Structural response probability density prediction method based on interpretable depth generation network

Abstract

The invention discloses a structural response probability density prediction method based on an interpretable depth generative network, and the method comprises the steps: 1, constructing an IDGN-PDF neural network which comprises a distribution fitting generative network G1, a smooth monotonic network and a probability density prediction generative network G2; step 2, based on the constructed IDGN-PDF neural network, training is carried out to obtain a trained IDGN-PDF neural network; and step 3, inputting a structure system response g (x) needing to be predicted into the probability density prediction generation network G2 of the trained IDGN-PDF neural network for reverse prediction, and obtaining a probability density function PDF and a cumulative distribution function CDF of the structure system response g (x) according to the probability density prediction capability of the G2 network. According to the method, a monotone mapping relation between Gaussian input and structural response of the probability density prediction generation network G2 is established, and high-efficiency and high-precision structural response probability density prediction is realized by utilizing reversibility and probability density prediction capability of the probability density prediction generation network G2.

Description

Technical Field

[0001] The present invention belongs to the field of information technology, and in particular relates to a structural response probability density prediction method based on an interpretable deep generative network. Background Art

[0002] Randomness, a form of uncertainty, is widely present in engineering structural systems. Due to the influence of random structural parameters (material properties, geometry) and environmental excitations (wind, earthquakes), the structural responses that determine structural safety also exhibit inherent randomness. To ensure the safe operation of engineering structural systems, it is crucial to consider the impact of randomness on structural performance during design and operation and conduct reliability assessments. Over the past half century, extensive research has been conducted on uncertainty quantification and reliability analysis of engineering structural systems, driving the development of related fields. Probabilistic analysis, which can obtain the probability density function (PDF) of the structural response, is a key component of uncertainty quantification and reliability assessment of engineering structural systems. After determining the PDF of the structural response, the reliability of the structure can be estimated by integrating the PDF. However, existing probabilistic analysis methods for structural response still suffer from computational efficiency and accuracy issues for complex structural systems. This is because collecting structural response samples to establish the PDF requires repeated, computationally intensive structural system analysis. This has become a significant challenge that needs to be addressed in the field of structural uncertainty quantification and reliability assessment.

[0003] Considering the different randomness given by the research object, there are two common structural response probability analysis methods: random structural analysis method and random vibration analysis method. The random structural analysis method mainly considers the randomness of structural parameters and has limited applicability. The random vibration analysis method considers the randomness of structural parameters and external excitations when analyzing the randomness of structural response, and its applicability is wider than the former. Random vibration analysis methods include modal superposition method, complex modal decomposition, pseudo-excitation method, time domain explicit method, random simulation method, moment method, probability evolution method, etc. Among them, random simulation method, moment method and probability evolution method have received widespread attention. The probability analysis method based on random simulation generates a large number of samples for the structural system, then performs structural analysis on each sample to obtain the response, and finally calculates the probability information of the sample response. The analysis method based on random simulation is simple and effective, but for low-probability events, millions of samples are often required to obtain satisfactory tail probability accuracy, and the analysis and calculation cost is extremely high

[18] . Therefore, the random simulation method is usually used as a verification method for simple problems. Probabilistic analysis methods based on statistical moments infer probabilistic information about structural responses from statistical moments of the structural response. These methods include equivalent linearization, moment truncation, and fractional-order moment methods. Equivalent linearization methods typically use a linear system to represent a nonlinear stochastic dynamic system, and based on the equivalence criteria of the first few order moments, they obtain response statistics that approximate the nonlinear system. Moment truncation methods calculate the evolution of second-order and higher-order moments by solving a series of ordinary differential equations related to the evolution of the nonlinear system's moments. Compared to equivalent linearization methods, moment truncation methods offer higher accuracy, but computational efficiency and accuracy are limited for high-dimensional nonlinear systems. Fractional-order moment methods numerically solve a series of complex fractional moments of the response to calculate the probability density, but their computational efficiency needs to be improved for strongly nonlinear problems. Furthermore, probabilistic analysis methods based on statistical moments provide statistical moments of the structural response but fail to provide detailed probability distribution information. Probabilistic analysis methods based on probability evolution determine the PDF of the stochastic structural response by calculating the dynamic system equations. Such methods include the Fokker-Planck-Kolmogorov (FPK) equation method, the path integral method, the probability density evolution method, and the Liouville equation method. The FPK equation method calculates the joint PDF of the structural response by solving the FPK equation. Path integral is also generally considered a numerical method for solving the FPK equation. By introducing a short-time Gaussian assumption into the integral form of the FPK equation, numerical discretization is performed, and the structural response probability density is gradually solved. To calculate the response probability density using the probability density evolution method, the random structural parameters and environmental excitations are considered as the basic random variables of the structural system. The basic random variables and time are then used to model the response of the structural system under random excitation.Probabilistic evolution methods are often combined with variance reduction techniques, proxy models, deep learning algorithms, etc. to improve their computational efficiency and accuracy. However, there are still certain computational difficulties in using probabilistic evolution methods to solve high-dimensional strongly nonlinear problems.

[0004] In summary, traditional methods still find it difficult to effectively balance computational efficiency and accuracy in the probability analysis of the response of large-scale engineering structural systems, and it is difficult to achieve efficient and accurate prediction of the probability density of structural system responses.

[0005] Glossary:

[0006] PDF: probability density function.

[0007] CDF: cumulative distribution function.

[0008] LSF: limit state function. Summary of the Invention

[0009] To solve the above problems, the present invention discloses a structural response probability density prediction method based on an interpretable deep generative network.

[0010] To achieve the above object, the technical solution of the present invention is:

[0011] A method for predicting structural response probability density based on an interpretable deep generative network includes the following steps:

[0012] Step 1: Construct an IDGN-PDF neural network, which includes a distribution fitting generation network G1, a smoothing monotonic network and a probability density prediction generation network G2;

[0013] The distribution fitting generation network G1 is used to achieve reversible prediction of the Gaussian variable z and the random variable x of the structural system:

[0014]

[0015] Among them, G1 and Generate forward and reverse predictions of network G1 for distribution fitting, respectively;

[0016] The smooth monotonic network is used to constrain the monotonicity of the structural system response of the output samples of the distribution fitting generator network G1:

[0017] The probability density prediction generation network G2 is used to achieve the following reversible prediction:

[0018]

[0019] Among them, G2 and are the forward and reverse predictions of the probability density prediction generation network G2; z1 is the first-dimensional input of the input layer of the probability density prediction generation network G2; λ is the second-dimensional input of the two-dimensional input layer of the probability density prediction generation network G2; the two-dimensional input is obtained by combining λ with z1, thereby meeting the requirement of paired input data of the probability density prediction generation network G2, and the value of λ is a one-dimensional Gaussian variable unrelated to z1; g2 is the first-dimensional output of the output layer of the probability density prediction generation network G2, that is, the forward prediction of z1; The second dimension output of the output layer of the probability density prediction generation network G2 is the forward prediction of λ; Step 2, train the IDGN-PDF neural network: first, train the distribution fitting generation network G1 and the smooth monotone network at the same time; obtain the trained distribution fitting generation network G1 and the trained smooth monotone network; input the sample of the random variable x into the trained distribution fitting generation network G1 to obtain the first dimension of the reverse prediction of the trained distribution fitting generation network G1 As the z1 sample, the z1 sample is input into the trained smooth monotonic network to obtain the predicted value M(z1); then the z1 sample and λ are used as the input of the probability density prediction generation network G2, M(z1) and λ are used as the true output of the probability density prediction generation network G2, and the probability density prediction generation network G2 is trained to obtain the trained probability density prediction generation network G2;

[0020] Step 3: Input the structural system response g(x) into the trained probability density prediction generation network G2 for reverse prediction to obtain the probability density function PDF and cumulative distribution function CDF of the structural system response g(x).

[0021] As a further improvement, the distribution fitting generation network G1 includes an input layer, m coupling layers and an output layer in sequence according to the data processing direction;

[0022] The smooth monotone network includes, in order according to the data processing direction: a single-node input layer, a first hidden layer, a second hidden layer, and a single-node output layer. The input of the smooth monotone network is the first dimension z1 of the Gaussian variable z, and the output is M(z1); M() represents the smooth monotone network;

[0023] The probability density prediction generation network G2 includes a two-dimensional input layer, n coupling layers and an output layer in sequence according to the data processing direction.

[0024] As a further improvement, in step 2, the steps of training the IDGN-PDF neural network are as follows:

[0025] 2.1) Pre-training a smooth monotonic network to approximate the linear function f(z1)=z1;

[0026] 2.2) Simultaneously train the distribution fitting generative network G1 and the smooth monotonic network to minimize the total loss function, thereby obtaining the trained distribution fitting generative network G1 and the trained smooth monotonic network;

[0027] 2.3) Input the sample of random variable x into the trained distribution fitting generative network G1 to obtain the first dimension of the reverse prediction of the distribution fitting generative network G1 As the z1 sample, the z1 sample is input into the trained smooth monotonic network to obtain the predicted value M(z1); then z1 and λ are used as the input of the probability density prediction generation network G2, M(z1) and λ are used as the true output of the probability density prediction generation network G2, and the probability density prediction generation network G2 is trained so that the loss function of the probability density prediction generation network G2 is minimized to obtain the trained probability density prediction generation network G2.

[0028] Further improvement, in step 2.2), the total loss function is loss 1+2 :

[0029] loss 1+2 =α1loss1+α2loss2;

[0030] α1 and α2 are weights; loss1 is the loss function of the distribution fitting generation network G1, and loss2 is the loss function of the smooth monotonic network;

[0031]

[0032] in, is the PDF prediction of a sample of random variable x, for The Jacobian matrix operator, det is the determinant operator; π() is the Gaussian distribution probability density function; Generate the first dimension of the inverse prediction of network G1 for distribution fitting;

[0033] Using the distribution fitting to generate the inverse network G1, we can get the reversibility of the network:

[0034]

[0035] As a further improvement, the loss function loss3 of the probability density prediction generation network G2 is as follows:

[0036]

[0037] in, Generate the prediction of λ by the network G2 for probability density prediction, Represents the first dimension of the output layer of the probability density prediction generation network G2, represents the output of a smooth monotonic network;

[0038] To improve training efficiency, the training data is rescaled by its mean and standard deviation:

[0039]

[0040] Among them, x o is a random variable in the original coordinate space, g o (x) is the structural system response in the original coordinate space, and are the mean values of random variables and structural system responses, respectively, and are the standard deviations of the random variables and the structural system response, respectively.

[0041] For further improvement, the learning rate used by the distribution fitting generation network G1 and the probability density prediction generation network G2 is 1×10 -4 , the learning rate of the smooth monotonic network is 1×10 -3 .

[0042] As a further improvement, the data processing method of the smooth monotone network is as follows:

[0043] The first hidden layer of the smooth monotonic network is calculated by adding weights and biases to the input, and then dividing the output of the first hidden layer into several groups, where the j-th output of the k-th group is for,

[0044]

[0045] w (k,j) is the weight, b (k,j) It is a deviation;

[0046] The second hidden layer uses the output of each group in the first hidden layer to calculate its output

[0047]

[0048] Among them, h k is the number of outputs in the kth group, LSE β is the LogSumExp function used for scaling;

[0049] Then we get the output of the smooth monotonic network:

[0050]

[0051] K is the total number of groups;

[0052] The smooth monotonic network is trained so that:

[0053] M(z1)=g(x)=g(G1(z))

[0054] g() represents the structural system response function.

[0055] Further improvement, the probability density prediction generation network G2 consists of a two-dimensional input layer, an output layer and n coupling layers, and the dimension of each layer input and output is set to 2. In the step 3, z1 is The first dimension of the output layer of the probability density prediction generation network G2 is the prediction g2 of g(x). The second dimension of the input layer of the probability density prediction generation network G2 is used to construct a two-dimensional input vector to meet the paired input requirements of the G2 network, represented by λ; λ is selected as a one-dimensional Gaussian variable unrelated to z1; the second item of the output of the probability density prediction generation network G2 is the prediction of λ itself, which is unrelated to g2;

[0056] The PDF output by the probability density prediction generation network G2 is obtained by the following formula:

[0057]

[0058] in, is the PDF predicted by the probability density prediction generator network G2, yes The Jacobian matrix of .

[0059] Further improvement, the CDF of the structural system response g(x) in the standard coordinate space is calculated by the following formula:

[0060]

[0061] in, It is the g in G2 network x The corresponding input, g x is the structural response value of the CDF to be calculated, yes The first dimension of the reverse prediction output;

[0062] is the CDF of the predicted structural response value g2, P() represents the probability, P(g2≤g x ) for g2≤g x The probability of Φ() is the CDF of the one-dimensional Gaussian distribution; F g(x) (g x ) is the CDF of g(x);

[0063] The PDF of the structural system response g(x) in the standard coordinate space is calculated by the following formula:

[0064]

[0065] For gx and The two-dimensional joint probability density function, that is, the two-dimensional joint PDF, p g(x) (g x ) is the PDF of g(x).

[0066] As a further improvement, the CDF and PDF of the structural system response g(x) in the original coordinate space are calculated by the following formulas:

[0067]

[0068] Advantages of the present invention:

[0069] The IDGN-PDF method of the present invention integrates a flow-based generative network (G1 network) to fit the probability distribution of random variables, a smooth monotonic network to restrict the interpretable one-to-one mapping relationship between the input Gaussian variables of the G1 network and the structural response, and a probability density prediction generative network (G2 network) to use this mapping relationship to predict the PDF and CDF of the structural output. When training the network, the learning process of the structural response distribution is reconverted into a multi-objective learning task, including the probability distribution fitting learning of random variables and the interpretable learning of the structural response of the generator network output samples. Case studies have shown that the IDGN-PDF method effectively establishes a monotonic relationship between Gaussian input and structural response, while achieving efficient and high-precision structural response probability density prediction. Specifically, this method uses 5,000 structural response samples for analysis and obtains a prediction accuracy similar to that of the Monte Carlo simulation method using 100,000 samples for analysis. BRIEF DESCRIPTION OF THE DRAWINGS

[0070] Figure 1 It is the IDGN-PDF neural network structure diagram;

[0071] Figure 2 It is a photo of the bridge and a finite element model diagram;

[0072] Figure 3 is a training sample graph;

[0073] Figure 4 are all input and output samples of the G1 network;

[0074] Figure 5 is the input and output sample grouping of the G1 network;

[0075] Figure 6 is the correlation between the input Gaussian variable z1 and the true LSF value g(x) of the output sample, the prediction g2 of the G2 network, and the prediction M(z1) of the smooth monotonic network;

[0076] Figure 7 (a) is the probability density function diagram of the training sample prediction;

[0077] Figure 7 (b) is the cumulative distribution function graph. DETAILED DESCRIPTION

[0078] The present invention will be further described below with reference to the accompanying drawings and embodiments.

[0079] Neural network structure of IDGN-PDF method:

[0080] The neural network structure of the IDGN-PDF method of the present invention is as follows Figure 1 As shown in Figure 3, it consists of a distribution fitting generative network (G1) for random variables, a smooth monotonic network for interpretability constraints of the G1 network, and a probability density prediction generative network (G2) for structural system responses.

[0081] The G1 network is used to fit the probability distribution of structural random variables. It adopts a flow-based generative network architecture consisting of m coupled layers, which can provide a reversible mapping between its input and output. Typically, the input and output layers of a G1 network have equal dimensions. The input of G1 is samples from a Gaussian distribution π(z), and the output of G1 is samples from the target distribution q(x). In this paper, the output layer of G1 is samples of the random variables involved in the structural system. The following reversible mapping can be achieved using the G1 network.

[0082]

[0083] Among them, x is the random variable of the structural system, z represents the Gaussian distribution variable, G1 and are the forward and reverse predictions of the G1 network respectively. Using the reversibility of the G1 network, the PDF of the output sample can be calculated

[0084]

[0085] in, is the PDF prediction of the random variable sample x, For reverse prediction The Jacobian matrix operator is det, and the determinant operator is det. After training the G1 network with the random variable x data set, G1 can map the Gaussian distribution π(z) to the distribution q(x) of the structural random variable. In addition, in theory, the estimated probability distribution of the structural system response of the output sample is It is consistent with the true value p(g(x)). However, p(g(x)) cannot be directly predicted using the G1 network.

[0086] The purpose of constructing a smooth monotonic network is to constrain the monotonicity of the structural system response g(x) generated by the G1 network and to make the G1 network interpretable. It has a single-node input layer, two hidden layers, and a single-node output layer. The output of the smooth monotonic network increases or decreases monotonically as the input increases. In this paper, the input of the smooth monotonic network is the first dimension of the Gaussian variable z in the G1 network, defined as z1. The first hidden layer is calculated by adding weights and biases to the input. The output of the first hidden layer is then divided into several groups. The j-th output of the k-th group is defined as

[0087]

[0088] where w (k,j) is the weight, b (k,j) is the bias. The second hidden layer uses the output of each group in the first hidden layer to calculate its output. For example, the second hidden layer calculates the k-th output by

[0089]

[0090] Among them, h k is the number of outputs in the kth group. LSE β is a scaling LogSumExp function defined as

[0091]

[0092] Among them, a n Indicates LSE β The input of , c is a constant, and β is a learnable positive scaling parameter in the monotonically increasing network. Finally, the smooth monotonic network obtains the following output

[0093]

[0094] By synchronously training the G1 network and the smooth monotonic network, the smooth monotonic network can be used to predict the structural system response of the sample.

[0095] M(z1)=g(x)=g(G1(z)) (7)

[0096] Therefore, the structural system response g(x) of the output sample increases monotonically with the increase of the input Gaussian variable z1. This property is referred to as the interpretability of the G1 network. Due to the interpretability of the G1 network, the cumulative distribution function (CDF) of g(x) can theoretically be calculated using the following formula:

[0097]

[0098] Among them, z xis the input corresponding to the output x in the G1 network, Φ is the CDF of the one-dimensional Gaussian distribution, is the first dimension of the reverse prediction of the G1 network. However, due to the irreversibility of g(x), it is impossible to determine the x corresponding to a given g(x). It is not easy to use the G1 network to calculate F g(x) (g x ).

[0099] The G2 network is also a flow-based generative network, but its architecture is very different from that of G1. Its purpose is to directly predict the probability density of the structural system response g(x). The G2 network consists of a two-dimensional input layer, an output layer, and n coupling layers. The dimensions of each input and output layer are set to 2. The first dimension of the input layer of the G2 network is z1, and the first dimension of the output layer is the prediction of g(x), represented by g2. The second dimension of the input layer of the G2 network is used to construct a coupling layer, represented by λ. In this paper, λ is selected as a one-dimensional Gaussian variable that is unrelated to z1. The second item of the output of the G2 network is the prediction of λ itself, which is unrelated to g2. Similar to the G1 network, the G2 network can provide the following reversible mapping

[0100]

[0101] Among them, G2 and are the forward and reverse predictions of the G2 network. The output PDF can be predicted by the following formula:

[0102]

[0103] in, is the output prediction PDF, yes The Jacobian matrix of the G2 network can be used to directly predict the PDF and CDF of the structural system response.

[0104] 5.2 Neural Network Model Training Strategy

[0105] The training of IDGN-PDF model is a multi-objective learning task. The loss function is an important part of the training of IDGN-PDF model. In order to train G1 network and smooth monotone network, two loss functions are defined. The first loss function of IDGN-PDF model calculates the likelihood function of the predicted probability distribution. 50 to confirm.

[0106]

[0107] Using the first loss function, the predicted distribution of the G1 network can approximate the x distribution; the second loss function is determined by calculating the relative error between the prediction of the smooth monotonic network and the structural system response of the samples generated by the G1 network.

[0108]

[0109] Using the second loss function, the G1 network can learn interpretability. In other words, the g(x) of the G1 network will increase as z1 increases. In order to balance the two loss functions, two weights are used to combine the loss functions.

[0110] loss 1+2 =α1loss1+α2loss2 (13)

[0111] Where α1 and α2 are weights. The loss function of the G2 network is defined as the relative error of its prediction

[0112]

[0113] The first term represents the relative error between the predicted structural system response and the true value, and the second term represents the prediction error of λ.

[0114] The training samples for both the G1 network and the smooth monotone network are x. To calculate the second loss function, the structural system response g(x) of the training samples must be precalculated. For G2 network training, the first-dimensional training samples λ are generated using random sampling from a Gaussian distribution and combined with the structural system response samples used for training the smooth monotone network. To improve training efficiency, the training data can be rescaled according to its mean and standard deviation.

[0115]

[0116] Among them, x o is a random variable in the original coordinate space, g o (x) is the structural system response in the original coordinate space, and are the mean values of random variables and structural system responses, respectively, and is their standard deviation.

[0117] Finally, after defining the loss function and preparing the training data, the IDGN-PDF network is trained. The three subnetworks of IDGN-PDF are trained separately at different stages. First, to obtain a good initialization, the smooth monotonic network is pre-trained to approximate the linear function f(z1) = z1 before training the full model of the G1 network and the smooth monotonic network. Second, the G1 network and the smooth monotonic network are trained simultaneously. Third, the predictions of the G1 network and the smooth monotonic network are used to train the G2 network. To ensure training stability, the input of the first coupled layer in the G1 network is constrained to a small interval, such as (-10, 10), and the output of the last coupled layer in the G1 network is also constrained to a interval that covers all training examples. To prevent the loss from being too small, the first loss function is clipped using a lower bound calculated by log(max(q(x))), where q(x) is the maximum probability density of the training examples. The network is trained using mini-batch gradient descent and the Adam optimizer. The sample size of the mini-batch is generally selected as 100, and the learning rate used by the G1 and G2 networks is 1×10 -4 , the learning rate of the smooth monotonic network can be set to a larger value of 1×10 -3 .

[0118] 5.3 Probability Density Prediction Principle

[0119] PDF prediction is implemented using the G2 network. After training the G2 network, the G2 network can be used to predict g(x) and λ. In order to use the G2 network to obtain higher PDF prediction accuracy for g(x), the input λ of the G2 network is set to zero, that is, the Gaussian variable has a larger probability density. When the input λ is fixed to zero, the first term g2 output by the G2 network increases monotonically with the increase of the input z1. Combined with the reversible characteristics of the G2 network, the CDF of the structural system response g(x) can be calculated by the following formula

[0120]

[0121] in, It is the g in G2 network x The corresponding input, yes Since g2 and λ are independent of each other, the PDF of the structural system response g(x) can be calculated by the following formula

[0122]

[0123] Considering the normalization of g(x), the CDF and PDF of the structural system response in the original coordinate space can be calculated by the following formulas:

[0124]

[0125] This example uses the IDGN-PDF method of the present invention to analyze the response probability of a cable-stayed bridge under vehicle loads, demonstrating its accuracy and efficiency in predicting the probability density of large-scale engineering structural systems. The bridge is a steel box girder with a span arrangement of 48m+204m+460m+204m+48m. The bridge photo and finite element model are shown in Figure 1. Figure 2 According to the allowable displacement in the middle of the main span, LSF is defined as

[0126] g(x)=1.15-M(x), (21)

[0127] Where M(x) represents the calculation model for the mid-span displacement response of the main span; 1.15 is the allowable displacement specified in the design code; and x represents the random variables considered in the calculation model. The random variables include the normally distributed elastic modulus of the main beam steel, the elastic modulus of the tower concrete, the elastic modulus of the cable, the Gumbel-distributed uniform load on the roadway, and the log-normally distributed cross-sectional area of the wire rope. The means of the random variables are 200 GPa, 30.37 GPa, 191 GPa, -10.5 kN / m, and 0.385 cm, respectively. 2 , and the standard deviations are 10 GPa, 1.519 GPa, 19.1 GPa, -1.05 kN / m, and 0.0385 cm 2 .

[0128] In the first step, 5000 training samples are generated according to the random variable distribution. Figure 3 The training samples are plotted in the x3 and x4 dimensions. To facilitate the training of the IDGN-PDF network, the training samples are normalized to approximately zero mean and one standard deviation. Figure 4 and Figure 5 The following plots show 10,000 input Gaussian samples and the corresponding generated samples for the trained G1 network. As can be seen from these plots, as the input Gaussian variable z1 increases, the set of output samples shifts toward the upper right quadrant of the coordinate space. This result demonstrates that the input z1 can be used to control the distribution of the output samples. Figure 6 The LSF of different input Gaussian variables z1 of the G2 network are plotted. Figure 6 As shown in Figure 2, when the input Gaussian variable z1 increases, the LSF increases monotonically, which indicates that the G2 network has good interpretability.

[0129] Finally, the trained G2 network is used to calculate the PDF and CDF of the LSF. The probability density prediction results are compared with the reference solution obtained by 100,000 Monte Carlo simulations. Figure 7 (a) and Figure 7 As shown in (b), the predicted PDF is an asymmetric probability distribution, but it is in good agreement with the actual distribution. Figure 7 (a) and Figure 7The high-precision prediction of the probability distribution shown in (b) demonstrates the high accuracy of the IDGN-PDF method of the present invention in predicting the structural response probability of cable-stayed bridges. The calculation results in Table 1 show that the method of the present invention has higher computational efficiency.

[0130] Table 1 Comparison of calculation amount of different methods

[0131]

[0132] Although the embodiments of the present invention have been disclosed above, they are not limited to the applications listed in the description and the embodiments. They can be fully applied to various fields suitable for the present invention. For those familiar with the art, additional modifications can be easily implemented. Therefore, without departing from the general concept defined by the claims and the scope of equivalents, the present invention is not limited to the specific details and shown here.

Claims

1. A structural response probability density prediction method based on an interpretable deep generative network, characterized in that: The steps include: Step 1: Construct an IDGN-PDF neural network, which includes a distribution fitting generation network G1, a smoothing monotonic network and a probability density prediction generation network G2; The distribution fitting generation network G1 is used to achieve reversible prediction of the Gaussian variable z and the random variable x of the structural system: Among them, G1 and Generate forward and reverse predictions of network G1 for distribution fitting, respectively; The smooth monotonic network is used to constrain the monotonicity of the structural system response of the output samples of the distribution fitting generator network G1: The probability density prediction generation network G2 is used to achieve the following reversible prediction: Among them, G2 and are the forward and reverse predictions of the probability density prediction generation network G2; z1 is the first-dimensional input of the input layer of the probability density prediction generation network G2; λ is the second-dimensional input of the two-dimensional input layer of the probability density prediction generation network G2; the two-dimensional input is obtained by combining λ with z1, thereby meeting the requirement of paired input data of the probability density prediction generation network G2, and the value of λ is a one-dimensional Gaussian variable unrelated to z1; g2 is the first-dimensional output of the output layer of the probability density prediction generation network G2, that is, the forward prediction of z1; Generate the second-dimensional output of the output layer of the network G2 for probability density prediction, that is, the forward prediction of λ; Step 2: Train the IDGN-PDF neural network: First, train the distribution fitting network G1 and the smooth monotone network simultaneously; obtain the trained distribution fitting network G1 and the trained smooth monotone network; input the sample of the random variable x into the trained distribution fitting network G1 to obtain the first dimension of the reverse prediction of the trained distribution fitting network G1 As the z1 sample, the z1 sample is input into the trained smooth monotonic network to obtain the predicted value M(z1); then the z1 sample and λ are used as the input of the probability density prediction generation network G2, M(z1) and λ are used as the true output of the probability density prediction generation network G2, and the probability density prediction generation network G2 is trained to obtain the trained probability density prediction generation network G2; Step 3: Input the structural system response g(x) into the trained probability density prediction generation network G2 for reverse prediction to obtain the probability density function PDF and cumulative distribution function CDF of the structural system response g(x).

2. The structural response probability density prediction method based on an interpretable deep generative network according to claim 1, characterized in that: The distribution fitting generation network G1 includes an input layer, m coupling layers and an output layer in sequence according to the data processing direction; The smooth monotone network includes, in order according to the data processing direction: a single-node input layer, a first hidden layer, a second hidden layer, and a single-node output layer. The input of the smooth monotone network is the first dimension z1 of the Gaussian variable z, and the output is M(z1); M() represents the smooth monotone network; The probability density prediction generation network G2 includes a two-dimensional input layer, n coupling layers and a two-dimensional output layer in sequence according to the data processing direction.

3. The structural response probability density prediction method based on an interpretable deep generative network according to claim 1, characterized in that: In step 2, the steps of training the IDGN-PDF neural network are as follows: 2.1) Pre-training a smooth monotonic network to approximate the linear function f(z1)=z1; 2.2) Simultaneously train the distribution fitting generative network G1 and the smooth monotonic network to minimize the total loss function, thereby obtaining the trained distribution fitting generative network G1 and the trained smooth monotonic network; 2.3) Input the sample of random variable x into the trained distribution fitting generative network G1 to obtain the first dimension of the reverse prediction of the distribution fitting generative network G1 As the z1 sample, the z1 sample is input into the trained smooth monotonic network to obtain the predicted value M(z1); then z1 and λ are used as the input of the probability density prediction generation network G2, M(z1) and λ are used as the true output of the probability density prediction generation network G2, and the probability density prediction generation network G2 is trained to minimize the loss function of the probability density prediction generation network G2 and obtain the trained probability density prediction generation network G2.

4. The structural response probability density prediction method based on an interpretable deep generative network according to claim 2, characterized in that: In step 2.2), the total loss function is loss 1+2 : loss 1+2 =α1loss1+α2loss2; α1 and α2 are weights; loss1 is the loss function of the distribution fitting generation network G1, and loss2 is the loss function of the smooth monotonic network; in, is the PDF prediction of a sample of random variable x, for The Jacobian matrix operator, det is the determinant operator; π() is the Gaussian distribution probability density function; Generate the first dimension of the inverse prediction of network G1 for distribution fitting; Using the distribution fitting to generate the inverse network G1, we can get the reversibility of the network:

5. The structural response probability density prediction method based on an interpretable deep generative network according to claim 2, characterized in that: The loss function loss3 of the probability density prediction generation network G2 is as follows: in, Generate the prediction of λ by the network G2 for probability density prediction, λ) (1) Represents the first dimension of the output layer of the probability density prediction generation network G2, represents the output of a smooth monotonic network; To improve training efficiency, the training data is rescaled by its mean and standard deviation: Among them, x o is a random variable in the original coordinate space, g o (x) is the structural system response in the original coordinate space, and are the mean values of random variables and structural system responses, respectively, and are the standard deviations of the random variables and the structural system response, respectively.

6. The structural response probability density prediction method based on an interpretable deep generative network according to claim 2, characterized in that: The learning rate used by the distribution fitting generation network G1 and the probability density prediction generation network G2 is 1×10 -4 , the learning rate of the smooth monotonic network is 1×10 -3 .

7. The structural response probability density prediction method based on an interpretable deep generative network according to claim 2, characterized in that: The data processing method of the smooth monotone network is as follows: The first hidden layer of the smooth monotonic network is calculated by adding weights and biases to the input, and then dividing the output of the first hidden layer into several groups, where the j-th output of the k-th group is for, w (k,j) is the weight, b (k,j) It is a deviation; The second hidden layer uses the output of each group in the first hidden layer to calculate its output Among them, h k is the number of outputs in the kth group, LSE β is the LogSumExp function used for scaling; Then we get the output of the smooth monotonic network: K is the total number of groups; The smooth monotonic network is trained so that: M(z1)=g(x)=g(G1(z)) g() represents the structural system response function.

8. The structural response probability density prediction method based on an interpretable deep generative network according to claim 1 is characterized in that: The probability density prediction generation network G2 consists of a two-dimensional input layer, an output layer and n coupling layers. The dimension of each layer input and output is set to 2. In step 3, z1 is The first dimension of the output layer of the probability density prediction generation network G2 is the prediction g2 of g(x). The second dimension of the input layer of the probability density prediction generation network G2 is used to construct a two-dimensional input vector to meet the paired input requirements of the G2 network, represented by λ; λ is selected as a one-dimensional Gaussian variable unrelated to z1; the second item of the output of the probability density prediction generation network G2 is the prediction of λ itself, which is unrelated to g2; The PDF output by the probability density prediction generation network G2 is obtained by the following formula: in, is the PDF predicted by the probability density prediction generator network G2, yes The Jacobian matrix of .

9. The structural response probability density prediction method based on an interpretable deep generative network according to claim 1 is characterized in that: The CDF of the structural system response g(x) in the standard coordinate space is calculated by the following formula: in, It is the g in G2 network x The corresponding input, g x is the structural response value of the CDF to be calculated, yes The first dimension of the reverse prediction output; is the CDF of the predicted structural response value g2, P() represents the probability, P(g2≤g x ) for g2≤g x The probability of Φ() is the CDF of the one-dimensional Gaussian distribution; F g(x) (g x ) is the CDF of g(x); The PDF of the structural system response g(x) in the standard coordinate space is calculated by the following formula: For gx and The two-dimensional joint probability density function, that is, the two-dimensional joint PDF, p g(x) (g x ) is the PDF of g(x).

10. The structural response probability density prediction method based on an interpretable deep generative network according to claim 9, characterized in that: The CDF and PDF of the structural system response g(x) in the original coordinate space are calculated by the following formulas:

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