Uncertainty quantification method based on derivative enhanced Gaussian process

Abstract

The invention discloses an uncertainty quantification method based on a derivative enhanced Gaussian process, and belongs to the technical field of uncertainty quantification, and the method comprises the steps: S1, obtaining a training sample set; s2, first-order derivatives and second-order derivatives of the aerodynamic performance parameters for the geometric parameters and the working conditions are obtained; s3, constructing a derivative enhanced Gaussian process model by using the training sample set obtained in the step S1; s4, verifying the prediction performance of the derivative enhanced Gaussian process model through cross validation, and evaluating the accuracy of uncertainty quantification; and S5, performing stochastic simulation in the neighborhood space of the initial sample point, predicting an objective function of a stochastic simulation sample by using the derivative enhanced Gaussian process model, and calculating a robustness index to complete uncertainty quantization. According to the uncertainty quantification method based on the derivative enhanced Gaussian process, the derivative information is fused into the Gaussian process, the global precision of the model and the local precision near the sample point are considered, and the uncertainty quantification precision can be improved.

Description

Technical Field

[0001] The present invention relates to the technical field of uncertainty quantification, and in particular to an uncertainty quantification method based on derivative enhanced Gaussian process. Background Art

[0002] In practical engineering problems, the analysis and research methods of uncertainty can be divided into two types: embedded and non-embedded. Embedded methods often quantify uncertainty through random simulation (such as Monte-Carlo method) and apply it to the modeling process to obtain a model that includes uncertainty. Non-embedded methods first build a more accurate deterministic model and then quantify uncertainty on this basis. This type of method essentially converts the robust optimization problem into a multi-objective optimization problem in which the objective function includes robustness indicators and the constraints include feasibility indicators. It is adopted by most engineering application fields.

[0003] Gaussian process regression is widely used in uncertainty quantification. It is a typical prediction algorithm based on covariance function, which has high nonlinear expression ability and good generalization performance. Its core advantage is that it can provide prediction results and confidence intervals of prediction results at the same time, thereby quantifying the uncertainty of the prediction results. However, the traditional Gaussian process model still has the following defects: (1) When faced with high-dimensional inputs or complex objective functions, a large amount of sampling data is usually required to achieve good prediction results, which leads to a significant increase in computational costs; (2) Uncertainty quantification has regional requirements for model accuracy, that is, the model must have high accuracy within the neighborhood of the sample point, while the Gaussian process model focuses on ensuring global accuracy within the design space. There are inherent fitting errors within the sample point and its neighborhood. These errors will be brought into the uncertainty quantification process and affect the uncertainty quantification results. Summary of the invention

[0004] The purpose of the present invention is to provide an uncertainty quantification method based on derivative enhanced Gaussian process to solve the problems existing in the above background technology.

[0005] To achieve the above object, the present invention provides an uncertainty quantification method based on derivative enhanced Gaussian process, comprising the following steps:

[0006] S1. For the RAE 2822 airfoil, a training sample set is obtained through an experimental design method. The number of samples is m, and each sample contains information on n working conditions and geometric parameters, and the corresponding airfoil geometric shape is generated;

[0007] S2. Perform CFD simulation on the airfoil of step S1 to obtain its aerodynamic performance parameter lift coefficient C l and the drag coefficient C dAt the same time, the automatic differentiation method is used to obtain the first and second derivatives of the aerodynamic performance parameters with respect to the working conditions and geometric parameters;

[0008] S3, constructing a derivative enhanced Gaussian process model using the training sample set obtained in step S1;

[0009] S4, verify the prediction performance of the derivative enhanced Gaussian process model by cross validation or leave-one-out method, that is, the influence of the change of airfoil geometric parameters on the aerodynamic performance, and evaluate the accuracy of uncertainty quantification; if the accuracy meets the requirements, proceed to the next step; if the accuracy does not meet the requirements, use the sequential design method to increase sample points and return to step S1;

[0010] S5. Perform random simulation in the neighborhood space of the initial sample point, use the derivative-enhanced Gaussian process model to predict the aerodynamic performance of the random simulation samples, and calculate the robustness indicators (such as mean, variance, etc.) at the sample points. This indicator is used to measure the impact of geometric uncertainty on the aerodynamic performance of the airfoil and complete the uncertainty quantification.

[0011] Preferably, the construction of the derivative enhanced Gaussian process model in step S2 includes setting and calculating a regression function and a correlation function.

[0012] Preferably, the regression function is a regression matrix calculated from m n-dimensional sample points and their derivatives, and the regression matrix of the training sample set in step S1 consists of a regression part and a derivative part, as follows:

[0013]

[0014] Where F(x) is the regression matrix of the derivative-enhanced Gaussian process; X is the geometric parameter sample; F0(X) is the regression matrix in the Gaussian process; d(F0(X)) and d 2 (F0(X)) are the first and second derivatives of F0(X) respectively.

[0015] Preferably, F0(X) is specifically expanded as follows:

[0016]

[0017] Where m is the number of samples; i = 1, 2, ..., m, which is used to distinguish different sample points; n is the number of dimensions, indicating that each sample point contains n-dimensional data; j = 1, 2, ..., n, which is used to distinguish the values ​​of different dimensions in the sample points. For ease of understanding, it can be expressed as

[0018] Preferably, the order of the regression function of the derivative-enhanced Gaussian process model is greater than the order of the derivative part in formula (1), otherwise the data in F(x) are all 0 and cannot play an enhancement role.

[0019] Preferably, for the Gaussian process model, the correlation function is a Gaussian kernel function, as follows:

[0020]

[0021] in, is the Euclidean distance between two sample points; θ is a hyperparameter; k and l represent two different samples respectively;

[0022] The first-order derivative and the second-order derivative are added to the correlation function of the derivative-enhanced Gaussian process model as follows:

[0023]

[0024] Where X represents a sample, and By adding derivative information to the regression function and the correlation function, the construction of the derivative-enhanced Gaussian process model is completed, and this model can express the mapping relationship between the airfoil performance parameters and the working conditions and geometric parameters, as follows:

[0025]

[0026] Among them, σ 2 is the variance.

[0027] Therefore, the present invention adopts the above-mentioned uncertainty quantification method based on derivative enhanced Gaussian process. By introducing derivative information (including first-order derivatives and second-order derivatives) at sample points, a high-precision derivative enhanced Gaussian process model can be constructed with fewer sample points, while ensuring the local accuracy within the neighborhood of the sample points, which not only reduces the calculation cost but also improves the accuracy of uncertainty quantification; and the uncertainty quantification results for the RAE 2822 airfoil can be used for aerodynamic robust optimization, and a robust model is constructed by combining sample data and robustness data, and an intelligent optimization algorithm is used for optimization, thereby obtaining an airfoil design scheme in which the geometric shape uncertainty has less influence on aerodynamic performance.

[0028] The technical solution of the present invention is further described in detail below through the accompanying drawings and embodiments. BRIEF DESCRIPTION OF THE DRAWINGS

[0029] Figure 1 A flowchart of an uncertainty quantification method based on derivative enhanced Gaussian process of the present invention;

[0030] Figure 2 Schematic diagram of the derivative-enhanced Gaussian process model in the present invention;

[0031] Figure 3 Schematic diagram of the comparison of the RAE 2822 airfoil before and after optimization of the present invention. DETAILED DESCRIPTION

[0032] Example

[0033] In order to make the purpose, technical solutions and advantages of the embodiments of the present invention clearer, the technical solutions in the embodiments of the present invention will be clearly and completely described below in conjunction with the drawings in the embodiments of the present invention. Obviously, the described embodiments are part of the embodiments of the present invention, not all of the embodiments. Generally, the components of the embodiments of the present invention described and shown in the drawings here can be arranged and designed in various different configurations.

[0034] like Figure 1-2 As shown in the figure, an uncertainty quantification method based on derivative-enhanced Gaussian process is applied to the aerodynamic robust design of RAE 2822 airfoil, which includes the following steps:

[0035] S1. Parameterize the RAE 2822 airfoil, select four parameters at its leading edge as design variables, use the experimental design method to obtain the initial sample points, and generate the airfoil shape based on the sample data;

[0036] S2. Perform CFD simulation on the airfoil obtained in step S1 to obtain the corresponding lift coefficient C l and the drag coefficient C d At the same time, the first and second derivatives of aerodynamic performance parameters with respect to geometric parameters are obtained by automatic differentiation method;

[0037] S3, constructing a derivative enhanced Gaussian process model using the data obtained in step S2;

[0038] S4, verify the derivative through cross-validation to enhance the prediction performance of the Gaussian process model; if the accuracy meets the requirements, proceed to the next step; if the accuracy does not meet the requirements, increase the sample points using the maximum MSE criterion and return to step S2;

[0039] S5. Perform random simulation in the neighborhood space of each sample point, and use the high-precision derivative-enhanced Gaussian process model to calculate the mean and variance of the lift coefficient and the drag coefficient respectively to complete the uncertainty quantification;

[0040] S6. Use the mean and variance data in step S5 to build a robust model, and use a genetic algorithm to find the optimal solution to obtain a robust optimization result of the airfoil that takes uncertainty into account, such as Figure 3 shown.

[0041] Therefore, the present invention adopts the above-mentioned uncertainty quantification method based on derivative enhanced Gaussian process, introduces derivative information into the regression matrix and correlation matrix of the Gaussian process, increases the information content of the sample points, and thus can obtain a high-precision model with fewer sample points; at the same time, the derivative information also ensures the model accuracy within the neighborhood of the sample point, which helps to improve the accuracy of uncertainty quantification.

[0042] Finally, it should be noted that the above embodiments are only used to illustrate the technical solution of the present invention rather than to limit it. Although the present invention has been described in detail with reference to the preferred embodiments, those skilled in the art should understand that they can still modify or replace the technical solution of the present invention with equivalents, and these modifications or equivalent replacements cannot cause the modified technical solution to deviate from the spirit and scope of the technical solution of the present invention.

Claims

1. An uncertainty quantification method based on derivative-enhanced Gaussian process, It is characterized in that The following steps are involved: S1. Obtain a training sample set, the number of samples is m, and each sample contains information on n working conditions and geometric parameters; S2. Obtain the lift coefficient C of the aerodynamic performance parameter l and the drag coefficient C d At the same time, the first and second derivatives of aerodynamic performance parameters with respect to geometric parameters are obtained by automatic differentiation method; S3, constructing a derivative enhanced Gaussian process model using the training sample set obtained in step S1; S4. Validate the prediction performance of the derivative-enhanced Gaussian process model by cross-validation or leave-one-out method and evaluate the accuracy of uncertainty quantification; if the accuracy meets the requirements, proceed to the next step; If the accuracy does not meet the requirements, the sequential design method is used to increase sample points and return to step S1; S5. Perform random simulation in the neighborhood space of the initial sample point, use the derivative-enhanced Gaussian process model to predict the objective function of the random simulation sample, and calculate the robustness index to complete the uncertainty quantification.

2. The uncertainty quantification method based on derivative enhanced Gaussian process according to claim 1, Features: The construction of the derivative enhanced Gaussian process model in step S2 includes the setting of regression function and correlation function.

3. The uncertainty quantification method based on derivative enhanced Gaussian process according to claim 2, It is characterized in that The regression function is a regression matrix calculated from m n-dimensional sample points and their derivatives, and the regression matrix of the training sample set in step S1 consists of a regression part and a derivative part, as follows: Where F(x) is the regression matrix of the derivative-enhanced Gaussian process; X is the geometric parameter sample; F 0 (X) is the regression matrix in the Gaussian process; d(F 0 (X)) and d 2 (F 0 (X)) are F 0 The first and second derivatives of (X).

4. The uncertainty quantification method based on derivative enhanced Gaussian process according to claim 3, It is characterized in that F 0 (X) is specifically expanded as follows: Among them, m is the number of samples; i = 1, 2, ..., m, is used to distinguish different sample points; n is the number of dimensions, indicating that each sample point contains n-dimensional data; j = 1, 2, ..., n, is used to distinguish the values ​​of different dimensions in the sample points.

5. The uncertainty quantification method based on derivative enhanced Gaussian process according to claim 3, Features: The order of the regression function of the derivative-augmented Gaussian process model is greater than the order of the derivative part in equation (1).

6. The uncertainty quantification method based on derivative enhanced Gaussian process according to claim 2, It is characterized in that For the Gaussian process model, the relevant function is the Gaussian kernel function, as follows: in, is the Euclidean distance between two sample points; θ is a hyperparameter; k and l represent two different samples respectively; The first-order derivative and the second-order derivative are added to the correlation function of the derivative-enhanced Gaussian process model as follows: Where X represents a sample, and

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