A method and system for fusion prediction of multi-fidelity CAE simulation data

Abstract

The present application relates to a kind of multi-fidelity CAE simulation data fusion prediction method and system, belong to CAE simulation technical field, solve the problem of high computational complexity and large memory consumption when handling massive multi-fidelity data in existing CAE simulation technology. Including: multi-fidelity data set is obtained by acquisition, data sample is selected from it to construct joint induced point set;By maximizing the edge likelihood function, the hyperparameters of Gaussian process model are learned, and a simulation prediction model is obtained;Wherein, based on the current hyperparameters, an adaptive weight matrix is constructed, and the cross-covariance matrix between the multi-fidelity data set and the joint induced point set is weighted;The autocovariance matrix of joint induced point set is triangularly decomposed;The simulation prediction model is used to simulate and predict the to-be-predicted data set, and a simulation response prediction result set is obtained. Efficient fusion and accurate prediction of simulation data are realized.

Description

Technical Field

[0001] This invention relates to the field of CAE simulation technology, and in particular to a method and system for fusion prediction of multi-fidelity CAE simulation data. Background Technology

[0002] In typical computer-aided engineering (CAE) simulation analysis, engineers usually need to build numerical models of products and predict their response under real-world conditions by solving physical equations. With the development of intelligent manufacturing and digital twin technologies, CAE simulation is evolving from traditional single-physics field and single-fidelity analysis to intelligent methods involving multi-physics field coupling and multi-source data fusion in application fields such as aerospace, automotive manufacturing, and energy equipment R&D.

[0003] While traditional Gaussian process regression methods can establish a nonlinear mapping between input and output, they face the bottleneck of excessively high computational complexity when dealing with large-scale datasets. Specifically, traditional methods require constructing and inverting the covariance matrix of the entire dataset, and their computational complexity increases cubically with the amount of data. When the data volume reaches tens of thousands or even millions, the computational burden becomes unbearable. Furthermore, traditional methods ignore the noise characteristics and quality differences existing in data from different sources, lack the ability to quantitatively evaluate the reliability of prediction results, and cannot meet the real-time and reliability requirements of intelligent CAE systems.

[0004] In recent years, with the development of machine learning technology, sparse Gaussian process methods have emerged. However, most existing sparse Gaussian process methods are designed for single-fidelity data and are difficult to effectively handle multi-fidelity data fusion problems. In multi-fidelity scenarios, they fail to fully consider the inherent differences in noise characteristics and quality levels of different fidelity data sources. This results in the model's inability to intelligently "play to its strengths and avoid its weaknesses," limiting prediction accuracy and lacking the ability to quantitatively evaluate the reliability of the fused prediction results, making it difficult to meet the high accuracy and high reliability requirements of intelligent CAE systems. Summary of the Invention

[0005] Based on the above analysis, the embodiments of the present invention aim to provide a method and system for fusion prediction of multi-fidelity CAE simulation data, in order to solve the problems of high computational complexity, large memory consumption, and insufficient data fusion accuracy in the processing of massive multi-fidelity data by existing CAE simulation technologies.

[0006] On one hand, embodiments of the present invention provide a method for fusion and prediction of multi-fidelity CAE simulation data, comprising the following steps: Low-fidelity and high-fidelity datasets from CAE simulations are collected to obtain multi-fidelity datasets; data samples are selected from the multi-fidelity datasets to construct a joint induced point set; Using the multi-fidelity dataset and the joint induced point set, the hyperparameters of the Gaussian process model are learned by maximizing the marginal likelihood function to obtain the simulation prediction model. Specifically, in maximizing the marginal likelihood function, an adaptive weight matrix is ​​constructed based on the current hyperparameters to weight the cross-covariance matrix between the multi-fidelity dataset and the joint induced point set. Furthermore, the autocovariance matrix of the joint induced point set is triangularly decomposed, and the marginal likelihood function value and its gradient are calculated based on the decomposition result and the weighted cross-covariance matrix. The simulation prediction model is used to perform simulation prediction on the acquired dataset to be predicted, and a set of simulation response prediction results is obtained.

[0007] Based on further improvements to the above method, the hyperparameters of the Gaussian process model include at least: the noise variance of the low-fidelity dataset, and the deviation variance between the low-fidelity dataset and the high-fidelity dataset; the adaptive weight matrix is ​​a block diagonal matrix, expressed as: , in, For adaptive weight matrix, For the number of data samples in a low-fidelity dataset, The number of data samples in the high-fidelity dataset. For the noise variance of the low-fidelity dataset, The variance of the bias between the low-fidelity dataset and the high-fidelity dataset. for An identity matrix of dimension 1 for An identity matrix of dimension 1.

[0008] A further improvement to the above method is to weight the cross-covariance matrix between the multi-fidelity dataset and the joint induced point set by multiplying the cross-covariance matrix on the right by the square root of the adaptive weight matrix to obtain the weighted cross-covariance matrix.

[0009] A further improvement to the above method involves calculating the marginal likelihood function values ​​and their gradients based on the decomposition results and the weighted cross-covariance matrix, including: The intermediate matrix is ​​obtained by transforming the weighted cross-covariance matrix using the transpose of the inverse of the lower triangular matrix obtained from the decomposition result. Construct a low-rank matrix based on the intermediate matrix, and solve for the inverse matrix of the sum of the low-rank matrix and the identity matrix; Using the inverse matrix, the intermediate matrix, and the weighted observation vector, the marginal likelihood function value and its gradient with respect to the current hyperparameter are calculated; the weighted observation vector is obtained by multiplying the square root of the adaptive weight matrix with the observation response value vector in the multifidelity dataset.

[0010] Based on the further improvement of the above method, the triangular decomposition is converted into the Cholesky decomposition, which decomposes the autocovariance matrix of the joint induced point set into the product of the lower triangular matrix and its transpose, and the resulting lower triangular matrix is ​​used as the decomposition result.

[0011] Based on further improvements to the above method, the simulation response prediction result set is obtained through the following formula: , in, For the dataset to be predicted The set of simulation response prediction results For the dataset to be predicted With joint induction point set The covariance matrix between them It is a lower triangular matrix. For the intermediate matrix, For adaptive weight matrix, For the observation vector, for The identity matrix, The number of data samples in the joint induction point set. This is a transpose operation.

[0012] Based on a further improvement of the above method, the method further includes: calculating the prediction variance corresponding to each simulation response prediction value in the simulation response prediction result set to obtain a prediction variance vector; each prediction variance includes: a model uncertainty component and a data noise component; the model uncertainty component automatically increases as the distance between the dataset to be predicted and the joint induced point set increases; the data noise component is the noise variance of the low-fidelity dataset and the sum of the deviation variance between the low-fidelity dataset and the high-fidelity dataset.

[0013] Based on a further improvement of the above method, in the process of maximizing the marginal likelihood function, the autocovariance matrix of the multifidelity dataset is approximated in a low rank based on the joint induced point set and the inverse of its autocovariance matrix.

[0014] Further improvements to the above method involve selecting data samples from the multi-fidelity dataset to construct a joint induced point set, which is obtained using a clustering algorithm or a random sampling algorithm.

[0015] On the other hand, embodiments of the present invention provide a fusion prediction system for multi-fidelity CAE simulation data, comprising: The data processing module is used to collect low-fidelity and high-fidelity datasets from CAE simulations to obtain multi-fidelity datasets; and to select data samples from the multi-fidelity datasets to construct a joint induced point set. The model training module is used to learn the hyperparameters of the Gaussian process model by maximizing the marginal likelihood function using the multi-fidelity dataset and the joint induced point set, thereby obtaining a simulation prediction model. Specifically, during the process of maximizing the marginal likelihood function, an adaptive weight matrix is ​​constructed based on the current hyperparameters to weight the cross-covariance matrix between the multi-fidelity dataset and the joint induced point set. Furthermore, the autocovariance matrix of the joint induced point set is triangularly decomposed, and the marginal likelihood function value and its gradient are calculated based on the decomposition result and the weighted cross-covariance matrix. The simulation prediction module is used to perform simulation prediction on the acquired dataset to be predicted using the simulation prediction model, and obtain a set of simulation response prediction results.

[0016] Compared with the prior art, the present invention can achieve at least one of the following beneficial effects: 1. By selecting a joint induced point set from the original multi-fidelity dataset, large-scale matrix operations are transformed into efficient computations in a low-dimensional space, reducing the computational complexity from cubic to a controllable range related to the number of joint induced points, thus significantly reducing the computational complexity and memory consumption of CAE simulation data fusion.

[0017] 2. By constructing an adaptive weight matrix, different weights are automatically assigned to data of different qualities based on the noise variance of low-fidelity data and high-fidelity data and the deviation variance between them. During model training, the weight of low-precision data is adaptively reduced, while the weight of high-precision data is increased, thereby effectively suppressing the negative impact of low-precision data on the model and improving the robustness and prediction accuracy of the Gaussian process model.

[0018] 3. By replacing direct inversion with triangular decomposition, the numerical overflow or precision loss caused by direct inversion when the autocovariance matrix of the joint induced point set is close to singular is avoided, which significantly improves the stability of numerical computation.

[0019] In this invention, the above-described technical solutions can be combined with each other to achieve more preferred combinations. Other features and advantages of this invention will be set forth in the following description, and some advantages may become apparent from the description or be learned by practicing the invention. The objects and other advantages of this invention can be realized and obtained from what is particularly pointed out in the description and drawings. Attached Figure Description

[0020] The accompanying drawings are for illustrative purposes only and are not intended to limit the invention. Throughout the drawings, the same reference numerals denote the same parts. Figure 1 This is a flowchart of a method for fusion and prediction of multi-fidelity CAE simulation data in Embodiment 1 of the present invention. Detailed Implementation

[0021] Preferred embodiments of the present invention will now be described in detail with reference to the accompanying drawings, which form part of this application and are used together with the embodiments of the present invention to illustrate the principles of the present invention, but are not intended to limit the scope of the present invention.

[0022] Example 1 A specific embodiment of the present invention discloses a method for fusion and prediction of multi-fidelity CAE simulation data, such as... Figure 1 As shown, it includes steps S1-S3.

[0023] S1. Collect low-fidelity and high-fidelity datasets from CAE simulations to obtain multi-fidelity datasets; select data samples from the multi-fidelity datasets to construct a joint induced point set.

[0024] It should be noted that multi-fidelity datasets refer to CAE simulation datasets obtained for the same engineering object or physical process under different simulation accuracy levels (including differences in mesh density, physical model complexity, time step, etc.). Each data sample includes a simulation parameter vector and a simulation response value.

[0025] Low-fidelity datasets typically originate from fast but limited-accuracy simulation models, such as the low-order surface element method used in aerodynamic analysis or the coarse-grid finite element model used in structural analysis. Low-fidelity datasets are represented as... ,in, and These are the first in the low-fidelity dataset. A vector of simulation parameters and its corresponding simulation response values. The number of data samples in the low-fidelity dataset.

[0026] High-fidelity datasets typically originate from high-precision but computationally expensive simulations or physics experiments, such as computational fluid dynamics simulations based on the Reynolds-averaged Navier-Stokes equations or high-precision sensor measurements. High-fidelity datasets are represented as... ,in, and These are the first in the high-fidelity dataset. A vector of simulation parameters and its corresponding simulation response values. This represents the number of data samples in the high-fidelity dataset.

[0027] Merging these two datasets yields a multi-fidelity dataset. ,in , Sample data in multi-fidelity datasets .

[0028] It should be noted that the simulation parameter vector includes at least one of the following: geometric dimension parameters, material property parameters, load condition parameters, or boundary condition parameters.

[0029] For example, geometric parameters include: thickness, length, diameter, fillet radius, etc. of the component; material property parameters include: elastic modulus, Poisson's ratio, density, thermal conductivity, etc.; and load and boundary condition parameters include: magnitude, direction, pressure value, constraint location, etc.

[0030] The simulation response values ​​include at least one of the following: structural mass properties, mechanical response, thermal response, fluid dynamics response, or multidisciplinary coupled response.

[0031] For example, the mass properties of the structure include: total mass of the vehicle body and total mass of the wing; the mechanical response includes: maximum equivalent stress, maximum deformation displacement and natural frequency; the thermal response includes: maximum temperature and average heat flux density; the fluid dynamic response includes: drag coefficient and lift coefficient; and the multidisciplinary coupled response includes: thermal stress under thermal-structural coupling.

[0032] Furthermore, using either the K-means clustering algorithm or a random sampling algorithm, a small number of highly representative data samples are selected from both the low-fidelity and high-fidelity datasets to form a joint inducement point set. Number of data samples in the joint induction point set It should be much smaller than the total number of samples in the original multifidelity dataset. The number of samples is typically kept to within a few thousand (e.g., no more than 2,000), which achieves significant dimensionality reduction. At the same time, these few samples retain the input space features and statistical regularities contained in the original dataset to the greatest extent possible, laying a solid foundation for building efficient and accurate approximate models in the future.

[0033] Taking the aerodynamic design of a certain type of aircraft wing as an example, the simulation parameter vector consists of the key geometric parameters of the airfoil, such as: maximum relative thickness, camber, leading edge radius, trailing edge angle, location of maximum thickness, and location of maximum camber. The simulation response value is the lift coefficient of the airfoil. A low-fidelity dataset of 50,000 data samples was obtained using the surface element method simulation software; a high-fidelity dataset of 2,000 data samples was obtained using CFD simulation software based on RANS equations. K-means clustering was used to select 800 induced points from the low-fidelity dataset and 200 induced points from the high-fidelity dataset, merging them to obtain 1,000 joint induced points.

[0034] It should be noted that this embodiment can also be applied to other CAE simulation prediction scenarios, such as automobile collision performance prediction and wind turbine blade structural response prediction.

[0035] S2. Using the multi-fidelity dataset and the joint induced point set, the hyperparameters of the Gaussian process model are learned by maximizing the marginal likelihood function to obtain the simulation prediction model. In the process of maximizing the marginal likelihood function, an adaptive weight matrix is ​​constructed based on the current hyperparameters to weight the cross-covariance matrix between the multi-fidelity dataset and the joint induced point set. Furthermore, the autocovariance matrix of the joint induced point set is decomposed into triangular components, and the marginal likelihood function value and its gradient are calculated based on the decomposition result and the weighted cross-covariance matrix.

[0036] It should be noted that the hyperparameters of the Gaussian process model At least include: noise variance of the low-fidelity dataset It is used to characterize the degree of measurement error or computational uncertainty of low-fidelity data itself, as well as the variance of the bias between low-fidelity and high-fidelity datasets. It is used to capture systematic differences in the process of mapping from low fidelity to high fidelity; it also includes hyperparameters of the kernel function, such as length scale and signal variance.

[0037] Traditional Gaussian processes require calculating and inverting the autocovariance matrix of the complete dataset, i.e., the autocovariance matrix of the multifidelity dataset obtained in step S1. Its dimensions are The computational complexity is as high as ,when When the number of points reaches tens of thousands or even millions (which is common in CAE simulations and big data scenarios), this computational burden becomes unbearable. To overcome this bottleneck, this embodiment is based on the joint induced point set and the inverse matrix of its autocovariance matrix. The low-rank approximation of the autocovariance matrix of a multi-fidelity dataset is shown in the following formula: , in, The cross-covariance matrix between the multi-fidelity dataset and the joint induced point set. Let be the autocovariance matrix of the joint induced point set. This is the transpose operation; the -1 in the upper right corner is for calculating the inverse matrix.

[0038] The aforementioned low-rank approximation transforms large-scale matrices into small-scale matrices. The inverse of the expression is used to reduce the computational complexity from... Reduced to .because much smaller This makes it possible to process massive amounts of data. Memory consumption has also been greatly optimized, eliminating the need to allocate huge amounts of storage. Matrix, only needs to handle size and The memory requirements are significantly reduced for matrices. At the same time, numerical stability is significantly improved because the inversion operation is only performed on small matrices that typically have good condition numbers, avoiding the numerical ill-conditioned problems that may occur in large-scale matrix inversion.

[0039] Furthermore, during each iteration of maximizing the marginal likelihood function, the following operations are performed: (1) Construct an adaptive weight matrix based on the current hyperparameters and weight the cross-covariance matrix between the multi-fidelity dataset and the joint induced point set.

[0040] First, based on the hyperparameters in the current iteration and Construct an adaptive weight matrix that reflects the differences in data quality at different fidelity levels. This matrix is ​​a block diagonal matrix, with its diagonal elements set to the reciprocal of the variance of the low-fidelity data noise and the reciprocal of the sum of the variances of the low-fidelity data noise and the variances of the deviation between the low-fidelity and high-fidelity data, respectively, according to the data source. It is expressed as: , in, for An identity matrix of dimension 1 for An identity matrix of dimension 1.

[0041] The adaptive weight matrix represents intelligent differentiated processing of data with different precision. The weight of low-precision data will be automatically reduced, while the weight of high-precision data will be increased accordingly.

[0042] Then, calculate the multi-fidelity dataset. With joint induction point set The cross-covariance matrix between Each element of this matrix is ​​calculated using a kernel function (typically a squared exponential kernel) from a Gaussian process model, representing the correlation between multi-fidelity data points and joint induced points. The weighted cross-covariance matrix is ​​obtained by right-multiplying the cross-covariance matrix by the square root of the adaptive weight matrix. The formula is shown below: , in, The square root of the adaptive weight matrix is ​​obtained by taking the square root of its diagonal elements.

[0043] It's important to note that the adaptive weighting mechanism enables the model to automatically identify and handle differences in data quality. When low-fidelity data contains significant noise, its corresponding weights are automatically reduced, thus mitigating the negative impact of low-precision data on model training. For high-fidelity data, the model considers both its inherent measurement noise and the systematic biases between it and low-fidelity data, resulting in a more reasonable weight allocation. This intelligent weighting strategy significantly improves the robustness and generalization ability of the multi-fidelity Gaussian process model, leading to more accurate and reliable prediction results.

[0044] (2) Perform triangular decomposition on the autocovariance matrix of the joint induced point set.

[0045] It should be noted that the triangular decomposition is a Cholesky decomposition, and the autocovariance matrix of the joint induced point set is obtained using the following formula. The product of a lower triangular matrix and its transpose is decomposed into a lower triangular matrix, and the resulting lower triangular matrix is ​​taken as the decomposition result: , in, It is a lower triangular matrix, and all its diagonal elements are positive, which ensures the subsequent numerical stability.

[0046] (3) Calculate the marginal likelihood and its gradient based on the decomposition results and the weighted cross-covariance matrix.

[0047] It should be noted that, theoretically, the marginal likelihood function requires calculating the inverse matrix and determinant of the derived matrix. Based on the low-rank approximation and the weighted cross-covariance matrix, the derived matrix is ​​defined by the following formula. : .

[0048] This derived matrix replaces the traditional Gaussian process. This unifies data weighting and low-rank approximation within a single framework, in which the marginal likelihood function is expressed as: , in, The weighted observation vector is the square root of the adaptive weight matrix. With the observed response vector in the multifidelity dataset Multiplying them together, we get That is, the simulation response values ​​from the low-fidelity dataset. Simulation response values ​​in high-fidelity datasets It is pieced together, with dimensions of .

[0049] It should be noted that the derived matrix It is still Matrix, if calculated directly and The computational complexity is still O(n). Therefore, this embodiment further employs the following numerical calculation techniques to calculate the derived matrix. The operation is transformed into operations on small-scale matrices.

[0050] Specifically, the marginal likelihood function value and its gradient are calculated through the following steps: ① Using the transpose of the inverse of the lower triangular matrix obtained from the decomposition result, transform the weighted cross-covariance matrix to obtain the intermediate matrix. The formula is shown below: .

[0051] It should be noted that, It is a triangular matrix, and its inverse matrix can be solved efficiently and numerically stably using back substitution without explicit inversion. Specifically, the operation involves constructing an identity matrix as the right-hand side and solving the system of trigonometric equations. ,get ,Then This process avoids direct calculation. This eliminates the problem of numerical overflow or precision loss that may be caused by ill-conditioned matrices. (Intermediate matrix) The dimension is ,because Both storage and computation are feasible, and the computational complexity of this step is: .

[0052] ② Construct a low-rank matrix based on the intermediate matrix, and solve for the inverse matrix of the sum of the low-rank matrix and the identity matrix.

[0053] Specifically, based on the intermediate matrix The constructed low-rank matrix is Its dimensions are The inverse matrix of the sum of the low-rank matrix and the identity matrix is: ,in for The computational complexity of finding the inverse matrix of the identity matrix is ​​O(n log n). That's perfectly acceptable.

[0054] ③ Using the inverse matrix, the intermediate matrix, and the weighted observation vector, calculate the marginal likelihood function value and its gradient with respect to the current hyperparameter; the weighted observation vector is obtained by multiplying the square root of the adaptive weight matrix with the observation response value vector in the multifidelity dataset.

[0055] Using Woodbury's identity, and The calculation is transformed into a This avoids direct processing by performing calculations. The matrix, and its specific formula are shown below: , , in, , For adaptive weight matrix The determinant, .

[0056] Finally, through variable substitution, the marginal likelihood function is simplified to the following form for efficient calculation, involving only... Matrix inversion and determinant, and matrix-vector multiplication: , in, It is a constant. , .

[0057] The hyperparameters are iteratively updated using optimization algorithms such as the conjugate gradient method or L-BFGS until the marginal likelihood function converges, thus learning the optimal hyperparameter values ​​and obtaining a trained simulation prediction model. Based on the optimal hyperparameter values, the final determined adaptive weight matrix, lower triangular matrix, and intermediate matrix are calculated and stored for use in the simulation prediction stage.

[0058] Compared to existing technologies, this step replaces the autocovariance matrix of the multi-fidelity dataset with an expression based on the joint induced point set through low-rank approximation, eliminating the need to handle large-scale dense matrices in marginal likelihood calculation. An adaptive weight matrix is ​​introduced to achieve differentiated weighting for data of different precisions, cleverly integrating weighting and low-rank approximation into the same framework. Using Cholesky decomposition, back substitution, and the Woodbury identity, all inversion operations are restricted to... On small-scale matrices, it significantly improves numerical stability and computational efficiency.

[0059] S3. Use the simulation prediction model to perform simulation prediction on the acquired dataset to be predicted, and obtain the simulation response prediction result set.

[0060] Acquired dataset to be predicted Its dimensions are ,in, The number of simulation parameter vectors to be predicted. This refers to the dimension of the simulation parameter vector. For example, in the aerodynamic design of a certain type of aircraft wing, It involves a combination of multiple new airfoil geometric parameters, and it is necessary to predict the corresponding lift coefficient.

[0061] The simulation response prediction result set is obtained using the following formula: , in, For the dataset to be predicted The set of simulation response prediction results For the dataset to be predicted With joint induction point set The covariance matrix between them has dimensions of Each element is calculated using a kernel function, representing the correlation between the point to be predicted and the joint induced point. This matrix can be calculated directly without the need for a low-rank approximation method. It is a lower triangular matrix. This is the intermediate matrix.

[0062] The above calculation method employs a numerically stable computational strategy, resulting in predictions with good numerical reliability. This intelligent fusion method effectively balances computational efficiency and prediction accuracy, making it possible to achieve fast and accurate multi-fidelity predictions in massive data environments.

[0063] Furthermore, the method in this embodiment also includes: calculating the prediction variance corresponding to each simulation response prediction value in the simulation response prediction result set to obtain a prediction variance vector; each prediction variance includes: a model uncertainty component and a data noise component; the model uncertainty component automatically increases as the distance between the dataset to be predicted and the joint induced point set increases; the data noise component is the sum of the noise variance of the low-fidelity dataset and the deviation variance between the low-fidelity dataset and the high-fidelity dataset.

[0064] Specifically, the formula for calculating the prediction variance is as follows: , , , in, To predict the variance vector, For the uncertainty components of the model, The noise component is a constant and represents the measurement error and fidelity deviation that cannot be eliminated in high-fidelity data, which is an inherent source of uncertainty. Let be the autocovariance matrix of the dataset to be predicted, with dimension . ; Let be the covariance matrix between the joint induced point set and the dataset to be predicted, with dimension . ,yes Transpose of; This indicates retrieving the diagonal elements of the matrix, and the output result is... The prediction variance vector.

[0065] It should be noted that the model uncertainty components This reflects the estimation error introduced by using a sparse approximation with a finite set of joint induced points. Its key characteristic is spatial adaptability: when the data point to be predicted is far from all joint induced points, The uncertainty component of the model approaches zero, therefore it approaches zero. (Prior variance) automatically increases; when the data point to be predicted is close to the joint induction point, the model uncertainty component automatically decreases. This allows users to intuitively judge the reliability of the prediction results—in sparse data regions, the model will proactively provide a larger uncertainty warning.

[0066] Furthermore, the predicted standard deviation is obtained by taking the square root of the predicted variance. Based on the simulated response prediction value and prediction standard deviation, the confidence interval is obtained, providing a reliable basis for engineering decision-making.

[0067] For example, in practical engineering such as airfoil optimization, the 95% confidence interval can be approximated as the simulated response prediction value ± 2 standard deviations, i.e. This range is used for reliability assessment in engineering design: in data-intensive areas, the confidence interval is narrower and the prediction results are more reliable; in data-sparse areas, the confidence interval automatically widens to remind users that the prediction results in these areas need to be used with caution.

[0068] In this embodiment, during the training phase of step S2, key quantities such as the intermediate matrix are calculated through low-rank approximation, weighted summation, and triangular decomposition. These stored quantities are directly reused during the prediction phase, reducing computational complexity compared to the number of points to be predicted. and the number of joint induction points Linear correlation, and related to the number of data samples in the original multifidelity dataset. The method is independent of the target data point; only a few small-scale matrix multiplications and additions are needed to calculate the covariance matrix between the data point to be predicted and the joint induced point set to obtain the predicted mean and variance. The high one-time cost of the training phase is amortized over countless rapid predictions, making it particularly suitable for engineering scenarios such as digital twins and real-time simulation optimization that require frequent evaluation of new design solutions. Simultaneously, the spatially adaptive uncertainty quantification capability provides users with valuable reliability information, which is lacking in traditional surrogate model methods.

[0069] Compared with existing technologies, this embodiment provides a method for fusion and prediction of multi-fidelity CAE simulation data. By selecting a joint induced point set from the original multi-fidelity dataset, it transforms large-scale matrix operations into efficient computations in a low-dimensional space, reducing the computational complexity from cubic to a controllable range related to the number of joint induced points. This significantly reduces the computational complexity and memory consumption of CAE simulation data fusion. By constructing an adaptive weight matrix, different weights are automatically assigned to data of different qualities based on the noise variance of low-fidelity and high-fidelity data and the variance of their deviations. During model training, the weights of low-precision data are adaptively reduced, while the weights of high-precision data are increased, effectively suppressing the negative impact of low-precision data on the model and improving the robustness and prediction accuracy of the Gaussian process model. By replacing direct inversion with triangular decomposition, it avoids the numerical overflow or precision loss problems caused by direct inversion when the autocovariance matrix of the joint induced point set is close to singular, significantly improving the stability of numerical computation.

[0070] Example 2 Another embodiment of the present invention discloses a fusion prediction system for multi-fidelity CAE simulation data, thereby implementing the fusion prediction method for multi-fidelity CAE simulation data in Embodiment 1. The specific implementation of each module is described in the corresponding description in Embodiment 1. The system includes: The data processing module is used to collect low-fidelity and high-fidelity datasets from CAE simulations to obtain multi-fidelity datasets; and to select data samples from the multi-fidelity datasets to construct a joint induced point set. The model training module is used to learn the hyperparameters of the Gaussian process model by maximizing the marginal likelihood function using the multi-fidelity dataset and the joint induced point set, thereby obtaining a simulation prediction model. Specifically, during the process of maximizing the marginal likelihood function, an adaptive weight matrix is ​​constructed based on the current hyperparameters to weight the cross-covariance matrix between the multi-fidelity dataset and the joint induced point set. Furthermore, the autocovariance matrix of the joint induced point set is triangularly decomposed, and the marginal likelihood function value and its gradient are calculated based on the decomposition result and the weighted cross-covariance matrix. The simulation prediction module is used to perform simulation prediction on the acquired dataset to be predicted using the simulation prediction model, and obtain a set of simulation response prediction results.

[0071] Since the multi-fidelity CAE simulation data fusion prediction system of this embodiment and the aforementioned multi-fidelity CAE simulation data fusion prediction method are related and can be mutually referenced, this description is redundant and will not be repeated here. Because the principle of this system embodiment is the same as that of the above method embodiment, this system embodiment also has the corresponding technical effects of the above method embodiment.

[0072] Those skilled in the art will understand that all or part of the processes of the methods described in the above embodiments can be implemented by a computer program instructing related hardware, and the program can be stored in a computer-readable storage medium. The computer-readable storage medium may be a disk, optical disk, read-only memory, or random access memory, etc.

[0073] The above description is only a preferred embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any changes or substitutions that can be easily conceived by those skilled in the art within the scope of the technology disclosed in the present invention should be included within the scope of protection of the present invention.

Claims

1. A method for fusing and predicting multi-fidelity CAE simulation data, characterized in that, Includes the following steps: Collect low-fidelity and high-fidelity datasets from CAE simulations to obtain multi-fidelity datasets; Data samples are selected from the multi-fidelity dataset to construct a joint induced point set; Using the multi-fidelity dataset and the joint induced point set, the hyperparameters of the Gaussian process model are learned by maximizing the marginal likelihood function to obtain the simulation prediction model. Specifically, in maximizing the marginal likelihood function, an adaptive weight matrix is ​​constructed based on the current hyperparameters to weight the cross-covariance matrix between the multi-fidelity dataset and the joint induced point set. Furthermore, the autocovariance matrix of the joint induced point set is triangularly decomposed, and the marginal likelihood function value and its gradient are calculated based on the decomposition result and the weighted cross-covariance matrix. The simulation prediction model is used to perform simulation prediction on the acquired dataset to be predicted, and a set of simulation response prediction results is obtained.

2. The method for fusion and prediction of multi-fidelity CAE simulation data according to claim 1, characterized in that, The hyperparameters of the Gaussian process model include at least: the noise variance of the low-fidelity dataset and the deviation variance between the low-fidelity dataset and the high-fidelity dataset; the adaptive weight matrix is ​​a block diagonal matrix, expressed as: ， in, For adaptive weight matrix, For the number of data samples in a low-fidelity dataset, The number of data samples in the high-fidelity dataset. For the noise variance of the low-fidelity dataset, The variance of the bias between the low-fidelity dataset and the high-fidelity dataset. for An identity matrix of dimension 1 for An identity matrix of dimension 1.

3. The method for fusion and prediction of multi-fidelity CAE simulation data according to claim 1, characterized in that, The weighting of the cross-covariance matrix between the multi-fidelity dataset and the joint induced point set is achieved by multiplying the cross-covariance matrix on the right by the square root of the adaptive weight matrix to obtain the weighted cross-covariance matrix.

4. The method for fusion and prediction of multi-fidelity CAE simulation data according to claim 1, characterized in that, The calculation of the marginal likelihood function value and its gradient based on the decomposition results and the weighted cross-covariance matrix includes: The intermediate matrix is ​​obtained by transforming the weighted cross-covariance matrix using the transpose of the inverse of the lower triangular matrix obtained from the decomposition result. Construct a low-rank matrix based on the intermediate matrix, and solve for the inverse matrix of the sum of the low-rank matrix and the identity matrix; Using the inverse matrix, the intermediate matrix, and the weighted observation vector, the marginal likelihood function value and its gradient with respect to the current hyperparameter are calculated; the weighted observation vector is obtained by multiplying the square root of the adaptive weight matrix with the observation response value vector in the multifidelity dataset.

5. The method for fusion and prediction of multi-fidelity CAE simulation data according to claim 1, characterized in that, The triangular decomposition is the Cholesky decomposition, which decomposes the autocovariance matrix of the joint induced point set into the product of the lower triangular matrix and its transpose, and the resulting lower triangular matrix is ​​used as the decomposition result.

6. The method for fusing and predicting multi-fidelity CAE simulation data according to claim 4, characterized in that, The simulation response prediction result set is obtained using the following formula: ， in, For the dataset to be predicted The set of simulation response prediction results For the dataset to be predicted With joint induction point set The covariance matrix between them It is a lower triangular matrix. For the intermediate matrix, For adaptive weight matrix, For the observation vector, for The identity matrix, The number of data samples in the joint induction point set. This is a transpose operation.

7. The method for fusion and prediction of multi-fidelity CAE simulation data according to claim 1, characterized in that, The method further includes: calculating the prediction variance corresponding to each simulation response prediction value in the simulation response prediction result set to obtain a prediction variance vector; each prediction variance includes: a model uncertainty component and a data noise component; the model uncertainty component automatically increases as the distance between the dataset to be predicted and the joint induced point set increases; the data noise component is the sum of the noise variance of the low-fidelity dataset and the deviation variance between the low-fidelity dataset and the high-fidelity dataset.

8. The method for fusion and prediction of multi-fidelity CAE simulation data according to claim 1, characterized in that, In the process of maximizing the marginal likelihood function, the autocovariance matrix of the multifidelity dataset is approximated in a low rank based on the joint induced point set and the inverse of its autocovariance matrix.

9. The method for fusion and prediction of multi-fidelity CAE simulation data according to claim 1, characterized in that, The joint induced point set is constructed by selecting data samples from the multi-fidelity dataset using a clustering algorithm or a random sampling algorithm.

10. A fusion prediction system for multi-fidelity CAE simulation data, characterized in that, include: The data processing module is used to collect low-fidelity and high-fidelity datasets from CAE simulations to obtain multi-fidelity datasets. Data samples are selected from the multi-fidelity dataset to construct a joint induced point set; The model training module is used to learn the hyperparameters of the Gaussian process model by maximizing the marginal likelihood function using the multi-fidelity dataset and the joint induced point set, thereby obtaining a simulation prediction model. Specifically, during the process of maximizing the marginal likelihood function, an adaptive weight matrix is ​​constructed based on the current hyperparameters to weight the cross-covariance matrix between the multi-fidelity dataset and the joint induced point set. Furthermore, the autocovariance matrix of the joint induced point set is triangularly decomposed, and the marginal likelihood function value and its gradient are calculated based on the decomposition result and the weighted cross-covariance matrix. The simulation prediction module is used to perform simulation prediction on the acquired dataset to be predicted using the simulation prediction model, and obtain a set of simulation response prediction results.

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