Seismic response sample expansion method based on nonlinear kernel function and evolutionary strategy

Abstract

The invention discloses a seismic response sample expansion method based on a non-linear kernel function and an evolutionary strategy, which realizes expansion of small sample data by establishing and sampling a multi-dimensional normal distribution model of building seismic response data, and introduces the non-linear kernel function to calculate non-linear relevancy among features in a data set. According to the method, complex feature association of data is more accurately captured by a distribution model, an evolutionary strategy method is adopted, a nonlinear covariance matrix of the distribution model is optimized by maximizing the similarity between an expansion sample and an original data set, and therefore stable solution in the data expansion process is ensured, and the problem that an existing sample expansion algorithm is unstable is solved.

Description

Technical Field

[0001] The invention relates to the technical field of earthquake response of building groups, and in particular to an earthquake response sample expansion method based on nonlinear kernel function and evolutionary strategy. Background Art

[0002] With the advancement of urbanization, the scale of urban building complexes in my country continues to expand, functional coupling becomes increasingly close, and structural forms become increasingly complex. This has led to a high risk of exposure to earthquake disasters in urban building complexes in my country. Once an earthquake occurs, the destruction of building complexes may cause serious casualties and economic losses. Therefore, earthquake damage prediction for urban building complexes is particularly important. According to the P-58 report on the FEM earthquake response sample expansion method based on nonlinear kernel function and evolutionary strategy published by the Pacific Seismological Research Center of the United States, there are currently three earthquake damage prediction methods under the framework of earthquake engineering, namely, intensity-based prediction, scenario-based prediction, and time-based prediction. Among them, the intensity-based prediction method is the basis, and the other two are expanded on it. This method requires the specification of the earthquake intensity to be considered, and successively calculates various performance indicators such as structural seismic response, damage and loss under this intensity.

[0003] In order to calculate the above performance indicators, it is necessary to carry out structural modeling and dynamic time-history analysis of urban building complexes. However, considering the large number of buildings at the urban scale, it is obviously not feasible to carry out mechanical modeling and dynamic analysis for each building according to traditional methods. Fortunately, machine learning technology provides a potential solution to the above problems. This technology only needs to conduct dynamic time-history analysis on some buildings in the city to collect training data, and then use seismic intensity parameters and building characteristic parameters as input features, and the engineering demand parameters of the structure as output labels. By training the machine learning model, the seismic response prediction of all buildings in the city can be achieved. Since this technology only needs to carry out mechanical modeling and response calculation on some buildings in the city, it can realize the seismic response prediction of all buildings, thus greatly reducing the computational cost of earthquake damage prediction of urban building complexes.

[0004] However, in order to collect sufficient structural response data to ensure sufficient training of the machine learning model, a large number (hundreds or thousands) of seismic inputs and nonlinear time-history analysis are required, which is extremely time-consuming to calculate. In engineering practice, in order to save computing resources, usually only a limited number of nonlinear time-history analyses are carried out. This means that in the task of predicting the response of urban building complexes, machine learning models can only be trained based on small samples, which greatly increases the modeling difficulty of the task. However, in response to the above problems, there is currently a lack of a set of sample expansion methods for urban building complex seismic response data sets suitable for engineering practice. This method should be able to obtain sufficient training samples based on small sample building seismic response data through data expansion technology, so as to train a machine learning model that can effectively predict the seismic response of the entire building complex. In response to this technical gap, the present invention intends to expand the small sample building seismic response data set by establishing and sampling a high-dimensional joint normal distribution model of seismic response data. On this basis, since the building seismic response data has complex nonlinear characteristic associations, if the traditional modeling method is still used when establishing the joint distribution model of the data set, that is, using the linear kernel function to calculate the covariance matrix of the distribution model, the model will be distorted and unable to accurately reflect the original data distribution, thus affecting the accuracy of data expansion. Therefore, it is necessary to introduce a nonlinear kernel function when establishing the joint distribution model of seismic response data to accurately capture the nonlinear characteristics of the response data. In addition, when using the nonlinear kernel function to establish the above joint distribution model, a data-driven method is usually used to determine the optimal value of the kernel function hyperparameters using the maximum marginal likelihood optimization method. However, due to the non-convex characteristics of the marginal likelihood function for the nonlinear kernel function hyperparameters, the optimization process often falls into the local optimum and cannot reach the global optimal solution, thus affecting the stability of the data expansion algorithm.

[0005] Therefore, how to reasonably introduce nonlinear kernel functions to achieve accurate and stable sample expansion is also a key technical challenge faced by machine learning-based earthquake response prediction of building complexes. Summary of the invention

[0006] Based on the problems raised by the above background technology, the purpose of the present invention is to provide an earthquake response sample expansion method based on nonlinear kernel functions and evolutionary strategies. By establishing and sampling a multidimensional normal distribution model of building earthquake response data, the expansion of small sample data is achieved, and a nonlinear kernel function is introduced to calculate the nonlinear correlation between features in the data set, so that the distribution model can more accurately capture the complex feature associations of the data. The evolutionary strategy method is used to optimize the nonlinear covariance matrix of the distribution model by maximizing the similarity between the expanded sample and the original data set, thereby ensuring the stable solution of the data expansion process and solving the problem of instability of the existing sample expansion algorithm.

[0007] The present invention is achieved through the following technical solutions:

[0008] The present invention provides a seismic response sample expansion method based on nonlinear kernel function and evolution strategy, comprising the following steps:

[0009] Step 1: Obtain a building complex seismic response data set, and divide the building complex seismic response data into single building response data sets;

[0010] Step 2: Calculate the single building response data set using a nonlinear kernel function to obtain a covariance matrix;

[0011] Step 3: establishing a multidimensional joint normal distribution model according to the covariance matrix, sampling the multidimensional joint normal distribution model, and generating an expanded response sample;

[0012] Step 4: Use evolutionary strategies to iteratively optimize the multidimensional joint normal distribution model until the iteration stop condition is met, and output the expanded building seismic response data set.

[0013] In the above technical solution, each single building is used as a basic analysis unit in this method, so the earthquake response data set of the building complex is first divided into single building response data sets for subsequent expansion analysis.

[0014] Then, when establishing the multidimensional joint normal distribution model of the single building response data set, a nonlinear kernel function is introduced, and a data-driven method is used to optimize the model hyperparameters. Specifically, in this method, the covariance matrix of the single building response data set is first calculated using a nonlinear kernel function, and then a multidimensional joint normal distribution model is established based on the covariance matrix. This model can accurately capture the complex nonlinear characteristics in the earthquake response data, thereby improving the accuracy of sample expansion.

[0015] Finally, the evolutionary strategy method is introduced to maximize the similarity between the expanded samples and the original data set, and use iterative optimization to improve the nonlinear covariance matrix of the distribution model, thereby ensuring that it can always effectively reflect the true data distribution, thereby ensuring the stability of the sample expansion process.

[0016] In an optional embodiment, dividing the building complex seismic response data into single building response data sets comprises the following steps:

[0017] The earthquake response data of the building complex is sorted and divided into a three-dimensional tensor X L×N×M ; Among them, the first dimension L is the number of individual buildings in the building complex, the second dimension N is the sample size of the earthquake corresponding data of each individual building, and the third dimension M is the number of features;

[0018] For any single building l, l = 1, ..., L, from the three-dimensional tensor X L×N×M Extract the structural response matrix X N×MA single building response dataset as a single building l.

[0019] In an optional embodiment, calculating the single building response data set using a nonlinear kernel function comprises the following steps:

[0020] Calculate the structural response matrix X N×M The mean vector of

[0021] Establish a nonlinear Gaussian kernel function and solve the covariance matrix K through the nonlinear Gaussian kernel function M×M ; Wherein, the nonlinear Gaussian kernel is used to measure the correlation between feature vectors;

[0022] Using the mean vector And the covariance matrix K M×M Calculate the optimal bandwidth parameters and calculate the covariance matrix based on the optimal bandwidth parameters

[0023] In an optional embodiment, using the mean vector And the covariance matrix K M×M Calculate the optimal bandwidth parameters and calculate the covariance matrix based on the optimal bandwidth parameters The steps include:

[0024] For the mean vector Solve and obtain the decentralized structural response matrix

[0025] Based on the covariance matrix and the decentralized structural response matrix, a log-likelihood function of the seismic response data is constructed and maximized.

[0026] The log-likelihood function is optimized by gradient optimization method Perform fitting to obtain the optimal bandwidth parameters;

[0027] Bringing the optimal bandwidth parameter into the Gaussian kernel function for calculation to obtain an optimal kernel function;

[0028] The covariance matrix is ​​obtained by solving the optimal kernel function

[0029] In an optional embodiment, establishing a multidimensional joint normal distribution model according to the covariance matrix includes the following steps:

[0030] Performing eigenvalue decomposition on the covariance matrix to obtain an eigenvalue matrix and an eigenvector matrix;

[0031] If any response sample in the single building response data set is made to obey a multidimensional joint normal distribution, a multidimensional joint normal distribution model conforming to the single building response data set is established by using the eigenvalue matrix and the eigenvector matrix.

[0032] In an optional embodiment, sampling the multidimensional joint normal distribution model includes:

[0033]

[0034] Among them, Y N′×M To generate the expanded response sample, N′ is the sample size of the expanded response sample, U p×N′ is the noise data matrix, V M×p is the eigenvector matrix, Γ p×p is the eigenvalue matrix, 1 N′×1 is a column vector of length N′ filled with all ones.

[0035] In an optional embodiment, the multi-dimensional joint normal distribution model is iteratively optimized using an evolutionary strategy, comprising the following steps:

[0036] At each iteration step, the multi-dimensional joint normal distribution model of the single building response data set is established using the mean vector and covariance matrix;

[0037] The probability density function of the multi-dimensional joint normal distribution model is used to calculate the seismic response samples, and the single building response data set is expanded by the seismic response samples to generate an expanded data set;

[0038] A first similar sample group and a second similar sample group are selected from the expanded data set, the second similar sample group is used to update the mean vector of the next iteration step, and the first similar sample group and the second similar sample group are used to update the covariance matrix of the next iteration step.

[0039] In an optional embodiment, screening out the first similar sample group and the second similar sample group from the expanded data set includes the following steps:

[0040] Calculating the data similarity between the single building response data set and the expanded data set, and ranking the earthquake response samples in the expanded data set according to the data similarity;

[0041] Select the top-ranked earthquake response sample as the first similar sample group, and select the top μ earthquake response samples as the second similar sample group; where μ <N′。

[0042] In an optional embodiment, using the first similar sample group and the second similar sample group to update the covariance matrix of the next iteration step includes:

[0043] The covariance matrix of the next iteration step is quickly updated using the second similar sample group; and,

[0044] The first similar sample group is used to smoothly update the covariance matrix of the next iteration step.

[0045] In an optional embodiment, the condition for stopping iteration includes: the data similarity before and after the expansion of the single building response data set reaches a similarity threshold or the change of the mean vector and the covariance matrix before and after iteration is less than a change threshold.

[0046] Compared with the prior art, the present invention has the following advantages and beneficial effects:

[0047] 1. By establishing and sampling a multidimensional normal distribution model of building seismic response data, the expansion of small sample data can be achieved.

[0048] 2. Introduce nonlinear kernel functions to calculate the nonlinear correlation between features in the data set, so that the distribution model can more accurately capture the complex feature associations of the data.

[0049] 3. The evolutionary strategy method is used to optimize the nonlinear covariance matrix of the distribution model by maximizing the similarity between the expanded samples and the original data set, thereby ensuring a stable solution for the data expansion process. BRIEF DESCRIPTION OF THE DRAWINGS

[0050] In order to more clearly illustrate the technical solutions of the exemplary embodiments of the present invention, the following briefly introduces the drawings required for use in the embodiments. It should be understood that the following drawings only illustrate certain embodiments of the present invention and should not be regarded as limiting the scope. For those of ordinary skill in the art, other relevant drawings can be obtained based on these drawings without creative work. In the drawings:

[0051] Figure 1 A schematic flow chart of a method for expanding earthquake response samples based on nonlinear kernel functions and evolutionary strategies provided in Example 1 of the present invention;

[0052] Figure 2 A schematic diagram of a seismic response data set of a 4-story building structure provided in Example 2 of the present invention;

[0053] Figure 3 A schematic diagram of solving the nonlinear covariance matrix of a seismic response data set provided in Example 2 of the present invention;

[0054] Figure 4 A schematic diagram of iterative updating of a seismic response data set distribution model based on an evolutionary strategy provided in Example 2 of the present invention;

[0055] Figure 5 The probability distribution diagram of the original and expanded seismic response data of a building provided in Example 2 of the present invention. DETAILED DESCRIPTION

[0056] In order to make the objectives, technical solutions and advantages of the present invention more clearly understood, the present invention is further described in detail below in conjunction with embodiments and drawings. The exemplary embodiments of the present invention and their description are only used to explain the present invention and are not intended to limit the present invention.

[0057] Example 1

[0058] Figure 1 A schematic diagram of a flow chart of a method for expanding earthquake response samples based on a nonlinear kernel function and an evolutionary strategy provided in Example 1 of the present invention is shown in FIG. Figure 1 As shown in FIG. 1 , the seismic response sample expansion method based on nonlinear kernel function and evolution strategy includes the following steps:

[0059] Step 1: Obtain a building complex seismic response data set, and divide the building complex seismic response data into single building response data sets;

[0060] Step 2: Calculate the single building response data set using a nonlinear kernel function to obtain a covariance matrix;

[0061] Step 3: establishing a multidimensional joint normal distribution model according to the covariance matrix, sampling the multidimensional joint normal distribution model, and generating an expanded response sample;

[0062] Step 4: Use evolutionary strategies to iteratively optimize the multidimensional joint normal distribution model until the iteration stop condition is met, and output the expanded building seismic response data set.

[0063] It should be noted that in this method, each single building is used as the basic analysis unit, so the building complex seismic response dataset is first divided into single building response datasets for subsequent expansion analysis.

[0064] Then, when establishing the multidimensional joint normal distribution model of the single building response data set, a nonlinear kernel function is introduced, and a data-driven method is used to optimize the model hyperparameters. Specifically, in this method, the covariance matrix of the single building response data set is first calculated using a nonlinear kernel function, and then a multidimensional joint normal distribution model is established based on the covariance matrix. This model can accurately capture the complex nonlinear characteristics in the earthquake response data, thereby improving the accuracy of sample expansion.

[0065] Finally, the evolutionary strategy method is introduced to maximize the similarity between the expanded samples and the original data set, and use iterative optimization to improve the nonlinear covariance matrix of the distribution model, thereby ensuring that it can always effectively reflect the true data distribution, thereby ensuring the stability of the sample expansion process.

[0066] Furthermore, this method can also be applied to other similar data expansion tasks. For any original data set, assume that it can be assembled into a two-dimensional matrix Where N is the sample size of the data set, and M is the number of features in the data set. Accurate and stable data expansion can be implemented by using the sample expansion method based on nonlinear kernel function and evolutionary strategy proposed in the present invention.

[0067] In an optional embodiment, dividing the building complex seismic response data into single building response data sets comprises the following steps:

[0068] The earthquake response data of the building complex is sorted and divided into a three-dimensional tensor X L×N×M ; Among them, the first dimension L is the number of individual buildings in the building complex, the second dimension N is the sample size of the earthquake corresponding data of each individual building, and the third dimension M is the number of features;

[0069] For any single building l, l = 1, ..., L, from the three-dimensional tensor X L×N×M Extract the structural response matrix X N×M A single building response dataset as a single building l.

[0070] It should be noted that the earthquake response data of each building complex can be divided into a three-dimensional tensor X L×N×M , where the first dimension L is the number of individual buildings in the building complex, the second dimension N is the sample size of the seismic response data of each individual building, and the third dimension M is the number of characteristics of the considered seismic intensity parameters and engineering demand parameters.

[0071] For any single building l = 1, ..., L, the corresponding structural response matrix X can be extracted from the above three-dimensional tensor N×M =[x 1 x 2 … x M ], where the eigenvector x i The dimension of (i=1,2,…,M) is equal to N.

[0072] The corresponding structural response matrix extracted from the three-dimensional tensor is the basis for the data expansion of the present invention.

[0073] In an optional embodiment, calculating the single building response data set using a nonlinear kernel function comprises the following steps:

[0074] Calculate the structural response matrix X N×M The mean vector of

[0075] Establish a nonlinear Gaussian kernel function and solve the covariance matrix K through the nonlinear Gaussian kernel function M×M ; Wherein, the nonlinear Gaussian kernel is used to measure the correlation between feature vectors;

[0076] Using the mean vector And the covariance matrix K M×M Calculate the optimal bandwidth parameters and calculate the covariance matrix based on the optimal bandwidth parameters

[0077] It should be noted that the structural response matrix X can be calculated by formula (1): N×M The mean vector of

[0078]

[0079] Construct a nonlinear Gaussian kernel function, through which the covariance matrix K can be solved M×M :

[0080]

[0081] Among them, k(x i ,x j ) is the kernel function, which measures the eigenvector x i and x j The correlation between .

[0082] Furthermore, the nonlinear Gaussian kernel function is constructed as follows:

[0083]

[0084] Among them, ∥x i -x j ∥ 2 is the eigenvector x i and x j The square of the Euclidean distance between them.

[0085] In an optional embodiment, using the mean vector And the covariance matrix K M×M Calculate the optimal bandwidth parameters and calculate the covariance matrix based on the optimal bandwidth parameters The steps include:

[0086] For the mean vector Solve and obtain the decentralized structural response matrix

[0087] Based on the covariance matrix and the decentralized structural response matrix, a log-likelihood function of the seismic response data is constructed and maximized.

[0088] The log-likelihood function is optimized by gradient optimization method Perform fitting to obtain the optimal bandwidth parameters;

[0089] Bringing the optimal bandwidth parameter into the Gaussian kernel function for calculation to obtain an optimal kernel function;

[0090] The covariance matrix is ​​obtained by solving the optimal kernel function

[0091] It should be noted that there is a bandwidth parameter σ when constructing the nonlinear Gaussian kernel function. SE , which controls the influence of the Euclidean distance between two eigenvectors on their correlation. However, this parameter is not a fixed value. M×M When calculating it is a random variable, in order to solve the covariance matrix The bandwidth parameters need to be further calculated.

[0092] Specifically, according to the mean vector calculated in formula (1), the decentralized structural response matrix is ​​obtained:

[0093]

[0094] Among them, 1 N×1 is a column vector of length N filled with all ones.

[0095] According to the covariance matrix K calculated in equations (2) and (4), M×M and decentralized structural response matrix Construct and maximize the log-likelihood function of earthquake response data To fit the optimal bandwidth parameter value through the gradient optimization method

[0096]

[0097] in, Representation dataset The nth (n=1,2,…,N) sample in .

[0098] Substitute the calculated optimal bandwidth parameter value back into formula (3) to calculate the covariance matrix This covariance matrix is ​​the covariance matrix to be solved in step 2.

[0099] In an optional embodiment, establishing a multidimensional joint normal distribution model according to the covariance matrix includes the following steps:

[0100] Performing eigenvalue decomposition on the covariance matrix to obtain an eigenvalue matrix and an eigenvector matrix;

[0101] If any response sample in the single building response data set is made to obey a multidimensional joint normal distribution, a multidimensional joint normal distribution model conforming to the single building response data set is established by using the eigenvalue matrix and the eigenvector matrix.

[0102] It should be noted that for the covariance matrix Perform eigenvalue decomposition:

[0103]

[0104] Γ p×p =diag(γ 1 ,γ 2 ,...,γ p ), (7)

[0105] V M×p =[v 1 v 2 … v p ], (8)

[0106] Where p is the covariance matrix The rank of; eigenvalue matrix Γ p×p is a diagonal matrix with p elements γ on the diagonal i (i=1,…,p) is the eigenvalue, and all other elements are 0; the eigenvector matrix V M×p By the column vector v of length M i (i=1,…,p).

[0107] Assume that any earthquake response data sample X in the data set n:N,M (n=1,2,…,N) obeys multidimensional joint normal distribution:

[0108]

[0109] Then, according to the eigenvalue matrix and eigenvector matrix solved in equations (6), (7) and (8), a multidimensional joint normal distribution model that conforms to the distribution characteristics of the original data set can be established.

[0110] In an optional embodiment, sampling the multidimensional joint normal distribution model includes:

[0111]

[0112] Among them, Y N′×M To generate the expanded response sample, N′ is the sample size of the expanded response sample, U p×N′ is the noise data matrix, V M×p is the eigenvector matrix, Γ p×p is the eigenvalue matrix, 1 N′×1 is a column vector of length N′ filled with all ones.

[0113] It should be noted that all elements in the noise data matrix are from the standard normal distribution In the expansion process, the sample size N′>N of the expanded response sample after expansion.

[0114] In an optional embodiment, the multi-dimensional joint normal distribution model is iteratively optimized using an evolutionary strategy, comprising the following steps:

[0115] At each iteration step, the multi-dimensional joint normal distribution model of the single building response data set is established using the mean vector and covariance matrix;

[0116] The probability density function of the multi-dimensional joint normal distribution model is used to calculate the seismic response samples, and the single building response data set is expanded by the seismic response samples to generate an expanded data set;

[0117] A first similar sample group and a second similar sample group are selected from the expanded data set, the second similar sample group is used to update the mean vector of the next iteration step, and the first similar sample group and the second similar sample group are used to update the covariance matrix of the next iteration step.

[0118] It should be noted that, in order to ensure that the expanded response samples meet the distribution of the original data set, the present invention uses an evolutionary strategy to iteratively optimize the multi-joint normal distribution model.

[0119] Specifically, at each iteration step t, the mean vector and the covariance matrix To establish the corresponding building seismic response dataset The probability density function of the multidimensional joint normal distribution model is:

[0120]

[0121] in, Represents the nth sample in the data set (n=1,2,…,N′).

[0122] Especially when t = 0, the earthquake response data set in equation (11) consists of the original data samples, that is,

[0123] The mean vector and covariance matrix are the solution values in Step 2, that is

[0124] Sampling the distribution model in Equation (11) to obtain N' seismic response samples, forming an expanded dataset

[0125] where is an eigenvector of dimension N'.

[0126] In an alternative embodiment, screening out the first similar sample group and the second similar sample group from the expanded dataset includes the following steps:

[0127] Calculating the data similarity between the single building response dataset and the expanded dataset, and ranking the seismic response samples in the expanded dataset according to the data similarity;

[0128] Selecting the seismic response sample ranked first as the first similar sample group, and selecting the top μ seismic response samples as the second similar sample group; where μ < N'.

[0129] It should be noted that, on the basis of expansion, a fitness function is introduced to evaluate the data similarity between the expanded response samples and the original dataset, and the calculation process is as follows:

[0130]

[0131] where is the i-th (i = 1, 2,..., N') sample in the expanded seismic response dataset; X n:N,M is the n-th (n = 1, 2,..., N) sample in the original dataset; represents the sample and the dataset X N×M the data similarity between them.

[0132] According to the data similarity obtained in Equation (12), ranking the data samples in the expanded dataset and screening out the top μ (μ < N') similar samples as the second similar sample group and the most similar 1 sample as the first similar sample group

[0133] In an alternative embodiment, updating the covariance matrix of the next iteration step using the first similar sample group and the second similar sample group includes:

[0134] Using the second similar sample group to quickly update the covariance matrix of the next iteration step; and,

[0135] The first similar sample group is used to smoothly update the covariance matrix of the next iteration step.

[0136] It should be noted that, subsequently, the second similar sample group is used to update the mean vector at the iteration step t+1:

[0137]

[0138] in, is the updated mean vector, c m is the learning rate when updating the mean vector, is the nth (n=1, 2, …, μ) data sample in the second similar sample group.

[0139] The covariance matrix at iteration step t+1 is updated using the second similar sample group and the first similar sample group:

[0140]

[0141] Furthermore, two strategies are used in formula (14) to update the covariance matrix: First, using the first similar sample group To implement a fast update strategy to improve the accuracy of the update process; secondly, using the second similar sample group To implement a smooth update strategy to ensure the smoothness of the update process. * and c μ They represent the learning rates for fast update and smooth update strategies respectively; α is the attenuation factor in the fast update strategy, which determines the influence of the historical optimal sample information on the effect of the fast update strategy in the current iteration step.

[0142] In an optional embodiment, the condition for stopping iteration includes: the data similarity before and after the expansion of the single building response data set reaches a similarity threshold or the change of the mean vector and the covariance matrix before and after iteration is less than a change threshold.

[0143] Example 2

[0144] Based on Example 1, Example 2 of the present invention targets 64 buildings of a school and executes an earthquake response sample expansion method based on nonlinear kernel function and evolutionary strategy, and analyzes the execution results.

[0145] In the embodiment of the present invention, 64 buildings in the Wangjiang Campus of Sichuan University are taken as targets, 100 seismic waves as shown in Table 1 are selected, and a seismic response data set of the building complex is obtained through simplified structural modeling and nonlinear time history analysis.

[0146] Step 1: Split the earthquake response dataset of the building complex to obtain the response dataset of each individual building

[0147] 100 seismic waves were selected, and the seismic response dataset of the building complex was obtained through simplified structural modeling and nonlinear time history analysis.

[0148] Assemble the collected building group seismic response data into a three-dimensional tensor X L×N×M , where the first dimension L=64 is the number of individual buildings in the building complex; the second dimension N=100 is the sample size of the seismic response data of each individual building; and the third dimension M is the number of characteristics of the considered seismic intensity parameters and engineering demand parameters.

[0149] For any single building l = 1, ..., L, the corresponding structural response matrix X can be extracted from the above three-dimensional tensor N×M =[x 1 x 2 … x M ], The value of the third dimension M varies depending on the specific building selected. The method for determining this dimension is as follows: Figure 2 shown.

[0150] Specifically, the peak ground acceleration (PGA) of the earthquake is considered as the seismic intensity parameter, and the maximum inter-story drift ratio (MIDR) of each layer of the building structure is considered as the engineering demand parameter.

[0151] Therefore, for a 4-story prototype structure as shown in the figure, its seismic response data set is characterized by X 100×4 , where the first column corresponds to the PGA of each ground motion, and the second to fifth columns correspond to the MIDR of each floor of the building under the earthquake. The seismic response datasets of all individual buildings in the building complex are divided and extracted according to the above method.

[0152] Step 2: Based on the building's seismic response data set, use the nonlinear kernel function to calculate the covariance matrix of the response data

[0153] In an embodiment of the present invention, given a building group earthquake response data set X 64×100×M , extract a seismic response dataset X of a 4-story prototype structure 100×5 :

[0154]

[0155] Solve the mean vector corresponding to the data set in equation (1) And the covariance matrix K 5×5 :

[0156]

[0157] Among them, k i,j is the kernel function, which is used to measure X 100×5 The correlation between the eigenvectors in the i-th and j-th columns. For example, k 1,2 That means X 100×5 The eigenvector of the first column in and the eigenvector of column 2 The correlation between the features.

[0158] Specifically, a nonlinear Gaussian kernel function is used to calculate formula (3):

[0159]

[0160] Among them, ∥x i -x j ∥ 2 is the eigenvector x i and x j The square of the Euclidean distance between SE represents the bandwidth parameter, which controls the effect of the Euclidean distance between two eigenvectors on their correlation. This parameter is not a fixed value, but a random variable. Its parameter calibration process is as follows: Figure 3 shown.

[0161] Specifically, according to the mean vector calculated in equation (2), the decentralized structural response matrix is ​​obtained:

[0162]

[0163] Among them, 1 100×1 is a column vector of length 100 filled with all ones.

[0164] According to the covariance matrix K calculated in equations (3) and (5), 5×5 and decentralized structural response matrix Construct and maximize the log-likelihood function of earthquake response data To fit the optimal bandwidth parameter value through the gradient optimization method

[0165]

[0166] in, Representation dataset The nth (n=1,2,…,100) sample in . Then, according to formula (3), we can get the covariance matrix

[0167] Step 3: Based on the covariance matrix, a joint normal distribution model of building seismic response data is established, and the sampling model generates an expanded response sample

[0168] In the example of the present invention, consider the optimal covariance matrix solved in step 2 is a full rank matrix. Therefore, its eigenvalue decomposition is performed as follows:

[0169]

[0170] Γ 5×5 =diag(γ 1 ,γ 2 ,...,γ 5 ) (8)

[0171] V M×5 =[v 1 v 2 … v 5 ] (9)

[0172] Among them, the eigenvalue matrix Γ 5×5 is a diagonal matrix with 5 elements γ on the diagonal i (i=1,…,5) is the eigenvalue, and all other elements are 0; the eigenvector matrix V 5×5 By a column vector v of length 5 i (i=1,…,p).

[0173] Assume that all earthquake response data samples in the dataset are X n:100,5 (n=1,2,…,100) obeys multidimensional joint normal distribution:

[0174]

[0175] Then, according to the eigenvalue matrix and eigenvector matrix solved in equations (7), (8) and (9), a multidimensional joint normal distribution model that conforms to the distribution characteristics of the original data set can be established.

[0176] In the example of the present invention, it is assumed that there are N′=2000 new samples in the expanded building seismic response data set, and

[0177] The expanded structural seismic response dataset Y is obtained by sampling 2000×5 :

[0178]

[0179] Among them, U 5×2000 is a noise data matrix, all elements of which are distributed by the standard normal distribution Sampling obtained; 1 2000×1 is a column vector of length 2000 filled with all ones.

[0180] Step 4: Using evolutionary strategies to iteratively optimize the multidimensional joint normal distribution model established in step 3 In the example of the present invention, a multidimensional joint normal distribution model for sampling and expanding seismic response data is established in step 3. To improve the expanded structural seismic response dataset Y 2000×5 Compared with the original data set X 100×5 The data similarity is calculated by using evolutionary strategies to iteratively optimize the distribution model.

[0181] The iterative process of the distribution model is as follows Figure 4 Specifically, at each iteration step t, the mean vector and the covariance matrix To establish the corresponding building seismic response dataset The probability density function of the multidimensional joint normal distribution model is:

[0182]

[0183] in, represents the nth sample in the data set (n = 1, 2, ..., 2000). In particular, when t = 0, the earthquake response data set in equation (12) consists of the original data samples, that is, Yt = 0 = X 100×5 ; The mean vector and covariance matrix are the solution values ​​in step 2, that is,

[0184] The distribution model in sampling formula (12) obtains 2000 earthquake response samples to form the expanded data set in is a feature vector with a dimension of 2000. On this basis, a fitness function is introduced to evaluate the data similarity between the expanded data sample and the original data set:

[0185]

[0186] in, is the i-th (i=1,2,…,2000) sample in the expanded earthquake response data set; X n:100,5 is the nth (n=1,2,…,100) sample in the original data set; Representation sample With the original data set X 100×5 According to the data similarity obtained in formula (13), the expanded data set Rank the data samples in the dataset and select the first μ = 100 similar samples as the “elite group” And the most similar candidate sample as the "optimal group" Then, the elite group is used to update the mean vector at iteration step t+1:

[0187]

[0188] in, is the updated mean vector, c m is the learning rate when updating the mean vector, is the nth (n=1,2,…,100) data sample in the elite group.

[0189] Use the elite group and the optimal group to update the covariance matrix at iteration step t+1:

[0190]

[0191] Among them, formula (15) adopts two strategies to update the covariance matrix: First, using the optimal group sample To implement a fast update strategy to improve the accuracy of the update process; secondly, use the elite group sample To implement a smooth update strategy to ensure the smoothness of the update process. * and c μ They represent the learning rates for fast update and smooth update strategies respectively; α is the attenuation factor in the fast update strategy, which determines the influence of the historical optimal sample information on the effect of the fast update strategy in the current iteration step.

[0192] Step 5: Repeat step 4 until any of the following conditions is met, then stop the iteration and output the expanded building seismic response dataset: (1) the data similarity between the seismic response datasets before and after expansion is high enough; (2) the mean vector With the covariance matrix The changes before and after the update are small enough.

[0193] The steps of the building group earthquake response data sample expansion method based on nonlinear kernel function and evolution strategy are adopted for all 64 buildings in the example of the present invention, and the expanded earthquake response data set of each building in the building group can be obtained. Figure 5As shown, taking any building in the building complex as an example, three data expansion methods are implemented for its original earthquake response data set (labeled as "Ori"), namely: the method of the present invention (labeled as "AdpKer"), the expansion method based on the linear kernel function and without evolutionary strategy optimization (labeled as "Lin"), and the expansion method based on the linear kernel function combined with evolutionary strategy optimization (labeled as "AdpNoKer"). By comparison, it is found that the distribution of the expanded data based on the present invention is most consistent with the distribution of the original response data. The above results are valid for all buildings in the examples of the present invention, which proves the effectiveness of the present invention in introducing nonlinear kernel functions and evolutionary strategy optimization in the data expansion process.

[0194] The specific implementation methods described above further illustrate the objectives, technical solutions and beneficial effects of the present invention in detail. It should be understood that the above description is only a specific implementation method of the present invention and is not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the present invention should be included in the scope of protection of the present invention.

Claims

1. A seismic response sample expansion method based on nonlinear kernel function and evolutionary strategy, characterized in that: The steps include: Step 1: Obtain a building complex seismic response data set, and divide the building complex seismic response data into single building response data sets; Step 2: Calculate the single building response data set using a nonlinear kernel function to obtain a covariance matrix; Step 3: establishing a multidimensional joint normal distribution model according to the covariance matrix, sampling the multidimensional joint normal distribution model, and generating an expanded response sample; Step 4: Use evolutionary strategies to iteratively optimize the multidimensional joint normal distribution model until the iteration stop condition is met, and output the expanded building seismic response data set.

2. The earthquake response sample expansion method based on nonlinear kernel function and evolution strategy according to claim 1 is characterized in that: Dividing the building complex seismic response data into single building response data sets comprises the following steps: The earthquake response data of the building complex is sorted and divided into a three-dimensional tensor X L×N×M ; Among them, the first dimension L is the number of individual buildings in the building complex, the second dimension N is the sample size of the earthquake corresponding data of each individual building, and the third dimension M is the number of features; For any single building l, l = 1, ..., L, from the three-dimensional tensor X L×N×M Extract the structural response matrix X N×M A single building response dataset as a single building l.

3. The earthquake response sample expansion method based on nonlinear kernel function and evolution strategy according to claim 2 is characterized in that: The nonlinear kernel function is used to calculate the single building response data set, including the following steps: Calculate the structural response matrix X N×M The mean vector of Establish a nonlinear Gaussian kernel function and solve the covariance matrix K through the nonlinear Gaussian kernel function M×M ; Wherein, the nonlinear Gaussian kernel is used to measure the correlation between feature vectors; Using the mean vector And the covariance matrix K M×M Calculate the optimal bandwidth parameters and calculate the covariance matrix based on the optimal bandwidth parameters 4. The earthquake response sample expansion method based on nonlinear kernel function and evolution strategy according to claim 3 is characterized in that: Using the mean vector And the covariance matrix K M×M Calculate the optimal bandwidth parameters and calculate the covariance matrix based on the optimal bandwidth parameters The steps include: For the mean vector Solve and obtain the decentralized structural response matrix Based on the covariance matrix and the decentralized structural response matrix, a log-likelihood function of the seismic response data is constructed and maximized. The log-likelihood function is optimized by gradient optimization method. Perform fitting to obtain the optimal bandwidth parameters; Bringing the optimal bandwidth parameter into the Gaussian kernel function for calculation to obtain an optimal kernel function; The covariance matrix is ​​obtained by solving the optimal kernel function 5. The earthquake response sample expansion method based on nonlinear kernel function and evolution strategy according to claim 1, characterized in that: Establishing a multidimensional joint normal distribution model according to the covariance matrix includes the following steps: Performing eigenvalue decomposition on the covariance matrix to obtain an eigenvalue matrix and an eigenvector matrix; If any response sample in the single building response data set is made to obey a multidimensional joint normal distribution, a multidimensional joint normal distribution model conforming to the single building response data set is established by using the eigenvalue matrix and the eigenvector matrix.

6. The earthquake response sample expansion method based on nonlinear kernel function and evolution strategy according to claim 5 is characterized in that: Sampling the multidimensional joint normal distribution model includes: Among them, Y N′×M To generate the expanded response sample, N′ is the sample size of the expanded response sample, U p×N′ is the noise data matrix, V M×p is the eigenvector matrix, Γ p×p is the eigenvalue matrix, is the mean vector, 1 N′×1 is a column vector of length N′ filled with all ones.

7. The earthquake response sample expansion method based on nonlinear kernel function and evolution strategy according to claim 1, characterized in that: The multi-dimensional joint normal distribution model is iteratively optimized using an evolutionary strategy, comprising the following steps: At each iteration step, the multi-dimensional joint normal distribution model of the single building response data set is established using the mean vector and covariance matrix; The probability density function of the multi-dimensional joint normal distribution model is used to calculate a seismic response sample, and the single building response data set is expanded by the seismic response sample to generate an expanded data set; A first similar sample group and a second similar sample group are selected from the expanded data set, the second similar sample group is used to update the mean vector of the next iteration step, and the first similar sample group and the second similar sample group are used to update the covariance matrix of the next iteration step.

8. The earthquake response sample expansion method based on nonlinear kernel function and evolution strategy according to claim 7, characterized in that: Screening out a first similar sample group and a second similar sample group from the expanded data set includes the following steps: Calculating the data similarity between the single building response data set and the expanded data set, and ranking the earthquake response samples in the expanded data set according to the data similarity; The first-ranked earthquake response sample is selected as the first similar sample group, and the first μ earthquake response samples are selected as the second similar sample group; wherein μ<N′.

9. The earthquake response sample expansion method based on nonlinear kernel function and evolution strategy according to claim 7, characterized in that: Updating the covariance matrix of the next iteration step by using the first similar sample group and the second similar sample group includes: The covariance matrix of the next iteration step is quickly updated using the second similar sample group; and, The first similar sample group is used to smoothly update the covariance matrix of the next iteration step.

10. The earthquake response sample expansion method based on nonlinear kernel function and evolution strategy according to claim 1, characterized in that: The conditions for stopping iteration include: the data similarity before and after the expansion of the single building response data set reaches a similarity threshold or the change of the mean vector and the covariance matrix before and after iteration is less than a change threshold.

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