Multi-fidelity matrix completion method based on generative adversarial network

Abstract

The invention discloses a multi-fidelity matrix completion method based on a generative adversarial network, and belongs to the technical field of data fusion. The method comprises the following steps: firstly, performing matrix data enhancement on a complete low-fidelity matrix and a high-fidelity matrix containing missing items to obtain a training set; secondly, selecting training data of which the batch number is B from the training set, fixing the weight of a generator, and updating the weight of a discriminator; selecting new training data of which the batch number is B from the training set, fixing the weight of the generator, and updating the weight of the generator; and in a certain round T, traversing the training set, and minimizing discrimination loss and generation loss through the above weight cross updating method. And finally, inputting the complete low-fidelity matrix and the high-fidelity matrix containing the missing item into the generative adversarial network after weight updating to obtain a complete high-fidelity matrix. According to the method, the missing items in the high-fidelity matrix are complemented by fusing the available items of the complete low-fidelity matrix and the high-fidelity matrix, and the method of complementing and generating the adversarial network through the multi-fidelity matrix is combined, so that the operation is simple and convenient, the practical operability is high, and the application is wide.

Description

Technical Field

[0001] This invention belongs to the field of data fusion technology and relates to a multi-fidelity matrix completion method based on generative adversarial networks. Background Technology

[0002] Multi-fidelity matrix completion (MMC) is a task that completes the missing terms of a high-fidelity matrix by fusing the available terms of both low-fidelity and high-fidelity matrices. It has wide applications in aerospace engineering (such as aircraft aerodynamic measurement), mechanical design (such as ship shape optimization), and digital twins (such as structural health monitoring). For example, in aircraft aerodynamic measurement, high-fidelity data obtained from wind tunnel tests is often incomplete due to high costs, while low-fidelity data obtained through partial differential equation calculations, although complete, lacks sufficient accuracy. Therefore, the MMC method is needed to fuse these two types of data to complete the high-fidelity matrix.

[0003] The current MMC needs to solve three problems: (1) There is only one complete low-fidelity matrix and one high-fidelity matrix with missing terms, and the amount of available training data is limited; (2) In some practical engineering problems, the high-fidelity matrix obtained usually has many missing terms; (3) In some cases, there may be significant differences between adjacent terms.

[0004] In existing technologies, the co-kriging model proposed by Forrester et al. constructs a fusion model based on the assumption of a linear trend relationship between high-fidelity and low-fidelity data. However, this assumption is difficult to hold in scenarios with significant nonlinear relationships, leading to a decrease in completion accuracy. The generative adversarial interpolation network (GAIN) proposed by Yoon et al. completes the matrix by marking missing positions with a mask matrix, but it requires a large amount of matrix data from the same scene for training and cannot be directly applied to MMC scenarios containing only a single pair of high-fidelity and low-fidelity matrices. These methods are either limited by the assumption of data relationships or require a large number of training samples, making it difficult to handle multi-fidelity matrix completion tasks with high missing rates and small sample sizes.

[0005] To address these issues, this paper proposes a multi-fidelity matrix completion method based on generative adversarial networks. This method expands the training samples through matrix data augmentation and introduces correlation coefficient loss to capture data correlation, thus solving the aforementioned problems to some extent.

[0006] Matrix completion is the task of filling in the missing items in a matrix. Matrix completion can be achieved by decomposing a matrix X containing missing values ​​into two (or more) matrices using matrix factorization. The multiplication of these decomposed matrices yields an approximation X' of the original matrix, and the values ​​of this approximation matrix X' are used to fill in the missing parts of the original matrix X.

[0007] Matrix completion has various optimization algorithms, such as greedy algorithms, alternating minimization, and Riemann optimization. Recent research has also involved non-convex optimization strategies, including gradient descent and variants, which can provide better restoration guarantees and computational efficiency in certain situations. Furthermore, matrix completion has many applications in practical problems, such as wireless communication, traffic sensing, image inpainting, and system recognition. This research has not only enhanced the theoretical foundation of matrix completion but also promoted its development in practical applications.

[0008] The goal of multi-fidelity modeling methods is to provide the highest possible data fit at the lowest possible cost, based on data sources of varying fidelity. High-fidelity data is characterized by high accuracy, limited quantity, and high acquisition cost, while low-fidelity data is characterized by lower availability, abundant data volume, and low acquisition cost. Existing multi-fidelity modeling methods can be broadly categorized into two types: those based on interpolation and those employing machine learning models. The former offers high model interpretability but typically requires a strong linear correlation between high- and low-fidelity data and certain assumptions about the data distribution (e.g., Gaussian distribution), significantly limiting its application scenarios. The latter has lower requirements for the linear relationship and data distribution between high- and low-fidelity data, but due to the large number of parameters in the model, it demands a large amount of data, which contradicts the problem's requirements. Therefore, it is necessary to develop a new method that, while ensuring modeling effectiveness, has broad application prospects and reduces the need for large amounts of data, especially high-fidelity data.

[0009] In existing technologies related to Generative Adversarial Networks (GANs), models for matrix completion tasks have some limitations. For example, the Generative Adversarial Interpolation Network (GAIN) proposed by Yoon et al., while completing matrix completion by recording the positions of missing items through a mask matrix, employs a prompting mechanism to re-evaluate the entries in the mask matrix, failing to preserve the original form of the mask matrix, which may affect the accurate utilization of missing position information. The MisGAN model designed by Li et al. includes two pairs of generators and discriminators: one pair generates a spurious complete data matrix, and the other pair generates a mask matrix to capture the missing process of real matrix entries. While this complex network structure can handle matrices with high missing rates, it requires a larger number of training samples, making it difficult to apply in multifidelity matrix completion (MMC) scenarios with only a single incomplete high-fidelity matrix or a single complete low-fidelity matrix. Furthermore, the Generative Adversarial Multiple Interpolation Network (GAMIN) proposed by Yoon et al. also requires a large number of training samples, making it unsuitable for situations with limited training data in MMC tasks. The problems with these existing GAN-based matrix completion models make them significantly inadequate when facing the requirements of MMC tasks, such as scarce training samples and the need to integrate high- and low-fidelity data.

[0010] Generative Adversarial Networks (GANs) were first proposed in 2014. Compared to traditional neural networks, a GAN model consists of two neural networks: a generator and a discriminator. The generator generates the desired target (in different problem scenarios, the generator might generate images from noisy data or fit corresponding data labels from data points). The discriminator determines whether the content generated by the generator conforms to the real data distribution. The training process follows the minmax rule, with the generator and discriminator training alternately until a Nash equilibrium is reached. Because of the discriminative mechanism, GANs can more fully extract effective information from the training data, effectively reducing the model's need for large amounts of data. This effectively solves the drawbacks of multi-fidelity modeling methods based on machine learning while retaining their advantages. Summary of the Invention

[0011] To address the problems of existing technologies, this invention provides a multi-fidelity matrix completion method based on generative adversarial networks (GAN-MMC). This method is implemented using a newly designed GAN-MMC. The GAN-MMC generator takes a complete low-fidelity matrix as input and outputs an approximate high-fidelity matrix. Simultaneously, a discriminator determines whether the high-fidelity approximation matrix from the generator matches the actual high-fidelity data distribution. The multi-fidelity matrix completion (MMC) task completes missing terms in the high-fidelity matrix by fusing the available terms from the complete low-fidelity matrix and the high-fidelity matrix. The completed high-fidelity matrix retains the correlation information between adjacent terms.

[0012] To achieve the above objectives, the technical solution adopted by the present invention is as follows:

[0013] A multi-fidelity matrix completion method based on generative adversarial networks (GANs) is disclosed. The method first augments the input data by combining a complete low-fidelity matrix with a high-fidelity matrix containing missing terms, resulting in an augmented training set. Second, B batches of training data are selected from the training set, and the generator weights are fixed. Update the discriminator weights using the Adam method Then, select a new batch of training data with batch number B from the training set, and fix the discriminator weights. Update generator weights using the Adam method Within a certain number of rounds T, the training set is traversed, and the discrimination loss and generation loss are minimized using the aforementioned method of cross-updating weights. Finally, the complete low-fidelity matrix and the high-fidelity matrix containing missing terms are input into the weight-updated generative adversarial network (GAN), which consists of a generator and a discriminator, to obtain the complete high-fidelity matrix. The specific steps include:

[0014] The first step is to perform matrix data augmentation on the complete low-fidelity matrix L and the high-fidelity matrix H containing missing terms in the input data, resulting in the augmented training set. The details are as follows:

[0015] Step 1.1: Create a new mask matrix M with the same shape as the high-fidelity matrix. Set the mask matrix entries corresponding to the missing entries in the original high-fidelity matrix to 0, and set the other entries to 1. The mask matrix for masking the high-fidelity matrix H is denoted as M(H).

[0016] Step 1.2, through dot product The operation combines the mask matrix M and the high-fidelity matrix H to obtain an approximate matrix of the high-fidelity matrix containing missing terms. The elements at the positions corresponding to the missing items in the original high-fidelity matrix H are 0, while the elements at other positions are the high-fidelity data in the original high-fidelity matrix H.

[0017] Step 1.3, by synchronously arranging the L matrix and The rows and columns of the matrix can generate a sufficient number of training samples (L row H row ) and (L col H col These rearranged low-fidelity matrices and high-fidelity approximation matrices Use it as a new training set.

[0018] The second step involves selecting B batches of training data from the training set generated in the first step, and optimizing the generator and discriminator weights of the GAN-MMC through minimax game training with a set number of epochs T. The details are as follows:

[0019] Step 2.1: Fix the generator weights during each training cycle. Update the discriminator weights using the Adam method

[0020] Define the discriminator loss as

[0021]

[0022] Among them, H n L represents the high-fidelity matrix from the first step training set. n L represents a low-fidelity matrix. ce L represents the cross-entropy loss function ce (x,y)=-ylogx-(1-y)log(1-x), where G[·] represents the generator and D[·] represents the discriminator. It's a dot product operation, M(H) n ) represents the high-fidelity matrix H n The mask matrix for masking.

[0023] The Adam method partly refers to minimizing the discriminator loss L through multiple iterations. dis (H n ,L n To achieve the optimization goal

[0024]

[0025] Then update the discriminator weights. Among them W d (t+1) W represents the updated discriminator weight matrix. d (t) Let B represent the discriminator weight matrix before the update, and let η represent the total number of training data selected in the current batch. d This represents the pre-set discriminator learning rate. This represents gradient operation.

[0026] Step 2.2: Within the same training cycle, fix the discriminator weights. Update generator weights using the Adam method

[0027] Define the generator loss as

[0028]

[0029] Among them, H n L represents the high-fidelity matrix from the first step training set. n L represents a low-fidelity matrix. ce L represents the cross-entropy loss function ce (x,y)=-ylogx-(1-y)log(1-x), G[·] denotes the generator, D[·] denotes the discriminator, M(H n ) represents the high-fidelity matrix H n The mask matrix for masking. It is a dot product operation.

[0030] Define the mean squared error (MSE) loss function as follows:

[0031]

[0032] in, The operation of squaring the F-norm. It is a dot product operation.

[0033] Define correlation loss as

[0034]

[0035] Where ρ(·,·) is the correlation coefficient, calculated as follows:

[0036]

[0037] Where I represents the number of rows in the matrix, and J represents the number of columns in the matrix.

[0038] This part of the Adam method refers to minimizing the generator loss L through multiple iterations. g (H n ,L n To achieve the optimization goal

[0039]

[0040] Then update the generator weights Among them W g (t+1) W represents the updated discriminator weight matrix. g (t) Let B represent the discriminator weight matrix before the update, and let η represent the total number of training data selected in the current batch. g This represents the pre-set discriminator learning rate. This indicates the calculation of the gradient, where κ and τ represent constant parameters, and L... g L represents the generator loss. mse L represents the mean squared error (MSE) loss function. ρ This indicates the loss of correlation.

[0041] Step 2.3: Iterate through T cycles, implementing the above minimax game training strategy in each cycle, and finally optimizing the weights of the generator and discriminator of GAN-MMC.

[0042] The third step involves inputting the complete low-fidelity matrix and the high-fidelity matrix (containing missing terms) into the generator of the generative adversarial network (GAN-MMC) after weight updates (after training, only the generator is retained during the testing phase to generate approximate HF matrix values) to obtain the complete high-fidelity matrix. The calculation method for the complete high-fidelity matrix is ​​as follows:

[0043]

[0044] in, H represents the generated complete high-fidelity matrix. n M(H) represents the high-fidelity matrix from the training set in the first step. n ) represents the high-fidelity matrix H n The mask matrix for masking. It's a dot product operation. It is an intermediate result of a high-fidelity matrix generated by the generator.

[0045] The beneficial effects of this invention are as follows:

[0046] (1) This invention provides a multi-fidelity matrix completion method based on generative adversarial networks. By employing generative adversarial networks, the generator takes a complete low-fidelity matrix as input and outputs an approximate high-fidelity matrix. A discriminator determines whether the high-fidelity approximate matrix from the generator is consistent with the actual high-fidelity data distribution. This effectively improves the performance of the multi-fidelity matrix completion (MMC) task.

[0047] (2) In traditional multi-fidelity matrix completion (MMC) tasks, due to the large number of missing terms in the high-fidelity (HF) matrix (i.e., although the traditional high-fidelity HF matrix can provide an accurate description of physical quantities, it is costly to obtain and has a large number of missing terms), the completion task can usually only be completed using the complete low-fidelity (LF) matrix and a small number (usually very few) of available terms in the original high-fidelity matrix. However, the accuracy of the terms in the matrix cannot meet practical requirements. This invention proposes multi-fidelity matrix completion (MMC) by introducing generative adversarial networks (GANs). By fusing the available terms of the complete low-fidelity matrix and the high-fidelity matrix, the missing terms in the high-fidelity matrix are completed. Combining multi-fidelity matrix completion and generative adversarial networks, GAN-MMC is proposed, which significantly improves this problem. At the same time, the correlation information between adjacent terms in the matrix is ​​considered.

[0048] In summary, the method proposed in this invention is simple to operate, highly practical, and widely applicable. Attached Figure Description

[0049] Figure 1 A diagram of a generative adversarial network structure for multifidelity matrix completion;

[0050] Figure 2 Augmented graphs for matrix data;

[0051] Figure 3 This is a schematic diagram of the stress field of a steel plate;

[0052] Figure 4 This is a diagram showing the actual stress field of the steel plate.

[0053] Figure 5 This is a simulation diagram of the stress field of the steel plate;

[0054] Figure 6 A schematic diagram of the velocity field of Oldroyd-B fluid flow in a cylinder between two parallel plates;

[0055] Figure 7 A true plot of the velocity field of the Oldroyd-B fluid flow;

[0056] Figure 8 This is a simulation diagram of the velocity field of fluid flow in the Oldroyd-B series. Detailed Implementation

[0057] The embodiments of the present invention will be further described below with reference to the accompanying drawings.

[0058] Depend on Figure 1 As shown, this invention follows the framework of the original generative adversarial network. The entire model structure is divided into two parts: a generator and a discriminator. The generator takes the complete low-fidelity matrix as input and outputs an approximate high-fidelity matrix, marking the positions using a mask matrix. The discriminator's role is to determine whether the distribution of the real high-fidelity data at the positions marked by the high-fidelity matrix from the generator and the mask matrix is ​​consistent. To obtain the correlation information between adjacent items in the matrix, both the generator and the discriminator employ multiple convolutional hidden layers.

[0059] First, matrix data augmentation is performed on the complete low-fidelity matrix and the high-fidelity matrix containing missing terms in the input data to obtain the augmented training set. Second, training data with batch size B is selected from the training set, and the generator weights are fixed. Update the discriminator weights using the Adam method Then, select a new batch of training data with batch number B from the training set, and fix the generator weights. Update generator weights using the Adam method Within a certain number of rounds T, the training set is traversed, and the discrimination loss and generation loss are minimized by the aforementioned method of cross-updating weights. Finally, the complete low-fidelity matrix and the high-fidelity matrix containing missing terms are input into the generative adversarial network after weight updates to obtain the complete high-fidelity matrix.

[0060] Specifically, the following steps are included:

[0061] The first step is to perform matrix data augmentation on the complete low-fidelity matrix L and the high-fidelity matrix H containing missing terms in the input data, resulting in the augmented training set. The details are as follows:

[0062] Step 1.1: Create a new mask matrix M with the same shape as the high-fidelity matrix. Set the mask matrix entries corresponding to the missing entries in the original high-fidelity matrix to 0, and set the other entries to 1. The mask matrix for masking the high-fidelity matrix H is denoted as M(H).

[0063] Step 1.2, through dot product The operation combines the mask matrix M and the high-fidelity matrix H to obtain an approximate matrix of the high-fidelity matrix containing missing terms. The elements at the positions corresponding to the missing items in the original high-fidelity matrix H are 0, while the elements at other positions are the high-fidelity data in the original high-fidelity matrix H.

[0064] Step 1.3 as follows Figure 2 As shown, by synchronously arranging the L matrix and The rows and columns of the matrix can generate a sufficient number of training samples (L row H row ) and (L col H col These rearranged low-fidelity matrices and high-fidelity approximation matrices Use it as a new training set.

[0065] The second step involves selecting B batches of training data from the training set generated in the first step, and optimizing the generator and discriminator weights of the GAN-MMC through minimax game training with a set number of epochs T. The details are as follows:

[0066] Step 2.1: Fix the generator weights during each training cycle. Update the discriminator weights using the Adam method

[0067] Define the discriminator loss as

[0068]

[0069] Among them, H n L represents the high-fidelity matrix from the first step training set.n L represents a low-fidelity matrix. ce L represents the cross-entropy loss function ce (x,y)=-ylogx-(1-y)log(1-x), where G[·] represents the generator and D[·] represents the discriminator. It's a dot product operation, M(H) n ) represents the high-fidelity matrix H n The mask matrix for masking.

[0070] The Adam method partly refers to minimizing the discriminator loss L through multiple iterations. dis (H n ,L n To achieve the optimization goal

[0071]

[0072] Then update the discriminator weights. Among them W d (t+1) W represents the updated discriminator weight matrix. d (t) Let B represent the discriminator weight matrix before the update, and let η represent the total number of training data selected in the current batch. d This represents the pre-set discriminator learning rate. This represents gradient operation.

[0073] Step 2.2: Within the same training cycle, fix the discriminator weights. Update generator weights using the Adam method

[0074] Define the generator loss as

[0075]

[0076] Among them, H n L represents the high-fidelity matrix from the first step training set. n L represents a low-fidelity matrix. ce L represents the cross-entropy loss function ce (x,y)=-ylogx-(1-y)log(1-x), G[·] denotes the generator, D[·] denotes the discriminator, M(H n ) represents the high-fidelity matrix H n The mask matrix for masking. It is a dot product operation.

[0077] Define the mean squared error (MSE) loss function as follows:

[0078]

[0079] in, The operation of squaring the F-norm. It is a dot product operation.

[0080] Define correlation loss as

[0081]

[0082] Where ρ(·,·) is the correlation coefficient, calculated as follows:

[0083]

[0084] Where I represents the number of rows in the matrix, and J represents the number of columns in the matrix.

[0085] This part of the Adam method refers to minimizing the generator loss L through multiple iterations. g (H n ,L n To achieve the optimization goal

[0086]

[0087] Then update the generator weights Among them W g (t+1) W represents the updated discriminator weight matrix. g (t) Let B represent the discriminator weight matrix before the update, and let η represent the total number of training data selected in the current batch. g This represents the pre-set discriminator learning rate. This indicates the calculation of the gradient, where κ and τ represent constant parameters, and L... g L represents the generator loss. mse L represents the mean squared error (MSE) loss function. ρ This indicates the loss of correlation.

[0088] Step 2.3: Iterate through T cycles, implementing the above minimax game training strategy in each cycle, and finally optimizing the weights of the generator and discriminator of GAN-MMC.

[0089] The third step involves inputting the complete low-fidelity matrix and the high-fidelity matrix (containing missing terms) into the generator of the generative adversarial network (GAN-MMC) after weight updates (after training, only the generator is retained during the testing phase to generate approximate HF matrix values) to obtain the complete high-fidelity matrix. The calculation method for the complete high-fidelity matrix is ​​as follows:

[0090]

[0091] in, H represents the generated complete high-fidelity matrix. n M(H) represents the high-fidelity matrix from the training set in the first step. n ) represents the high-fidelity matrix H n The mask matrix for masking. It's a dot product operation. It is an intermediate result of a high-fidelity matrix generated by the generator.

[0092] To verify the effectiveness of GAN-MMC, experiments were set up in two applications of real physical field simulation: one was to generate the stress field of a steel plate; the other was to simulate the velocity field of fluid flow in Oldroyd-B.

[0093] Regarding the stress field simulation of steel plates, consider the stress field of a steel plate with a hole in the middle (e.g., Figure 3 Point S on the right is subjected to a concentrated force F. s The included angle is α, and the left side is fixed.

[0094] First, a linear interpolation method is used to realize the stress field of the steel plate under high-fidelity data settings (e.g., Figure 4 ).

[0095] Then, based on the complete low-fidelity matrix and the high-fidelity matrix with only 4 available terms, the proposed GAN-MMC method is used to complete the high-fidelity matrix. The results show that the GAN-MMC method accurately approximates the true stress field under the high-fidelity data setting. The simulation diagram of the GAN-MMC method is shown below. Figure 5 .

[0096] Regarding the velocity field simulation of Oldroyd-B fluid flow, the velocity field simulation of Oldroyd-B fluid flow through a cylinder between two parallel plates (e.g.) Figure 6 ).

[0097] First, the finite element method was used to calculate the fluid velocity, resulting in a 10×125-dimensional low-fidelity matrix. The corresponding velocity field simulation (e.g., obtained through linear interpolation of the low-fidelity matrix) was then performed. Figure 7 ).

[0098] Secondly, the flow velocity values ​​at three grid points are selected, and some measurement methods, such as particle image velocity measurement, are used. Then, these three points are used as available items in the high-fidelity matrix.

[0099] Finally, by combining the complete low-fidelity matrix and the high-fidelity matrix containing three available terms, the GAN-MMC method was used to complete the high-fidelity matrix. The results show that GAN-MMC also accurately approximates the velocity field simulation of real Oldroyd-B fluid flow in this task (e.g., ...). Figure 8 ).

[0100] Finally, it should be noted that the above embodiments are only used to illustrate the method solutions of the present invention, and are not intended to limit them. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications to the method solutions described in the foregoing embodiments, or equivalent substitutions for some or all of the method features, do not cause the essence of the corresponding method solutions to deviate from the scope of the method solutions of the embodiments of the present invention.

Claims

1. A multi-fidelity matrix completion method based on generative adversarial networks, characterized in that, The multi-fidelity matrix completion method includes the following steps: The first step is to perform matrix data augmentation on the complete low-fidelity matrix L and the high-fidelity matrix H containing missing terms in the input data to obtain the augmented training set. The second step is to select training data of batch size B from the training set and fix the generator weights. And update the discriminator weights Select new batches of training data with size B from the training set, and fix the discriminator weights. And update the weights Within a certain number of rounds T, the training set is traversed, and the discrimination loss and generation loss are minimized by the above method of cross-updating weights. The third step involves inputting the complete low-fidelity matrix and the high-fidelity matrix containing missing terms into the weighted generative adversarial network (GAN), which consists of a generator and a discriminator, to ultimately obtain the complete high-fidelity matrix.

2. The multi-fidelity matrix completion method based on generative adversarial networks according to claim 1, characterized in that, The first step is specifically as follows: Step 1.1: Create a new mask matrix M with the same shape as the high-fidelity matrix. Set the mask matrix entries corresponding to the missing entries in the original high-fidelity matrix to 0, and set the other entries to 1. The masking matrix for masking a high-fidelity matrix H is denoted as M(H); Step 1.2, through dot product The operation combines the mask matrix M and the high-fidelity matrix H to obtain an approximate matrix of the high-fidelity matrix containing missing terms. The elements at the positions corresponding to the missing items in the original high-fidelity matrix H are 0, and the elements at other positions are the high-fidelity data in the original high-fidelity matrix H. Step 1.3, by synchronously arranging the L matrix and The rows and columns of the matrix are used to generate training samples (L row H row ) and (L col H col These rearranged low-fidelity matrices and high-fidelity approximation matrices Use it as a new training set.

3. The multi-fidelity matrix completion method based on generative adversarial networks according to claim 2, characterized in that, The second step is as follows: Step 2.1: Fix the generator weights during each training cycle. Update the discriminator weights using the Adam method Define the discriminator loss as: Among them, H n L represents the high-fidelity matrix from the first step training set. n L represents a low-fidelity matrix. ce L represents the cross-entropy loss function ce (x,y)=-ylogx-(1-y)log(1-x), where G[·] represents the generator and D[·] represents the discriminator. It's a dot product operation, M(H) n ) represents the high-fidelity matrix H n The mask matrix for masking; The Adam method partly refers to minimizing the discriminator loss L through multiple iterations. dis (H n ,L n This is done to achieve the optimization goal, and then update the discriminator weights. Step 2.2: Within the same training cycle, fix the discriminator weights. Update generator weights using the Adam method Define the generator loss as: Among them, H n L represents the high-fidelity matrix from the first step training set. n L represents a low-fidelity matrix. ce L represents the cross-entropy loss function ce (x,y)=-ylogx-(1-y)log(1-x), G[·] denotes the generator, D[·] denotes the discriminator, M(H n ) represents the high-fidelity matrix H n The mask matrix for masking. It is a dot product operation; If we define the mean squared error (MSE) loss function and the correlation loss function respectively, then the Adam method refers to minimizing the generator loss L through multiple iterations. g (H n ,L n To achieve the optimization goal, the generator weights are then updated. Step 2.3: Iterate through T cycles, implementing the above minimax game training strategy in each cycle, and finally optimizing the generator and discriminator weights of GAN-MMC.

4. The multi-fidelity matrix completion method based on generative adversarial networks according to claim 3, characterized in that, The optimization objective in step 2.1 is as follows: Among them, W d (t+1) W represents the updated discriminator weight matrix. d (t) Let B represent the discriminator weight matrix before the update, and let η represent the total number of training data selected in the current batch. d This represents the pre-set discriminator learning rate. This represents gradient operation.

5. The multi-fidelity matrix completion method based on generative adversarial networks according to claim 3, characterized in that, In step 2.2: The mean square error (MSE) loss function is: in, The operation of squaring the F-norm. It is a dot product operation; The correlation loss is: Wherein, ρ(·,·) is the correlation coefficient, calculated as follows: Where I represents the number of rows in the matrix, and J represents the number of columns in the matrix; The optimization objective is: in, W represents the updated discriminator weight matrix. g (t) Let B represent the discriminator weight matrix before the update, and let η represent the total number of training data selected in the current batch. g This represents the pre-set discriminator learning rate. This indicates the calculation of the gradient, where κ and τ represent constant parameters, and L... g L represents the generator loss. mse L represents the mean squared error (MSE) loss function. ρ This indicates the loss of correlation.

6. The multi-fidelity matrix completion method based on generative adversarial networks according to claim 3, characterized in that, In the third step, the complete high-fidelity matrix is ​​calculated as follows: in, H represents the generated complete high-fidelity matrix. n M(H) represents the high-fidelity matrix from the training set in the first step. n ) represents the high-fidelity matrix H n The mask matrix for masking. It's a dot product operation. It is an intermediate result of a high-fidelity matrix generated by the generator.

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