Small-probability multi-failure mode system reliability analysis method fusing ICE-SYS and MRGP

Abstract

The invention discloses a small-probability multi-failure mode system reliability analysis method fusing ICE-SYS and MRGP. The small-probability multi-failure mode system reliability analysis method comprises the following steps: S1, generating an initial sample construction training set; s2, constructing a multi-response Gaussian process MRGP model according to the training set, executing a cross entropy important sampling method ICE-SYS of the system on the MRGP model, and taking the last layer of samples of the ICE-SYS as a candidate sample pool S; s3, calculating a system prediction response mu * (x) and a prediction standard deviation sigma * (x); s4, calculating a value [epsilon] r of an error-based stop criterion based on [mu] * (x) and [sigma] * (x); s5, if epsilon r is smaller than or equal to 0.01, executing the step S6; otherwise, selecting a new training sample # imgabs0 # from the S according to an error reduction learning function, calculating the response of the new training sample # imgabs0 #, adding the new training sample # imgabs0 # into the training set, and returning to the step S2; and S6, calculating a failure probability and a variable coefficient. The method solves the problems that an existing method is insufficient in calculation precision and low in calculation efficiency under multiple failure modes.

Description

Technical Field

[0001] The present invention relates to the technical field of reliability analysis, and in particular to a reliability analysis method for a system with low probability and multiple failure modes that integrates ICE-SYS and MRGP. Background Art

[0002] In the reliability analysis of engineering equipment, equipment failure is usually the combined result of multiple failure modes and involves small failure probabilities. Although the existing cross entropy importance sampling method (ICE-IS) can effectively evaluate small failure probabilities through sequential sampling, its calculation results are prone to deviation when faced with multiple failure modes, resulting in insufficient accuracy of reliability evaluation. Summary of the invention

[0003] The technical problem to be solved by the present invention is to provide a reliability analysis method for a system with low probability and multiple failure modes by integrating ICE-SYS and MRGP, so as to solve the problems of insufficient calculation accuracy and low calculation efficiency of the existing methods under multiple failure modes.

[0004] In order to solve the above technical problems, the technical solutions adopted by the present invention are as follows.

[0005] A reliability analysis method for a system with low probability and multiple failure modes integrating ICE-SYS and MRGP includes the following steps:

[0006] S1. Generate initial samples to construct training sets;

[0007] S2. Construct a multi-response Gaussian process MRGP model based on the training set, and perform the systematic cross entropy important sampling method ICE-SYS on the MRGP model, and use the last layer of samples of ICE-SYS as the candidate sample pool S;

[0008] S3. Calculate the system's predicted response μ * (x) and the prediction standard deviation σ * (x);

[0009] S4. Based on μ * (x) and σ * (x) Calculate the value of the stopping criterion ε based on the error r ;

[0010] S5. If ε r If it is less than or equal to 0.01, execute step S6; otherwise, select a new training sample from S according to the error reduction learning function. Calculate its response and add it to the training set and return to step S2;

[0011] S6. Calculate the failure probability and coefficient of variation.

[0012] Preferably, the step S1 is specifically as follows: using the 5-standard deviation criterion to generate N0 initial samples S train ; Call the function to calculate the response of the initial sample {g (1) (S train ), g (2) (S train ), …, g (M) (S train )}, and g (j) Represents the performance function of the jth component; constructs a training set based on the initial samples and their responses.

[0013] Preferably, in step S2, the new smooth indicator function is introduced into the existing cross entropy important sampling method ICE-IS to construct the cross entropy important sampling method ICE-SYS of the system, and the cross entropy important sampling method ICE-SYS of the system is executed on the MRGP model, which specifically includes the following steps:

[0014] S21. Initialization; specifically comprising the following steps:

[0015] S211. Set l=1, where l is the number of iterations of ICE-SYS, starting from 1;

[0016] S212. Select the distribution family and sampling density h(x; w l )’s initial parameter w l ;

[0017] S213. Set sample size N and target value δ target , the target value range is δ target ∈[1,4.5]; set σ0 = +∞ and Among them, σ0 is the initial value of the system smoothing parameter, is the initial value of the smoothing parameter of the jth component;

[0018] S22. Sampling; specifically includes the following steps:

[0019] S221. According to h(x; w l ) Draw N random samples;

[0020] S222. Call the function to calculate the response g of the component (j) (x)(j=1,2,…,M) and the system response g(x); for a series system, g(x)=min{g (1) (x),g (2) (x)…,g (M) (x)}; for a parallel system, g(x) = max{g (1) (x),g (2) (x)…,g (M) (x)};

[0021] S223. Solve the smoothing parameter σ corresponding to the system l , and solve for the smoothing parameter σ l The optimization problem is in, is Φ(-g(x i ) / σ)W l (x i ;v l The likelihood ratio of the standard deviation and mean of )(i=1,2,…,N); W l (x i ;v l ) is the weight of the sample; σ is a variable, which is the smoothing parameter to be solved;

[0022] S224. Evaluate sample weights If the coefficient of variation is greater than δ target Then execute step S23, otherwise execute step S24;

[0023] S23. Parameter update; specifically includes the following steps:

[0024] S231. Solve the smoothing parameters corresponding to each component, the smoothing parameters corresponding to the jth component The optimization problem is in, is Φ(-g (j) (x i ) / σ)W l (x i ;v l The likelihood ratio of the standard deviation and mean of (i=1,2,…,N);

[0025] S232. Constructing system smoothness indicator function h(x; σ l ); if the components are connected in series, then If components are connected in parallel,

[0026] S233. Estimate the optimization problem v in the lth iteration by the system smooth indicator function l The solution of the optimization problem is Among them, h(x i ; v) is the quasi-optimal important sampling density function; h(x i ; v) is the weight, which is the ratio of the original distribution and the quasi-optimal important sampling density function;

[0027] S234.Set w l+1 =v l and l=l+1, return to step S22;

[0028] S24. Estimate the failure probability based on the last layer of samples. The formula for estimating the failure probability is: Among them, I F (x) is the failure indicator function, when g(x)≤0, I F (x) is equal to 1, otherwise I F (x) is equal to 0; W(x i ;v T ) is the final weight, which is the ratio of the original distribution to the final estimated quasi-optimal important sampling density function.

[0029] Preferably, the specific steps of step S3 are: j (x) is the predicted response of the jth component, σ j (x) is the predicted standard deviation of the jth component;

[0030] In the case of a series system, if but if but Where M' is μ j (x)≤0(j=1,2,…,M'); the final μ * (x) = μ jj (x) and σ * (x) = σ jj (x);

[0031] In the case of a parallel system, if but if but Where M' is μ j (x)>0(j=1,2,…,M'); the final μ * (x) = μ jj (x) and σ * (x) = σ jj (x).

[0032] Preferably, the stopping criterion derivation in step S4 is specifically as follows:

[0033] According to μ * (x) and σ * (x), random variables are divided into the following four categories:

[0034] The first category: The predicted response and the true response are both greater than 0, and its mathematical expression is X1={x|μ * (x)>0;|μ * (x) / σ * (x)|>τ}; The second category: the predicted response and the true response are both less than 0, and its mathematical expression is X2={x|μ * (x)<0;|μ* (x) / σ * (x)|>τ}; The third category: the predicted response is greater than 0, but the actual response is less than 0, and its mathematical expression is X3={x|μ * (x)>0;|μ * (x) / σ * (x)|<τ}; The fourth category: the predicted response is less than 0, but the true response is greater than 0, and its mathematical expression is X4={x|μ * (x)<0;|μ * (x) / σ * (x)|<τ}; where τ is a constant that controls the confidence interval, which is 1.96;

[0035] The number of failed samples Expressed as Where I(·) is the counting factor; N2 is the number of samples in the second category, N3 is the number of samples in the third category, and N4 is the number of samples in the fourth category; the prediction accuracy of MRGP is defined as Therefore ε r It is expressed as:

[0036]

[0037] in, and It is the upper bound of N3 and N4.

[0038] Preferably, the derivation of the error reduction learning function in step S5 is specifically as follows:

[0039] The MRGP model constructed by the current training set is g GP (x DoE ), the response of each component follows a Gaussian distribution The expression of the error reduction learning function is Among them, x new is a new sample; Ω w is the candidate sample set consisting of the third and fourth category samples; r (x DoE ;x new ) is to minimize the prediction error of the updated MRGP; the predicted response of MRGP is used instead of the real response. If x is changed without changing the MRGP hyperparameters new The predicted response of is added to the training set, and the predicted response of MRGP remains unchanged, that is,

[0040]

[0041] The MRGP is reconstructed while keeping the hyperparameters unchanged to obtain the updated variance and then obtain new training samples.

[0042] Preferably, in step S6:

[0043] The failure probability is calculated as:

[0044]

[0045] The formula for calculating the coefficient of variation is:

[0046]

[0047] if Then output the failure probability, otherwise increase and return to step S2.

[0048] Due to the adoption of the above technical scheme, the technical progress achieved by the present invention is as follows.

[0049] The present invention achieves the following significant effects by proposing a systematic cross entropy importance sampling method (ICE-SYS) and a reliability analysis method combined with a multi-response Gaussian process (MRGP):

[0050] 1. Improve the calculation accuracy under multiple failure modes: By introducing a new smooth indicator function, ICE-SYS effectively improves the calculation deviation problem of the traditional cross entropy important sampling method under multiple failure modes, thereby significantly improving the reliability analysis accuracy of small failure probabilities.

[0051] 2. Improve computational efficiency: By combining ICE-SYS with MRGP and designing an error-based stopping criterion (EBSC), unnecessary surrogate model updates are avoided and the number of function calls is reduced, thus significantly reducing the overall computational resource requirements and time consumption. BRIEF DESCRIPTION OF THE DRAWINGS

[0052] Figure 1 is a flow chart of the present invention;

[0053] Figure 2 It is a cantilever beam model diagram of the present invention. DETAILED DESCRIPTION

[0054] The present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0055] A reliability analysis method for systems with low probability and multiple failure modes that integrates ICE-SYS and MRGP is proposed. The smooth indicator function is introduced into the existing cross entropy important sampling method ICE-IS to propose a systematic cross entropy important sampling method ICE-SYS, thereby improving the calculation accuracy. In order to further improve the solution efficiency, ICE-SYS is combined with the multi-response Gaussian process (MRGP), and the corresponding learning function and stopping criterion are designed. Figure 1As shown, the following steps are included:

[0056] S1. Generate initial samples to construct a training set.

[0057] Generate N0 initial samples S train In order to fully explore the failure domain, the 5-standard deviation criterion should be used to generate the initial sample. The function is called to calculate the response of the initial sample {g (1) (S train ), g (2) (S train ), …, g (M) (S train )},g (j) Represents the performance function of the jth component, and constructs a training set based on the initial samples and their responses.

[0058] S2. Construct a multi-response Gaussian process MRGP model based on the training set, and perform the systematic cross-entropy importance sampling method ICE-SYS on the MRGP model, and use the last layer of samples of ICE-SYS as the candidate sample pool S.

[0059] The new smooth indicator function is introduced into the existing cross entropy important sampling method ICE-IS to construct the systematic cross entropy important sampling method ICE-SYS. The implementation of the systematic cross entropy important sampling method ICE-SYS on the MRGP model specifically includes the following steps:

[0060] S21. Initialization.

[0061] The specific steps include:

[0062] S211. Set l=1, where l is the number of iterations of ICE-SYS, starting from 1.

[0063] S212. Select the distribution family and sampling density h(x; w l )’s initial parameter w l .

[0064] S213. Set sample size N and target value δ target In most cases, the target value range is δ target ∈[1,4.5]; set σ0 = +∞ and Among them, σ0 is the initial value of the system smoothing parameter, is the initial value of the smoothing parameter for the jth component.

[0065] S22. Sampling.

[0066] The specific steps include:

[0067] S221. According to h(x; w l ) Draw N random samples.

[0068] S222. Call the function to calculate the response g of the component (j) (x)(j=1,2,…,M) and the system response g(x); for a series system, g(x)=min{g (1) (x),g (2) (x)…,g (M) (x)}; for a parallel system, g(x) = max{g (1) (x),g (2) (x)…,g (M) (x)}.

[0069] S223. Solve the smoothing parameter σ corresponding to the system l . Solve for the smoothing parameter σ l The optimization problem is in, is Φ(-g(x i ) / σ)W l (x i ;v l The likelihood ratio of the standard deviation and mean of )(i=1,2,…,N); W l (x i ;v l ) is the weight of the sample; σ is a variable, which is the smoothing parameter to be solved.

[0070] S224. Evaluate sample weights If the coefficient of variation is greater than δ target Then execute step S23, otherwise execute step S24.

[0071] S23. Parameter update.

[0072] The specific steps include:

[0073] S231. Solve the smoothing parameters corresponding to each component, the smoothing parameters corresponding to the jth component The optimization problem is in, is Φ(-g (j) (x i ) / σ)W l (x i ;v l The likelihood ratio of the standard deviation and the mean of )(i=1,2,…,N).

[0074] S232. Constructing system smoothness indicator function h(x; σ l ). If the components are connected in series, then If components are connected in parallel,

[0075] S233. Estimate the optimization problem v in the lth iteration by the system smooth indicator function l The optimization problem is Among them, h(x i ; v) is the quasi-optimal important sampling density function; h(x i ; v) is the weight, which is the ratio of the original distribution and the quasi-optimal importance sampling density function.

[0076] S234.Set w l+1 =v l And l=l+1, return to step S22.

[0077] S24. Estimate the failure probability based on the last layer of samples. The formula for estimating the failure probability is: Among them, I F (x) is the failure indicator function, when g(x)≤0, I F (x) is equal to 1, otherwise I F (x) is equal to 0; W(x i ;v T ) is the final weight, which is the ratio of the original distribution to the final estimated quasi-optimal important sampling density function.

[0078] S3. Calculate the system's predicted response μ * (x) and the prediction standard deviation σ * (x).

[0079] The specific steps of this step are:

[0080] μ j (x) is the predicted response of the jth component, σ j (x) is the predicted standard deviation of the jth component.

[0081] In the case of a series system, if but if but Where M' is μ j (x)≤0(j=1,2,…,M'); the final μ * (x) = μ jj (x) and σ * (x) = σ jj (x).

[0082] In the case of a parallel system, if but if but Where M' is μ j (x)>0(j=1,2,…,M'); the final μ *(x) = μ jj (x) and σ * (x) = σ jj (x).

[0083] S4. Based on μ * (x) and σ * (x) Calculate the value of the error-based stopping criterion (EBSC) ε r .

[0084] The specific derivation of EBSC ​​in this step is:

[0085] According to μ * (x) and σ * (x), random variables are divided into the following four categories:

[0086] The first category: The predicted response and the true response are both greater than 0, and its mathematical expression is X1={x|μ * (x)>0;|μ * (x) / σ * (x)|>τ}; The second category: the predicted response and the true response are both less than 0, and its mathematical expression is X2={x|μ * (x)<0;|μ * (x) / σ * (x)|>τ}; The third category: the predicted response is greater than 0, but the actual response is less than 0, and its mathematical expression is X3={x|μ * (x)>0;|μ * (x) / σ * (x)|<τ}; The fourth category: the predicted response is less than 0, but the true response is greater than 0, and its mathematical expression is X4={x|μ * (x)<0;|μ * (x) / σ * (x)|<τ}; where τ is a constant that controls the confidence interval and is 1.96.

[0087] Therefore, the number of failed samples Expressed as Where I(·) is the counting factor; N2 is the number of samples in the second category, N3 is the number of samples in the third category, and N4 is the number of samples in the fourth category; the prediction accuracy of MRGP is defined as Therefore ε r It is expressed as:

[0088]

[0089] in, and It is the upper bound of N3 and N4.

[0090] S5. If ε r If it is less than or equal to 0.01, execute step S6; otherwise, select a new training sample from S according to the error reduction (ER) learning function Calculate its response And add it to the training set and return to step S2.

[0091] The derivation of the ER learning function in this step is as follows:

[0092] The MRGP model constructed by the current training set is g GP (x DoE ), so the response of each component follows a Gaussian distribution The purpose of the ER learning function is to add new samples x to the training set new , thereby minimizing the prediction error ε of the updated MRGP r (x DoE ;x new ), therefore, the expression of ER learning function is Among them, x new is a new sample; Ω w is the candidate sample set consisting of the third and fourth category samples; r (x DoE ;x new ) is to minimize the prediction error of the updated MRGP. The estimation of necessitates the use of the real function, which will cause computational burden. Therefore, the predicted response of MRGP is used instead of the real response. If x is changed without changing the MRGP hyperparameters, new The predicted response of is added to the training set, and the predicted response of MRGP remains unchanged, that is,

[0093]

[0094] However, the variance decreases in the region where new samples are added. Rebuilding the MRGP while keeping the hyperparameters unchanged can obtain the updated variance and thus obtain new training samples.

[0095] S6. Calculate the failure probability and coefficient of variation.

[0096] The failure probability is calculated as:

[0097]

[0098] The formula for calculating the coefficient of variation is:

[0099]

[0100] if Then output the failure probability, otherwise increase and return to step S2.

[0101] The following two specific examples demonstrate the implementation process and results.

[0102] Case 1: Four failure domains case

[0103] This case is characterized by four failure domains, which are composed of four components. The functional functions of the four components are:

[0104]

[0105] Among them, both random variables obey the standard normal distribution; the four components form a system in series, so when any component fails, the system fails. The calculation results of different methods are shown in Table 1:

[0106]

[0107] Table 1

[0108] As can be seen from Table 1, the Monte Carlo simulation (MCS) method calls 2×10 7 The function obtains an accurate value, and the calculated failure probability is used as a reference value. The calculation results of ICE-IS and ICE-SYS are both the average of 20 calculations. Compared with the existing ICE-IS, ICE-SYS has high calculation efficiency and high calculation accuracy, so the ICE-SYS proposed in the present invention is effective. Subsequently, compared with other existing system reliability analysis methods, it can be seen that the function function call number of the method proposed in the present invention is the least, only 19.60 times, and the calculation error is only 0.22%. Therefore, the method proposed in the present invention has high calculation accuracy and efficiency.

[0109] Case 2: Cantilever beam case

[0110] The cantilever beam model is Figure 2 As shown, it has three failure modes. The first failure mode is that the maximum stress of the cantilever beam exceeds the yield strength, and its function function is in:

[0111]

[0112] The second failure mode is when the maximum deflection of the cantilever beam exceeds a predefined threshold, and its function is g2 = 35-δ, where:

[0113]

[0114] I=wh 3 / 12

[0115]

[0116] The third failure mode is that the shear stress of the cantilever beam exceeds the shear strength, and its function is g3 = τ a -3R / 2wh.

[0117] When any of the three failure modes occurs, the system fails. The parameters of the cantilever beam are: a1 = 1.5, a2 = 4.5, b1 = 0.75, b2 = 2.5, c1 = 0.25, c2 = 1.75, d1 = 1.25, d2 = 4.75, L = 5.1 and E = 200 × 10 9 The distribution information of random variables is shown in Table 2:

[0118]

[0119]

[0120] Table 2

[0121] The calculation results of different methods are shown in Table 3:

[0122]

[0123] Table 3

[0124] As can be seen from Table 3, MCS calls 5×10 9 The function obtains an accurate value. Compared with the existing methods EMO-MMO-IS and FERUM-IS, the method ICE-SYS proposed in this application has higher calculation accuracy. Compared with FORM, ICE-SYS calculates the accurate failure probability. The failure probability error calculated by the existing ICE-IS is 84.74%, while the calculation error of ICE-SYS is only 0.02%. In addition, the function function call times of the method proposed in the present invention are less than that of ALK-MAIS-TCR. Therefore, the method proposed in the present invention has higher calculation accuracy and efficiency.

Claims

1. A reliability analysis method for a system with low probability and multiple failure modes that integrates ICE-SYS and MRGP, characterized by: The following steps are involved: S1. Generate initial samples to construct training sets; S2. Construct a multi-response Gaussian process MRGP model based on the training set, and perform the systematic cross entropy important sampling method ICE-SYS on the MRGP model, and use the last layer of samples of ICE-SYS as the candidate sample pool S; S3. Calculate the system's predicted response μ * (x) and the prediction standard deviation σ * (x); S4. Based on μ * (x) and σ * (x) Calculate the value of the stopping criterion ε based on the error r ; S5. If ε r If it is less than or equal to 0.01, execute step S6; otherwise, select a new training sample from S according to the error reduction learning function. Calculate its response and add it to the training set and return to step S2; S6. Calculate the failure probability and coefficient of variation.

2. According to claim 1, a reliability analysis method for a system with low probability and multiple failure modes integrating ICE-SYS and MRGP is characterized by: The step S1 is specifically as follows: using the 5-standard deviation criterion to generate N0 initial samples S train ; Call the function to calculate the response of the initial sample {g (1) (S train ), g (2) (S train ), …, g (M) (S train )}, and g (j) Represents the performance function of the jth component; constructs a training set based on the initial samples and their responses.

3. According to claim 2, a reliability analysis method for a system with low probability and multiple failure modes integrating ICE-SYS and MRGP is characterized by: In step S2, the new smooth indicator function is introduced into the existing cross entropy important sampling method ICE-IS to construct the cross entropy important sampling method ICE-SYS of the system. The cross entropy important sampling method ICE-SYS of the system is executed on the MRGP model, which specifically includes the following steps: S21. Initialization; specifically comprising the following steps: S211. Set l=1, where l is the number of iterations of ICE-SYS, starting from 1; S212. Select the distribution family and sampling density h(x; w l )’s initial parameter w l ; S213. Set sample size N and target value δ target , the target value range is δ target ∈[1,4.5]; set σ0 = +∞ and Among them, σ0 is the initial value of the system smoothing parameter, is the initial value of the smoothing parameter of the jth component; S22. Sampling; specifically includes the following steps: S221. According to h(x; w l ) Draw N random samples; S222. Call the function to calculate the response g of the component (j) (x)(j=1,2,…,M) and the system response g(x); for a series system, g(x)=min{g (1) (x),g (2) (x)…,g (M) (x)}; for a parallel system, g(x) = max{g (1) (x),g (2) (x)…,g (M) (x)}; S223. Solve the smoothing parameter σ corresponding to the system l , and solve for the smoothing parameter σ l The optimization problem is in, is Φ(-g(x i ) / σ)W l (x i ;v l The likelihood ratio of the standard deviation and mean of )(i=1,2,…,N); W l (x i ;v l ) is the weight of the sample; σ is a variable, which is the smoothing parameter to be solved; S224. Evaluate sample weights If the coefficient of variation is greater than δ target Then execute step S23, otherwise execute step S24; S23. Parameter update; specifically includes the following steps: S231. Solve the smoothing parameters corresponding to each component, the smoothing parameters corresponding to the jth component The optimization problem is in, is Φ(-g (j) (x i ) / σ)W l (x i ;v l The likelihood ratio of the standard deviation and mean of (i=1,2,…,N); S232. Constructing system smoothness indicator function h(x; σ l ); if the components are connected in series, then If components are connected in parallel, S233. Estimate the optimization problem v in the lth iteration by the system smooth indicator function l The solution of the optimization problem is Among them, h(x i ; v) is the quasi-optimal important sampling density function; h(x i ; v) is the weight, which is the ratio of the original distribution and the quasi-optimal important sampling density function; S234.Set w l+1 =v l and l=l+1, return to step S22; S24. Estimate the failure probability based on the last layer of samples. The formula for estimating the failure probability is: Among them, I F (x) is the failure indicator function, when g(x)≤0, I F (x) is equal to 1, otherwise I F (x) is equal to 0; W(x i ;v T ) is the final weight, which is the ratio of the original distribution to the final estimated quasi-optimal important sampling density function.

4. According to claim 3, a reliability analysis method for a system with low probability and multiple failure modes integrating ICE-SYS and MRGP is characterized in that: The specific steps of step S3 are: j (x) is the predicted response of the jth component, σ j (x) is the predicted standard deviation of the jth component; In the case of a series system, if but if but Where M' is μ j (x)≤0(j=1,2,…,M'); the final μ * (x) = μ jj (x) and σ * (x) = σ jj (x); In the case of a parallel system, if but if but Where M' is μ j (x)>0(j=1,2,…,M'); the final μ * (x) = μ jj (x) and σ * (x) = σ jj (x).

5. The reliability analysis method of a system with low probability and multiple failure modes integrating ICE-SYS and MRGP according to claim 4 is characterized by: The stop criterion derivation in step S4 is specifically as follows: According to μ * (x) and σ * (x), random variables are divided into the following four categories: The first category: The predicted response and the true response are both greater than 0, and its mathematical expression is X1 = {xμ * (x)>0;μ * (x) / σ * (x)>τ}; The second category: the predicted response and the true response are both less than 0, and its mathematical expression is X2={xμ * (x)<0;μ * (x) / σ * (x)|>τ}; The third category: the predicted response is greater than 0, but the true response is less than 0, and its mathematical expression is X3={xμ * (x)>0;μ * (x) / σ * (x)<τ}; The fourth category: the predicted response is less than 0, but the actual response is greater than 0, and its mathematical expression is X4={xμ * (x)<0;μ * (x) / σ * (x)<τ}; where τ is a constant that controls the confidence interval, which is 1.96; The number of failed samples Expressed as Where I(·) is the counting factor; N2 is the number of samples in the second category, N3 is the number of samples in the third category, and N4 is the number of samples in the fourth category; the prediction accuracy of MRGP is defined as Therefore ε r It is expressed as: in, and It is the upper bound of N3 and N4.

6. The reliability analysis method of a system with low probability and multiple failure modes integrating ICE-SYS and MRGP according to claim 5 is characterized by: The derivation of the error reduction learning function in step S5 is specifically as follows: The MRGP model constructed by the current training set is g GP (x DoE ), the response of each component follows a Gaussian distribution The expression of the error reduction learning function is Among them, x new is a new sample; Ω w is the candidate sample set consisting of the third and fourth category samples; r (x DoE ;x new ) is to minimize the prediction error of the updated MRGP; the predicted response of MRGP is used instead of the real response. If x is changed without changing the MRGP hyperparameters new The predicted response of is added to the training set, and the predicted response of MRGP remains unchanged, that is, The MRGP is reconstructed while keeping the hyperparameters unchanged to obtain the updated variance and then obtain new training samples.

7. The reliability analysis method of a system with low probability and multiple failure modes integrating ICE-SYS and MRGP according to claim 6 is characterized by: In step S6: The failure probability is calculated as: The formula for calculating the coefficient of variation is: if Then output the failure probability, otherwise increase and return to step S2.

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