Bridge probability finite element model updating method based on BayesFlow
Through the BayesFlow model combined with feature networks and conditional reversible neural networks, the problems of high computational cost and high uncertainty in bridge finite element model updates are solved, and fast and accurate model updates and online evaluation are achieved.
Patent Information
- Application Number
- CN202510435214.X
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-04-08
- Publication Date
- 2025-07-18
AI Technical Summary
When dealing with complex civil engineering structures, the existing bridge finite element model update method has problems such as high computational cost, insufficient sampling method, insufficient generalization ability of proxy models, and high uncertainty in online updates, especially the difficulty in solving likelihood functions and obvious uncertainty in neural network training in Bayesian theorem.
The BayesFlow model is adopted, combining feature networks and conditional reversible neural networks, and the maximum statistical information is automatically extracted by scaling high-dimensional data to low dimensions and automatically extracting the maximum statistical information, so as to realize posterior distribution inference of parameters, avoid direct calculation of likelihood functions and training uncertainty of traditional neural networks, and use Monte Carlo simulation and stochastic gradient descent to optimize the training process.
Fast and accurate bridge probability finite element model updates are achieved, reducing computational costs and training uncertainty, and supporting online structural status assessment and health monitoring.
Smart Images
Figure CN120337653A_ABST
Abstract
Description
Technical Field
[0001] The present invention relates to the field of structural health monitoring, and particularly to a method for updating the probabilistic finite element model of a bridge based on BayesFlow. Background Art
[0002] The finite element model of a bridge is an important tool for predicting the response and performance of the bridge structure. The update of the finite element model can identify the weak links and potential risk points in the bridge structure; by comparing and analyzing the models before and after the update, the damage or degradation in the bridge structure can be detected in a timely manner, providing a scientific basis for the maintenance of the bridge. This helps to formulate a more scientific and reasonable maintenance strategy, extend the service life of the bridge, and reduce the maintenance cost.
[0003] Traditionally, according to whether uncertainty is considered, the finite element model update methods can be divided into two categories: deterministic and probabilistic. The deterministic methods mainly include iterative methods based on gradients or sensitivities, metaheuristic algorithms, and artificial neural networks, etc. The deterministic finite element model update methods generally can only consider the point estimation of the finite element model parameters and cannot provide any confidence or uncertainty bounds. Bayes' theorem is often used for the update of the finite element model with uncertainty. This type of method combines the prior distribution of the finite element model parameters representing the engineer's experience and the likelihood function representing the observed data, and then uses Bayes' theorem to obtain the posterior distribution of the finite element model parameters. For complex civil engineering structures, the main problem faced by the Bayes-based finite element model update method is the difficulty in solving the likelihood function. The expression of the posterior distribution of the finite element model parameters is non-standard and high-dimensional, making it difficult to directly integrate and solve. Therefore, random sampling methods are usually adopted to approximate the posterior distribution of the parameters, such as Markov Chain Monte Carlo (MCMC), Transitional Markov Chain Monte Carlo (TMCMC), and Hamiltonian Monte Carlo (HMC). Wan et al. updated the finite element model of a pedestrian cable-stayed bridge using the MCMC algorithm and used the Gaussian process as a surrogate model to reduce the computational cost. Fang et al. proposed a practical Bayesian inference method based on the MCMC algorithm and software interaction and applied the proposed method to the finite element model update of the Ting Kau cable-stayed bridge in Hong Kong, China. Asadollahi et al. applied the TMCMC algorithm to the cable-stayed bridge model and proposed a method for quantifying the uncertainty in the prediction error. Mao et al. used Kriging as a surrogate model and compared the performance of the MH-MCMC algorithm and the HMC algorithm in the finite element model update of a suspension bridge. Baisthakur and Chakraborty proposed an improved HMC algorithm and applied the proposed method to the finite element model update of a steel truss bridge. However, the problem with the random sampling algorithm is that a large number of samples will bring high computational costs. Although surrogate models can be used to reduce the computational burden, the surrogate models face problems such as insufficient generalization ability and unreliable extrapolation results.
[0004] To avoid dealing with the complex likelihood function in Bayes' theorem, likelihood-free methods emerged, among which approximate Bayesian computation is the most mature. Approximate Bayesian computation approximates the posterior by repeatedly sampling from the prior distribution of the parameters and then running the forward model simulation to obtain multiple data sets. If the simulated data set is similar enough to the actually observed data set, the corresponding data set is accepted as a posterior sample; otherwise, it is rejected. However, the tolerance of the "accept-reject" mechanism in approximate Bayesian computation largely determines the estimation accuracy of the posterior distribution. A lower tolerance can improve the accuracy of the posterior approximation distribution, but it will also significantly increase the rejection rate, resulting in a substantial increase in computational costs. Additionally, like the aforementioned Bayesian methods, approximate Bayesian computation only infers the posterior distribution of parameters for single measurement data. When the measurement data changes, the entire process of posterior inference of parameters needs to be re-executed, which is not conducive to online structural state assessment or SHM.
[0005] To simultaneously meet the requirements of not directly calculating the likelihood function and being applicable to multiple measurement data or online finite element model updating, it is a feasible idea to combine Bayesian theory and neural networks. Existing research has proposed using Bayesian neural networks for finite element model updating of bridge structures and has adopted relevant technologies to reduce network training time and optimize the number of neurons and transfer functions in the network hidden layer; however, this itself will bring uncertainties in model training.
[0006] In summary, when using sampling approximation methods such as the MCMC method to solve the posterior distribution of updated parameters, thousands of finite element model analyses need to be carried out. Although surrogate models (such as Gaussian process models, Kriging) can be used to avoid time-consuming finite element model calculations. However, these methods are faced with problems such as an insufficiently comprehensive sample space and insufficient generalization ability of the surrogate model, resulting in unreliable extrapolation results of the surrogate model. The performance of likelihood-free methods such as approximate Bayesian computation is severely affected by the selection of key parameters of the method. When using neural network methods such as Bayesian neural networks for finite element model updating, correction targets such as modal shapes or modal frequencies are used as the input of the neural network, and the updated parameters of the finite element model are used as the output of the neural network, which itself will bring uncertainties in model training.
[0007] In view of this, the present invention proposes a method for updating the probabilistic finite element model of a bridge based on BayesFlow. Summary of the Invention
[0008] The purpose of the present invention is to propose a method for updating the probabilistic finite element model of a bridge based on BayesFlow, which can quickly and accurately update the probabilistic finite element model of the bridge.
[0009] To achieve the above object, the technical solution of the present invention is: a method for updating the probability finite element model of a bridge based on BayesFlow, specifically including the following steps:
[0010] S1: Determine the finite model update parameters and determine the change range for training the update parameters using BayesFlow;
[0011] S2: Generate training samples for the BayesFlow model;
[0012] S3: Design the BayesFlow model; the BayesFlow model includes a feature network and an inference network h ω (·), the feature network scales the high-dimensional data to a low dimension and automatically extracts the maximum statistical information, and the inference network h ω (·) infers the posterior distribution of the update parameters based on the statistical information;
[0013] S4: Jointly train the parameters in the BayesFlow model and h ω (·);
[0014] S5: Accuracy verification of the BayesFlow model
[0015] S6: Posterior inference of the update parameters.
[0016] Preferably, the S2 is specifically as follows:
[0017] First, assume that the distribution of the update parameters is a uniform distribution, and then use the Latin hypercube sampling method in the MTLAB software to randomly sample the update parameters within the change range of the update parameters to obtain update parameter samples; parameterize the finite element model of the bridge, including using the APDL language of ANSYS; use the MTLAB software to call the ANSYS software to generate training samples according to the update parameter samples; the training samples include two parts: the update parameter vector θ and the update target quantity y, and the update target quantity y is the simulated observed value, including modal frequency, vibration mode, and cable force of the cable-stayed bridge; finally, divide the training samples into a training set and a validation set.
[0018] Preferably, the feature network adopts a long short-term memory network or a convolutional neural network, and the inference network h ω (·) adopts a conditional invertible neural network cINN.
[0019] Preferably, the BayesFlow model uses the simulated observed value y as the input of the feature network and uses the feature network The output is passed to the cINN as conditional observations, based on which the cINN realizes a bidirectional invertible mapping between the updated parameter vector θ and the latent variable z, where the latent variable z follows a standard Gaussian distribution.
[0020] Preferably, the S4 is specifically as follows:
[0021] During the training of the BayesFlow model, h(·) and ω (·) and the model parameters ω and γ of are jointly trained, and the training objective is to minimize the KL divergence value between the true posterior f(θ|y) of the simulated observations y and the estimated posterior , that is:
[0022]
[0023] where f(y) is the probability density function of y, E(·) is the expectation operator, is the posterior estimate of θ, and KL(·) is the KL divergence function;
[0024] According to the rule of mapping non-Gaussian distributions to Gaussian distributions in normalizing flows, is expressed as:
[0025]
[0026] where p z represents the probability density function of the latent variable z, then the training objective function of the BayesFlow model is rewritten as:
[0027]
[0028] An approximate solution of Equation (3) is obtained through Monte Carlo simulation;
[0029] For a training data set containing M samples the estimation of the model parameters is expressed as:
[0030]
[0031] Equation (4) is the training loss function of the BayesFlow model, which is minimized using the stochastic gradient descent method.
[0032] Preferably, the S5 is specifically as follows:
[0033] When the training of the BayesFlow model is completed, the accuracy of the model is verified using the validation set data; the accuracy metrics use the coefficient of determination R 2 and the normalized root mean square error NRMSE. The closer R 2 is to 1 and the closer NRMSE is to zero, the higher the accuracy of the BayesFlow model.
[0034] Preferably, S6 is specifically as follows:
[0035] Obtain the actual measurement observation value y obs , and the actual measurement observation value y obs is input into the feature network for feature extraction, and the extracted features are input into the inference network h ω (·). The inference network h ω (·) performs multiple inferences on the posterior distribution of the update parameters θ of the finite element model according to the latent variable z. The average value of the inferred values of θ will be used as the updated value of the finite element model parameters, and the variance of θ is used to evaluate the update uncertainty.
[0036] Compared with the prior art, the present invention has the following beneficial effects:
[0037] The present invention avoids the solution of the likelihood function in the Bayesian finite element model update and avoids the finite element model update method based on the traditional neural network, and can realize the online finite element model update and the training uncertainty problem during model training. BRIEF DESCRIPTION OF THE DRAWINGS
[0038] Figure 1 It is a schematic diagram of the bridge probabilistic finite element model based on BayesFlow of the present invention. DETAILED DESCRIPTION OF THE EMBODIMENTS
[0039] The following combines the attached Figure 1 , and specifically describes the technical solution of the present invention.
[0040] The present invention proposes a method for updating a bridge probabilistic finite element model based on BayesFlow; the overall idea is to use a feature network to extract features from the simulated observation data y of the finite element model, and input the extracted features into the cINN. Then, the cINN is used to realize the reversible mapping between the finite element model parameter vector and the latent variable z. Finally, when the actual measurement observation data is obtained, BayesFlow performs reverse inference to obtain the posterior distribution of the finite element model parameter vector. The implementation of the present invention can be divided into an offline training stage (S1 to S5) and an online finite element model update stage (S6). The specific implementation process is as follows:
[0041] S1: Determine the finite model update parameters and determine the range of changes for the update parameters used in BayesFlow training.
[0042] S2: Generate training samples for the BayesFlow model.
[0043] First, assume that the distribution of the update parameters is a uniform distribution, and then use the Latin hypercube sampling method in the MTLAB software to randomly sample the update parameters to obtain the update parameter samples. The finite element model of the bridge is parametrically modeled. For example, using the APDL language of ANSYSd, the MTLAB software can be used to call the ANSYS software to generate training samples according to the update parameter samples. The training samples consist of two parts. One is the update parameter vector θ, and the other is the update target quantity y, such as modal frequency, vibration mode, cable force of cable-stayed bridge, etc. y is also the simulated observation value. Finally, for the convenience of verifying the accuracy of the BayesFlow model, the training samples are divided into a training set and a validation set.
[0044] S3: Design of the BayesFlow model.
[0045] The BayesFlow model contains a feature network and an inference network h ω (·). The role of is to scale the high-dimensional data to a low dimension and automatically extract the maximum statistical information. h ω (·) infers the posterior distribution of the update parameters based on the statistical information. According to the temporal or spatial characteristics of the observed data y, can be selected as a long short-term memory network or a convolutional neural network, etc. h ω (·) is implemented by a conditional invertible neural network (cINN). In the BayesFlow model, the simulated observation value y is the input of, and the output of will be passed as a conditional observation value to the cINN. On this basis, the cINN realizes the bidirectional invertible mapping between the update parameter vector θ and the latent variable z, and the latent variable z is a standard Gaussian distribution.
[0046] S4: Jointly train the parameters of and h and h ω (·) in the BayesFlow model.
[0047] When training the BayesFlow model, jointly train the model parameters ω and γ of h ω (·) and . The training objective is to minimize the KL divergence value between the true posterior f(θ|y) of the simulated observation value y and the estimated posterior , that is:
[0048]
[0049] In the formula, f(y) is the probability density function of y, E(·) is the expectation operator, is the posterior estimate of θ, and KL(·) is the KL divergence function.
[0050] According to the rule of mapping the non-Gaussian distribution in the standardization flow to the Gaussian distribution, it can be expressed as:
[0051]
[0052] p z represents the probability density function of the latent variable z; further, the training objective function of BayesFlow can be rewritten as:
[0053]
[0054] The approximate solution of Equation (3) can be obtained by Monte Carlo simulation.
[0055] For a training data set containing M samples the estimation of the model parameters can be expressed as:
[0056]
[0057] Equation (4) is the training loss function of the BayesFlow model, which can be minimized by the stochastic gradient descent method.
[0058] S5: Precision verification of the BayesFlow model
[0059] When the training of the BayesFlow model is completed, the precision of the model is verified using the data in the validation set. The precision metrics can use the coefficient of determination (R 2 ) and the normalized root mean square error (NRMSE). The closer R 2 is to 1 and the closer NRMSE is to zero, the higher the accuracy of the BayesFlow model.
[0060] S6: Posterior inference of updated parameters.
[0061] When the precision of the BayesFlow model is sufficient, it can be used for the posterior inference of the updated parameters of the model. Specifically, when the actual measured observation value y obs is obtained, and y obs is also input into the feature network for feature extraction, and the extracted features are input into the inference network h ω (·). The inference network h ω (·) makes multiple inferences on the posterior distribution of the updated parameters θ of the finite element model according to the latent variable z. The average value of the inferred values of θ will be used as the updated value of the finite element model parameters, and the variance of θ will be used to evaluate the update uncertainty.
[0062] The application method of the present invention is as follows:
[0063] Use finite element software such as ANSYS to perform parametric modeling of the bridge, and use the PYTHON language to implement the programming of the BayesFlow model. Then select the parameters for updating the finite element model, and use MATALB to call ANSYS to generate the training data for the BayesFlow model. Finally, complete the training of the BayesFlow model and update the model according to the actual measured observations.
[0064] The above are the preferred embodiments of the present invention. All changes made according to the technical solution of the present invention, when the functions and effects produced do not exceed the scope of the technical solution of the present invention, fall within the protection scope of the present invention.
Claims
1. A method for updating the probabilistic finite element model of a bridge based on BayesFlow, characterized in that, Specifically, it includes the following steps: S1: Determine the finite model update parameters and determine the range of changes for the BayesFlow training update parameters; S2: Generate training samples for the BayesFlow model; S3: Design the BayesFlow model; the BayesFlow model contains a feature network and an inference network h ω (·), the feature network scales high-dimensional data to a low dimension and automatically extracts the maximum statistical information, and the inference network h ω (·) infers the posterior distribution of the updated parameters based on the statistical information; S4: Jointly train the parameters of the BayesFlow model and h ω (·); S5: Precision verification of the BayesFlow model S6: Posterior inference of the update parameters.
2. The method for updating the probabilistic finite element model of a bridge based on BayesFlow according to claim 1, wherein The specific content of S2 is as follows: First, assume that the distribution of the update parameters is a uniform distribution. Then, in the MTLAB software, use the Latin Hypercube Sampling method to randomly sample the update parameters within the range of changes of the update parameters to obtain update parameter samples; Parametric modeling of the finite element model of the bridge, including using the APDL language of ANSYS; According to the update parameter samples, use the MTLAB software to call the ANSYS software to generate training samples; The training samples include two parts: the update parameter vector θ and the update target quantity y, and the update target quantity y is the simulated observation value, including modal frequency, vibration mode, and cable force of the cable-stayed bridge; Finally, divide the training samples into a training set and a validation set.
3. The method for updating the bridge probability finite element model based on BayesFlow according to claim 2, characterized in that The feature network employs a long short-term memory network or a convolutional neural network, and the inference network h ω (·)employs a conditional invertible neural network cINN.
4. A method for updating the probabilistic finite element model of a bridge based on BayesFlow according to claim 3, characterized in that The BayesFlow model uses the simulated observed value y as the input of the feature network and passes the output of the feature network as the conditional observed value to the cINN. On this basis, the cINN realizes a bidirectional reversible mapping between the updated parameter vector θ and the latent variable z, where the latent variable z follows a standard Gaussian distribution.
5. A method for updating the probabilistic finite element model of a bridge based on BayesFlow according to claim 4, characterized in that, The specific content of S4 is as follows: Joint training of h during the training of the BayesFlow model ω (·) and the model parameters ω and γ of, and the training objective is to minimize the KL divergence value between the true posterior f(θ|y) of the simulated observation value y and the estimated posterior That is: where \(f(y)\) is the probability density function of \(y\), \(E(\cdot)\) is the expectation operator, is the posterior estimate of \(\theta\), and \(KL(\cdot)\) is the KL divergence function; According to the rule of mapping non-Gaussian distribution in the standardization flow to Gaussian distribution, It is expressed as: where p z represents the probability density function of the latent variable z, the training objective function of the BayesFlow model is rewritten as: Obtain an approximate solution of Equation (3) through Monte Carlo simulation; For a training data set containing M samples The estimation of the model parameters is expressed as: Equation (4) is the training loss function of the BayesFlow model, which is minimized using the stochastic gradient descent method.
6. A method for updating the probabilistic finite element model of a bridge based on BayesFlow according to claim 1, characterized in that, The specific content of S5 is as follows: When the BayesFlow model training is completed, the accuracy of the model is verified using the validation set data; the accuracy metrics use the coefficient of determination R 2 and the normalized root mean square error NRMSE. The closer R 2 is to 1 and the closer NRMSE is to zero, the higher the accuracy of the BayesFlow model indicates.
7. A method for updating the probabilistic finite element model of a bridge based on BayesFlow according to claim 1, characterized in that The specific content of S6 is as follows: Obtain the actual measured observation value y obs , the actual measured observation value y obs is input into the feature network for feature extraction, and the extracted features are input into the inference network h ω (·), the inference network h ω (·) makes multiple inferences on the posterior distribution of the parameters θ for updating the finite element model according to the latent variable z, and the average value of the inferred values of θ is used as the updated value of the finite element model parameters, and the variance of θ is used to evaluate the update uncertainty.