Multi-fidelity Bayesian optimization method based on multi-level expectation improvement criterion

Abstract

The invention discloses a multi-fidelity Bayesian optimization method based on a multi-level expectation improvement criterion. The method comprises the following steps: S1, initializing a sample set; s2, constructing an MLGP model by using the data set; s3, sequential point collection is carried out by using a multi-level EI criterion, and iterative optimization is carried out; s4, judging whether a stop condition is met or not, and if yes, outputting a result; if the stop condition is not met, updating the proxy model, returning to S2, and continuing circulation; the optimization precision of an existing method depends on the sample size of high-fidelity data, and the auxiliary functions of a plurality of low-fidelity systems cannot be played to the maximum extent. Meanwhile, an existing method generally deviates to a heuristic mode in trade-off selection, namely, the fidelity level of a next sampling point does not have very high interpretability. The optimization algorithm provided by the invention can select fewer high-precision sample points to achieve the purpose of global optimization, the sequential criterion LEI provided by the invention has an analytic expression form, the contribution of the fidelity of each level to the whole is clear, and the method has very strong interpretability.

Description

Technical Field

[0001] The present invention relates to the technical field of function optimization, and more specifically, to a multi-fidelity Bayesian optimization method based on a multi-level expected improvement criterion. Background Art

[0002] In global optimization problems, the optimized black box function is often a non-convex, multi-peaked, and difficult-to-evaluate objective function. In this case, global optimization becomes difficult. This situation often occurs in problems such as engineering optimization and hyperparameter optimization in machine learning.

[0003] Bayesian optimization uses the Gaussian process model as a proxy model to fit the black box function of the true target, and selects the most promising evaluation points based on the information provided by the proxy model. It is widely used to solve the optimization problems of complex black box functions and is an effective global optimization algorithm that is very suitable for the optimization solution of such problems in various fields.

[0004] In order to further save computing costs, in Bayesian optimization, a small amount of high-fidelity computer simulations can be assisted by more low-fidelity computer simulation experiments, which will have the potential to achieve a balance between "resource consumption and optimization accuracy". Multi-fidelity Bayesian optimization (MBO) has emerged. Its basic idea is to efficiently select the fidelity level and position of the next sampling through the multi-fidelity sequential criterion under the existing Bayesian optimization framework, improve the multi-fidelity proxy model, and then find the global optimum.

[0005] The sequential criteria of existing MBO methods are divided into two categories: the first category directly uses single-fidelity sequential criteria. A typical representative is to use EI criteria, LCB criteria, etc. to directly combine them with multi-fidelity proxy models to complete the optimization process.

[0006] The second category is the improvement of the EI criterion. There are two types of efficient multi-fidelity sequential optimization criteria, namely enhanced EI (AEI) and variable fidelity EI (VFEI).

[0007] However, the optimization accuracy of existing methods depends on the sample size of high-fidelity data, and cannot maximize the auxiliary role of several low-fidelity systems. At the same time, existing methods generally tend to be heuristic in choosing trade-offs, that is, the determination of the fidelity level of the next sampling point is not very explanatory. Summary of the invention

[0008] The present invention provides a multi-fidelity Bayesian optimization method based on a multi-level expected improvement criterion to solve the problem that the optimization accuracy of the existing methods depends on the sample size of high-fidelity data and cannot maximize the auxiliary role of several low-fidelity systems; at the same time, the existing methods are generally heuristic in the selection of trade-offs, that is, the determination of the fidelity level of the next sampling point is not very explanatory.

[0009] In order to achieve the above object, the present invention provides the following technical solutions:

[0010] The multi-fidelity Bayesian optimization method based on the multi-level expected improvement criterion includes the following steps:

[0011] S1: Initialization sample set;

[0012] S2: Build the MLGP model using the dataset;

[0013] S3: Use multi-level EI criteria to sequentially collect points and iteratively optimize;

[0014] S4: Determine whether the stopping condition is met. If so, output the result; if not, update the proxy model, return to S2, and continue the loop.

[0015] Preferably, the S1 specifically comprises the following steps:

[0016] S1.1: Use nested Latin square design to obtain initial sample locations;

[0017] S1.2: Run the multi-fidelity system and obtain response data;

[0018] S1.3: Calculate the difference of the optimized target data;

[0019] S1.4: Run the multi-fidelity constraint system and obtain response data;

[0020] S1.5: Calculate the difference of constraint data and obtain training data;

[0021] The S1.1 specifically includes:

[0022] {X1,X2,...,X n}=NLHS(Ω)

[0023] Ω is the experimental design space, {X1,X2,...,X n} respectively represent the sample points of the first layer, the second layer, ..., the highest layer fidelity and

[0024] Preferably, S1.2 and S1.3 specifically include the following steps:

[0025] S1.2: Run the multi-fidelity system and get response data

[0026] Y i =f i (X i ),i=1,2,...,n

[0027] S1.3: Calculate the difference of the optimization target data

[0028] Z i =Y i -f i-1 (X i ),i=1,2,...,n

[0029] like There is no need to re-run f i-1 , f0≡0

[0030] Preferably, S1.4 and S1.5 specifically include the following steps:

[0031] S1.4: Run the multi-fidelity constraint system and obtain response data

[0032] G i =g i (X i ),i=1,2,...,n

[0033] S1.5: Calculate the difference of constraint data

[0034] M i =G i -g i-1 (X i ),i=1,2,...,n

[0035] And get training data

[0036] D={X1,G1,M1}∪{X2,G2,M2}∪…∪{X n ,G n ,M n}

[0037] Preferably, the S2 specifically includes the following steps:

[0038] S2.1: Build multiple independent Gaussian process regression models;

[0039] S2.2: summing up the parameters to obtain the MLGP model that optimizes the objective function;

[0040] S2.3: summing up the parameters to obtain the MLGP model of the constraint function;

[0041] Wherein, the S2.1 specifically includes:

[0042] Create a proxy model:

[0043] f i (x)~GP(μ i (x),K(x,x))

[0044] The model is controlled by a mean function μ(x) and a covariance function K(x,x') which defines the correlation of the function between different input points.

[0045] The predicted mean and variance are calculated from it:

[0046]

[0047] Among them, μ(x*) is the prior mean at the input point x*, K(X,x*) is the covariance matrix between the input points X and x*, σ 2 n is the variance of the observation noise, and I is the identity matrix.

[0048]

[0049] Preferably, the S2.2 specifically includes the following steps:

[0050] Sum the parameters and get the MLGP model that optimizes the objective function

[0051] in:

[0052] Preferably, the S2.3 specifically includes the following steps:

[0053] Sum the parameters to get the MLGP model of the constraint function

[0054] in:

[0055] Preferably, the S3 specifically includes the following steps:

[0056] S3.1: Propose LEI expression;

[0057] S3.2: Sequential optimization using LEI criteria;

[0058] Wherein, the LEI expression is:

[0059] LEI(x,l)=EI(x)α1(x,l)α2(l)α3(x,l)

[0060] Specifically,

[0061]

[0062] in:

[0063] (1) The EI acquisition function is:

[0064]

[0065] In the formula, and σ(x) are the mean and standard deviation at x in the Gaussian process model, and ymin is the optimal value observed so far. In addition, Φ represents the cumulative distribution function (CDF) of the multivariate Gaussian distribution, and φ represents the probability density function (PDF).

[0066] (5)α1(x) is the Pearson correlation coefficient:

[0067]

[0068] (6)α2 is the cost ratio of running the high-precision and low-precision systems once:

[0069]

[0070] (7)α3(x) is the constraint processing term

[0071]

[0072] Preferably, the sequential optimization process using the LEI criterion comprises:

[0073] S3.2.1: Use genetic algorithm to maximize LEI to determine the next experimental design point location and fidelity level;

[0074] S3.2.2: Update the input portion of the training set;

[0075] S3.2.3: Run the multi-fidelity system and obtain response data;

[0076] S3.2.4: Calculate the difference of the optimization target data;

[0077] S3.2.5: Run the multi-fidelity constraint system and obtain response data;

[0078] S3.2.6: Calculate the difference of constraint data and update training data;

[0079] S3.2.7: Determine whether to update the current optimal point; if l * =n, then y min =min(Y n ); otherwise y min constant.

[0080] Preferably, S3.2.1 and S3.2.2 specifically include the following steps:

[0081] S3.2.1:

[0082]

[0083] S3.2.2:

[0084]

[0085] Preferably, S3.2.3:

[0086] Y i =f i (X i ),i=1,2,...,l * ;

[0087] S3.2.4:

[0088] Z i =Y i -f i-1 (X i ),i=1,2,...,l * ;

[0089] Preferably, S3.2.5 and S3.2.6 specifically include the following steps:

[0090] S3.2.5:

[0091] G i =g i (X i ),i=1,2,...,l * ;

[0092] S3.2.6:

[0093] Calculate the difference: M i =G i -g i-1 (X i ),i=1,2,...,l * ;

[0094] Update data: D = {X1, M1, Z1} ∪ {X2, M2, Z2} ∪… ∪ {X n ,M n ,Z n};

[0095] Preferably, the stopping condition in S4 is one of the following situations:

[0096] 1. The test resources have reached the preset value;

[0097] 2. The optimal value explored meets the preset accuracy;

[0098] 3. The maximum number of iterations is reached;

[0099] 4. The operation time reaches the preset value;

[0100] 5. Other reasonable conditions.

[0101] The principle and beneficial effects of this technical solution:

[0102] (1) The sequential criterion proposed in the present invention will first focus on searching the low-fidelity system so that the proxy model has a global approximate property, and then focus on digging deep into the high-fidelity sample points to solve the extreme value of the optimization problem; therefore, this method is not affected by the initial design and has relatively good robustness.

[0103] (2) Since the present invention first considers modeling of low-fidelity systems, and the convergence rate of MLGP is jointly determined by the number of high-fidelity and low-fidelity sample points, the present invention can save more high-fidelity sample points.

[0104] (3) Mathematically speaking, the method involved in the present invention can be naturally extended to the optimization problem of multiple fidelity. Compared with the algorithm based on Co-kriging, the proxy model modeling efficiency of this method is very high. Experimental verification shows that this method has significant advantages in the optimization problem of multiple fidelity. BRIEF DESCRIPTION OF THE DRAWINGS

[0105] Figure 1 It is the algorithm flow chart of the present invention;

[0106] Figure 2 A comparison chart of four multi-fidelity Bayesian optimization algorithms;

[0107] Figure 3 This is a schematic diagram of the distance towing anti-interference test scenario;

[0108] Figure 4 This is a graph of the algorithm convergence process for a practical application case; DETAILED DESCRIPTION

[0109] The present invention is further described in detail below in conjunction with the accompanying drawings and embodiments:

[0110] Example:

[0111] like Figure 3 As shown, the digital simulation of a radar cross section (RCS) is calculated using physical optics methods, which includes high-dimensional integration. Both modeling and simulation require a lot of calculations and time.

[0112] The present invention is used to study the influence of distance R, bait towing distance l, speed v of bait and ship and signal-to-interference ratio β of bait release interference signal on the angle tracking accuracy angle φ of radar, but R, l, v, β are the main factors affecting the radar interference effect; therefore, this experiment will study their influence on φ, set two fidelity systems, and the stopping condition is the number of high-precision system calls, reaching the maximum number of iterations 100, and explore the combination of experimental factors that make the radar angle tracking accuracy angle φ.

[0113] That is, solve the following mathematical problem:

[0114]

[0115] It can be seen that this scenario does not contain constraint function conditions, so the constraint-related solution process can be omitted;

[0116] The steps of solving the problem using the method of the present invention are as follows:

[0117] S1: Initialization sample set

[0118] S1.1: Initial sample locations using nested Latin square design

[0119] {X l ,X h}=NLHS(Ω)

[0120] where Ω is the experimental design space, and

[0121]

[0122] Where X l and X h Represent low-fidelity and high-fidelity sample points respectively.

[0123] S1.2: Run the multi-fidelity system and get response data

[0124] Y h =f h (X h ),Y l =f l (X l )

[0125] S1.3: Calculate the difference of the optimization target data

[0126] Z h =Y h -f l (X h )

[0127] S2: Building the MLGP model using dataset D

[0128] S2.1: Constructing multiple independent Gaussian process regression models

[0129] The established proxy model is:

[0130] f i (x)~GP(μ i (x),K(x,x′))

[0131] The model is controlled by a mean function μ(x) and a covariance function K(x,x') which defines the correlation of the function between different input points.

[0132] The predicted mean and variance are calculated from it:

[0133]

[0134] Where μ(x*) is the prior mean at the input point x*, K(X,x*) is the covariance matrix between the input points X and x*, σ2n is the variance of the observation noise, and I is the identity matrix.

[0135]

[0136] S2.2: Summing the parameters to obtain the MLGP model that optimizes the objective function

[0137] in:

[0138] S3: Use multi-level EI criteria to sequentially sample points and iteratively optimize

[0139] The proposed LEI expression is:

[0140] Specifically,

[0141] The EI acquisition function is:

[0142]

[0143] In the formula, and σ(x) are the mean and standard deviation at x in the Gaussian process model, and ymin is the optimal value observed so far. In addition, Φ represents the cumulative distribution function (CDF) of the multivariate Gaussian distribution, and φ represents the probability density function (PDF).

[0144] S3.1: Use genetic algorithm to maximize LEI to determine the next experimental design point location and fidelity level;

[0145]

[0146] S3.2: Update the input part of the training set

[0147] X l =X l ∪x * ,X h =X h ∪x *

[0148] S3.3: Run the multi-fidelity system and get response data

[0149] If l * ==1

[0150] Y l =f l (X l )

[0151] Else

[0152] Y l =f l (X l ),Y h =f h (X h )

[0153] endIF

[0154] S3.4: Calculate the difference of optimization target data

[0155] Z h =Y h -f l (X h )

[0156] D={X l ,Z l}∪{X h ,Z h}

[0157] Where: Z l =Y l

[0158] S3.5: Determine whether to update the current optimal point. * =n, then y min =min(Y h ); otherwise y min constant.

[0159] S4: Determine whether the stop condition is reached. If so, output the result y min If the stop condition is not met, update the proxy model and return to S2 to continue the cycle.

[0160] The specific usage and function of this embodiment are as follows:

[0161] like Figure 1 As shown, the present invention provides a multi-fidelity Bayesian optimization method based on a multi-level expected improvement criterion, and the multi-fidelity Bayesian optimization method based on a multi-level expected improvement criterion includes the following steps:

[0162] S1: Initialization sample set;

[0163] S2: Build the MLGP model using the dataset;

[0164] S3: Use multi-level EI criteria to sequentially collect points and iteratively optimize;

[0165] S4: Determine whether the stopping condition is met. If so, output the result; if not, update the proxy model, return to S2, and continue the loop.

[0166] like Figure 1 As shown, S1 specifically includes the following steps:

[0167] S1.1: Use nested Latin square design to obtain initial sample locations;

[0168] S1.2: Run the multi-fidelity system and obtain response data;

[0169] S1.3: Calculate the difference of the optimized target data;

[0170] S1.4: Run the multi-fidelity constraint system and obtain response data;

[0171] S1.5: Calculate the difference of constraint data and obtain training data;

[0172] S1.1 specifically includes:

[0173] {X1,X2,...,X n}=NLHS(Ω)

[0174] Ω is the experimental design space, {X1,X2,...,X n} respectively represent the sample points of the first layer, the second layer, ..., the highest layer fidelity and

[0175] like Figure 1 As shown, S1.2 and S1.3 specifically include the following steps:

[0176] S1.2: Run the multi-fidelity system and get response data

[0177] Y i =f i (X i ),i=1,2,...,n

[0178] S1.3: Calculate the difference of the optimization target data

[0179] Z i =Y i -f i-1 (X i ),i=1,2,...,n

[0180] like There is no need to re-run f i-1 , f0≡0

[0181] like Figure 1 As shown, S1.4 and S1.5 specifically include the following steps:

[0182] S1.4: Run the multi-fidelity constraint system and obtain response data

[0183] G i =g i (X i ),i=1,2,...,n

[0184] S1.5: Calculate the difference of constraint data

[0185] M i =G i -g i-1 (X i ),i=1,2,...,n

[0186] And get training data

[0187] D={X1,G1,M1}∪{X2,G2,M2}∪…∪{X n ,G n ,M n}

[0188] like Figure 1 As shown, S2 specifically includes the following steps:

[0189] S2.1: Build multiple independent Gaussian process regression models;

[0190] S2.2: summing up the parameters to obtain the MLGP model that optimizes the objective function;

[0191] S2.3: summing up the parameters to obtain the MLGP model of the constraint function;

[0192] Among them, S2.1 specifically includes:

[0193] Create a proxy model:

[0194]

[0195] The model is controlled by a mean function μ(x) and a covariance function K(x,x') which defines the correlation of the function between different input points.

[0196] The predicted mean and variance are calculated from it:

[0197]

[0198] Among them, μ(x*) is the prior mean at the input point x*, K(X,x*) is the covariance matrix between the input points X and x*, σ 2 n is the variance of the observation noise, and I is the identity matrix.

[0199]

[0200] like Figure 1 As shown, S2.2 specifically includes the following steps:

[0201] Sum the parameters and get the MLGP model that optimizes the objective function

[0202] in:

[0203] like Figure 1 As shown, S2.3 specifically includes the following steps:

[0204] Sum the parameters to get the MLGP model of the constraint function

[0205] in:

[0206] like Figure 1 As shown, S3 specifically includes the following steps:

[0207] S3.1: Propose LEI expression;

[0208] S3.2: Sequential optimization using LEI criteria;

[0209] The LEI expression is:

[0210] LEI(x,l)=EI(x)α1(x,l)α2(l)α3(x,l)

[0211] Specifically,

[0212]

[0213] in:

[0214] (1) The EI acquisition function is:

[0215]

[0216] In the formula, and σ(x) are the mean and standard deviation at x in the Gaussian process model, and ymin is the optimal value observed so far. In addition, Φ represents the cumulative distribution function (CDF) of the multivariate Gaussian distribution, and φ represents the probability density function (PDF).

[0217] (8)α1(x) is the Pearson correlation coefficient:

[0218]

[0219] (9)α2 is the cost ratio of running the high-precision and low-precision systems once:

[0220]

[0221] (10)α3(x) is the constraint processing term

[0222]

[0223] like Figure 1 As shown in Figure 2, the sequential optimization process using the LEI criterion includes:

[0224] S3.2.1: Use genetic algorithm to maximize LEI to determine the next experimental design point location and fidelity level;

[0225] S3.2.2: Update the input portion of the training set;

[0226] S3.2.3: Run the multi-fidelity system and obtain response data;

[0227] S3.2.4: Calculate the difference of the optimization target data;

[0228] S3.2.5: Run the multi-fidelity constraint system and obtain response data;

[0229] S3.2.6: Calculate the difference of constraint data and update training data;

[0230] S3.2.7: Determine whether to update the current optimal point; if l * =n, then y min =min(Y n ); otherwise y min constant.

[0231] like Figure 1 As shown, S3.2.1 and S3.2.2 specifically include the following steps:

[0232] S3.2.1:

[0233]

[0234] S3.2.2:

[0235]

[0236] like Figure 1 As shown, S3.2.3 and S3.2.4 specifically include the following steps:

[0237] S3.2.3:

[0238] Y i =f i (X i ),i=1,2,...,l * ;

[0239] S3.2.4:

[0240] Z i =Y i -f i-1 (X i ),i=1,2,...,l * ;

[0241] like Figure 1 As shown, S3.2.5 and S3.2.6 specifically include the following steps:

[0242] S3.2.5:

[0243] G i =g i (X i ),i=1,2,...,l * ;

[0244] S3.2.6:

[0245] Calculate the difference: M i =G i -g i-1 (X i ),i=1,2,...,l * ;

[0246] Update data: D = {X1, M1, Z1} ∪ {X2, M2, Z2} ∪… ∪ {X n ,M n ,Z n};

[0247] like Figure 1 As shown, the stop condition in S4 is one of the following situations:

[0248] 1. The test resources have reached the preset value;

[0249] 2. The optimal value explored meets the preset accuracy;

[0250] 3. The maximum number of iterations is reached;

[0251] 4. The operation time reaches the preset value;

[0252] 5. Other reasonable conditions.

[0253] The above is only an embodiment of the present invention, and the common knowledge such as the known specific technical solutions and / or characteristics in the solution is not described in detail here. It should be pointed out that for those skilled in the art, without departing from the technical solution of the present invention, several modifications and improvements can be made, which should also be regarded as the protection scope of the present invention, and these will not affect the effect of the implementation of the present invention and the practicality of the patent. The scope of protection required by this application shall be based on the content of its claims, and the specific implementation methods and other records in the specification can be used to interpret the content of the claims.

Claims

1. A multi-fidelity Bayesian optimization method based on a multi-level expected improvement criterion, characterized in that: The following steps are involved: S1: Initialization sample set; S2: Build the MLGP model using the dataset; S3: Use multi-level EI criteria to sequentially collect points and iteratively optimize; S4: Determine whether the stopping condition is met. If so, output the result; if not, update the proxy model, return to S2, and continue the loop.

2. The multi-fidelity Bayesian optimization method based on the multi-level expected improvement criterion according to claim 1, characterized in that: The S1 specifically includes the following steps: S1.1: Use nested Latin square design to obtain initial sample locations; S1.2: Run the multi-fidelity system and obtain response data; S1.3: Calculate the difference of the optimized target data; S1.4: Run the multi-fidelity constraint system and obtain response data; S1.5: Calculate the difference of constraint data and obtain training data; The S1.1 specifically includes: {X1,X2,...,X n }=NLHS(Ω) Ω is the experimental design space, {X1,X2,...,X n } respectively represent the sample points of the first layer, the second layer, ..., the highest layer fidelity and 3. The multi-fidelity Bayesian optimization method based on the multi-level expected improvement criterion according to claim 2, characterized in that: S1.2 and S1.3 specifically include the following steps: S1.2: Run the multi-fidelity system and get response data Y i =f i (X i ),i=1,2,...,n S1.3: Calculate the difference of the optimization target data Z i =Y i -f i-1 (X i ),i=1,2,...,n like There is no need to re-run f i-1 , f0≡0.

4. The multi-fidelity Bayesian optimization method based on the multi-level expected improvement criterion according to claim 3, characterized in that: S1.4 and S1.5 specifically include the following steps: S1.4: Run the multi-fidelity constraint system and obtain response data G i =g i (X i ),i=1,2,...,n S1.5: Calculate the difference of constraint data M i =G i -g i-1 (X i ),i=1,2,...,n And get training data D={X1,G1,M1}∪{X2,G2,M2}∪…∪{X n ,G n ,M n }。 5. The multi-fidelity Bayesian optimization method based on the multi-level expected improvement criterion according to claim 4, characterized in that: The S2 specifically includes the following steps: S2.1: Build multiple independent Gaussian process regression models; S2.2: summing up the parameters to obtain the MLGP model that optimizes the objective function; S2.3: summing up the parameters to obtain the MLGP model of the constraint function; Wherein, the S2.1 specifically includes: Create a proxy model: F i (x)~GP(μ i (x),K(x,x′)) The model is controlled by a mean function μ(x) and a covariance function K(x, x') which defines the correlation of the function between different input points. The predicted mean and variance are calculated from it: Among them, μ(x*) is the prior mean at the input point x*, K(X, x*) is the covariance matrix between the input points X and x*, σ 2 n is the variance of the observation noise, and I is the identity matrix.

6. The multi-fidelity Bayesian optimization method based on the multi-level expected improvement criterion according to claim 5, characterized in that: The S2.2 specifically includes the following steps: The parameters are summed to obtain the MLGP model that optimizes the objective function. f n (x)~N(μ(x),s 2 (x)) in:

7. The multi-fidelity Bayesian optimization method based on the multi-level expected improvement criterion according to claim 6, characterized in that: The S2.3 specifically includes the following steps: Sum the parameters to get the MLGP model of the constraint function in:

8. The multi-fidelity Bayesian optimization method based on the multi-level expected improvement criterion according to claim 7, characterized in that: The S3 specifically includes the following steps: S3.1: Propose LEI expression; S3.2: Sequential optimization using LEI criteria; Wherein, the LEI expression is: LEI(x,l)=EI(x)α1(x,l)α2(l)α3(x,l) Specifically, in: (1) The EI acquisition function is: In the formula, and σ(x) are the mean and standard deviation at x in the Gaussian process model, and ymin is the optimal value observed so far. In addition, Φ represents the cumulative distribution function (CDF) of the multivariate Gaussian distribution, and φ represents the probability density function (PDF). (2)α1(x) is the Pearson correlation coefficient: (3) α2 is the cost ratio of running the high-precision and low-precision systems once: (4)α3(x) is the constraint processing term 9. The multi-fidelity Bayesian optimization method based on the multi-level expected improvement criterion according to claim 8, characterized in that: The sequential optimization process using the LEI criteria includes: S3.2.1: Use genetic algorithm to maximize LEI to determine the next experimental design point location and fidelity level; S3.2.2: Update the input portion of the training set; S3.2.3: Run the multi-fidelity system and obtain response data; S3.2.4: Calculate the difference of the optimization target data; S3.2.5: Run the multi-fidelity constraint system and obtain response data; S3.2.6: Calculate the difference of constraint data and update training data; S3.2.7: Determine whether to update the current optimal point; if l * =n, then y min =min(Y n ); otherwise y min constant.

10. The multi-fidelity Bayesian optimization method based on the multi-level expected improvement criterion according to claim 9, characterized in that: S3.2.1 and S3.2.2 specifically include the following steps: S3.2.1： S3.2.2： 11. The multi-fidelity Bayesian optimization method based on the multi-level expected improvement criterion according to claim 10, characterized in that: S3.2.3 and S3.2.4 specifically include the following steps: S3.2.3： Y i =f i (X i ),i=1,2,...,l * ; S3.2.4： Z i =Y i -f i-1 (X i ),i=1,2,...,l * 。 12. The multi-fidelity Bayesian optimization method based on the multi-level expected improvement criterion according to claim 11, characterized in that: S3.2.5 and S3.2.6 specifically include the following steps: S3.2.5： G i =g i (X i ),i=1,2,...,l * ; S3.2.6： Calculate the difference: M i =G i -g i-1 (X i ), i = 1, 2, ..., l * ; Update data: D = {X1, M1, Z1} ∪ {X2, M2, Z2} ∪ ... ∪ {X n , M n , Z n }.

13. The multi-fidelity Bayesian optimization method based on the multi-level expected improvement criterion according to claim 12, characterized in that: The stop condition in S4 is one of the following situations:

1. The test resources have reached the preset value; 2. The optimal value explored meets the preset accuracy; 3. The maximum number of iterations is reached; 4. The operation time reaches the preset value; 5. Other reasonable conditions.

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