A model design method for assisting optimization design of automobile intake and exhaust pipes

Abstract

The application discloses a model design method for assisting in the optimization design of an automobile intake and exhaust pipe, and comprises the following steps: characteristic variables of a design agent model are designed; a data set for training the agent model is constructed; a training set is analyzed, and main intermediate characteristics are selected; PCA dimension reduction algorithm is used to reduce the dimension of the main intermediate characteristics; a mapping relationship between geometric characteristics and m-dimensional intermediate characteristics after dimension reduction is established; and a multistage agent model for the optimization design of the automobile intake and exhaust pipe is established. The intermediate characteristics are introduced into the agent model for assisting in the optimization design of the automobile intake and exhaust pipe, the richness of characteristics is increased, the precision of the agent model is effectively improved, and the optimization design of the intake and exhaust pipe can be effectively assisted, so that the production efficiency of automobile manufacturers is improved, and the production cost is reduced.

Description

Technical Field

[0001] This invention relates to the field of automotive exhaust pipes, and in particular to a model design method for assisting in the optimization design of automotive intake and exhaust pipes. Background Technology

[0002] The structural parameters of the intake and exhaust pipes, intake and exhaust passages, and intake and exhaust valves, along with their matching with the engine, affect the formation of the air-fuel mixture and the scavenging process. Improving the engine's scavenging and mixture formation processes plays a crucial role in enhancing engine power and fuel economy. A well-designed engine intake system can effectively reduce intake resistance and uneven intake between cylinders, and fully utilize the inertia and pressure fluctuation effects of the gas. A well-designed exhaust system can reduce exhaust resistance, minimize exhaust energy loss, and improve exhaust gas energy utilization. Therefore, the intake and exhaust systems are crucial components of an engine, and the optimized design of the intake and exhaust pipes has significant practical implications for energy conservation and emission reduction.

[0003] With the rapid development of computer technology, finite element analysis (FEA) and computational fluid dynamics (CFD) have become a solid foundation for the widespread application of computer-aided engineering (CAE) and computer-aided design (CAD) in various engineering fields. Using simulation methods based on numerical simulation for engine design allows for various simulation experiments to be conducted within the created numerical model of a gasoline engine. This enables the understanding and analysis of the impact of key parameters and control coefficients of various components on engine performance, providing guidance for the design and optimization of the engine's intake and exhaust systems.

[0004] However, traditional numerical simulation methods based on solving partial differential equations to predict specific design schemes can take several hours or even days, which hinders the application of optimization methods based on numerous calculations. In contrast, surrogate models based on simulation results can respond quickly to input, providing predictions in just a few seconds. This places higher demands on the accuracy of surrogate models. Summary of the Invention

[0005] The purpose of this section is to outline some aspects of embodiments of the present invention and to briefly describe some preferred embodiments. Simplifications or omissions may be made in this section, as well as in the abstract and title of this application, to avoid obscuring the purpose of these documents; however, such simplifications or omissions should not be construed as limiting the scope of the invention.

[0006] In view of the problems existing in the prior art, the present invention is proposed.

[0007] Therefore, the technical problem to be solved by this invention is to better assist in the optimized design of automobile intake and exhaust pipes and improve the accuracy of the surrogate model.

[0008] To solve the above-mentioned technical problems, the present invention provides the following technical solution: a model design method for assisting in the optimization design of automobile intake and exhaust pipes, characterized in that: Design the characteristic variables of the agent model; Construct a dataset for training the agent model; Analyze the training set and select the main mediation features; The main mediation features were reduced in dimensionality using the PCA dimensionality reduction algorithm. Establish a mapping relationship between geometric features and the reduced m-dimensional intermediate features; Establish a multi-level proxy model for the optimized design of automotive intake and exhaust pipes.

[0009] As a preferred embodiment of the model design method for assisting in the optimization design of automotive intake and exhaust pipes according to the present invention, the feature variables include geometric parameters (GP) and mediation parameters (MP) of the spatial configuration. Based on the spatial configuration of the automotive intake and exhaust pipes, geometric feature parameters characterizing the spatial configuration are extracted. ,in Represent the number of geometric features; and extract the intermediate features of the spatial configuration of the car's intake and exhaust pipes. ,in The number of mediating features is represented by the geometric features and the mediating features, which together constitute the feature variables of the surrogate model.

[0010] As a preferred embodiment of the model design method for assisting in the optimization design of automotive intake and exhaust pipes according to the present invention, the intermediate feature is a flow field feature characterizing the gas flow state in the pipe, and its calculation method is: the centerline of the intake and exhaust pipes... Divide the space into equal parts, and calculate the direction vector of the tangent at each division point as the flow field characteristic of the spatial configuration. .

[0011] As a preferred embodiment of the model design method for assisting in the optimization design of automotive intake and exhaust pipes as described in this invention, a training dataset Train is constructed by collecting paired data of geometric feature variables (GP) and optimization design objective T for different intake and exhaust pipe design schemes using simulation tools.

[0012] As a preferred embodiment of the model design method for assisting in the optimization design of automotive intake and exhaust pipes as described in this invention, the method utilizes a tree-based feature importance analysis method to calculate the importance score of each mediating feature for the target variable, thereby obtaining... ; Select the highest rated Dimensional Mediator Features .

[0013] As a preferred embodiment of the model design method for assisting in the optimization design of automotive intake and exhaust pipes as described in this invention, the general method for calculating the feature importance based on a tree model is as follows: the global importance of a feature is measured by the average importance of the feature in a single tree.

[0014]

[0015] in, Indicates the number of tree models. Representation of features In a single tree The importance of features on, The number of leaf nodes in the tree. That is, the number of non-leaf nodes in the tree. Is and node Related features It is a node The reduction in squared loss after splitting.

[0016] As a preferred embodiment of the model design method for assisting in the optimization design of automotive intake and exhaust pipes as described in this invention, the main intermediate features are reduced using the PCA (Principal Component Analysis) dimensionality reduction algorithm. Dimensionality reduction is performed to obtain the reduced intermediate features. ,in This represents the dimension of the intermediate features after dimensionality reduction.

[0017] As a preferred embodiment of the model design method for assisting in the optimization design of automotive intake and exhaust pipes as described in this invention, the calculation process of the PCA dimensionality reduction algorithm is as follows: First, center all samples:

[0018]

[0019] in Indicates the first Values ​​of flow field characteristic variables of the sample Then calculate the covariance matrix of the sample. ; For covariance matrix Perform eigenvalue decomposition:

[0020] in It is a matrix composed of the eigenvectors of the covariance matrix. It is a diagonal matrix, and the elements on the diagonal are the eigenvalues ​​of the covariance matrix. ; Take the largest Eigenvectors corresponding to each eigenvalue .

[0021] As a preferred embodiment of the model design method for assisting in the optimization design of automotive intake and exhaust pipes as described in this invention, wherein: the geometric features of the automotive intake and exhaust pipes are used... As the independent variable, the mediating features after dimensionality reduction are used. For the dependent variable, establish a data model between the independent and dependent variables:

[0022] in, For the first A reduced-dimensional mediation feature express and The regression model between them.

[0023] As a preferred embodiment of the model design method for assisting in the optimization design of automotive intake and exhaust pipes according to the present invention, a secondary proxy model for the optimization design of automotive intake and exhaust pipes is established using a data modeling method. Specifically, a data model is established with the optimization design objective T as the dependent variable.

[0024] in, For target variable The regression model between the aforementioned feature variables, For the mth , .

[0025] The beneficial effects of this invention are as follows: By introducing intermediate features into the surrogate model for optimizing the design of automotive intake and exhaust pipes, the feature richness is increased, and the accuracy of the surrogate model is effectively improved. This can effectively assist in the optimization design of intake and exhaust pipes, helping automobile manufacturers improve production efficiency and reduce production costs. Attached Figure Description

[0026] To more clearly illustrate the technical solutions of the embodiments of the present invention, the drawings used in the description of the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort. Wherein: Figure 1 This is a flowchart of the method for designing a multi-level agent model provided in an embodiment of the present invention.

[0027] Figure 2 This is a schematic diagram of the spatial configuration of the vehicle's intake and exhaust pipes in this embodiment.

[0028] Figure 3 This is a schematic diagram of the curvature characteristics of the first bend in the pipeline.

[0029] Figure 4 This is a schematic diagram of the curvature characteristics of the pipeline after the first bend.

[0030] Figure 5 This is a schematic diagram showing the width characteristics of the pipeline after the first bend.

[0031] Figure 6 This is a schematic diagram of the curvature characteristics of the second bend in the pipeline.

[0032] Figure 7 This is a schematic diagram of the curvature characteristics of the third bend in the pipeline.

[0033] Figure 8 This is a schematic diagram showing the width characteristics of the third bend in the pipeline.

[0034] Figure 9 This is a schematic diagram showing the height characteristics of the pipeline after the first bend.

[0035] Figure 10 This is a diagram illustrating the importance of mediator features. Detailed Implementation

[0036] To make the above-mentioned objects, features and advantages of the present invention more apparent and understandable, the specific embodiments of the present invention will be described in detail below with reference to the accompanying drawings.

[0037] Many specific details are set forth in the following description in order to provide a full understanding of the invention. However, the invention may also be practiced in other ways different from those described herein, and those skilled in the art can make similar extensions without departing from the spirit of the invention. Therefore, the invention is not limited to the specific embodiments disclosed below.

[0038] Secondly, the term "one embodiment" or "embodiment" as used herein refers to a specific feature, structure, or characteristic that may be included in at least one implementation of the present invention. The phrase "in one embodiment" appearing in different places in this specification does not necessarily refer to the same embodiment, nor is it a single or selective embodiment that is mutually exclusive with other embodiments.

[0039] Example 1 Reference Figures 1-10 This is the first embodiment of the present invention, which provides a model design method for assisting in the optimization design of automotive intake and exhaust pipes, characterized in that: Design the characteristic variables of the agent model; Construct a dataset for training the agent model; Analyze the training set and select the main mediation features; The PCA dimensionality reduction algorithm is used to reduce the dimensionality of the main mediation features; Establish a mapping relationship between geometric features and the reduced m-dimensional intermediate features; Establish a multi-level proxy model for the optimized design of automotive intake and exhaust pipes.

[0040] The feature variables include geometric parameters (GP) and mediation parameters (MP) of the spatial configuration. Based on the spatial configuration of the vehicle's intake and exhaust pipes, geometric feature parameters are extracted to characterize the spatial configuration. ,in Represent the number of geometric features; and extract the intermediate features of the spatial configuration of the car's intake and exhaust pipes. ,in The number of mediation features is represented by geometric features and mediation features, which together constitute the feature variables of the surrogate model.

[0041] Intermediate features are flow field characteristics that characterize the flow state of gas in a pipe. Their calculation method is as follows: [The text abruptly shifts to a different topic] ...the centerline of the inlet and outlet pipes... Divide the space into equal parts and calculate the direction vector of the tangent at each division point as the flow field characteristic of the spatial configuration. .

[0042] Based on the target variable T of the automotive intake and exhaust pipe optimization design, a training dataset Train is constructed by collecting paired data of geometric feature variables (GP) and optimization design target T for different intake and exhaust pipe design schemes using simulation tools.

[0043] Using a tree-based feature importance analysis method, the importance score of each mediating feature to the target variable is calculated, resulting in... ; Select the highest rated Dimensional Mediator Features .

[0044] The general method for calculating feature importance based on tree models is as follows: the global importance of a feature is measured by the average importance of the feature within a single tree.

[0045]

[0046] in, Indicates the number of tree models. Representation of features In a single tree The importance of features on, The number of leaf nodes in the tree. That is, the number of non-leaf nodes in the tree. Is and node Related features It is a node The reduction in squared loss after splitting.

[0047] The PCA (Principal Component Analysis) dimensionality reduction algorithm was used to analyze the main mediation features. Dimensionality reduction is performed to obtain the reduced intermediate features. ,in This represents the dimension of the intermediate features after dimensionality reduction.

[0048] The calculation process of the PCA dimensionality reduction algorithm is as follows: First, center all samples:

[0049]

[0050] in Indicates the first Values ​​of flow field characteristic variables of the sample Then calculate the covariance matrix of the sample. ; For covariance matrix Perform eigenvalue decomposition:

[0051] in It is a matrix composed of the eigenvectors of the covariance matrix. It is a diagonal matrix, and the elements on the diagonal are the eigenvalues ​​of the covariance matrix. ; Take the largest Eigenvectors corresponding to each eigenvalue .

[0052] Based on the geometric features of automobile intake and exhaust pipes As the independent variable, the mediating features after dimensionality reduction are used. For the dependent variable, establish a data model between the independent and dependent variables:

[0053] in, For the first A reduced-dimensional mediation feature express and The regression model between them.

[0054] A secondary proxy model for the optimization design of automotive intake and exhaust pipes is established using data modeling methods. Specifically, a data model is built with the optimization design objective T as the dependent variable.

[0055] in, For target variable Regression model between characteristic variables, For the mth , .

[0056] Example 2 Reference Figures 1-10 This is the second embodiment of the present invention, which is based on the previous embodiment. S1: Based on the spatial configuration of the car's intake and exhaust pipes, design parameters to characterize the spatial configuration as feature variables of the surrogate model. The feature variables include the geometric parameters GP of the spatial configuration and the mediation parameters MP of the spatial configuration. S2: Based on the optimization design goals of automobile intake and exhaust pipes, use simulation tools to collect paired data of geometric feature variables (GP) and optimization design goals of different intake and exhaust pipe design schemes as training sets for subsequent machine learning operations; S3: Based on the training dataset, calculate the importance score of each mediating feature using feature importance analysis, and select the top-scoring features. Dimensional features serve as the primary mediating features; S4: Using the PCA feature generation algorithm to... The main mediating features of the dimensional reduction Dimensionality is reduced to obtain the intermediate features; S5: Establishing geometric features and dimensionality reduction The mapping relationship between the features of the intermediate dimension is obtained. A mapping model; S6: Establish a multi-level surrogate model for the optimization design of automotive intake and exhaust pipes based on the geometric features, the mapping model, and the optimization design objective.

[0057] Furthermore, in step one, based on the spatial configuration of the vehicle's intake and exhaust pipes, geometric feature parameters are extracted to characterize the spatial configuration. ,in Represent the number of geometric features; and extract the intermediate features of the spatial configuration of the car's intake and exhaust pipes. ,in This represents the number of mediation features. Geometric features and mediation features constitute the feature variables of the surrogate model.

[0058] Furthermore, the intermediate characteristics of the spatial configuration of the automobile's intake and exhaust pipes are the flow field characteristics characterizing the gas flow state within the pipes. The calculation method is as follows: [The text abruptly shifts to a different topic] ...the centerline of the intake and exhaust pipes... Divide the space into equal parts, and calculate the direction vector of the tangent at each division point as the flow field characteristic of the spatial configuration. .

[0059] Furthermore, step two involves optimizing the design of the vehicle's intake and exhaust pipes based on the target variables. By combining simulation tools, geometric characteristic variables (GPs) and optimization design objectives of different intake and exhaust pipe design schemes are collected. The training dataset is constructed using paired data. .

[0060] Furthermore, step three utilizes feature importance analysis to calculate the importance score of each mediating feature for the target variable, thus obtaining... Select the highest-rated Intermediate features .

[0061] Furthermore, the general method for calculating feature importance based on tree models is as follows: the global importance of a feature is measured by the average importance of the feature in each individual tree.

[0062]

[0063] in, Indicates the number of tree models. Representation of features In a single tree The importance of features. The number of leaf nodes in the tree. That is, the number of non-leaf nodes in the tree. Is and node Related features It is a node The reduction in squared loss after splitting.

[0064] Furthermore, step four uses the PCA (Principal Component Analysis) dimensionality reduction algorithm to analyze the main mediation features. Dimensionality reduction is performed to obtain the reduced intermediate features. ,in This represents the dimension of the intermediate features after dimensionality reduction.

[0065] Furthermore, the computational process of the PCA dimensionality reduction algorithm is as follows: (1) First, center all samples:

[0066]

[0067] in Indicates the first The values ​​of the flow field characteristic variables of the sample. (2) Then calculate the covariance matrix of the sample. ; (3) For the covariance matrix Perform eigenvalue decomposition:

[0068] in It is a matrix composed of the eigenvectors of the covariance matrix. It is a diagonal matrix, and the elements on the diagonal are the eigenvalues ​​of the covariance matrix. .

[0069] (4) Take the largest one Eigenvectors corresponding to each eigenvalue .

[0070] Furthermore, step five uses the geometric characteristics of the vehicle's intake and exhaust pipes as independent variables. Based on the reduced mediation features Using machine learning methods to build a data model between independent and dependent variables, with the dependent variable as the dependent variable:

[0071] in, For the first A reduced-dimensional mediation feature express and The regression model between them.

[0072] Furthermore, step six employs data modeling methods to establish a secondary proxy model for the optimization design of the vehicle's intake and exhaust pipes. Specifically, based on the aforementioned optimization design objectives... Establish a data model for the dependent variable:

[0073] in, For the target variable The regression model between the aforementioned feature variables, The regression model described in step five, .

[0074] Example 3 Reference Figures 1-10 This is the third embodiment of the present invention, which is based on the previous embodiment. The present invention will now be described in detail with reference to actual examples and related figures.

[0075] A manufacturer needs to design the intake and exhaust pipes of a car, taking into account the total pressure drop. Minimum is the optimization objective. For example... Figure 1 As shown, the multi-level agency model design method based on mediation analysis provided by this invention includes the following steps: (1) Design the characteristic variables of the proxy model: Combine the specific spatial configuration of the car's intake and exhaust pipes, such as Figure 2 As shown in Table 1, seven geometric feature variables are designed (e.g., Figures 3-9 (As shown).

[0076] Table 1 Statistical Table of Geometric Characteristic Variables

[0077] The method for extracting mediating feature variables involves dividing the centerline of the intake and exhaust pipes into 100 equal parts and calculating the direction vector of the tangent at each division point as the mediating feature of the spatial configuration. .

[0078] (2) Use simulation tools to construct a dataset for model training. The feature variables of the dataset are the geometric feature variables and the mediating feature variables in step (1). The number of samples in the dataset is 800.

[0079] (3) Considering that the mediating feature variables have high dimensionality (containing 100 dimensions) and cannot be directly used for surrogate model training, a tree-based feature importance analysis method is used to analyze the importance of the mediating feature variables. The results of the feature importance analysis are as follows: Figure 10 As shown, through Figure 10 It can be observed that, The target variable is of high importance, therefore it is selected. It is used as the main mediating variable in surrogate model training; (4) Considering that the dataset has 800 samples, and that the 6 mediator features and 7 geometric features selected in step (3) are used together for training the surrogate model, it is highly likely to cause the curse of dimensionality. Therefore, the PCA dimensionality reduction algorithm is used to reduce the dimensionality of the dataset. The six mediation features were further compressed and reduced to 3 dimensions to obtain the dimensionality-reduced mediation variables. .

[0080] (5) Mediating variables It is generally not directly applicable to the design and reconstruction of intake and exhaust pipes, and is difficult to measure directly; therefore, it is necessary to use mediating variables. Modeling analysis is performed, using easily measurable geometric features applicable to the design and reconstruction of intake and exhaust ducts to model and fit the mediating variables. This implementation scheme uses seven geometric feature variables as input variables and employs the GBDT algorithm to... Three mediating variables were used to model and fit the model.

[0081]

[0082]

[0083]

[0084] in They represent A data model with three mediating variables.

[0085] (6) Adopt The intermediary agency model Establish a proxy model for the optimized design of automotive intake and exhaust pipes:

[0086] in For the proxy model in this embodiment, GBDT is selected as the training algorithm for the proxy model. These represent the geometric characteristic parameters of the intake and exhaust pipes. Indicates selection One or more of the three mediation models.

[0087] Experimental results: The coefficient of determination (R²) and mean absolute percentage error (MAPE) are chosen as evaluation metrics for model performance. R² and MAPE are defined as follows:

[0088]

[0089] Where n is the number of test datasets, For real data, For predicted values, This represents the average value of the actual data. The larger the R² value and the smaller the MAPE value, the higher the accuracy of the model.

[0090] Table 2. Results of the Proxy Model Modeling Experiment

[0091] Based on the experimental results in Table 2, it can be found that when only geometric feature parameters are used for surrogate model training (serial number 1), the MAPE index is 0.49%. However, by introducing mediating features... The MAPE index of the surrogate model gradually decreased to 0.15%, which indicates that introducing mediation features can effectively improve model accuracy and reduce prediction error. Considering that intermediate characteristic variables cannot be directly used for the design and reconstruction of intake and exhaust pipes, geometric characteristic variables are adopted. Regression fitting of the mediating variables and training of the surrogate model using the fitted results introduce some fitting error. However, experimental results show that even with this fitting error, the accuracy of the multi-level surrogate model based on mediation analysis is higher than that of the surrogate model using only geometric feature parameters (compare No. 6 and No. 1). Furthermore, the R² scores of all experimental results are above 99%, ensuring the usability of all surrogate models.

[0092] In summary, the multi-level surrogate model design method based on mediation analysis can effectively improve the accuracy of the surrogate model, thereby effectively assisting in the optimization design of intake and exhaust pipes. Therefore, this invention can improve the design efficiency of intake and exhaust pipes, better assist automobile manufacturers in exhaust pipe design, improve production efficiency, and reduce production costs. It should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and are not intended to limit it. Although the present invention has been described in detail with reference to preferred embodiments, those skilled in the art should understand that modifications or equivalent substitutions can be made to the technical solutions of the present invention without departing from the spirit and scope of the invention.

Claims

1. A model design method for assisting in the optimization design of automotive intake and exhaust pipes, characterized in that: Design the characteristic variables of the agent model; Construct a dataset for training the agent model; Analyze the training set and select the main mediation features; The main mediation features were reduced in dimensionality using the PCA dimensionality reduction algorithm. Establish a mapping relationship between geometric features and the reduced m-dimensional intermediate features; Establish a multi-level proxy model for the optimized design of automotive intake and exhaust pipes; The feature variables include the geometric features GP of the spatial configuration and the intermediate features MP of the spatial configuration. Based on the spatial configuration of the vehicle's intake and exhaust pipes, geometric feature parameters used to characterize the spatial configuration are extracted. ,in Represent the number of geometric features; and extract the intermediate features of the spatial configuration of the car's intake and exhaust pipes. ,in The number of mediation features is represented, and the geometric features and the mediation features constitute the feature variables of the surrogate model; The intermediate feature is a flow field feature characterizing the flow state of gas in a pipe, and its calculation method is as follows: [The text abruptly shifts to a different topic] ...the centerline of the inlet and outlet pipes... Divide the space into equal parts, and calculate the direction vector of the tangent at each division point as the flow field characteristic of the spatial configuration. ; Based on the geometric features of automobile intake and exhaust pipes As the independent variable, the mediating features after dimensionality reduction are used. For the dependent variable, establish a data model between the independent and dependent variables: ； in, For the first A reduced-dimensional mediation feature express and Regression model between them; A secondary proxy model for the optimization design of automotive intake and exhaust pipes is established using data modeling methods. Specifically, a data model is established with the optimization design objective T as the dependent variable. ； in, For target variable The regression model between the aforementioned feature variables, For the mth .

2. The model design method for assisting in the optimization design of automotive intake and exhaust pipes as described in claim 1, characterized in that: Based on the target variable T of the automotive intake and exhaust pipe optimization design, a training dataset Train is constructed by collecting paired data of geometric feature variables GP and optimization design target T for different intake and exhaust pipe design schemes using simulation tools.

3. The model design method for assisting in the optimization design of automotive intake and exhaust pipes as described in claim 2, characterized in that: Using a tree-based feature importance analysis method, the importance score of each mediating feature to the target variable is calculated, resulting in... ; Select the highest rated Dimensional Mediator Features .

4. The model design method for assisting in the optimization design of automotive intake and exhaust pipes as described in claim 3, characterized in that: The general method for calculating feature importance based on tree models is as follows: the global importance of a feature is measured by the average importance of the feature within a single tree. ； ； in, Indicates the number of tree models. Representation of features In a single tree The importance of features on The number of leaf nodes in the tree. That is, the number of non-leaf nodes in the tree. Is and node Related features It is a node The reduction in squared loss after splitting.

5. The model design method for assisting in the optimization design of automotive intake and exhaust pipes as described in claim 4, characterized in that: The main mediation features were analyzed using the PCA dimensionality reduction algorithm. Dimensionality reduction is performed to obtain the reduced mediating features. ,in This represents the dimension of the intermediate features after dimensionality reduction.

6. The model design method for assisting in the optimization design of automotive intake and exhaust pipes as described in claim 5, characterized in that: The calculation process of the PCA dimensionality reduction algorithm is as follows: First, center all samples: ； ； in Indicates the first Values ​​of flow field characteristic variables of the sample Then calculate the covariance matrix of the sample. ; For covariance matrix Perform eigenvalue decomposition: ； in It is a matrix composed of the eigenvectors of the covariance matrix. It is a diagonal matrix, and the elements on the diagonal are the eigenvalues ​​of the covariance matrix. ; Take the largest Eigenvectors corresponding to each eigenvalue .

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