Complex structure active learning reliability analysis method based on RBF (Radial Basis Function) model under minimum failure probability

Abstract

The invention discloses a complex structure active learning reliability analysis method based on an RBF (Radial Basis Function) model under a minimum failure probability, which is characterized in that a radial basis function (RBF) model is combined with active learning, the RBF model is used as an agent model to approximate a real performance function, and an important sampling technology is used to improve the sampling efficiency near a failure area. By combining a multi-modal optimization technology, the method can identify and utilize a plurality of most probable failure points (MPPs), thereby more accurately estimating the failure probability. According to the AL-RBF-I S method provided by the invention, the number of required training points and the calculation time can be remarkably reduced while the accuracy is ensured, particularly, the performance is excellent when the problem of small failure probability is handled, and the analysis on the static strength reliability of the bogie frame is more convenient.

Description

Technical Field

[0001] The invention relates to the technical field of active learning radial basis function (RBF), in particular to an active learning reliability analysis method for complex structures based on an RBF model under extremely small failure probability. Background Art

[0002] The bogie frame is an important component of locomotives and the main load-bearing component of the running gear of rail vehicles. The safety and reliability of the bogie frame directly affect the safety of rail vehicles. Therefore, the accurate evaluation of the reliability of the bogie frame is of great significance to ensure the safety of railway transportation. Most domestic scholars focus on the research of bogie frames in terms of static strength, fatigue life, fatigue strength analysis, etc. In this type of analysis, the basic parameters are regarded as definite values, and the calculated results are also deterministic results. However, in engineering practice, the structural strength of the bogie frame is often affected by many uncertain factors, such as material properties, geometric parameters, and loads. Therefore, combining reliability analysis theory with finite element method can more appropriately and reasonably judge the static performance of the bogie frame in the design stage.

[0003] So far, structural reliability analysis methods can be roughly divided into three categories. The first category is the analytical method represented by the first-order reliability method (FORM) and the second-order reliability method (SORM), that is, the Taylor expansion approximates the limit state surface (LSS) at the point closest to the origin (MPP) in the standard normal space. The second category of methods is the digital sampling method, such as the Monte Carlo (MCS) simulation method, subset simulation (SS method, importance sampling (IS) method), etc. The third category is the reliability method based on the surrogate model. Common surrogate models include the response surface method (RSM), the polynomial chaos expansion (PCE) model, the neural network model, the Kriging model, the radial basis function (RBF) model, etc. In recent years, researchers have continuously deepened the research on the AL-KRG model, thereby improving the efficiency of the AL-KRG model and enriching the application scenarios of the AL-KRG model. These studies are mainly divided into the following aspects: ① Point addition criteria. By constructing a more reasonable point addition criterion, better training points can be selected. ② Demand-oriented convergence criteria.

[0004] In the field of approximate models, there is no proxy model whose performance always exceeds that of other proxy models. Therefore, in addition to the Kriging model, other types of proxy models have also been widely studied in the field of structural reliability analysis. Some studies have shown that in fields with high nonlinearity, the RBF model performs better than the Kriging model. Considering that the sample size of important sampling for small probability failure problems is much smaller than that of the traditional Monte Carlo method, the present invention proposes an active learning method based on radial basis function RBF for the static strength reliability analysis of bogie frames. Summary of the invention

[0005] In order to solve the problems existing in the prior art, the purpose of the present invention is to provide an active learning reliability analysis method for complex structures based on RBF model under extremely small failure probability. The present invention first uses the Jackknife cross-validation method to obtain the uncertain parameters of the RBF model, obtains the agent's limit state plane (LSS) according to the uncertain parameters of the RBF model, and then uses the evolutionary multi-objective optimization multi-modal optimization (EMO-MMO) algorithm to obtain the MPP, and establishes an auxiliary probability density function (iPDF) considering the weight of the MPP; finally, according to the learning function, new training points are continuously added to make the RBF model sufficiently accurate, and the RBF model that reaches the convergence threshold is used to solve the structural failure probability.

[0006] To achieve the above object, the technical solution adopted by the present invention is: a complex structure active learning reliability analysis method based on RBF model under extremely small failure probability, comprising the following steps:

[0007] Step 1: Convert the random variable into the standard normal space, use the Latin hypercube sampling method LHS to generate a small number of initial training points, calculate the true value of the performance function, and form the initial experimental design DoE;

[0008] Step 2: Train the initial RBF model according to the initial experimental design DoE;

[0009] Step 3: Obtain the proxy limit state plane LSS according to the current RBF model prediction information;

[0010] Step 4, obtaining the point MPP closest to the origin in multiple standard normal spaces according to the multi-modal multi-objective optimization algorithm EMO-MMO;

[0011] Step 5: Calculate the weight of the point MPP closest to the origin in the standard normal space and establish the auxiliary probability density function iPDF to generate N IS An important sample;

[0012] Step 6: Determine whether the RBF model reaches a preset convergence threshold;

[0013] Step 7: When the RBF model is not less than the preset convergence threshold, the active learning function is used to obtain the optimal training point;

[0014] Step 8: Add the optimal training points to the initial experimental design DoE, update the RBF model, and use the RBF model that reaches the convergence criterion to predict the failure probability.

[0015] As a further improvement of the present invention, the step 1 is specifically as follows:

[0016] Let the random variable be x=[x1,…,x k ], after being transformed into the standard normal space, it becomes u=[u1,…,u k ], calculate the true value of the performance function and form the initial experimental design DoE (u (i) ,G(u (i) ))(i=1,…,k).

[0017] As a further improvement of the present invention, in step 2, the performance function G(u) is expressed by the RBF model as follows:

[0018]

[0019] in, represents the basis function vector; β=[β1,…,β k ] T represents the weight coefficient vector; k is the number of radial basis functions; That is the radial basis function; r u,i =||Uu (i) || represents any point u and u (i) The spatial distance between them; c is the shape parameter of the radial basis function;

[0020] Substituting the initial experimental design DoE into (1) yields:

[0021]

[0022] Its matrix form is:

[0023] G=Φβ (3)

[0024] Among them, G=[(G(u (1) ),…,G(u k )] T ; Φ is the RBF matrix of the training point position; then the weight coefficient vector β is:

[0025] β=Φ -1 G(4)

[0026] The predicted response at the unknown point can be estimated using the weight coefficient vector β and formula (1); the shape parameter c of the radial basis function has a great influence on the accuracy of the RBF model, and cross-validation technology is usually required for correction.

[0027] As a further improvement of the present invention, the step 3 is specifically as follows:

[0028] The prediction mean and variance of the current RBF model are obtained using the cross-validation method; the current DoE is (u (i) ,G(u (i)))(i=1,…,k), if a training point (u (i) ,G(u (i) )), then the RBF model becomes:

[0029] G (-i) =Φ (-i) β (-i) (5)

[0030] Among them, G (-i) means G removes G(u (i) ) after the vector; Φ (-i) is the matrix obtained by deleting the elements of the i-th row and i-th column of Φ; β (-i) is the new coefficient vector; remove a training point (u (i) ,G(u (i) ), the predicted value of the RBF model at the unknown point is:

[0031]

[0032] in, By delete The i-th column element is obtained; by using the block inversion operation, the overall time complexity is reduced to o(m 3 ), as shown below:

[0033]

[0034] After obtaining the predicted values ​​of k sub-models, the predicted mean and predicted standard deviation of u are defined according to the Jackknife method:

[0035]

[0036] and

[0037] The limit state plane LSS divides the random space into failure zone and safety zone; the uncertainty of prediction makes it impossible for the RBF model to distinguish the failure zone from the safety zone; considering the uncertainty, the area with a certain probability or confidence to become the failure zone is expressed as:

[0038]

[0039] Among them, α is a constant that controls the confidence level; the area where the positive and negative signs of the function have a certain probability of being wrongly predicted is:

[0040]

[0041] use and The set of ensures that no potential failure area is missed, expressed as:

[0042]

[0043] Proxy failure area The control function is:

[0044]

[0045] That is the proxy limit state plane LSS.

[0046] As a further improvement of the present invention, the step 4 is specifically as follows:

[0047] In the standard normal space, the optimal solution to the constrained optimization problem is MPP, which is expressed as:

[0048]

[0049] in, is the specified search area;

[0050] Equation (14) is solved based on the multi-modal multi-objective optimization algorithm EMO-MMO.

[0051] As a further improvement of the present invention, the step 5 is specifically as follows:

[0052] use represents the MPPs obtained using the multimodal multi-objective optimization algorithm EMO-MMO. The weight of each MPP is defined as:

[0053]

[0054] in, is the cumulative function of the standard normal distribution. The function value calculated at , then the auxiliary probability density function i PDF is expressed as:

[0055]

[0056] in:

[0057]

[0058] Generate N according to the importance sampling method IS Obey important sample.

[0059] As a further improvement of the present invention, in step 6, the convergence condition is defined as:

[0060]

[0061] in, and are the rationality measure and belief measure of the system failure region respectively; γ is the convergence threshold; and The calculation formula is:

[0062]

[0063] Among them, I F (u) is the failure indicator function; φ(u) is the probability density function PDF of the standard normal space.

[0064] As a further improvement of the present invention, the step 7 is specifically as follows:

[0065] The active learning function is:

[0066]

[0067] The variance of |G(u)| is:

[0068]

[0069] Considering both expectation and variance, we get the coefficient of variation CV as the learning criterion:

[0070]

[0071] Formula (23) is named CV |G| Function, CV |G| The maximum value of means that the point is close to the agent LSS, that is, the value of E(|G(u)|) is small, or has a large prediction uncertainty, that is, the value of V(|G(u)|) is large, or both; then the best training point for each iteration is:

[0072] u*=argmaxCV |G| (u) (24).

[0073] As a further improvement of the present invention, in step 8, the failure probability is estimated as:

[0074]

[0075] The coefficient of variation is expressed as:

[0076]

[0077] The variance of the failure probability is expressed as:

[0078]

[0079] The present invention also provides an application of an active learning reliability analysis method for complex structures based on an RBF model under extremely small failure probability, and the static strength of the bogie frame is analyzed according to the active learning reliability analysis method for complex structures based on an RBF model under extremely small failure probability as described above.

[0080] The beneficial effects of the present invention are:

[0081] 1. The present invention combines the RBF model with active learning and uses the importance sampling technique to improve the sampling efficiency near the failure area. By combining the multimodal optimization technique, the method can identify and utilize multiple most likely failure points (MPPs) to estimate the failure probability more accurately.

[0082] 2. The present invention uses the coefficient of variation (CV) of the functional function |G(u)| as a learning standard, which can significantly reduce the number of required training points and calculation time while ensuring accuracy, thereby improving the calculation efficiency of the static strength reliability analysis problem of the bogie frame. BRIEF DESCRIPTION OF THE DRAWINGS

[0083] Figure 1 is a flow chart of an embodiment of the present invention;

[0084] Figure 2 This is the area with a high probability of failure in the embodiment of the present invention. and the region where the sign is incorrectly predicted Schematic diagram of

[0085] Figure 3 This is a schematic diagram of a 23-bar truss structure in an embodiment of the present invention;

[0086] Figure 4 It is a schematic diagram of the finite element model of the bogie frame in an embodiment of the present invention. DETAILED DESCRIPTION

[0087] The embodiments of the present invention are described in detail below with reference to the accompanying drawings.

[0088] Example

[0089] This embodiment is described by taking the reliability analysis of the static strength of the bogie frame as an example. Figure 1 As shown, a bogie frame static strength reliability analysis method based on active learning radial basis function RBF includes:

[0090] Step S1: transform the random variables into the standard normal space, use the Latin hypercube (LHS) sampling method to generate a small number of initial training points, calculate the true value of the performance function, and form an initial experimental design (DoE);

[0091] The random variable is x=[x1,…,x k], after being transformed into the standard normal space, it becomes u=[u1,…,u k ], calculate the true value of the performance function and form the initial experimental design (DoE) (u (i) ,G(u (i) ))(i=1,…,k).

[0092] Step S2: training the initial RBF model according to the initial DoE;

[0093] The performance function G(u) is expressed by the RBF model as:

[0094]

[0095] in, represents the basis function vector; β=[β1,…,β k ] T represents the weight coefficient vector; k is the number of radial basis functions; This is the radial basis function. u,i =||uu (i) || represents any point u and u (i) The spatial distance between them; c is the shape parameter of the radial basis function.

[0096] Substituting the initial DoE into equation (1) yields:

[0097]

[0098] Its matrix form is:

[0099] G=Φβ (3)

[0100] Among them, G=[G(u (1) ),…,G(u k )] T ; Φ is the RBF matrix of the training point. Then the coefficient vector β is:

[0101] β=Φ -1 G(4)

[0102] The predicted response at the unknown point can be estimated using the weight coefficient vector β and formula (1). The shape parameter c has a great influence on the accuracy of the RBF model and usually requires cross-validation technology to correct it.

[0103] Step S3: Obtain the proxy limit state plane LSS according to the current RBF model prediction information;

[0104] The prediction mean and variance of the current RBF model can be obtained using the cross-validation method. The current DoE is for:

[0106] G (-i) =Φ (-i) β (-i) (5)

[0107] Among them, G (-i) means G removes G(u (i) ) after the vector; Φ (-i) is the matrix obtained by deleting the elements of the i-th row and i-th column of Φ; β (-i) is the new coefficient vector. Remove a training point (u (i) ,G(u (i) )), the prediction value of the RBF model at the unknown point is:

[0108]

[0109] in, By delete The i-th column element of is obtained. In formula (6), there exists Φ (-i) The inverse solution of 3 ). The Jackknife method requires m inverse operations to estimate the predicted mean and standard deviation of the model. Therefore, the total time complexity is o(m 4 ). By using the block inversion operation, the overall time complexity can be reduced to o(m 3 ), as shown in the following formula.

[0110]

[0111] After obtaining the predicted values ​​of k sub-models, the predicted mean and predicted standard deviation of u can be defined according to the Jackknife method:

[0112]

[0113] and

[0114] like Figure 2 As shown in Figure 1, the limit state plane LSS can divide the random space into failure zone and safety zone. However, the uncertainty of the prediction makes it impossible for the RBF model to distinguish the failure zone from the safety zone. Considering the uncertainty, the area with a greater probability or confidence to become the failure area is expressed as:

[0115]

[0116] Where α is a constant that controls the confidence level. If α is set to 1.96, the confidence level will be 95%. The area where the function sign is more likely to be mispredicted is:

[0117]

[0118] use and The set of can ensure that any potential failure area is not missed, which is expressed as:

[0119]

[0120] Proxy failure area The control function is:

[0121]

[0122] It is the agent LSS.

[0123] Step S4: Obtain multiple MPPs according to the EMO-MMO algorithm.

[0124] In the standard normal space, the optimal solution to the constrained optimization problem is the MPP, which is expressed as:

[0125]

[0126] in, is the specified search area; in practical applications, G(u) is likely to be a multimodal function, so It is also multimodal.

[0127] In addition, the LSS obtained from the initial RBF model will be very inaccurate. At this time, although G(u) is unimodal, it does not mean that It is also a single peak. It is very likely to be multimodal, so there may be multiple global optimal solutions for equation (14). Even if there is only one global optimal solution for equation (14), the contribution of some local optimal solutions to the failure probability integral cannot be directly ignored. If the iPDF is constructed with only one MPP as the center, the error of biased estimation will become very large. Therefore, searching for all MPPs on the proxy LSS is meaningful for improving the accuracy of model estimation. Many optimization problems in actual engineering belong to multimodal optimization (MMO), which is to find multiple optimal solutions in one run. The multimodal optimization (EMO-MMO) algorithm based on evolutionary multi-objective optimization is used to solve equation (14).

[0128] Step S5: Calculate the MPP weight and establish iPDF to generate N IS An important sample;

[0129] The EMO-MMO algorithm can find multiple solutions that are mixed global optimal solutions, local optimal solutions, and even inferior solutions. This requires the establishment of an iPDF that can take into account the weight of each solution, so as to avoid wasting training points by allocating them to unimportant failure areas.

[0130] use represents the MPPs obtained by the EMO-MMO algorithm, and the weight of each MPP is defined as:

[0131]

[0132] in, is the cumulative function of the standard normal distribution. The function value calculated at , then iPDF can be expressed as:

[0133]

[0134] in:

[0135]

[0136] Generate N according to the importance sampling method IS Obey important sample.

[0137] Step S6: Determine whether the RBF model is accurate enough and reaches a preset convergence threshold;

[0138] The convergence condition is defined as:

[0139]

[0140] in, and are the rationality measure and confidence measure of the system failure area respectively; γ=0.95 is the convergence threshold specified in this paper. and The calculation formula is:

[0141]

[0142] Among them, I F (u) is the failure indicator function; φ(u) is the probability density function PDF of the standard normal space.

[0143] Step S7: When the RBF model is not less than the preset convergence threshold, the active learning function is used to obtain the optimal training point;

[0144] The active learning function is as follows:

[0145]

[0146] The variance of |G(u)| is:

[0147]

[0148] Taking both expectation and variance into account, we get the coefficient of variation (CV) as the learning criterion:

[0149]

[0150] Formula (23) is named CV |G| Function, CV |G| The maximum value of means that the point is very close to the agent LSS (E(|G(u)|) has a small value) or has a large prediction uncertainty (V(|G(u)|) has a large value), or both. Therefore, the optimal training point for each iteration is:

[0151] u * =argmaxCV |G| (u) (24)

[0152] Step S8: Add the optimal training point to the DoE, update the RBF model, and use the RBF model that reaches the convergence criterion to predict the failure probability.

[0153] The failure probability can be estimated as:

[0154]

[0155] The coefficient of variation can be expressed as:

[0156]

[0157] The variance of the failure probability can be expressed as:

[0158]

[0159] The 23-bar truss structure is used as a test case to verify the above steps. Figure 3 As shown, the structure is subjected to 6 vertical forces. There are 10 independent random variables in this case, which are the elastic modulus (E1 and E2) and cross-sectional area (A1 and A2) of the horizontal and diagonal bars and the 6 vertical forces. The specific distribution is shown in Table 1. The function is defined as:

[0160] G(u)=160-Δ(u) (28)

[0161] where Δ is the deflection at the midpoint of the truss base and u is a vector of random variables.

[0162] Table 1

[0163]

[0164] Table 2 shows the calculation results of different methods. It can be seen that the AL-RBF-IS method proposed in this embodiment requires the least number of function calls, which reflects the advantages of the AL-RBF-IS method.

[0165] Table 2

[0166]

[0167] The reliability analysis method of this application is verified using a bogie frame static strength reliability analysis case.

[0168] The bogie frame finite element model was established in HyperMesh software, using COMBIN14 units to simulate the primary and secondary suspensions; BEAM188 beam units to simulate components such as wheelsets, swing arm positioning, and axle boxes; and MASS21 mass units to simulate the mass of components such as motors and gearboxes. The bogie frame is mainly meshed using Shell181 shell units, and some complex components use Solid185 entity units. The total number of nodes in the frame finite element model is 178,622, and the total number of units is 771,466. The specific finite element model is as follows: Figure 4 shown.

[0169] This application selects the worst abnormal load condition as the reliability analysis condition, considering the influence of 9 uncertain factors such as load, plate thickness, material properties, etc. The main parameters of the framework are shown in Table 3. The functional function of the framework is defined as:

[0170] G(u)=σ s -σ(u) max (29)

[0171] Among them, σ s is the yield strength of the material, σ max is the equivalent stress on the frame under abnormal load conditions, u=(u1,u2,…,u n ) represents the input random variable, and G(u)≤0 indicates structural failure.

[0172] Table 3

[0173]

[0174] The random input variable settings of the frame are shown in Table 4, and the calculation results are shown in Table 5. The reliability analysis of the bogie frame belongs to the problem of small failure probability. The AL-RBF-IS method is used for calculation. Only 44 real functional functions are called, and the initial sample points are 12. Then 32 new sample points are added in the learning process. The failure probability is solved as P f =3.40×10 -7, the coefficient of variation is CoV = 3.49%.

[0175] Table 4

[0176]

[0177] Table 5

[0178]

[0179] The above case results show that the AL-RBF-IS method proposed in this embodiment can well solve the problem of static strength reliability of the bogie frame. From the results, it can be seen that even in the face of extremely small failure probability problems, the AL-RBF-IS method can obtain accurate results and has certain advantages over other algorithms. When solving implicit problems in actual engineering, the AL-RBF-IS method also performs well.

[0180] The above-mentioned embodiments only express the specific implementation of the present invention, and the description thereof is relatively specific and detailed, but it cannot be understood as limiting the scope of the present invention. It should be pointed out that, for ordinary technicians in this field, several variations and improvements can be made without departing from the concept of the present invention, which all belong to the protection scope of the present invention.

Claims

1. A complex structure active learning reliability analysis method based on RBF model under extremely small failure probability, characterized by: The following steps are involved: Step 1: Convert the random variable into the standard normal space, use the Latin hypercube sampling method LHS to generate a small number of initial training points, calculate the true value of the performance function, and form the initial experimental design DoE; Step 2: Train the initial RBF model according to the initial experimental design DoE; Step 3: Obtain the proxy limit state plane LSS according to the current RBF model prediction information; Step 4, obtaining the point MPP closest to the origin in multiple standard normal spaces according to the multi-modal multi-objective optimization algorithm EMO-MMO; Step 5: Calculate the weight of the point MPP closest to the origin in the standard normal space and establish the auxiliary probability density function iPDF to generate N IS An important sample; Step 6: Determine whether the RBF model reaches a preset convergence threshold; Step 7: When the RBF model is not less than the preset convergence threshold, the active learning function is used to obtain the optimal training point; Step 8: Add the optimal training points to the initial experimental design DoE, update the RBF model, and use the RBF model that reaches the convergence criterion to predict the failure probability.

2. The active learning reliability analysis method for complex structures based on RBF model under minimal failure probability according to claim 1 is characterized in that: The step 1 is specifically as follows: Let the random variable be x=[x1,…,x k ], after being transformed into the standard normal space, it becomes u=[u1,…,u k ], calculate the true value of the performance function and form the initial experimental design DoE (u (i) ,G(u (i) ))(i=1,…,k).

3. The active learning reliability analysis method for complex structures based on RBF model under minimal failure probability according to claim 2 is characterized in that: In step 2, the performance function G(u) is expressed by the RBF model as: in, represents the basis function vector; β=[β1,…,β k ] T represents the weight coefficient vector; k is the number of radial basis functions; That is the radial basis function; r u,i =||uu (i) || represents any point u and u (i) The spatial distance between them; c is the shape parameter of the radial basis function; Substituting the initial experimental design DoE into (1) yields: Its matrix form is: G=φβ (3) Among them, G=[G(u( 1 )),…,G(u k )] T ; Φ is the RBF matrix of the training point position; then the weight coefficient vector β is: β=Φ -1 G (4) The predicted response at the unknown point can be estimated using the weight coefficient vector β and formula (1); the shape parameter c of the radial basis function has a great influence on the accuracy of the RBF model, and cross-validation technology is usually required for correction.

4. The active learning reliability analysis method for complex structures based on RBF model under minimal failure probability according to claim 3 is characterized in that: The step 3 is as follows: The prediction mean and variance of the current RBF model are obtained using the cross-validation method; the current DoE is (u (i) ,G(u (i) ))(i=1,…,k), if a training point (u (i) ,G(u (i) )), then the RBF model becomes: G (-i) =Φ (-i) b (-i) (5) Among them, G (-i) means G removes G(u (i) ) after the vector; Φ (-i) is the matrix obtained by deleting the elements of the i-th row and i-th column of Φ; β (-i) is the new coefficient vector; remove a training point (u (i) ,G(u (i) )), the prediction value of the RBF model at the unknown point is: in, By delete The i-th column element is obtained; by using the block inversion operation, the overall time complexity is reduced to o(m 3 ), as shown below: After obtaining the predicted values ​​of k sub-models, the predicted mean and predicted standard deviation of u are defined according to the Jackknife method: and The limit state plane LSS divides the random space into failure zone and safety zone; the uncertainty of prediction makes it impossible for the RBF model to distinguish the failure zone from the safety zone; considering the uncertainty, the area with a certain probability or confidence to become the failure zone is expressed as: Among them, α is a constant that controls the confidence level; the area where the positive and negative signs of the function have a certain probability of being mispredicted is: use and The set of ensures that no potential failure area is missed, which is expressed as: Proxy failure area The control function is: That is the proxy limit state plane LSS.

5. The complex structure active learning reliability analysis method based on RBF model under minimal failure probability according to claim 4 is characterized in that: The step 4 is specifically as follows: In the standard normal space, the optimal solution to the constrained optimization problem is MPP, which is expressed as: in, is the specified search area; Equation (14) is solved based on the multi-modal multi-objective optimization algorithm EMO-MMO.

6. The active learning reliability analysis method for complex structures based on RBF model under minimal failure probability according to claim 5 is characterized in that: The step 5 is specifically as follows: use represents the MPPs obtained using the multimodal multi-objective optimization algorithm EMO-MMO. The weight of each MPP is defined as: in, is the cumulative function of the standard normal distribution. The function value calculated at , then the auxiliary probability density function iPDF is expressed as: in: Generate N according to the importance sampling method IS Obey important sample.

7. The complex structure active learning reliability analysis method based on RBF model under minimal failure probability according to claim 6 is characterized in that: In step 6, the convergence condition is defined as: in, and are the rationality measure and belief measure of the system failure region respectively; γ is the convergence threshold; and The calculation formula is: Among them, I F (u) is the failure indicator function; φ(u) is the probability density function PDF of the standard normal space.

8. The complex structure active learning reliability analysis method based on RBF model under minimal failure probability according to claim 7 is characterized in that: The step 7 is specifically as follows: The active learning function is: The variance of |G(u)| is: Considering both expectation and variance, we get the coefficient of variation CV as the learning criterion: Formula (23) is named CV |G| Function, CV |G| The maximum value of means that the point is close to the agent LSS, that is, the value of E(|G(u)|) is small, or has a large prediction uncertainty, that is, the value of V(|G(u)|) is large, or both; then the best training point for each iteration is: u * =argmax CV |G| (u) (24)。 9. The complex structure active learning reliability analysis method based on RBF model under minimal failure probability according to claim 8, characterized in that: In step 8, the failure probability is estimated as: The coefficient of variation is expressed as: The variance of the failure probability is expressed as:

10. An application of an active learning reliability analysis method for complex structures based on RBF model under extremely small failure probability, characterized in that: The static strength of the bogie frame is analyzed according to the complex structure active learning reliability analysis method based on the RBF model under the extremely small failure probability as described in any one of claims 1-9.

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