nuclear fusion has the following difficulties: the amount of measurement data is usually very limited (low coverage); the measurement data has large systematic and random errors; and the extremely high acquisition frequency affects the signal-to-noise ratio of the measurement data.

[0004] Therefore, in the existing technology, traditional methods such as Abel, Fourier analysis and Maximum Entropy used in the field of nuclear fusion plasma generally have problems such as low computational efficiency and poor accuracy, and cannot effectively meet the needs of real-time computing applications. Summary of the invention

[0005] The purpose of this application is to provide a tomographic reconstruction method based on Gaussian process, which solves the problems of poor reconstruction accuracy and low computational efficiency of traditional methods.

[0006] The present application provides a tomographic reconstruction method based on Gaussian process, the method comprising:

[0007] Discretize the physical quantity region of the plasma into N rectangular grids; wherein the rectangular grids represent the physical quantity to be reconstructed; N is a positive integer;

[0008] A forward calculation model for multi-dimensional measurement data is established according to the measurement principle of the diagnostic system, the physical quantity to be reconstructed and the preset data error;

[0009] The prior probability of the distribution of the physical quantity to be reconstructed is established by using a non-stationary Gaussian process with a hierarchical model;

[0010] When the data error satisfies an independent normal distribution, a multidimensional normal distribution is used to establish the conditional probability of the measured data.

[0011] According to the conditional probability and the prior probability, the posterior probability of the physical quantity distribution to be reconstructed is determined;

[0012] The hyperparameters in the covariance function are optimized, and the probability of the hyperparameters is maximized using the gradient method to determine the maximum probability value of the hyperparameters. Under the maximum probability value, the maximum a posteriori probability estimate and its confidence interval of the physical quantity to be reconstructed are determined by the posterior probability as the spatial distribution of the physical quantity to be reconstructed.

[0013] In one implementation, the calculation model of the physical quantity is established based on line integral data of a plasma diagnostic system.

[0014] In one implementation, the forward calculation model of the multi-dimensional measurement data is expressed as: in, represents multidimensional measurement data, M represents the dimension of multidimensional measurement data, represents the physical quantity to be reconstructed of the Nth rectangular grid, Indicates the preset data error, A matrix representing the mapping from physical quantities to measurement data.

[0015] In one implementation, a non-stationary Gaussian process with a hierarchical model is used to establish a priori probability of the distribution of the physical quantity to be reconstructed, specifically:

[0016] The hierarchical model includes a first layer for calculating the length scale distribution of the kernel function of the stationary Gaussian process and a second layer for calculating the distribution of the physical quantity to be reconstructed of the non-stationary Gaussian process;

[0017] The first layer of the hierarchical model uses a stationary Gaussian covariance function to calculate the optimal value of the length scale distribution of N rectangular grid positions;

[0018] The second layer of the hierarchical model is conditioned on the optimal value of the length scale distribution calculated by the first layer, and uses a non-stationary Gaussian covariance function to calculate the prior probability of the distribution of the physical quantity to be reconstructed.

[0019] In one implementation, the first layer of the hierarchical model uses a stationary Gaussian covariance function to calculate the optimal value of the length scale distribution of N rectangular grid positions, including:

[0020] Selecting length scale distributions of n rectangular grids from N rectangular grids as known quantities of low dimension, n is less than N, and n is a positive integer;

[0021] The optimal value of the length scale distribution of N rectangular grids is inferred from known quantities of low dimensionality using a stationary Gaussian process.

[0022] The second layer of the hierarchical model is conditioned on the optimal value of the length scale distribution calculated by the first layer, and uses a non-stationary Gaussian covariance function to calculate the prior probability of the distribution of the physical quantity to be reconstructed, including:

[0023] According to the covariance matrix of the length scale distribution of the n rectangular grids, the covariance matrix of the optimal value of the length scale distribution of the N rectangular grids, and a submatrix used to characterize the correlation between the two covariance matrices, a joint probability between the length scale distribution of the n rectangular grids and the optimal value of the length scale distribution of the N rectangular grids is established;

[0024] The joint probability is converted into the prior probability of the distribution of the physical quantity to be reconstructed.

[0025] In one implementation, the conditional probability is expressed as:

[0026] in, represents multidimensional measurement data, M represents the dimension of multidimensional measurement data, Represents the mapping matrix from physical quantities to measurement data, represents the physical quantity to be reconstructed, represents the hyperparameters of the Gaussian process model, T represents the matrix transpose, Represents a diagonal matrix whose diagonal elements are data errors.

[0027] In one implementation, the posterior probability of the physical quantity distribution to be reconstructed is determined in the following manner: the conditional probability is multiplied by the prior probability, and the product is used as the posterior probability of the physical quantity distribution to be reconstructed.

[0028] In one implementation, the expression of the posterior probability is:

[0029]

[0030] in, represents multidimensional measurement data, M represents the dimension of multidimensional measurement data, A computational model that represents a physical quantity, represents the physical quantity to be reconstructed, represents the hyperparameters of the Gaussian process, represents the predicted data obtained by calculating the physical quantity, T represents the matrix transpose, f represents the physical quantity, Represents a diagonal matrix whose diagonal elements are data errors. .

[0031] In one implementation, the probability of the hyperparameter is maximized using the gradient method to determine the maximum probability value of the hyperparameter. Specifically, the marginal likelihood probability of the hyperparameter is maximized using the conjugate gradient method to determine the maximum probability value of the hyperparameter.

[0032] In one implementation, the expression for maximizing the probability of the hyperparameter using the conjugate gradient method is:

[0033] in, represents multidimensional measurement data, M represents the dimension of multidimensional measurement data, Represents the mapping matrix from physical quantities to measurement data, represents the predicted data calculated by physical quantities, represents the hyperparameters of the Gaussian process, represents the average of the prior probability, and T represents the matrix transpose.

[0034] Compared with the prior art, this application has the following beneficial effects:

[0035] A tomographic reconstruction method based on Gaussian process is provided in the present application, and a prior probability is established by using Gaussian process under the framework of Bayesian probability theory. Gaussian process is a general Bayesian non-parametric model method. Because it is not restricted by a specific function form, it has good flexibility and generalization, and can avoid excessive constraints on error data. First, the hyperparameters in the kernel function of the Gaussian process are used to define the correlation between a large number of different discrete spatial points; and the marginal likelihood probability about the hyperparameter is maximized by the conjugate gradient method to achieve reconstruction optimization. In particular, when the smoothness of the spatial distribution varies significantly in different regions, a non-stationary Gaussian process can be used to achieve local adaptive regularization of the smoothness. During the reconstruction process, the smoothness of different spatial points is adjusted accordingly according to the layout characteristics, that is, the prior probability, as follows: For a large number of discrete points generated after regional discretization, the local smoothness of these points is first calculated by a stationary Gaussian process, and the multidimensional normal probability distribution obtained will be used for the reconstruction of the physical quantity itself. The non-stationary Gaussian process further improves the accuracy of reconstruction by achieving global optimization. Under the condition of ensuring the overall appropriate smoothness, it can accurately reconstruct the fine structure and the prior probability of the physical quantity distribution to be reconstructed based on the hyperparameters and multidimensional normal probability distribution. When the conditional probability satisfies the multidimensional normal distribution, the hyperparameters are optimized using the conjugate condition, so that the posterior probability has a closed solution, that is, the optimal solution, which can improve the computational efficiency of tomographic reconstruction and enable this method to meet real-time computing applications. BRIEF DESCRIPTION OF THE DRAWINGS

[0036] The drawings described herein are used to provide a further understanding of the embodiments of the present application, constitute a part of the present application, and do not constitute a limitation on the embodiments of the present application. In the drawings:

[0037] Figure 1 A schematic diagram of a process flow of a tomographic reconstruction method based on a Gaussian process provided in an embodiment of the present application;

[0038] Figure 2 An example diagram of simulation calculation of a stationary Gaussian process in one-dimensional space provided by an embodiment of the present application;

[0039] Figure 3 This is an example diagram of the simulation calculation of a non-stationary Gaussian process in one-dimensional space provided in an embodiment of the present application. DETAILED DESCRIPTION

[0040] In order to make the objectives, technical solutions and advantages of the present application more clearly understood, the present application is further described in detail below in conjunction with examples and drawings. The illustrative implementation scheme of the present application and its description are only used to explain the present application and are not intended to limit the present application.

[0041] It should be noted that the term "including" or "may include" used in various embodiments of the present application indicates the existence of the function, operation or element applied for, and does not limit the increase of one or more functions, operations or elements. In addition, as used in various embodiments of the present application, the terms "including", "having" and their cognates are only intended to represent a specific feature, number, step, operation, element, assembly or a combination of the foregoing items, and should not be understood as first excluding the existence of one or more other features, numbers, steps, operations, elements, assemblies or a combination of the foregoing items or increasing the possibility of one or more features, numbers, steps, operations, elements, assemblies or a combination of the foregoing items. In the description of the embodiments of the present application, the meaning of "multiple" is two or more, unless otherwise clearly and specifically limited.

[0042] At present, there are hundreds of different diagnostic systems in the field of magnetic confinement nuclear fusion for measuring physical quantities such as temperature, density, and radiation distribution of fusion plasma.

[0043] Due to the high temperature characteristics of fusion plasma, the diagnostic system must be installed outside the plasma, which constitutes a typical measurement method of integrating along the observation line of sight. For this multi-channel line integral data, the spatial distribution of the corresponding physical quantity must be obtained through the tomographic reconstruction method. The tomographic reconstruction problem in the field of nuclear fusion has the following difficulties: the amount of measurement data is usually very limited (low coverage); the measurement data has large systematic and random errors; the extremely high acquisition frequency affects the signal-to-noise ratio of the measurement data.

[0044] Therefore, a hierarchical reconstruction method is needed to address these difficulties. In the existing technology, traditional methods such as Abel, Fourier analysis and Maximum Entropy used in the field of nuclear fusion generally have problems such as low computational efficiency and poor accuracy, and cannot effectively meet the needs of real-time computing applications.

[0045] In view of the above-mentioned problem of inversion calculation of line integral data with limited number and errors, the embodiment of the present application has developed a tomographic reconstruction method based on Gaussian process under the framework of Bayesian probability theory. This method uses the kernel function of Gaussian process as an effective way to achieve smoothness regularization, and further realizes local adaptive regularization of smoothness through non-stationary Gaussian process, which significantly improves the accuracy of reconstruction compared with traditional methods. In view of the scarcity and error characteristics of actual measurement data, the posterior probability obtained by this method can not only give an optimal solution through the maximum posterior probability estimation, but also realize uncertainty quantification through its confidence interval. In addition, the prior probability established by Gaussian process makes the posterior probability have a closed form optimal solution, thereby improving the calculation efficiency, making this method have great potential in real-time computing applications.

[0046] The tomographic reconstruction method provided in this embodiment will be described in detail below in conjunction with specific implementation methods. Figure 1 , Figure 1 A schematic diagram of a tomographic reconstruction method based on a Gaussian process provided in an embodiment of the present application is shown in FIG. Figure 1 As shown, the method includes:

[0047] S101, discretizing the physical quantity region of the plasma into N rectangular grids, wherein the rectangular grids represent the physical quantity to be reconstructed, and N is a positive integer.

[0048] In this embodiment, the physical quantity may be the temperature, density, radiation distribution, etc. of the plasma, and no redundant description is given in this embodiment. Secondly, the discrete number of the physical quantity region into N rectangular grids is a conventional technical means at present, and no redundant description is given in this embodiment. For example, when the physical quantity is a radiation value, the radiation value f represented by each rectangular grid is i (i=1,...,N) needs to be inverted from the measured line integral data. Accordingly, the radiation distribution can be expressed as a vector

[0049] S102, establishing a forward calculation model for multi-dimensional measurement data according to a calculation model of physical quantities, the physical quantities to be reconstructed and preset data errors.

[0050] In this embodiment, a corresponding calculation model R is established according to the measurement principle of the diagnostic system. Get ideal forecast data Superimposing a reasonable error ε on the predicted data is equivalent to the actual measured data For measurement data with dimension M, the process can be expressed as the following forward model:

[0051] Specifically, the calculation model of the physical quantity is established based on the line integral data of the plasma diagnostic system. The forward model of the diagnostic system is expressed in matrix form: The physical quantity to be inferred (e.g. radiation distribution) vector and measurement data It can be expressed as the following relationship: in, represents multidimensional measurement data, M represents the dimension of multidimensional measurement data, represents the physical quantity to be reconstructed of N rectangular grids, Indicates the preset data error, It represents the forward calculation model established according to the diagnostic system, that is, the mapping matrix from physical quantities to measurement data.

[0052] S103, using a non-stationary Gaussian process with a hierarchical model to establish a priori probability of the distribution of the physical quantity to be reconstructed.

[0053] In this embodiment, a non-stationary Gaussian process with a hierarchical model is used to establish a priori probability of the distribution of the physical quantity to be reconstructed, specifically:

[0054] The hierarchical model includes a first layer for calculating the length scale distribution of the kernel function of the stationary Gaussian process and a second layer for calculating the distribution of the physical quantity to be reconstructed of the non-stationary Gaussian process;

[0055] The first layer of the hierarchical model uses a stationary Gaussian covariance function to calculate the optimal value of the length scale distribution of N rectangular grid positions;

[0056] The second layer of the hierarchical model is conditioned on the optimal value of the length scale distribution calculated by the first layer, and uses a non-stationary Gaussian covariance function to calculate the prior probability of the distribution of the physical quantity to be reconstructed.

[0057] It should be noted that the hyperparameters of the kernel function provided in this embodiment are described using the length scale as an example. Of course, other hyperparameters in the kernel function of the Gaussian process can also be selected, and this embodiment does not make specific limitations.

[0058] According to the Bayesian principle, we first use the Gaussian process to establish the prior probability of the physical quantity to be inferred, and then use the hyperparameters in the covariance function to (e.g. length scale, etc.) to achieve smoothness regularization. In addition, for non-stationary Gaussian processes with adaptive capabilities, a hierarchical model is needed to implement the non-stationary Gaussian process Establishing prior probability, the first layer is mainly through a stationary Gaussian process GP l To achieve all grid points (coordinates ) length scale The second layer is to use The non-stationary Gaussian process GP is a parameter f To achieve radiation distribution Inference of prior probability.

[0059] More specifically, the calculation of the prior probability is implemented as follows:

[0060] The first layer of the hierarchical model includes: selecting length scale distributions of n rectangular grids from N rectangular grids as known quantities of low dimension, where n is less than N and is a positive integer; using a stationary Gaussian process to infer optimal values ​​of the length scale distributions of the N rectangular grids from the known quantities of low dimension;

[0061] Regarding the second layer of the hierarchical model, it includes: establishing a joint probability between the length scale distribution of n rectangular grids and the optimal value of the length scale distribution of N rectangular grids based on the covariance matrix of the length scale distribution of n rectangular grids, the covariance matrix of the optimal value of the length scale distribution of N rectangular grids, and a submatrix used to characterize the correlation between the two covariance matrices; and converting the joint probability into a priori probability of the distribution of physical quantities to be reconstructed.

[0062] Specifically, we use Gaussian process to Establish prior probability, under the Gaussian process model Satisfies the multidimensional normal probability distribution: In the formula, the covariance matrix It can be derived through the following squared exponential covariance function: The length scale l and variance σ in the squared exponential covariance function determine the smoothness and amplitude of the random process, respectively. Therefore, as the first layer of the hierarchical model, it can be used to determine the hyperparameters of the covariance function of the stationary Gaussian process, and use the stationary Gaussian process to calculate the length scale distribution of N rectangular grid positions, which is used to implement the non-stationary Gaussian process model of the next layer.

[0063] The key to the tomographic reconstruction method provided in this embodiment is to establish the prior probability of the non-stationary Gaussian process through a hierarchical model. Specifically, the non-stationary covariance function used by the algorithm is:

[0064]

[0065] The local kernel matrix in the above formula is is the spatial location is a function of the local length scale, which determines the In the specific calculation, for all rectangular grids The inference is mainly achieved through the following process: First, select the length scales within several grids As known quantities (see Table 1).

[0066] Table 1

[0067]

[0068] Then use the Gaussian process to get the known quantity from the low dimension Inferring length scales in other grids In the Gaussian process model, the length scale of all grid cells is Satisfies normal distribution It can be expressed as the following joint probability:

[0069] The submatrix in the above formula and They are and The covariance matrix of is the correlation between the two, which has the following forms:

[0070] Among them, P and Q represent the dimensions of known and unknown quantities respectively; the elements of the three matrices in the above formula are Indicates about The squared exponential covariance function value at two locations.

[0071] Convert the joint probability to a known quantity For two variables that conform to the normal probability distribution The conditional probability of a variable can be derived from its joint probability as follows:

[0072]

[0073] According to the above conditions, the and The joint probability can be converted into the form of prior probability:

[0074] Among them, the expected value of the form of the prior probability is Corresponding to the maximum posterior probability, the realization is from Infer That is, through the first layer of stationary Gaussian process GP l Realized from low dimensional Inferring high-dimensional

[0075] like Figure 2As shown, when the covariance matrix has (a) a larger or (b) a smaller length scale, (c) the random function generated by the Gaussian process has a larger smoothness (black curve) or a smaller smoothness (red curve) accordingly.

[0076] like Figure 3 As shown, Figure 3 The non-stationary covariance matrix and the random process implemented by it with varying smoothness at different spatial locations are shown, where Figure 3 (a) Its non-stationary covariance matrix has a length scale that varies with spatial position, which is used to achieve local adaptive regularization of smoothness; (b) When based on the stationary Gaussian process GP l The covariance matrix of the established prior probability has a decreasing length scale distribution, and (c) the random function generated by the non-stationary Gaussian process has a correspondingly decreasing smoothness.

[0077] S104, when the data error satisfies an independent normal distribution, a multidimensional normal distribution is used to establish a conditional probability of the multidimensional measurement data distribution.

[0078] In this embodiment, the expression of conditional probability is:

[0079] in, represents multidimensional measurement data, M represents the dimension of multidimensional measurement data, Represents the mapping matrix from physical quantities to measurement data, represents the physical quantity to be reconstructed, represents the hyperparameters of the Gaussian process, T represents the matrix transpose, Represents a diagonal matrix whose diagonal elements are data errors.

[0080] S105, according to the Bayesian formula, the conditional probability and the prior probability are combined to determine the posterior probability of the physical quantity distribution to be reconstructed.

[0081] In this embodiment, in view of the scarcity and error characteristics of the actual measurement data, the posterior probability obtained by the algorithm can not only give an optimal solution through the maximum a posteriori probability estimate (Maximum a Posteriori estimate), but also realize uncertainty quantification through its confidence interval, and determine the posterior probability of the physical quantity distribution to be reconstructed in the following way: multiply the conditional probability and the prior probability, and use the product as the posterior probability of the physical quantity distribution to be reconstructed.

[0082] Specifically, the expression of the posterior probability is:

[0083]

[0084] in, represents multidimensional measurement data, M represents the dimension of multidimensional measurement data, Represents the mapping matrix from physical quantities to measurement data, represents the physical quantity to be reconstructed, T represents the matrix transpose, represents the hyperparameters of the Gaussian process, represents the average of the prior probabilities.

[0085] S106, optimize the hyperparameters in the covariance function, maximize the probability of the hyperparameters using the gradient method, determine the maximum probability value of the hyperparameters, and under the maximum probability value, determine the maximum a posteriori probability estimate and its confidence interval of the physical quantity to be reconstructed by the posterior probability as the spatial distribution of the physical quantity to be reconstructed.

[0086] In this embodiment, the hyperparameters are optimized to determine the maximum probability value of the hyperparameters. Specifically, the conjugate gradient method is used to maximize the probability of the hyperparameters to determine the maximum probability value of the hyperparameters.

[0087] Specifically, the expression for maximizing the probability of the hyperparameter using the conjugate gradient method is:

[0088] in, represents multidimensional measurement data, M represents the dimension of multidimensional measurement data, Represents the mapping matrix from physical quantities to measurement data, represents the predicted data calculated by physical quantities, represents the hyperparameters of the Gaussian process, represents the average of the prior probability, and T represents the matrix transpose.

[0089] Since R in the expression of the posterior probability represents a linear relationship, the expected value and covariance matrix of the posterior probability have the following analytical solutions:

[0090] at last, It gives The optimal solution of The diagonal elements of give confidence intervals for the solution.

[0091] Based on the embodiments described above, it can be seen that in a tomographic reconstruction method based on Gaussian process provided in the embodiments of the present application, the method uses Gaussian process to establish prior probability under the framework of Bayesian probability theory. Gaussian process is a general Bayesian non-parametric model method. Because it is not restricted by a specific function form, it has good flexibility and generalization, and can avoid excessive constraints on error data. First, the hyperparameters in the kernel function of the Gaussian process are used to define the correlation between a large number of different discrete spatial points; and the marginal likelihood probability about the hyperparameter is maximized by the conjugate gradient method to achieve reconstruction optimization. In particular, when the smoothness of the spatial distribution varies significantly in different regions, a non-stationary Gaussian process can be used to achieve local adaptive regularization of the smoothness. During the reconstruction process, the smoothness of different spatial points is adjusted accordingly according to the layout characteristics, that is, the prior probability, as follows: For a large number of discrete points generated after the regional discretization, the local smoothness of these points is first calculated by a stationary Gaussian process, and the non-stationary Gaussian process is obtained to be used for the reconstruction of the physical quantity itself. The non-stationary Gaussian process further improves the accuracy of reconstruction by achieving global optimization. While ensuring the overall appropriate smoothness, it can also accurately reconstruct the local fine structure. When the conditional probability satisfies the multidimensional normal distribution, the Gaussian process is used to establish the prior probability to meet the conjugate condition, so that the posterior probability has a closed solution, thereby improving the computational efficiency of tomographic reconstruction, making this method suitable for real-time computing applications.

[0092] The specific implementation methods described above further illustrate the purpose, technical solutions and beneficial effects of the present application in detail. It should be understood that the above description is only the specific implementation method of the present application and is not intended to limit the scope of protection of the present application. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the present application should be included in the scope of protection of the present application.

Claims

1. A tomographic reconstruction method based on Gaussian process, characterized in that the method include: Discretize the physical quantity region of the plasma into N rectangular grids; wherein the rectangular grids represent the physical quantity to be reconstructed; N is a positive integer; Establishing a forward calculation model for multi-dimensional measurement data according to a calculation model of physical quantities, physical quantities to be reconstructed and preset data errors; The prior probability of the distribution of the physical quantity to be reconstructed is established by using a non-stationary Gaussian process with a hierarchical model; When the data errors satisfy independent normal distribution, multidimensional normal distribution is used to establish the conditional probability of multidimensional measurement data; According to the Bayesian formula, the conditional probability and the prior probability are combined to determine the posterior probability of the distribution of the physical quantity to be reconstructed; The hyperparameters in the covariance function are optimized, and the probability of the hyperparameters is maximized using the gradient method to determine the maximum probability value of the hyperparameters. Under the maximum probability value, the maximum a posteriori probability estimate and its confidence interval of the physical quantity to be reconstructed are determined by the posterior probability as the spatial distribution of the physical quantity to be reconstructed.

2. The method according to claim 1, characterized in that The calculation model of the physical quantity is established according to the line integral data measured by the plasma diagnosis system.

3. The method according to claim 2, characterized in that The expression of the forward calculation model of the multi-dimensional measurement data is: in, represents multidimensional measurement data, M represents the dimension of multidimensional measurement data, represents the physical quantity to be reconstructed of N rectangular grids, Indicates the preset data error, It represents the forward calculation model established according to the diagnostic system, that is, the mapping matrix from physical quantities to measurement data.

4. The method according to claim 1, characterized in that: The non-stationary Gaussian process with a hierarchical model is used to establish the prior probability of the distribution of the physical quantity to be reconstructed, specifically: The hierarchical model includes a first layer for calculating the length scale distribution of the kernel function of the stationary Gaussian process and a second layer for calculating the distribution of the physical quantity to be reconstructed of the non-stationary Gaussian process; The first layer of the hierarchical model uses a stationary Gaussian covariance function to calculate the optimal value of the length scale distribution of N rectangular grid positions; The second layer of the hierarchical model is conditioned on the optimal value of the length scale distribution calculated by the first layer, and uses a non-stationary Gaussian covariance function to calculate the prior probability of the distribution of the physical quantity to be reconstructed.

5. The method according to claim 4, characterized in that The first layer of the hierarchical model uses a stationary Gaussian covariance function to calculate the optimal value of the length scale distribution of N rectangular grid positions, including: Selecting length scale distributions of n rectangular grids from N rectangular grids as known quantities of low dimension, n is less than N, and n is a positive integer; The optimal value of the length scale distribution of N rectangular grids is inferred from known quantities of low dimensionality using a stationary Gaussian process. The second layer of the hierarchical model is conditioned on the optimal value of the length scale distribution calculated by the first layer, and uses a non-stationary Gaussian covariance function to calculate the prior probability of the distribution of the physical quantity to be reconstructed, including: According to the covariance matrix of the length scale distribution of the n rectangular grids, the covariance matrix of the optimal value of the length scale distribution of the N rectangular grids, and a submatrix used to characterize the correlation between the two covariance matrices, a joint probability between the length scale distribution of the n rectangular grids and the optimal value of the length scale distribution of the N rectangular grids is established; The joint probability is converted into the prior probability of the distribution of the physical quantity to be reconstructed.

6. The method according to claim 1, characterized in that The expression of the conditional probability is: in, represents multidimensional measurement data, M represents the dimension of multidimensional measurement data, Represents the mapping matrix from physical quantities to measurement data, represents the physical quantity to be reconstructed, represents the hyperparameters of the Gaussian process, T represents the matrix transpose, Represents a diagonal matrix whose diagonal elements are data errors.

7. The method according to claim 1, characterized in that The posterior probability of the physical quantity distribution to be reconstructed is determined in the following manner: the conditional probability is multiplied by the prior probability, and the product is used as the posterior probability of the physical quantity distribution to be reconstructed.

8. The method according to claim 7, characterized in that The expression of the posterior probability is: in, represents multidimensional measurement data, M represents the dimension of multidimensional measurement data, Represents the mapping matrix from physical quantities to measurement data, represents the physical quantity to be reconstructed, T represents the matrix transpose, represents the hyperparameters of the Gaussian process, represents the average of the prior probabilities.

9. The method according to claim 1, characterized in that: The gradient method is used to maximize the probability of the hyperparameters to determine the maximum probability value of the hyperparameters. Specifically, the conjugate gradient method is used to maximize the marginal likelihood probability of the hyperparameters to determine the maximum probability value of the hyperparameters.

10. The method according to claim 9, characterized in that The expression for maximizing the probability of the hyperparameter using the conjugate gradient method is: in, represents multidimensional measurement data, M represents the dimension of multidimensional measurement data, Represents the mapping matrix from physical quantities to measurement data, represents the predicted data calculated by physical quantities, represents the hyperparameters of the Gaussian process, represents the average of the prior probability, and T represents the matrix transpose.

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