A cross-task multi-fidelity probability modeling method fusing feature mapping and meta-learning

Abstract

This invention discloses a cross-task multi-fidelity probabilistic modeling method integrating feature mapping and meta-learning, belonging to the field of simulation modeling technology. It addresses the problem of rapidly constructing high-precision prediction models across multiple simulation tasks. The invention includes: constructing a multi-task multi-fidelity dataset for each task; establishing a deep feature mapping network structure to perform feature mapping on the input data in the multi-task multi-fidelity dataset, generating feature vectors; constructing a multi-fidelity Gaussian process to complete the construction and training of a single-task multi-fidelity model; establishing a meta-learning cross-task optimization method to achieve cross-task optimization across multiple historical simulation tasks based on the single-task multi-fidelity model, obtaining optimal initialization parameters; and using the optimal initialization parameters to perform adaptive multi-fidelity modeling on new high-cost simulation tasks, outputting the prediction results for the new high-cost simulation tasks. This invention significantly reduces the number of high-fidelity simulation calls and the cost of repeated training.

Description

Technical Field

[0001] This invention belongs to the field of simulation modeling technology, specifically involving a cross-task multi-fidelity probabilistic modeling method that integrates feature mapping and meta-learning. Background Technology

[0002] With the rapid development of aerospace, advanced manufacturing, energy equipment, and materials science, the design and optimization of complex engineering systems increasingly rely on high-precision physical simulation technologies. High-fidelity numerical simulation models, represented by computational fluid dynamics (CFD), multiphysics finite element analysis, and high-precision electromagnetic simulation, can accurately characterize complex physical processes and are important tools for engineering design verification and performance evaluation. However, these high-fidelity simulation models are typically built upon complex systems of partial differential equations, involve massive computational scales, and have extremely high requirements for mesh resolution and numerical accuracy. They often need to be run on high-performance computing platforms, and a single simulation can take hours or even days. Therefore, in actual engineering design iterations, directly relying on high-fidelity models for parameter optimization or large-scale design space searches will result in extremely high computational costs and time consumption.

[0003] To reduce computational overhead, low-fidelity models are often introduced in engineering practice, such as simplified physical models, coarse-mesh simulation models, or empirical approximation models. While low-fidelity models have lower computational costs, their prediction accuracy is relatively limited. To balance accuracy and efficiency, multi-fidelity modeling methods have gradually become a research hotspot. Existing multi-fidelity modeling methods typically assume a deterministic mapping relationship between high-fidelity and low-fidelity models, for example, by establishing the connection through linear correlation models or residual correction models. In a typical Co-Kriging framework, the high-fidelity output is represented as the sum of a linear scaling of the low-fidelity output and a residual Gaussian process, achieving high-precision predictions by jointly modeling data of different fidelities. In recent years, some studies have also utilized neural networks to perform nonlinear corrections on low-fidelity prediction results, or employed deep kernel learning methods to combine neural network feature mappings with Gaussian processes, thereby enhancing the model's expressive power.

[0004] While existing methods have improved the prediction accuracy of multi-fidelity modeling to some extent, they still have significant shortcomings in practical complex engineering applications. First, most existing methods are based on building models for a single task, and their training process typically relies on high-fidelity sample data collected for the current task. When engineering design parameters, geometric structures, or operating conditions change, the original model is often difficult to directly transfer to the new task scenario, requiring the collection of new high-fidelity samples and retraining of the model. This "task-isolated" modeling approach leads to repetitive modeling and computation in multi-design scenario or multi-condition optimization problems, making it difficult to meet the practical needs of rapid iterative design.

[0005] Secondly, existing multi-fidelity methods typically assume a fixed functional relationship or a stationary error structure between low-fidelity and high-fidelity models. However, in complex physical systems, error distributions often differ significantly between different design tasks. For example, under different geometries or flow states, the systematic deviations of low-fidelity models may exhibit completely different patterns. Traditional methods struggle to characterize such cross-task-varying error patterns, thus often requiring relearning of error mapping relationships in new task scenarios, making it difficult for models to adapt quickly.

[0006] More importantly, existing methods lack the ability to transfer knowledge across tasks. In real-world engineering environments, there are often multiple similar but not identical simulation tasks, such as aerodynamic analysis of different structural models, performance prediction under different material parameters, or heat conduction simulation under different boundary conditions. These tasks share certain physical laws and error patterns, but existing multi-fidelity modeling methods typically model each task independently, failing to utilize structural information and parameter initialization experience learned from previous tasks. In other words, existing methods belong to a typical single-task learning paradigm and are not designed from the perspective of "how to quickly learn new tasks."

[0007] Therefore, in high-cost simulation scenarios, existing multi-fidelity modeling methods still face the following pressing technical challenges: difficulty in achieving rapid cross-task generalization, inability to effectively characterize complex nonlinear error structures, and high computational costs due to repeated training in multi-task environments. Thus, it is necessary to propose a multi-fidelity modeling method capable of learning a universal representation structure across multiple simulation tasks and rapidly constructing high-precision prediction models in new task scenarios using only a small number of high-fidelity samples, thereby significantly reducing the data dependency and computational burden of high-cost simulations. Summary of the Invention

[0008] The problem this invention aims to solve is to rapidly construct high-precision prediction models across multiple simulation tasks. It proposes a cross-task multi-fidelity probabilistic modeling method that integrates feature mapping and meta-learning.

[0009] To achieve the above objectives, the present invention provides the following technical solution:

[0010] A cross-task multi-fidelity probabilistic modeling method integrating feature mapping and meta-learning includes the following steps:

[0011] S1. For each task, collect low-fidelity sample datasets and high-fidelity sample datasets respectively, perform min-max normalization on all input data, and standardize all output data to construct a multi-task multi-fidelity dataset;

[0012] S2. Establish a deep feature mapping network structure to perform feature mapping on the input data in the multi-task multi-fidelity dataset and generate feature vectors;

[0013] S3. Constructing a multi-fidelity Gaussian process: First, a low-fidelity Gaussian process is constructed in the feature space for the low-fidelity samples after feature mapping. Then, an autoregressive multi-fidelity structure is used to construct a high-fidelity Gaussian process based on the low-fidelity Gaussian process. Finally, a joint Gaussian distribution of low-fidelity and high-fidelity is constructed. All parameters are optimized by maximizing the joint log-marginal likelihood function to complete the construction and training of a multi-fidelity model for a single task.

[0014] S4. Establish a meta-learning cross-task optimization method. Based on the multi-fidelity model of a single task obtained in step S3, realize cross-task optimization between multiple historical simulation tasks and obtain the optimal initialization parameters.

[0015] S5. Using the optimal initialization parameters obtained in step S4, perform adaptive multifidelity modeling on the new high-cost simulation task and output the prediction results of the new high-cost simulation task.

[0016] Furthermore, the specific implementation method of step S1 includes the following steps:

[0017] S1.1. Let the set of historical simulation tasks be... Each task This corresponds to a type of engineering design scenario;

[0018] For the i-th task Low-fidelity sample datasets and high-fidelity sample datasets were collected separately.

[0019] Low-fidelity data sources include one of the following: simplified physical models, coarse-mesh simulations, low-order models, or empirical models; high-fidelity data sources include one of the following: fine-mesh simulations, multiphysics coupled simulations, or high-precision experimental measurements, denoted as:

[0020]

[0021]

[0022] in, This represents the low-fidelity data for the i-th task. This represents the original input vector in the low-fidelity data for the i-th task. The simulation output data in the low-fidelity data of the i-th task. This represents the high-fidelity data for the i-th task. Let represent the original input vector in the high-fidelity data of the i-th task. Let represent the simulation output data in the high-fidelity data of the i-th task, where the number of low-fidelity data is much greater than the number of high-fidelity data. ;

[0023] S1.2. Perform min-max normalization on all original input vectors x to obtain the normalized original input vectors. ;

[0024] S1.3. Standardize all simulation output data y within each task to obtain standardized simulation output data. The expression is:

[0025]

[0026] in, , These are the mean and standard deviation of the high-fidelity output for the corresponding task, respectively.

[0027] Furthermore, the specific implementation method of step S2 includes the following steps:

[0028] S2.1. Construct the mapping function:

[0029] ,

[0030] in, For feature vectors, For the feature space dimension, For mapping functions, For feature mapping parameters, It is the space of real numbers;

[0031] S2.2. The deep feature mapping network structure for determining the mapping function is a multi-layer feedforward fully connected neural network, with the input vector being the normalized original input vector. , Given the dimension of the original input vector, the first layer of the network performs a linear transformation on the input parameters and applies a non-linear activation, expressed as:

[0032]

[0033] in, and These are the first and second training parameters of the first layer of the network. This is the output of the first layer network. For activation functions;

[0034] The second layer performs a nonlinear transformation on the output of the first layer:

[0035]

[0036] in, and These are the first and second training parameters of the second layer network. This is the output of the second layer network;

[0037] The final output layer generates feature vectors:

[0038]

[0039] in, and These are the first and second training parameters of the output layer network.

[0040] Furthermore, the specific implementation method of step S3 includes the following steps:

[0041] S3.1. Construct a low-fidelity Gaussian process in the feature space for the low-fidelity samples after feature mapping, expressed as:

[0042]

[0043] Among them, low-fidelity function It is modeled as a Gaussian process. Represents a Gaussian process. For kernel functions;

[0044] S3.2. An autoregressive multifidelity structure is used to construct a highfidelity Gaussian process based on a lowfidelity Gaussian process. The expression is:

[0045]

[0046] in, For high-fidelity functions, The correlation coefficient is... It is a residual Gaussian process;

[0047] S3.3. Construct a joint Gaussian distribution of low-fidelity and high-fidelity, and optimize all parameters by maximizing the joint log-marginal likelihood function, expressed as:

[0048]

[0049] in, Output data for high-fidelity and low-fidelity simulations. Let covariance matrix be the variance matrix. The amount of training data; to obtain all optimized parameters, including feature mapping parameters. Kernel function hyperparameters, correlation coefficient and noise parameters.

[0050] Furthermore, the specific implementation method of step S4 includes the following steps:

[0051] S4.1. The meta-learning objective is designed as follows:

[0052] ,

[0053] in, This represents a set of parameters shared across tasks; Adapt indicates that in a task... Perform a small number of sample update operations. Represents the task-level loss function; Including feature mapping network parameters Correlation coefficient Kernel function hyperparameters and noise parameters;

[0054] S4.2. The design incorporates tasks during the meta-training phase. The data is divided into support sets and query set The set consists of two parts: a support set used to simulate a small number of high-fidelity samples and a query set used to evaluate the generalization ability after adaptation.

[0055] From the task Randomly selected from high-fidelity samples Using 10 samples as the support set, we obtain:

[0056] ,

[0057] in, For the first The first task Each supporting input feature vector For the first The first task One output is supported. for Any one of them, To support the total number of samples in the set, The values ​​range from 5 to 20, and the remaining high-fidelity samples are used as the query set.

[0058]

[0059] in, This represents the high-fidelity data for the i-th task;

[0060] S4.3. Meta-learning is divided into two parts: inner update and outer update. The inner update is performed for each task. In supporting sets Perform one or more gradient updates:

[0061] ,

[0062] in, For the updated parameter set, Represents the loss function For the set of parameters shared across tasks Find the partial derivative. The inner learning rate, The loss is negative log-marginal likelihood; the inner layer updates in 1 to 5 steps.

[0063] After completing the inner layer update for all tasks, the outer layer update is performed. The outer layer update uses a query set to evaluate the performance of the updated model.

[0064]

[0065] in, The meta-learning rate is used; after multiple rounds of iterative training, the optimal initialization parameters are obtained. .

[0066] Furthermore, the specific implementation method of step S5 includes the following steps:

[0067] S5.1. For newly established tasks First, low-fidelity sample data and high-fidelity sample data are collected. All input data are normalized and output data are standardized.

[0068] S5.2. Apply the optimal initialization parameters obtained during the meta-learning phase. As initial parameters for modeling the new task, high-fidelity samples are first input into the feature mapping network for feature mapping.

[0069] Then, a corresponding multifidelity Gaussian process model is constructed, and the highfidelity negative log-likelihood loss is calculated. ;

[0070] Then execute Gradient update yields hyperparameters of the multifidelity model :

[0071] ;

[0072] Using hyperparameters of multifidelity models We obtain a multi-fidelity prediction model for the new task, for any input to be predicted. The process involves normalization and standardization, feature mapping, high-fidelity prediction, denormalization and standardization, and finally, the prediction output.

[0073] The beneficial effects of this invention are:

[0074] This invention presents a cross-task multi-fidelity probabilistic modeling method integrating feature mapping and meta-learning. Addressing the challenges of scarce high-fidelity samples, repetitive modeling during task switching, and complex relationships between low-fidelity and high-fidelity errors in high-cost simulation scenarios, this method proposes a multi-fidelity probabilistic modeling approach that integrates meta-learning and feature mapping. This method constructs a cross-task shared feature space through deep feature mapping, establishes a multi-fidelity Gaussian process model within this feature space to fuse low-fidelity trends and high-fidelity residuals, and outputs prediction uncertainty. Furthermore, meta-learning is introduced for cross-task optimization to obtain transferable parameter initialization, enabling the model to achieve rapid calibration and high-precision prediction in new tasks with only a small number of high-fidelity samples, thereby significantly reducing the number of high-fidelity simulation calls and the cost of repeated training.

[0075] The present invention describes a cross-task multi-fidelity probabilistic modeling method that integrates feature mapping and meta-learning. The overall approach is "deep feature mapping - multi-fidelity Gaussian process modeling - meta-learning cross-task optimization - rapid adaptation with small samples for new tasks". By learning a transferable feature representation space and initialization parameters on a set of historical simulation tasks, the model can achieve high prediction accuracy with only a small number of high-fidelity samples when facing new simulation tasks, thereby significantly reducing the number of high-fidelity simulation calls and computational costs. Attached Figure Description

[0076] Figure 1 This is a flowchart of a cross-task multi-fidelity probabilistic modeling method that integrates feature mapping and meta-learning, as described in this invention.

[0077] Figure 2 This is a block diagram of a cross-task multi-fidelity probabilistic modeling method that integrates feature mapping and meta-learning, as described in this invention.

[0078] Figure 3 This is a diagram of the deep feature mapping network structure of the present invention;

[0079] Figure 4 This invention provides a meta-learning cross-task training graph;

[0080] Figure 5 This is a comparison chart of the monthly average zonal wind height and low fidelity data for January at 50°N latitude used in this invention. Detailed Implementation

[0081] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and specific embodiments. It should be understood that the specific embodiments described herein are only for explaining the invention and are not intended to limit the invention; that is, the described specific embodiments are merely a part of the embodiments of the invention, and not all of them. The components of the specific embodiments of the invention described and shown in the accompanying drawings can generally be arranged and designed in various different configurations, and the invention may also have other embodiments.

[0082] Therefore, the following detailed description of specific embodiments of the invention provided in the accompanying drawings is not intended to limit the scope of the claimed invention, but merely to illustrate selected specific embodiments of the invention. All other specific embodiments obtained by those skilled in the art based on these specific embodiments without inventive effort are within the scope of protection of this invention.

[0083] To further understand the invention's content, features, and effects, the following specific embodiments are provided, along with accompanying drawings. Figure 1 -Appendix Figure 5 Detailed explanation is as follows:

[0084] Example 1:

[0085] A cross-task multi-fidelity probabilistic modeling method integrating feature mapping and meta-learning includes the following steps:

[0086] S1. For each task, collect low-fidelity sample datasets and high-fidelity sample datasets respectively, perform min-max normalization on all input data, and standardize all output data to construct a multi-task multi-fidelity dataset;

[0087] In engineering practice, a "task" can correspond to a certain type of design object or operating condition, such as different geometric shapes, different combinations of material parameters, different boundary conditions, or different flow states. Let the set of historical simulation tasks be: Each task This corresponds to a specific type of engineering design scenario.

[0088] Furthermore, the specific implementation method of step S1 includes the following steps:

[0089] S1.1. Let the set of historical simulation tasks be... Each task This corresponds to a type of engineering design scenario;

[0090] For the i-th task Low-fidelity sample datasets and high-fidelity sample datasets were collected separately.

[0091] Low-fidelity data sources include one of the following: simplified physical models, coarse-mesh simulations, low-order models, or empirical models; high-fidelity data sources include one of the following: fine-mesh simulations, multiphysics coupled simulations, or high-precision experimental measurements, denoted as:

[0092]

[0093]

[0094] in, This represents the low-fidelity data for the i-th task. This represents the original input vector in the low-fidelity data for the i-th task. The simulation output data in the low-fidelity data of the i-th task. This represents the high-fidelity data for the i-th task. Let represent the original input vector in the high-fidelity data of the i-th task. Let represent the simulation output data in the high-fidelity data of the i-th task, where the number of low-fidelity data is much greater than the number of high-fidelity data. ;

[0095] S1.2. Perform min-max normalization on all original input vectors x to obtain the normalized original input vectors. ;

[0096] Furthermore, to ensure data alignment across different tasks and improve training stability, unified preprocessing is performed on both input and output. Min-max normalization is applied to all input parameters:

[0097] ,

[0098] in, , The minimum and maximum values ​​of each dimension can be taken from the entire set of historical tasks, or the range of engineering parameters can be given in advance.

[0099] The output data is standardized within each task to eliminate training difficulties caused by differences in output dimensions and replication between different tasks.

[0100] S1.3. Standardize all simulation output data y within each task to obtain standardized simulation output data. The expression is:

[0101]

[0102] in, , These are the mean and standard deviation of the high-fidelity output for the corresponding task, respectively.

[0103] Furthermore, when high-fidelity samples for a new task are extremely scarce, low-fidelity outputs can be used to estimate the initial values. , It will then be updated once more high-fidelity samples are collected.

[0104] For multi-output scenarios (such as simultaneously predicting lift, drag, and peak pressure), the following two implementation methods can be adopted: First, construct a separate method for each output dimension and train it independently; second, construct a multi-output Gaussian process (multi-task GP) and perform a weighted summation of each output dimension in the loss function. To facilitate engineering implementation, this invention preferably adopts the method of "splitting multiple outputs into multiple scalar outputs".

[0105] S2. Establish a deep feature mapping network structure to perform feature mapping on the input data in the multi-task multi-fidelity dataset and generate feature vectors;

[0106] Furthermore, the specific implementation method of step S2 includes the following steps:

[0107] S2.1. Construct the mapping function:

[0108] ,

[0109] in, For feature vectors, For the feature space dimension, For mapping functions, For feature mapping parameters, It is the space of real numbers;

[0110] Furthermore, through training This makes the inputs corresponding to similar outputs closer in the feature space; the structures shared between different tasks have the same representation mode; and the feature space is more suitable for modeling with stationary kernel functions. Finally, this invention decomposes the "complex physical nonlinear modeling problem" into two sub-problems: (1) learning the nonlinear transformation from input to structural features by a neural network; and (2) modeling statistical relationships and uncertainties in the feature space by a Gaussian process.

[0111] When determining the feature mapping network structure, this invention follows the following design principles:

[0112] First, the network should have sufficient expressive power to characterize complex nonlinear relationships, but it should not be too deep to avoid overfitting and unstable training.

[0113] Second, the network structure should be fully shared across different tasks to achieve unified representation across tasks;

[0114] Third, the dimension of the output feature should be significantly lower than or controlled within a reasonable range than the dimension of the original parameter space, in order to reduce the computational complexity of the subsequent Gaussian process.

[0115] S2.2. The deep feature mapping network structure for determining the mapping function is a multi-layer feedforward fully connected neural network, with the input vector being the normalized original input vector. , Given the dimension of the original input vector, the first layer of the network performs a linear transformation on the input vector and applies a non-linear activation, expressed as:

[0116]

[0117] in, and These are the first and second training parameters of the first layer of the network. This is the output of the first layer network. For activation functions;

[0118] The second layer performs a nonlinear transformation on the output of the first layer:

[0119]

[0120] in, and These are the first and second training parameters of the second layer network. This is the output of the second layer network;

[0121] The final output layer generates feature vectors:

[0122]

[0123] in, and Here are the first and second training parameters of the output layer network. The output layer does not use an activation function to maintain the continuity and differentiability of the feature space, allowing the Gaussian process kernel function to have favorable properties in this space.

[0124] Furthermore, feature dimensions The choice of [aspect name] directly affects the modeling complexity, feature representation ability, and cross-task sharing effect of Gaussian processes. If [aspect name] is chosen... If it is too small, important physical information may be lost. If the kernel size is too large, the size of the Gaussian process kernel matrix increases, leading to computational complexity. Therefore, this invention suggests: if , A value of 20-50 is acceptable; if , A value of 50-100 is acceptable; if You can choose 100~200; The final value is determined by considering both validation error and computational cost. Network parameters are initialized using the He initialization method to accommodate the characteristics of the ReLU activation function. The Adam optimizer is used during training, and its adaptive learning rate mechanism helps ensure stable training under different task data distributions.

[0125] In the overall framework of this embodiment, the feature mapping network not only undertakes the task of expressing nonlinearity, but also bears the core function of "cross-task structure sharing". During the meta-learning phase, all historical tasks share the same feature mapping parameters. When each task performs local updates on a small number of high-fidelity samples, it is essentially fine-tuning the local structure within the same feature space. Therefore, the goal of meta-learning is not to optimize feature mappings for a particular task, but rather to learn a "feature space with good initialization performance for all tasks." This design allows the feature mapping network to gradually develop a "structural prior," enabling rapid adaptation based on the existing structure with only a small number of samples when a new task arrives.

[0126] because After undergoing nonlinear transformation, the complex nonlinear relationships in the original space are transformed into relationships closer to stationary in the feature space, thus enabling subsequent use of the RBF kernel or... Kernel modeling can be effective.

[0127] In high-cost simulation modeling problems, there are often highly complex nonlinear relationships between design parameters and high-fidelity simulation outputs. This is particularly true in scenarios such as aerospace aerodynamic analysis, multiphysics coupled simulation, or material property prediction, where strong coupling effects may exist between input parameters, while the output response may exhibit significant nonstationary characteristics. If a multifidelity Gaussian process model is directly constructed in the original parameter space, the kernel function must simultaneously handle both "nonlinear expression" and "uncertainty modeling," leading to an overly complex kernel function structure, an increased number of hyperparameters, and difficulty in sharing the kernel across different tasks.

[0128] Furthermore, although different simulation tasks differ in specific geometric structures or operating parameters, their underlying physical laws usually share certain commonalities. For example, in aerodynamic analysis tasks of different aircraft models, the basic trends in the changes between lift and angle of attack have consistent physical mechanisms; under different combinations of material parameters, the heat conduction process still follows the same physical equations. Existing multi-fidelity methods, because they directly model in the original parameter space, find it difficult to abstract this cross-task shared "structural feature representation," resulting in each task needing to independently learn the mapping relationship between input and output, thus failing to achieve knowledge transfer. Therefore, this invention introduces, before multi-fidelity modeling, a method such as... Figure 2The deep feature mapping network shown maps the original design parameter space to a more structured feature space through the nonlinear expressive power of neural networks, making complex nonlinear relationships smoother in the feature space, thereby reducing the difficulty of subsequent Gaussian process modeling; it shares the same feature mapping structure on multiple historical tasks, enabling data from different tasks to be represented in the same feature space, thus building a cross-task shared representation foundation; and it allows the feature mapping parameters to be optimized to an "easy and fast-adapting" initial state under the meta-learning framework, so that when a new task appears, only a small number of high-fidelity samples are needed to complete effective calibration.

[0129] The original design parameter vector is known to be In traditional methods, the Gaussian process kernel function is usually applied directly to... ,Right now However, in complex nonlinear problems, two parameter points that are close in Euclidean distance may have significantly different high-fidelity outputs; conversely, two points that are far apart in parameter space may have similar output structures. This illustrates that distance metrics in the original space are not necessarily suitable for Gaussian process modeling.

[0130] S3. Constructing a multi-fidelity Gaussian process: First, a low-fidelity Gaussian process is constructed in the feature space for the low-fidelity samples after feature mapping. Then, an autoregressive multi-fidelity structure is used to construct a high-fidelity Gaussian process based on the low-fidelity Gaussian process. Finally, a joint Gaussian distribution of low-fidelity and high-fidelity is constructed. All parameters are optimized by maximizing the joint log-marginal likelihood function to complete the construction and training of a multi-fidelity model for a single task.

[0131] After deep feature mapping, the original design parameters are mapped to a unified feature space shared across multiple historical tasks, which can, to some extent, eliminate the nonlinear complexity of the original input and output spaces. However, feature mapping alone cannot solve the core problems in high-cost simulation scenarios: there is a systematic bias between low-fidelity and high-fidelity outputs; the number of high-fidelity samples is extremely small, making it difficult to build a stable prediction model independently; and the prediction uncertainty needs to be quantified in engineering. Therefore, this invention further constructs a multi-fidelity Gaussian process in the feature space to achieve information fusion between low-fidelity and high-fidelity outputs and uncertainty estimation.

[0132] In existing multi-fidelity methods, a common approach is to directly fit the low-fidelity to high-fidelity mapping using neural networks. However, these methods are typically deterministic models and cannot provide prediction confidence intervals, making them difficult to guide subsequent high-fidelity sampling in high-cost simulation scenarios. A Gaussian Process (GP) is a probability-based nonparametric regression model whose core idea is to treat the function itself as a stochastic process and model the joint distribution of function values. Unlike neural networks, GP not only provides the predicted mean but also outputs the predicted variance, thereby quantifying uncertainty. This characteristic is significant in high-cost simulations, as it can identify uncertain regions in the model, guide active sampling, and avoid blindly increasing the number of high-fidelity simulations. Therefore, this embodiment uses GP as the core of multi-fidelity modeling.

[0133] Furthermore, the specific implementation method of step S3 includes the following steps:

[0134] S3.1. Construct a low-fidelity Gaussian process in the feature space for the low-fidelity samples after feature mapping, expressed as:

[0135]

[0136] Among them, low-fidelity function It is modeled as a Gaussian process. Represents a Gaussian process. The kernel function; that is, the function values ​​on any finite set of sample points follow a multivariate Gaussian distribution. The kernel matrix elements are defined as follows: ;

[0137] Furthermore, suppose that after feature mapping, the low-fidelity sample is represented as... ;

[0138] Furthermore, in the feature space, since the neural network has already compressed the complex nonlinear relationships into a smoother structure, there is no need to use an overly complex kernel function. This embodiment uses the RBF kernel (squared exponential kernel):

[0139] ,

[0140] in, For signal variance, This is the length scale parameter. When the feature dimension is high, the ARD (Archived Distance) approach can be used, setting an independent length scale for each dimension.

[0141] ,

[0142] This enables the automatic identification of important feature dimensions, improves cross-task stability, and reduces the impact of invalid features.

[0143] S3.2. An autoregressive multifidelity structure is used to construct a highfidelity Gaussian process based on a lowfidelity Gaussian process. The expression is:

[0144]

[0145] in, For high-fidelity functions, The correlation coefficient is... It is a residual Gaussian process; defined as In this structure, low-fidelity learning is responsible for learning the overall trend, while the residual process learns systematic errors through... This helps to characterize the correlation between high and low fidelity data under different tasks, avoiding the problem of insufficient samples caused by directly modeling on high-fidelity data.

[0146] Furthermore, in engineering practice, low-fidelity models can typically reflect the overall trend of a system, but they exhibit systematic biases. Therefore, high-fidelity output can usually be represented as "low-fidelity prediction + error correction." Based on this physical intuition, an autoregressive multifidelity structure is adopted.

[0147] S3.3. Construct a joint Gaussian distribution of low-fidelity and high-fidelity, and optimize all parameters by maximizing the joint log-marginal likelihood function, expressed as:

[0148]

[0149] in, Output data for high-fidelity and low-fidelity simulations. Let covariance matrix be the variance matrix. The amount of training data; to obtain all optimized parameters, including feature mapping parameters. Kernel function hyperparameters, correlation coefficient and noise parameters.

[0150] Furthermore, the low-fidelity and high-fidelity observations constitute a joint Gaussian distribution. Let... The joint distribution is

[0151] ,

[0152] in:

[0153] ,

[0154] in, and For noise variance, Let the identity matrix be denoted by 'identity matrix'. This is achieved by maximizing the joint log-marginal likelihood function:

[0155] ,

[0156] It can simultaneously optimize feature mapping parameters Kernel function hyperparameters, correlation coefficient Including noise parameters. All parameters are updated jointly via backpropagation and gradient descent.

[0157] When making predictions, for new feature points Define test points The covariance vector with respect to the training observations is:

[0158] ,

[0159] in, Indicates will With low-fidelity training input set The vector obtained by performing a kernel function. and Similarly, the high-fidelity prediction distribution satisfies:

[0160] ,

[0161] in, To predict the mean, For predicting variance:

[0162] ,

[0163] In addition, high-fidelity output including observation noise The prediction variance is:

[0164] .

[0165] Prediction variance reflects model uncertainty and can be used to judge prediction reliability, select the next high-fidelity sampling point, and avoid overfitting risk. This is also an important technical advantage of this invention over pure neural network residual modeling methods.

[0166] The training complexity of standard GP is , The number represents the observed sample size. When the sample size is large, this invention can use induced point sparse GP to construct a multi-fidelity model. This part achieves trend and residual separation modeling by constructing a multi-fidelity GP in a shared feature space, enabling joint learning of low and high fidelity models and outputting prediction uncertainty. This part can be seamlessly integrated with the meta-learning mechanism.

[0167] S4. Establish a meta-learning cross-task optimization method. Based on the multi-fidelity model of a single task obtained in step S3, realize cross-task optimization between multiple historical simulation tasks and obtain the optimal initialization parameters.

[0168] While the model can achieve fusion prediction of low-fidelity and high-fidelity data within a single task, a key problem remains: when the engineering design scenario changes, the original model still needs to be retrained or the kernel function parameters re-estimated, and cannot be directly transferred to the new task. This "task-isolated modeling approach" is the fundamental reason why existing multi-fidelity methods are inefficient in high-cost simulation scenarios.

[0169] In real-world engineering environments, multiple simulation tasks often share certain physical laws. For example, although the specific parameters of aerodynamic analyses of different geometries differ, the flow control equations are consistent; and although the boundary conditions vary under different material combinations, the basic heat transfer mechanism remains the same. Therefore, the feature representations and error structures learned from multiple historical tasks necessarily contain transferable structural information.

[0170] Based on the above understanding, this invention introduces a meta-learning mechanism to perform cross-task optimization across multiple historical simulation tasks, enabling the model parameters to learn an initial state that is "easy to adapt to quickly" at the overall level. Its core idea is not to optimize the prediction error for a specific task, but rather to optimize the initialization of a parameter so that when facing a new task, only a small number of high-fidelity samples and a small number of gradient updates are needed to obtain good prediction performance.

[0171] Furthermore, the specific implementation method of step S4 includes the following steps:

[0172] S4.1. The meta-learning objective is designed as follows:

[0173] ,

[0174] in, This represents a set of parameters shared across tasks; Adapt indicates that in a task... Perform a small number of sample update operations. Represents the task-level loss function; Including feature mapping network parameters Correlation coefficient Kernel function hyperparameters and noise parameters;

[0175] S4.2. The design incorporates tasks during the meta-training phase. The data is divided into support sets and query set The set consists of two parts: a support set used to simulate a small number of high-fidelity samples and a query set used to evaluate the generalization ability after adaptation.

[0176] From the task Randomly selected from high-fidelity samples Using 10 samples as the support set, we obtain:

[0177] ,

[0178] in, For the first The first task Each supporting input feature vector For the first The first task One output is supported. for Any one of them, To support the total number of samples in the set, The values ​​range from 5 to 20, and the remaining high-fidelity samples are used as the query set.

[0179]

[0180] in, This represents the high-fidelity data for the i-th task;

[0181] S4.3. Meta-learning is divided into two parts: inner update and outer update. The inner update is performed for each task. In supporting sets Perform one or more gradient updates:

[0182] ,

[0183] in, For the updated parameter set, Represents the loss function For the set of parameters shared across tasks Find the partial derivative. The inner learning rate, The loss is negative log-marginal likelihood; the inner layer updates in 1 to 5 steps.

[0184] After completing the inner layer update for all tasks, the outer layer update is performed. The outer layer update uses a query set to evaluate the performance of the updated model.

[0185]

[0186] in, The meta-learning rate is used; after multiple rounds of iterative training, the optimal initialization parameters are obtained. .

[0187] Furthermore, in this embodiment, the feature mapping network parameters Belongs to the meta-learning optimization object, multiple truth relation numbers Meta-learning can also be incorporated, and hyperparameters such as kernel function length and scaling can be optimized separately within the task or incorporated into the meta-layer. Therefore, the entire system forms a three-layer structure: feature space learning, multi-fidelity statistical modeling, and cross-task parameter initialization optimization. This layered design ensures that the feature space has a cross-task shared structure, and multi-fidelity error relationships can be quickly calibrated, so that new tasks do not need to be trained from scratch.

[0188] S5. Using the optimal initialization parameters obtained in step S4, perform adaptive multifidelity modeling on the new high-cost simulation task and output the prediction results of the new high-cost simulation task.

[0189] After completing cross-task meta-learning training and obtaining the optimal initialization parameters Then, rapid predictive modeling can be performed for new high-cost simulation tasks. Unlike traditional multi-fidelity methods that require retraining the complete model on new tasks, this embodiment transforms the new task modeling process into a small-scale rapid calibration problem by utilizing learned cross-task initialization parameters. That is, a high-precision predictive model is built under the condition of collecting only a very small number of high-fidelity samples, and the number of high-fidelity simulation calls is significantly reduced.

[0190] Furthermore, the specific implementation method of step S5 includes the following steps:

[0191] S5.1. For newly established tasks First, low-fidelity sample data and high-fidelity sample data are collected. All input data are normalized and output data are standardized.

[0192] S5.2. Apply the optimal initialization parameters obtained during the meta-learning phase. As initial parameters for modeling the new task, high-fidelity samples are first input into the feature mapping network for feature mapping.

[0193] Then, a corresponding multifidelity Gaussian process model is constructed, and the highfidelity negative log-likelihood loss is calculated. ;

[0194] Then execute Gradient update yields hyperparameters of the multifidelity model :

[0195] ;

[0196] Using hyperparameters of multifidelity models Construct a multi-fidelity prediction model for a new task, for any input to be predicted. The process involves normalization and standardization, feature mapping, high-fidelity prediction, denormalization and standardization, and finally, the prediction output.

[0197] Furthermore, the predicted mean and predicted variance are calculated using the aforementioned formulas. Since the Gaussian process outputs the predicted variance, uncertainty information can be used to optimize the high-fidelity sampling strategy.

[0198] Compared with traditional methods, the computation process in the new task stage has undergone a fundamental change. This invention uses cross-task initialization parameters and updates them with a small number of gradients. It requires only 5 to 20 high-fidelity samples, reducing the prediction time by 60% to 80%, and achieving a significant reduction in the number of high-fidelity simulations and rapid design iteration.

[0199] The following is an application example of this embodiment:

[0200] To verify the effectiveness of the proposed meta-learning-based multi-fidelity modeling method in predicting complex physical fields, global mid-to-upper atmospheric equilibrium wind data was selected as the verification object. This data, obtained from atmospheric temperature and pressure information acquired by satellite remote sensing, is calculated using atmospheric dynamic equilibrium relationships and is used to describe the monthly mean zonal wind distribution in the global mid-to-upper regions. The data covers an altitude range of approximately 18 km to 100 km and a latitude range of 50°S to 50°N, and includes multi-year continuous observation records. Since the observation data from different years differ in statistical stability and interannual perturbations, datasets with different fidelities can be constructed using multi-year average results and single-year observation results, thus forming a multi-fidelity modeling verification scenario to evaluate the performance of the proposed method in complex physical field prediction tasks.

[0201] The monthly average zonal wind data within the range of 18km–100km altitude and 50°S–50°N latitude from January 2002 to January 2019 were selected as the research object. To construct a multi-fidelity structure: (1) the data of the same latitude from January 2002 to January 2019 were averaged as high-fidelity data, which has low statistical variance and high confidence; (2) the corresponding latitude data in January 2019 were used as low-fidelity data, which is greatly affected by interannual disturbances and has low statistical confidence; (3) the tasks in different latitude zones share similar large-scale dynamic structures, but the specific wind speed amplitude and height distribution are different, which constitutes a typical cross-task learning scenario.

[0202] Input variables are defined as The output variable is ,in Geometric height Indicates latitude, This represents latitudinal wind speed (east-west direction). All data has been normalized.

[0203] Three types of models are constructed here for comparison: (1) the traditional single-fidelity model M1, which uses only high-fidelity data to train a Gaussian process model for prediction; (2) the classic Co-kriging multifidelity model M2, which uses a linearly correlated multifidelity Gaussian process model for data fusion; and (3) the method of this invention M. our A deep feature mapping network is constructed to establish a multi-fidelity co-kriging model in the feature space. Initial parameters are optimized through meta-learning, enabling the model to converge quickly in new-dimensional tasks with only a small number of high-fidelity samples. The final output is the predicted mean and predicted variance. To quantify model performance, negative log-likelihood (NLL) is used as an evaluation metric; the smaller the value, the more accurately the model describes the probability of the real physical field.

[0204] Taking the monthly mean zonal wind at 50° North latitude as an example, such as Figure 4 The diagram shows a comparison between high-fidelity and low-fidelity data. With a latitude resolution of 2.5°, and including endpoints, the 50°S–50°N latitude range is divided into 41 latitude zones. Here, 45°N, 47.5°N, and 50°N are used as test tasks, and the remaining 38 latitude zones are used as training tasks. In the 18km–100km altitude range, 83 altitudes were sampled from the original data. No interpolation was performed. For each task, 63 altitudes were selected at equal intervals on the low-fidelity object as the support dataset, and the remaining 20 as the query dataset. On the high-fidelity object, 10 altitudes were selected as the support set, and 5 altitudes as the query set. NLL was finally calculated only on the high-fidelity model. The outer loop iteration count was set to 500, with 16 tasks randomly selected in each iteration. The inner loop iteration count was set to 10. The inner loop learning rate was 0.01, and the outer loop learning rate was 0.001. The second derivative of the outer loop optimization was ignored.

[0205] When the initialization parameters of model M1 and model M2 are randomly generated, model M our When using the initialization parameters obtained from meta-learning, as shown in Tables 1-3, the initial NLL and the NLL after 10 training iterations for different models are compared on three test tasks.

[0206] Table 1. NLL Comparison on the 45°N Task

[0207]

[0208] Table 2. NLL Comparison on the 47.5°N Mission

[0209]

[0210] Table 3. NLL Comparison on the 50°N Task

[0211]

[0212] The experimental results above show that the nonlinear convergence (NLL) of the method in this embodiment is significantly lower than that of the traditional single-task single-fidelity model, and it also has significant advantages over the single-task multi-fidelity model. It can be seen that during cross-task migration, the NLL convergence speed of the method in this invention on the new latitude task is significantly faster. Therefore, the method of this invention can effectively capture the nonlinear relationship between altitude and zonal wind and achieve knowledge sharing across latitude tasks.

[0213] It should be noted that relational terms such as "first" and "second" are used merely to distinguish one entity or operation from another, and do not necessarily require or imply any such actual relationship or order between these entities or operations. Furthermore, the terms "comprising," "including," or any other variations thereof are intended to cover non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such a process, method, article, or apparatus. Without further limitations, an element defined by the phrase "comprising one..." does not exclude the presence of other identical elements in the process, method, article, or apparatus that includes said element.

[0214] Although this application has been described above with reference to specific embodiments, various modifications can be made and components can be replaced with equivalents without departing from the scope of this application. In particular, as long as there is no structural conflict, the features in the specific embodiments disclosed in this application can be combined with each other in any way. The lack of an exhaustive description of these combinations in this specification is merely for the sake of brevity and resource conservation. Therefore, this application is not limited to the specific embodiments disclosed herein, but includes all technical solutions falling within the scope of the claims.

Claims

1. A cross-task multi-fidelity probabilistic modeling method integrating feature mapping and meta-learning, characterized in that, Includes the following steps: S1. For each task, collect low-fidelity sample datasets and high-fidelity sample datasets respectively, perform min-max normalization on all input data, and standardize all output data to construct a multi-task multi-fidelity dataset; S2. Establish a deep feature mapping network structure to perform feature mapping on the input data in the multi-task multi-fidelity dataset and generate feature vectors; S3. Constructing a multi-fidelity Gaussian process: First, a low-fidelity Gaussian process is constructed in the feature space for the low-fidelity samples after feature mapping. Then, an autoregressive multi-fidelity structure is used to construct a high-fidelity Gaussian process based on the low-fidelity Gaussian process. Finally, a joint Gaussian distribution of low-fidelity and high-fidelity is constructed. All parameters are optimized by maximizing the joint log-marginal likelihood function to complete the construction and training of a multi-fidelity model for a single task. S4. Establish a meta-learning cross-task optimization method. Based on the multi-fidelity model of a single task obtained in step S3, realize cross-task optimization between multiple historical simulation tasks and obtain the optimal initialization parameters. S5. Using the optimal initialization parameters obtained in step S4, perform adaptive multifidelity modeling on the new high-cost simulation task and output the prediction results of the new high-cost simulation task.

2. The method for cross-task multi-fidelity probabilistic modeling that integrates feature mapping and meta-learning according to claim 1, characterized in that, The specific implementation method of step S1 includes the following steps: S1.

1. Let the set of historical simulation tasks be... Each task This corresponds to a type of engineering design scenario; For the i-th task Low-fidelity sample datasets and high-fidelity sample datasets were collected separately. Low-fidelity data sources include one of the following: simplified physical models, coarse-mesh simulations, low-order models, or empirical models; high-fidelity data sources include one of the following: fine-mesh simulations, multiphysics coupled simulations, or high-precision experimental measurements, denoted as: ； ； in, This represents the low-fidelity data for the i-th task. This represents the original input vector in the low-fidelity data for the i-th task. The simulation output data in the low-fidelity data of the i-th task. This represents the high-fidelity data for the i-th task. Let represent the original input vector in the high-fidelity data of the i-th task. Let represent the simulation output data in the high-fidelity data of the i-th task, where the number of low-fidelity data is much greater than the number of high-fidelity data. ; S1.

2. Perform min-max normalization on all original input vectors x to obtain the normalized original input vectors. ; S1.

3. Standardize all simulation output data y within each task to obtain standardized simulation output data. The expression is: ； in, , These are the mean and standard deviation of the high-fidelity output for the corresponding task, respectively.

3. The method for cross-task multi-fidelity probabilistic modeling that integrates feature mapping and meta-learning according to claim 2, characterized in that, The specific implementation method of step S2 includes the following steps: S2.

1. Construct the mapping function: ， in, For feature vectors, For the feature space dimension, For mapping functions, For feature mapping parameters, It is the space of real numbers; S2.

2. The deep feature mapping network structure for determining the mapping function is a multi-layer feedforward fully connected neural network, with the input vector being the normalized original input vector. , Given the dimension of the original input vector, the first layer of the network performs a linear transformation on the input parameters and applies a non-linear activation, expressed as: ； in, and These are the first and second training parameters of the first layer of the network. This is the output of the first layer network. For activation functions; The second layer performs a nonlinear transformation on the output of the first layer: ； in, and These are the first and second training parameters of the second layer network. This is the output of the second layer network; The final output layer generates feature vectors: ； in, and These are the first and second training parameters of the output layer network.

4. The method for cross-task multi-fidelity probabilistic modeling that integrates feature mapping and meta-learning according to claim 3, characterized in that, The specific implementation method of step S3 includes the following steps: S3.

1. Construct a low-fidelity Gaussian process in the feature space for the low-fidelity samples after feature mapping, expressed as: ； Among them, low-fidelity function It is modeled as a Gaussian process. Represents a Gaussian process. For kernel functions; S3.

2. An autoregressive multifidelity structure is used to construct a highfidelity Gaussian process based on a lowfidelity Gaussian process. The expression is: ； in, For high-fidelity functions, The correlation coefficient, It is a residual Gaussian process; S3.

3. Construct a joint Gaussian distribution of low-fidelity and high-fidelity, and optimize all parameters by maximizing the joint log-marginal likelihood function, expressed as: ； in, Output data for high-fidelity and low-fidelity simulations. Let covariance matrix be the variance matrix. The amount of training data; to obtain all optimized parameters, including feature mapping parameters. Kernel function hyperparameters, correlation coefficient and noise parameters.

5. The method for cross-task multi-fidelity probabilistic modeling that integrates feature mapping and meta-learning according to claim 4, characterized in that, The specific implementation method of step S4 includes the following steps: S4.

1. The meta-learning objective is designed as follows: ， in, This represents a set of parameters shared across tasks; Adapt indicates that in a task... Perform a small number of sample update operations. Represents the task-level loss function; Including feature mapping network parameters Correlation coefficient Kernel function hyperparameters and noise parameters; S4.

2. The design incorporates tasks during the meta-training phase. The data is divided into support sets and query set The set consists of two parts: a support set used to simulate a small number of high-fidelity samples and a query set used to evaluate the generalization ability after adaptation. From the task Randomly selected from high-fidelity samples Using 10 samples as the support set, we obtain: ， in, For the first The first task Each supporting input feature vector For the first The first task One output is supported. for Any one of them, To support the total number of samples in the set, The values ​​range from 5 to 20, and the remaining high-fidelity samples are used as the query set. ； in, This represents the high-fidelity data for the i-th task; S4.

3. Meta-learning is divided into two parts: inner update and outer update. The inner update is performed for each task. In supporting sets Perform one or more gradient updates: ， in, For the updated parameter set, Represents the loss function For the set of parameters shared across tasks Find the partial derivative. The inner learning rate, The loss is negative log-marginal likelihood; the inner layer updates in 1 to 5 steps. After completing the inner layer update for all tasks, the outer layer update is performed. The outer layer update uses a query set to evaluate the performance of the updated model. ； in, The meta-learning rate is used; after multiple rounds of iterative training, the optimal initialization parameters are obtained. .

6. The method for cross-task multi-fidelity probabilistic modeling integrating feature mapping and meta-learning as described in claim 5, characterized in that, The specific implementation method of step S5 includes the following steps: S5.

1. For newly established tasks First, low-fidelity sample data and high-fidelity sample data are collected. All input data are normalized and output data are standardized. S5.

2. Apply the optimal initialization parameters obtained during the meta-learning phase. As initial parameters for modeling the new task, high-fidelity samples are first input into the feature mapping network for feature mapping. Then, a corresponding multifidelity Gaussian process model is constructed, and the highfidelity negative log-likelihood loss is calculated. ; Then execute Gradient update yields hyperparameters of the multifidelity model : ； Using hyperparameters of multifidelity models We obtain a multi-fidelity prediction model for the new task, for any input to be predicted. The process involves normalization and standardization, feature mapping, high-fidelity prediction, denormalization and standardization, and finally, the prediction output.

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