Bridge thermal strain prediction method under non-constant noise influence
By using a heteroscedastic Gaussian process regression model and variational Bayesian inference method, the problem of non-constant noise influence in bridge temperature-induced strain prediction is solved, achieving high-precision and efficient temperature-induced strain prediction and quantifying uncertainty.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- CHANGSHA UNIVERSITY OF SCIENCE AND TECHNOLOGY
- Filing Date
- 2025-07-24
- Publication Date
- 2026-07-21
AI Technical Summary
Existing methods for predicting temperature-induced strain in bridges fail to effectively handle the effects of non-constant noise, resulting in decreased prediction accuracy and insufficient uncertainty characterization, making it difficult to reflect the true service condition of bridges.
A heteroscedastic Gaussian process regression model was adopted. Data was obtained through a bridge health monitoring system to construct a joint prior distribution of the latent function and the noise log-variance function. The posterior distribution was optimized using the variational Bayesian inference method to construct a bridge temperature-induced strain prediction model adapted to non-constant noise.
It achieves high-precision prediction of temperature-induced strain in bridges, can explain measurement errors and uncertainties in model predictions, and improves prediction accuracy and computational efficiency.
Smart Images
Figure CN120930093B_ABST
Abstract
Description
Technical Field
[0001] This invention relates to the field of bridge service condition assessment technology, and in particular to a method for predicting temperature-induced strain in bridges under the influence of non-constant noise. Background Technology
[0002] The main girder structure undergoes thermal expansion and contraction under temperature changes. Influenced by factors such as structural form, material properties, and boundary constraints, it is prone to uneven deformation and stress concentration, leading to crack propagation and performance degradation. In severe cases, this can cause overall bridge instability or even collapse. Temperature-induced strain in the main girder is the relative deformation per unit length along the component's axis caused by temperature changes. It directly reflects the bridge's internal force distribution, structural deformation, and service status, serving as a crucial indicator for identifying abnormal conditions and diagnosing defects. Constructing a temperature-induced strain prediction model for the main girder based on long-term temperature and strain monitoring data enables multi-scale characterization of the main girder's temperature response characteristics and structural performance evaluation, providing support for identifying the main girder's health status and assessing risks.
[0003] Traditional main girder strain prediction relies primarily on deterministic methods, neglecting the influence of uncertainties such as traffic load, material parameters, and sensing errors. Probabilistic main girder strain prediction methods quantify the uncertainty of prediction results through probability density, quantiles, or prediction intervals. Gaussian process regression, a nonparametric probabilistic prediction model based on Bayesian theory, offers advantages such as fewer model parameters, strong generalization ability, and adaptive acquisition of hyperparameters, and is widely used in modeling the strain response of main girders under temperature effects. However, due to factors such as temperature fluctuations, traffic load, and the stability of the sensing system, observation noise exhibits significant heteroscedasticity. Existing temperature-induced strain prediction methods for main girders based on Gaussian process regression models often assume constant observation noise, leading to reduced uncertainty characterization and accuracy of the prediction results, making it difficult to reflect the time-varying characteristics and health status of the main girder in actual service. Therefore, there is an urgent need to propose a method for predicting bridge temperature-induced strain under the influence of non-constant noise. Summary of the Invention
[0004] To address the problems in existing technologies, this invention proposes a method for predicting the temperature-induced strain of bridges under the influence of non-constant noise. This method enables the prediction of the temperature-induced strain of the main beam, while also explaining the uncertainties caused by measurement errors and model predictions. By considering non-constant noise, the prediction accuracy is improved.
[0005] To achieve the above objectives, the present invention adopts the following technical solution:
[0006] A method for predicting temperature-induced strain in bridges under the influence of non-constant noise includes the following steps:
[0007] Step 1: Obtain historical monitoring data of the main girder temperature and temperature-induced strain through the bridge health monitoring system, and extract the main girder temperature X and temperature-induced strain y as the training set data D = (X, y); assume the test set data is D. * =(X * ,y * );
[0008] Step 2: Within the framework of heteroscedastic Gaussian processes, obtain the sample latent function f and the noise log-variance function g based on D = (X,y). Then, construct the joint prior distribution p(f,g) containing the undetermined hyperparameter θ. Based on p(f,g) and the likelihood function p(y|f,g), construct the true joint posterior distribution p(f,g|y). Then, construct the latent function f separately. * The conditional distribution p(f) * |f) and the noise log-variance function g * The conditional distribution p(g) * |g), and simultaneously based on f inferred from the posterior of y. * and g * Constructing a test set of temperature-induced strain y in the main beam * The conditional distribution p(y) * |f * ,g * According to p(f,g|y) and p(f) * |f), p(g * |g), p(y * |f * ,g * Derivation of the temperature-induced strain y of the main beam * The conditional distribution p(y) * |y), and p(y) * |y) is used as an undetermined heteroscedastic Gaussian regression model;
[0009] Step 3: Using the variational Bayesian inference method, construct the variational free energy bound F(q(f),q(g)) based on the log-marginal likelihood function logp(y) and the KL divergence between the joint posterior variational distribution q(f,g) and p(f,g|y); take the maximum value of F(q(f),q(g)) with respect to q(f) as the optimal posterior variational distribution q. * (f), q * (f) Substituting into F(q(f),q(g)) yields the edge variable boundary F(q(g));
[0010] Step 4: Assuming the posterior variational distribution q(g) follows a multivariate normal distribution N(μ,Σ), substitute q(g) into F(q(g)) to obtain the new marginal variational boundary F(μ,Σ); solve for the stationary points of F(μ,Σ), and construct a positive semi-definite diagonal matrix Λ based on the stationary points. Transform μ and Σ into functions μ(Λ) and ∑(Λ) in Λ. Simultaneously, based on the principle of Type-II maximum likelihood estimation, use the gradient algorithm to maximize F(μ(Λ),∑(Λ)) with respect to the hyperparameters θ and Λ, and solve to obtain... and hyperparameters
[0011] Step 5: According to get and Then according to and Obtain the posterior variational distribution Will Substituting into F(q(f),q(g)), the optimal posterior variational distribution is obtained by solving. Obtain new main beam temperature data Obtain the latent function value logarithmic variance function value of noise Will and Substitute p(f) into each * |f) and p(g * In |g), we get and Will and Substituting into the undetermined heteroscedastic Gaussian regression model, we obtain the final predicted distribution. according to For y * Make predictions.
[0012] It should be noted that the heteroscedastic Gaussian process framework is as follows:
[0013] The regression model for a heteroscedastic Gaussian process is y(x i )=f(x i )+ε(x i ), x i Let y(x) be the temperature monitoring value of the i-th main beam. i ) is x i Temperature-induced strain value of main beam, ε(x) i ) represents the observation noise function;
[0014] Wherein, the sample latent function and Σ f Let f(x) be the variance scaling parameter and covariance matrix, respectively.
[0015] Wherein, the observation noise function ε=ε(x)~N(0,exp(g(x))), and the noise log-variance function μ0 and 1 r Let g(x) be the prior mean of g(x) and an r×1 vector of all 1s, respectively. and Σ g Let g(x) be the variance scaling parameter and covariance matrix, respectively.
[0016] Furthermore, the two Gaussian processes f(x) and g(x) are independent of each other.
[0017] It should be noted that in step two, p(f,g) is determined according to equation (2.1):
[0018]
[0019] In the formula: f is the sample latent function of y; g is the noise log-variance function of y; p(f,g) is the joint prior distribution of f and g; p(f) is the prior distribution of f; p(g) is the prior distribution of g; x i x j Let be the i-th and j-th main beam temperature data in the training set, i≠j; 0 is the zero mean vector of f; μ01 is the mean vector of g; f is based on x i x j The covariance matrix between them; For g based on x i x j The covariance matrix between them;
[0020] Wherein, p(f) is determined according to equation (2.2):
[0021]
[0022] Wherein, p(g) is determined according to equation (2.4):
[0023]
[0024] In the formula: μ0 is the prior mean of g(x), which is an undetermined hyperparameter; 1 r Represents an r×1 vector of all 1s;
[0025] Wherein, the covariance matrix and Both adopt the form of a squared exponential kernel function and are determined according to equations (2.6) and (2.7) respectively:
[0026]
[0027]
[0028] In the formula: θ=[θ f ,θ g [μ0] is the set of hyperparameters. and l is The hyperparameters are denoted as θ. f , and l is The hyperparameters are denoted as θ. g ; l、 These represent the function variance, length dimension, and noise variance, respectively; ||x i -x j ||2 is x i x j The n-dimensional Euclidean distance between them.
[0029] It should be noted that in step two, within the framework of heteroscedastic Gaussian processes, based on the joint prior distribution p(f,g) and the likelihood function p(y|f,g), the true joint posterior distribution p(f,g|y) is obtained according to equation (2.8):
[0030]
[0031] In the formula: p(f,g|y) is the true joint posterior distribution of f and g given y; p(f,g) is the joint prior distribution of f and g; p(y|f,g) is the likelihood function of y; p(y) is the marginal likelihood function of y;
[0032] The likelihood function p(y|f,g) follows a multivariate normal distribution and is determined according to equation (2.9):
[0033]
[0034] In the formula: y represents the temperature-induced strain of the main beam in the training set. I r Let r×r be the identity matrix. For I r The transpose of y is given by , and R is the noise variance matrix of y.
[0035] Wherein, R is determined according to formula (2.10):
[0036] R=diag(exp(g(x1),exp(g(x2),...,exp(g(x r )))(2.10)
[0037] In the formula: diag represents the transformation of a vector into a diagonal matrix, whose diagonal elements are exp(g(x1)), exp(g(x2)), ..., exp(g(x...)). r )), all off-diagonal elements are 0.
[0038] It should be noted that in step two, based on the true joint posterior distribution p(f,g|y) and the three conditional distributions p(y) * |f * ,g * ), p(f * |f), p(g * |g), Test the temperature-induced strain of the main beam y * The conditional distribution p(y) * |y) is obtained according to equation (2.14):
[0039] p(y * |y)=∫p(y * |f * ,g * )·p(f * |f)·p(g * |g)·p(f,g|y)dfdgdf * dg * (2.14)
[0040] In the formula: p(y * |y) represents the temperature-induced strain y of the main beam in the test concentration. * conditional distribution; f * For y * The latent function; g * For y * The noise log-variance function; p(y * |f * ,g * f is obtained through posterior inference. * and g * Next y * The conditional distribution of p(f); * |f) is f under a given f * The conditional distribution of p(g); * |g) represents g given g. * The conditional distribution of ; p(f,g|y) is the true joint posterior distribution.
[0041] It should be noted that in step three, the variational free energy bound F(q(f),q(g)) is determined according to equation (3.1):
[0042] F(q(f),q(g))=logp(y)-KL(q(f,g)||p(f,g|y)) (3.1)
[0043] In the formula: F(q(f),q(g)) is the variational free energy bound; logp(y) is the log marginal likelihood function; KL(q(f,g)||p(f,g|y)) is the KL divergence between the joint posterior variational distributions q(f,g) and p(f,g|y);
[0044] Among them, based on the mean field assumption, solvable and mutually independent posterior variational distributions q(f) and q(g) are introduced, and q(f,g) is determined according to equation (3.2):
[0045] q(f,g)=q(f)q(g) (3.2)
[0046] In the formula: q(f,g) is the joint posterior variational distribution; q(f) is the posterior variational distribution of f; q(g) is the posterior variational distribution of g; q(f) and q(g) are independent of each other and both follow a multivariate Gaussian distribution;
[0047] The marginal likelihood function p(y) is obtained according to equation (3.3):
[0048] p(y)=∫∫p(y|f,g)p(f)p(g)dfdg (3.3)
[0049] In the formula: p(y) is the marginal likelihood function of y;
[0050] The KL divergence between q(f,g) and p(f,g|y) is obtained according to equation (3.4):
[0051]
[0052] In the formula: KL(q(f)q(g)||p(f,g|y)) is the KL divergence between the joint posterior variational distribution q(f,g) and the true joint posterior distribution p(f,g|y).
[0053] It should be noted that in step three, the optimal posterior variational distribution q * (f) Determined according to formula (3.5):
[0054]
[0055] In the formula: q * (f) represents the optimal posterior variational distribution; The variational free energy bound F(q(f),q(g)) is maximized with respect to q(f); Z(q(g)) is a normalization constant that ensures q * (f) The integral is 1; logp(y|f,g) is the log-conditional likelihood function of y given f and g;
[0056] F(q(g)) is obtained according to equation (3.6):
[0057] F(q(g))=logZ(q(g))-KL(q(g)||p(g)) (3.6)
[0058] In the formula: F(q(g)) is the edge variable boundary; KL(q(g)||p(g)) is the KL divergence between q(g) and p(g); p(g) is the prior distribution of the noise log-variance function g.
[0059] It should be noted that in step four, F(μ,Σ) is obtained according to equation (4.4):
[0060]
[0061] In the formula: μ is the mean vector of q(g); Σ is the covariance matrix of q(g); R' is the noise variance diagonal matrix, obtained by diagonal transformation of the noise variance matrix R; tr(Σ) is the trace of the covariance matrix Σ. That is, the sum of the elements on the main diagonal of the covariance matrix.
[0062] It should be noted that in step four, μ(Λ) and ∑(Λ) are determined according to equations (4.5) and (4.6), respectively;
[0063]
[0064]
[0065] In the formula: μ(Λ) is a function of μ with respect to Λ; Σ(Λ) is a function of Σ with respect to Λ; Λ is a positive semi-definite diagonal matrix; Covariance matrix An invertible matrix.
[0066] It should be noted that in step five, Determine according to formulas (5.1) and (5.2):
[0067]
[0068] Right now:
[0069] In the formula: α is the weight vector,
[0070] and posterior variational distribution The results obtained according to (5.3) and (5.4) respectively are as follows:
[0071]
[0072]
[0073] In the formula: and They are respectively and The posterior variational distribution; They are respectively The predicted mean and predicted variance; They are respectively The predicted mean and predicted variance;
[0074] in, Determine according to equations (5.5) and (5.6) respectively:
[0075]
[0076]
[0077] In the formula: For f and Covariance matrix between; for The transpose of the matrix; for and The covariance value;
[0078] in, Determine according to equations (5.7) and (5.8) respectively:
[0079]
[0080]
[0081] In the formula: For g and Covariance matrix between; for The transpose of the matrix; for and The covariance value.
[0082] It should be noted that in step five, the final predicted distribution Determine according to formulas (5.9) and (5.10):
[0083]
[0084] Right now:
[0085] In the formula: Main beam temperature-induced strain y * The final predicted distribution;
[0086] Therefore, the main beam temperature response y *The predicted mean and variance are obtained according to equation (5.11):
[0087]
[0088] In the formula: Based on the training set data D and the newly acquired main beam temperature data Temperature-induced strain y of lower main beam * The predicted mean; Based on the training set data D and the newly acquired main beam temperature data Temperature-induced strain y of lower main beam * The prediction variance.
[0089] Compared with the prior art, the beneficial effects of the present invention are as follows:
[0090] 1. This invention constructs a temperature-induced strain prediction model for main beams, which can realize the probabilistic prediction of the strain response of main beams under temperature load, explain the uncertainty caused by measurement errors and model prediction, and quantify the uncertainty of new prediction data.
[0091] 2. This invention establishes a framework for predicting the temperature-induced strain of bridge main beams based on a heteroscedastic Gaussian process regression model that adapts to non-constant noise. It introduces a noise variance function to handle the noise levels at different observation points in the input data, which can effectively ensure the model's prediction accuracy under time-varying noise conditions.
[0092] 3. This invention proposes an approximate analytical method based on variational Bayesian inference, which introduces an easily computable parameterized Gaussian distribution. By maximizing the lower bound of evidence, the optimizable approximate distribution approximates the true posterior distribution, which is difficult to compute directly. This reduces the computational complexity of the iterative process and improves computational efficiency. Attached Figure Description
[0093] Figure 1 This is a flowchart of the bridge temperature-induced strain prediction method under the influence of non-constant noise according to the present invention. Detailed Implementation
[0094] The technical solutions of the present invention will now be clearly and completely described in conjunction with the accompanying drawings.
[0095] like Figure 1 , Figure 1 This is a flowchart of the bridge temperature-induced strain prediction method under the influence of non-constant noise according to the present invention.
[0096] The Gaussian process regression used in this invention is a nonparametric regression model based on a Bayesian probabilistic framework. The observation model of a conventional Gaussian process regression is y(x i )=f(x i )+ε i , εi x i Let y(x) be the input feature value of the i-th sample point. i ) is the input x i The observed output value, i.e., the noisy observed value, f(x) i ) represents the true function value of the Gaussian process, ε i To conform to the observed noise function value, assume the noise variance. It is a constant, but in reality, the noise variance is... It is not constant; it depends on the input data x, hence the introduction of a heteroscedastic noise Gaussian regression model.
[0097] Within the framework of heteroscedastic Gaussian processes, the observation model is y(x i )=f(x i )+ε(x i The latent function f(x) follows a function with a mean of 0 and a covariance of . The Gaussian distribution, i.e. and Σ f Let f(x) be the variance scaling parameter and covariance matrix of the latent function f(x); assume that the observation noise function ε(x) follows a function with zero mean and variance ε = 0. The distribution is Gaussian, i.e., ε(x) ~ N(0, exp(g(x))), and the function g(x) controlling the noise variance follows a mean of μ01. r variance is The Gaussian distribution, i.e. μ0 and 1 r Let g(x) be the prior mean of g(x) and an r×1 vector of all 1s. and Σ g Let f(x) and g(x) be the variance scaling parameter and covariance matrix of the noise function g(x), respectively. The two Gaussian processes f(x) and g(x) are independent of each other.
[0098] This invention provides a method for predicting temperature-induced strain in bridges under the influence of non-constant noise, comprising the following steps:
[0099] Step 1: Obtain historical monitoring data of the main girder temperature and temperature-induced strain through the bridge health monitoring system, and extract the main girder temperature data X and the main girder temperature-induced strain data y as the training set data D = (X, y); assume the test set data is D. * =(X * ,y * );
[0100] Bridge health monitoring systems are a conventional technical means in this field. The temperature and strain of the main beam can be obtained from the bridge health monitoring system. The relative deformation per unit length of the main beam component due to thermal expansion and contraction along the component axis is called the temperature-induced strain of the main beam. That is, the temperature-induced strain of the main beam can be obtained by removing the main beam displacement and measuring the length of the main beam in the axial direction. The temperature-induced strain of the main beam is a part of the main beam strain. The method of extracting the temperature-induced strain of the main beam from the main beam strain is the same as the method of extracting the temperature-induced displacement of the support from the support displacement. The latter is obtained using Littlewood-Paley and Meyer wavelet theory. For details, please refer to the invention patent application number CN202410045611.1. It will not be elaborated here.
[0101] In actual main beam structural health monitoring, temperature-induced strain data are often susceptible to noise due to environmental vibration, electromagnetic interference, sensor errors, and uneven temperature changes. This noise mixes with the actual temperature-induced strain signal, resulting in a situation where signal and noise coexist. The noise variance in the data is not constant under different conditions but fluctuates with changes in environmental factors and structural response, exhibiting obvious heteroscedasticity. While Empirical Wavelet Transform (EWT) can capture the multi-scale characteristics of the signal, signals and noise of different frequency components may have similar time-frequency characteristics, making it difficult to distinguish whether the removed modal components are noise or signal. However, EWT decomposes the signal into multiple modal components corresponding to different frequency bands. By running a Variational Heteroscedastic Gaussian Regression Process (VHGPR) model on different modal components, the influence of different frequency components on the model's prediction variance can be observed. Therefore, the dataset after filtering out different modal components can be used for variational heteroscedastic Gaussian process (VHGP) model prediction. By comparing the stability and prediction accuracy of the model under different noise intensities before and after filtering, it can be revealed that the VHGP model has good heteroscedasticity modeling ability under different noise levels.
[0102] Step 2: Within the framework of heteroscedastic Gaussian processes, obtain the sample latent function f and the noise log-variance function g based on D = (X,y). Then, construct the joint prior distribution p(f,g), which contains the undetermined hyperparameter θ. Construct the true joint posterior distribution p(f,g|y) based on the joint prior distribution p(f,g) and the likelihood function p(y|f,g). Then, construct the latent function f separately. * The conditional distribution p(f) * |f) and the noise log-variance function g * The conditional distribution p(g) * |g), and simultaneously based on the latent function f obtained from the posterior inference of the temperature-induced strain y of the main beam in the training set. * And the logarithmic variance function of noise g *Construct a test set of temperature-induced strain y in the main beam * The conditional distribution p(y) * |f * ,g * Based on the true joint posterior distribution and the aforementioned three conditional distributions, the temperature-induced strain y is derived. * The conditional distribution p(y) * |y), and conditional distribution p(y) * |y) is used as an undetermined heteroscedastic Gaussian regression model; the temperature-induced strain y of the main beam * That is, the temperature-induced strain to be predicted;
[0103] It should be noted that in step one, based on the heteroscedastic Gaussian process framework, the joint prior distribution p(f,g) of the sample latent function f and the noise log-variance function g is determined according to equation (2.1):
[0104]
[0105] In the formula: f is the sample latent function of the temperature-induced strain y of the main beam in the training set; g is the noise log-variance function of the temperature-induced strain y of the main beam in the training set; p(f,g) is the joint prior distribution of f and g; p(f) is the prior distribution of the sample latent function f; p(g) is the prior distribution of the noise log-variance function g; x i x j Let represent the i-th and j-th main beam temperature data in the training set, i≠j; 0 represents the zero mean vector of the sample latent function f; μ01 represents the mean vector of the noise log-variance function g. The latent function f is based on the main beam temperature data x. i x j The covariance matrix between the samples is used to describe the correlation of the latent function f at different temperature points; The noise logarithmic variance function g is based on the main beam temperature data x. i x j The covariance matrix between them is used to describe the correlation between the noise log-variance function g at different temperature points;
[0106] Among them, the Gaussian process prior distribution p(f) of the sample latent function f of the temperature-induced strain of the main beam in the training set is determined according to equation (2.2):
[0107]
[0108] Furthermore, the functional expression of the sample latent function f is determined according to equation (2.3):
[0109] f = (f(x1), f(x2), ..., f(x) r )) T (2.3)
[0110] In the formula: f(x) r x represents the temperature data value of the r-th main beam. r The corresponding latent function values of the samples; r is the number of samples in the training set; (f(x1), f(x2), ..., f(x... r )) T For (f(x1),f(x2),...,f(x) r The transpose of ))
[0111] Among them, the Gaussian process prior distribution p(g) of the noise logarithmic variance function g of the main beam temperature-induced strain in the training set is determined according to equation (2.4):
[0112]
[0113] In the formula: μ0 is the prior mean of g(x), which is an undetermined hyperparameter; 1 r Represents an r×1 vector of all 1s;
[0114] Furthermore, the analytical expression of the noise log-variance function g is determined according to equation (2.5):
[0115] g=(g(x1),g(x2),...,g(x r )) T (2.5)
[0116] In the formula: g(x) r x represents the temperature data value of the r-th main beam. r The corresponding noise log-variance function values; (g(x1),g(x2),...,g(x r )) T For (g(x1), g(x2), ..., g(x) r The transpose of ))
[0117] Wherein, the covariance matrix and Both adopt the form of a squared exponential kernel function and are determined according to equations (2.6) and (2.7) respectively:
[0118]
[0119]
[0120] In the formula: θ=[θ f ,θ g [μ0] is the set of hyperparameters. And l is K θf The hyperparameters are denoted as θ. f , And l is K θgThe hyperparameters are denoted as θ. g ; l、 These represent the function variance, length dimension, and noise variance in the covariance matrix, respectively; ||x i -x j ||2 is x i x j The n-dimensional Euclidean distance between them, taking three-dimensional input feature values as an example.
[0121] It should be noted that in step one, within the framework of heteroscedastic Gaussian processes, given the temperature-induced strain data y of the main beam in the training set, the true joint posterior distribution p(f,g|y) of the sample latent function f and the noise log-variance function g is obtained according to equation (2.8) based on the joint prior distribution p(f,g) and the likelihood function p(y|f,g).
[0122]
[0123] In the formula: p(f,g|y) is the true joint posterior distribution of the sample latent function f and the noise log-variance function g under the given training set main beam temperature-induced strain data y; p(f,g) is the joint prior distribution of f and g; p(y|f,g) is the likelihood function; p(y) is the marginal likelihood function of y;
[0124] In the heteroscedastic Gaussian process framework, the likelihood function p(y|f,g) of the temperature-induced strain data y of the main beam in the training set follows a multivariate normal distribution and is determined according to equation (2.9):
[0125]
[0126] In the formula: y represents the temperature-induced strain data of the main beam in the training set. I r Let r×r be the identity matrix. For I r The transpose of , R is the noise variance matrix of the temperature-induced strain y of the main beam in the training set;
[0127] Wherein, R is determined according to formula (2.10):
[0128] R=diag(exp(g(x1),exp(g(x2),...,exp(g(x r )))(2.10)
[0129] In the formula: diag represents the transformation of a vector into a diagonal matrix, whose diagonal elements are exp(g(x1)), exp(g(x2)), ..., exp(g(x...)). r )), all off-diagonal elements are 0.
[0130] It should be noted that in step one, the conditional distribution p(f) * |f), p(g * |g), p(y * |f * ,g * We obtain the results using equations (2.11), (2.12), and (2.13) respectively:
[0131] p(y * |f * ,g * )N(y * |f * ,exp(g * ))(2.11)
[0132]
[0133]
[0134] Where: K ff Let f be the covariance matrix between f and f; For K ff The inverse matrix of K; f* For f and f * The covariance matrix between them; For K f* The transpose of K; f** f * with f * The covariance matrix between them; K gg Let g be the covariance matrix between g and g; For K gg The inverse matrix, K g* For g and g * The covariance matrix between them; For K g* The transpose of K; g** For g * With g * The covariance matrix between them.
[0135] It should be noted that in step one, based on the true joint posterior distribution and the three conditional distributions p(y) * |f * ,g * ), p(f * |f), p(g * |g), Test set main beam temperature data X * Temperature-induced strain of main beam y * The conditional distribution p(y) * |y) is obtained according to equation (2.14):
[0136] p(y * |y)=∫p(y * |f * ,g * )·p(f * |f)·p(g * |g)·p(f,g|y)dfdgdf * dg * (2.14)
[0137] In the formula: p(y * |y) represents the temperature data of the main beam in the test set. * Temperature-induced strain of main beam y * conditional distribution; y * To test the temperature-induced strain of the main beam; f * To test the temperature-induced strain y of the main beam * The latent function; g * To test the temperature-induced strain y of the main beam * The noise log-variance function; p(y * |f * ,g * f is the latent function f obtained by posterior inference from the temperature-induced strain y of the main beam in the training set. * And the logarithmic variance function of noise g * Constructing a test set of temperature-induced strain y in the main beam * The conditional distribution of p(f); * |f) is the latent function f under a given sample latent function f. * The conditional distribution of p(g); * |g) represents the noise log-variance function g under a given noise log-variance function g. * The conditional distribution of ; p(f,g|y) is the true joint posterior distribution.
[0138] In fact, the given main beam temperature data X in the test set derived in step two * Temperature-induced strain of main beam y * The conditional distribution is a set of training data D = (X, y) and test set main beam temperature data X. * The undetermined heteroscedastic Gaussian regression model with undetermined hyperparameter θ is then used in the subsequent step of using variational Bayesian inference to optimize the parameters of the posterior variational distribution and the hyperparameter θ to solve the heteroscedastic Gaussian regression model. This model is based on the newly acquired main beam temperature data. Predicting the temperature-induced strain of the main beam (with test set D) * =(X * ,y * Related symbols such as X * For unknown quantities, wavy lines (Represents the corresponding specific expression or value).
[0139] Step 3: Using the variational Bayesian inference method, construct the variational free energy bound F(q(f),q(g)) based on the log-marginal likelihood function logp(y) and the KL divergence between the joint posterior variational distribution q(f,g) and the true joint posterior distribution p(f,g|y). Take the maximum value of F(q(f),q(g)) with respect to q(f) as the optimal posterior variational distribution q. * (f), q * (f) Substituting into F(q(f),q(g)) yields the edge variable boundary F(q(g));
[0140] Since the true joint posterior distribution p(f,g|y) is difficult to solve directly, this invention employs a variational Bayesian inference method. Based on the mean field assumption, a solvable posterior variational distribution q(f,g) is introduced to approximate the true joint posterior distribution p(f,g|y). The difference between the logarithmic marginal likelihood function logp(y) (i.e., evidence) of the temperature-induced strain data y of the main beam in the training set and the KL divergence between the joint posterior variational distribution q(f,g) and the true posterior distribution p(f,g|y) is defined as the variational free energy bound F(q(f),q(g)). The KL divergence is minimized by maximizing the variational free energy bound F(q(f),q(g)) to obtain the optimal approximation q(f,g|y) of the true joint posterior distribution p(f,g|y). Given q(g), the optimal posterior variational distribution q is obtained by maximizing F(q(f),q(g)). * (f) replaces q(f), q * Since (f)≥q(f), the dependence of the lower bound of evidence F(q(f),q(g)) on q(f) is eliminated, yielding the marginal variation boundary F(q(g)). In the heteroscedastic Gaussian distribution (HGP), the marginal variation boundary is the upper bound of the standard variation boundary. Furthermore, as a special case of F(q(f),q(g)), it is also the lower bound of the evidence logp(y). Therefore, logp(y)≥F(q(g))=F(q) * (f),q(g))≥F(q(f),q(g)).
[0141] It should be noted that in step three, the mean field assumption introduces solvable posterior variational distributions q(f) and q(g) that are independent variational distributions. The variational free energy bound F(q(f), q(g)) is determined according to equation (3.1) based on the logarithmic marginal likelihood function logp(y) of the temperature-induced strain y of the main beam in the training set (i.e., evidence) and the difference in KL divergence between the joint posterior variational distribution q(f,g) and the true joint posterior distribution p(f,g|y).
[0142] F(q(f),q(g))=logp(y)-KL(q(f,g)||p(f,g|y))(3.1)
[0143] In the formula: F(q(f),q(g)) is the variational free energy bound, which is also the lower bound of evidence. The value of F depends on two r-dimensional posterior variational distributions q(f) and q(g); logp(y) is the log marginal likelihood function, which is usually called evidence. Its value is independent of the posterior variational distribution; KL(q(f,g)||p(f,g|y)) is the KL divergence between the joint posterior variational distribution q(f,g) and the true joint posterior distribution p(f,g|y);
[0144] Among them, the solvable independent posterior variational distributions q(f) and q(g) introduced based on the mean field are the product of independent factors decomposed from the joint posterior variational distribution q(f,g), and q(f,g) is determined according to equation (3.2):
[0145] q(f,g)=q(f)q(g)(3.2)
[0146] In the formula: q(f,g) is the joint posterior variational distribution; q(f) is the posterior variational distribution of the sample latent function f; q(g) is the posterior variational distribution of the noise log-variance function g; the posterior variational distributions q(f) and q(g) are independent of each other and are both assumed to follow a multivariate Gaussian distribution;
[0147] Among them, the marginal likelihood function in the logarithmic marginal likelihood function logp(y) of the temperature-induced strain data y of the main beam in the training set (i.e., the evidence) is obtained according to equation (3.3):
[0148] p(y)=∫∫p(y|f,g)p(f)p(g)dfdg(3.3)
[0149] In the formula: p(y) is the marginal likelihood function of the temperature-induced strain y of the main beam;
[0150] The KL divergence between the joint posterior variational distribution q(f,g) and the true joint posterior distribution p(f,g|y) is used to measure the difference between the two probability distributions, and is obtained from equation (3.4):
[0151]
[0152] In the formula: KL(q(f)q(g)||p(f,g|y)) is the KL divergence between the joint posterior variational distribution q(f,g) and the true joint posterior distribution p(f,g|y). Since the KL divergence is non-negative, for any variational distributions q(f) and q(g), logp(y)≥F(q(f),q(g)).
[0153] It should be noted that in step three, in order to eliminate the dependence of the lower bound of evidence F(q(f), q(g)) on q(f), the optimal posterior variational distribution q is obtained by maximizing the lower bound of evidence F(q(f), q(g)) based on the given q(g). * (f) is used to replace q(f), and the optimal posterior variational distribution q of the sample latent function f of the temperature-induced strain y of the main beam in the training set is obtained. * (f) Determined according to formula (3.5):
[0154]
[0155] In the formula: The lower bound of evidence, F(q(f), q(g)), takes the maximum value with respect to q(f); Z(q(g)) is a normalization constant that ensures q * (f) The integral is 1, i.e., Z(q(g))=∫e ∫q(g)log p(y|f,g)dg p(f)df; logp(y|f,g) is the conditional log-likelihood function of the temperature-induced strain y of the main beam under the given sample latent function f and the noise log-variance function g.
[0156] It should be noted that in step three, the optimal posterior variational distribution q of the sample latent function vector f is... * (f) Substituting back the lower bound of the evidence F(q(f), q(g)), we can simplify it to obtain the marginal boundary F(q(g)), which is obtained according to equation (3.6):
[0157] F(q(g))=logZ(q(g))-KL(q(g)||p(g))(3.6)
[0158] In the formula: F(q(g)) is the marginal boundary; KL(q(g)||p(g)) is the KL divergence between q(g) and p(g); p(g) is the prior distribution of g.
[0159] Step 4: Assuming the posterior variational distribution q(g) follows a multivariate normal distribution N(μ,Σ), substitute q(g) into the marginal variational bound F(q(g)) to simplify it and obtain a new marginal variational bound F(μ,Σ). Solve for the stationary points of F(μ,Σ) and construct a positive semidefinite diagonal matrix Λ based on the stationary points. Transform μ and Σ into functions μ(Λ) and ∑(Λ) in Λ. Simultaneously, based on the principle of Type-II maximum likelihood estimation, use the gradient algorithm to maximize the new marginal variational bound F(μ(Λ),∑(Λ)) with respect to the hyperparameters θ and Λ, and solve for the result. and hyperparameters
[0160] Assume that the posterior variational distribution q(g) of the temperature-induced strain y of the main beam in the training set follows a multivariate normal distribution with a mean of r×1 vector μ and a covariance matrix of r×r matrix Σ, i.e., q(g)N(μ,Σ). Then, we use the covariance matrix K based on the training set data... θf and K θg These represent the covariance matrices calculated at the input points using the corresponding covariance functions. The marginal variational boundary F(q(g)) is simplified to F(μ,Σ) after further substitution. The free variational parameters required to determine μ and Σ are: The number of free parameters increases quadratically with the number of observations. Then, an approximation of the Gaussian process regression model (GPR) is derived. By solving the stationary point equation, the mean vector μ and covariance matrix Σ of the posterior variational distribution q(g) of the temperature-induced strain of the main beam in the training set at the extreme point are found. At the extreme point, (μ, Σ) depends on the same positive semi-definite diagonal matrix Λ, which is determined only by r diagonal elements. Using r positive elements as the only free variational parameters, (μ, Σ) is reparameterized according to the lower bound of evidence F(μ, Σ). The r variational parameters in Λ are optimized by maximizing the lower bound of evidence F(μ(Λ), ∑(Λ)) = F(Λ), while simultaneously ensuring that F(Λ) is effective relative to the hyperparameter θ = (θ... f ,θ g Maximizing (μ0) to optimize the hyperparameter θ to achieve Type-II maximum likelihood estimation for model selection. The entire optimization process is nonlinear and can be implemented analytically using gradient algorithms to obtain the parameters. and hyperparameters The gradient algorithm can use the conjugate gradient algorithm;
[0161] It should be noted that in step four, the mean vector μ and covariance matrix Σ of the posterior variational distribution q(g) are substituted into the marginal variational boundary F(q(g)) and simplified to F(μ,Σ). F(μ,Σ) is obtained according to equations (4.1), (4.2), (4.3), and (4.4):
[0162]
[0163]
[0164]
[0165] Right now:
[0166] In the formula: μ is the mean vector of the posterior variational distribution q(g); Σ is the covariance matrix of the posterior variational distribution q(g); R' is the noise variance diagonal matrix, obtained by diagonal transformation of the noise variance matrix R; R'=diag(exp(μ1-Σ) 11 / 2),exp(μ2-Σ22 / 2),...,exp(μ r -Σ rr / 2)); tr(Σ) is the trace of the covariance matrix Σ. That is, the sum of the elements on the main diagonal of the covariance matrix;
[0167] It should be noted that in step four, a derivation approximating the GPR model is performed to reduce computational complexity. The stationary point equation is obtained to obtain the mean vector μ and covariance matrix Σ of the posterior variational distribution at the extreme point. μ and Σ are then transformed into expressions μ(Λ) and ∑(Λ) in terms of Λ. μ(Λ) and ∑(Λ) are determined according to equations (4.5) and (4.6), respectively.
[0168]
[0169] In the formula: μ(Λ) is a function of μ with respect to Λ; Σ(Λ) is a function of Σ with respect to Λ; Λ is a positive semi-definite diagonal matrix; Covariance matrix An invertible matrix.
[0170] Step 5: According to get and Then according to and Obtain the posterior variational distribution posterior variational distribution Substituting into the variational free energy bound F(q(f),q(g)), the optimal posterior variational distribution is obtained by solving. Approximate joint variational distribution is obtained Obtain new main beam temperature data Obtain the latent function value logarithmic variance function value of noise Will Value and Substitute the values into the conditional distribution p(f) respectively * |f) and p(g * In |g), we get and Will and Substituting into the undetermined heteroscedastic Gaussian regression model, we obtain the final predicted distribution. according to Temperature-induced strain y of the main beam * Make predictions.
[0171] The optimal posterior variational distribution obtained by the lower bound F of the maximum marginalization variation. and To approximate the true joint posterior distribution p(f,g|y). Based on the Bayesian inference framework, the test point X... * f * and g * The posterior variational distribution can be obtained through the conditional distribution. and optimal posterior variational distribution The mean and variance can also be calculated analytically. Substituting the inferred distribution function and optimized parameters into the undetermined heteroscedastic Gaussian regression model in step two, we obtain the variational heteroscedastic Gaussian process regression model of the main beam temperature-induced strain. The given main beam temperature data from the test set are then used... Substituting into the variational heteroscedastic Gaussian process regression model of the temperature-induced strain of the main beam p(y) * The temperature-induced strain of the main beam of the bridge is predicted in |y).
[0172] It should be noted that in step five, the optimal posterior variational distribution is obtained through the maximum marginalization variational lower bound F. We need to substitute equation (4.1) into equation (3.5) to obtain... Determine using equations (5.1) and (5.2):
[0173]
[0174] Right now:
[0175] In the formula: α is a weight vector used to linearly combine the posterior mean of the sample latent function f based on the temperature-induced strain y of the main beam in the training set.
[0176] It should be noted that in step five, the test point X is obtained under variational Bayesian inference. * Place and The posterior variational distributions are obtained from (5.3) and (5.4), respectively:
[0177]
[0178]
[0179] In the formula: and The latent function values are respectively Log-variance of noise The posterior variational distribution;
[0180] in, Test points in the main beam temperature data of the test set latent function value posterior variational distribution The predicted mean and predicted variance are determined by equations (5.5) and (5.6):
[0181]
[0182]
[0183] In the formula: For f and Covariance matrix between; for The transpose of the matrix; for and The covariance value;
[0184] in: To test the main beam temperature data Log-variance of noise posterior variational distribution The predicted mean and predicted variance are determined by equations (5.7) and (5.8):
[0185]
[0186]
[0187] In the formula: For g and Covariance matrix between; for The transpose of the matrix; for and The covariance value.
[0188] It should be noted that in step five, the conditions inferred above are distributed... and the optimized posterior variational distribution and hyperparameters Substituting into the undetermined Gaussian regression model from step one, through and and The integral yields the posterior variational distribution. and Reusing the obtained approximate joint variational distribution To approximate the true posterior distribution p(f,g|y), the test points are obtained by transforming the formula in equation (5.9). Temperature-induced strain of main beam y * The predicted distribution is determined according to equation (5.10):
[0189]
[0190] Right now:
[0191] In the formula: To test the temperature of the main beam Temperature-induced strain of main beam y * The final predicted distribution;
[0192] Therefore, the temperature of the main beam of the test set was measured. Temperature-induced strain y of the main beam at the location * The predicted mean and variance can be obtained from equation (5.11):
[0193]
[0194] In the formula: Based on the training set data D and the newly acquired main beam temperature data Temperature-induced strain y of lower main beam * The predicted mean; Based on the training set data D and the newly acquired main beam temperature data Temperature-induced strain y of lower main beam * The prediction variance.
[0195] Substituting the inferred conditional distribution, optimized variational distribution, and hyperparameters into the undetermined heteroscedastic Gaussian regression model in step two yields a deterministic heteroscedastic Gaussian regression model. The resulting model models heteroscedastic noise using variational Bayesian inference and exhibits good prediction accuracy and computational efficiency, making it highly valuable for application. Based on the given heteroscedastic Gaussian regression model, the current main beam temperature data can be analyzed. Temperature-induced strain of main beam y * Based on the prediction results and measured data, an evaluation index for the operating status of the bridge main beam is established. By comparing the evaluation index with the predetermined threshold, the operating status of the main beam can be determined, and a basis for the maintenance and safety assessment of the main beam structure can be provided.
[0196] The applicant further declares that while the above embodiments illustrate the implementation method of the present invention, the present invention is not limited to the above-described embodiments, meaning that the present invention does not necessarily rely on the above methods and structures for implementation. Those skilled in the art should understand that any improvements to the present invention, equivalent substitutions for the selected implementation methods, additions to steps, and selections of specific methods all fall within the protection and disclosure scope of the present invention.
[0197] This invention is not limited to the above-described embodiments. All embodiments that use similar structures and methods to achieve the purpose of this invention are within the scope of protection of this invention.
Claims
1. A method for predicting temperature-induced strain in bridges under the influence of non-constant noise, characterized in that: Step 1: Obtain historical monitoring data of the main girder temperature and temperature-induced strain through the bridge health monitoring system, and extract the main girder temperature X and temperature-induced strain y as the training set data D = (X, y); assume the test set data is D. * =(X * ,y * ); Step 2: Within the framework of heteroscedastic Gaussian processes, obtain the sample latent function f and the noise log-variance function g based on D = (X,y). Then, construct the joint prior distribution p(f,g) containing the undetermined hyperparameter θ. Based on p(f,g) and the likelihood function p(y|f,g), construct the true joint posterior distribution p(f,g|y). Then, construct the latent function f separately. * The conditional distribution p(f) * |f) and the noise log-variance function g * The conditional distribution p(g) * |g), and simultaneously based on f inferred from the posterior of y. * and g * Constructing a test set of temperature-induced strain y in the main beam * The conditional distribution p(y) * |f * ,g * According to p(f,g|y) and p(f) * |f), p(g * |g), p(y * |f * ,g * ), Derive the conditional distribution p(y) * |y), and p(y) * |y) is used as an undetermined heteroscedastic Gaussian regression model; Step 3: Using the variational Bayesian inference method, construct the variational free energy bound F(q(f),q(g)) based on the log-marginal likelihood function logp(y) and the KL divergence between the joint posterior variational distribution q(f,g) and p(f,g|y); take the maximum value of F(q(f),q(g)) with respect to q(f) as the optimal posterior variational distribution q. * (f), q * (f) Substituting into F(q(f),q(g)) yields the edge variable boundary F(q(g)); Step 4: Assuming the posterior variational distribution q(g) follows a multivariate normal distribution N(μ,Σ), substitute q(g) into F(q(g)) to obtain the new marginal variational boundary F(μ,Σ); solve for the stationary points of F(μ,Σ), and construct a positive semi-definite diagonal matrix Λ based on the stationary points. Transform μ and Σ into functions μ(Λ) and ∑(Λ) in Λ. Simultaneously, based on the principle of Type-II maximum likelihood estimation, use the gradient algorithm to maximize F(μ(Λ),∑(Λ)) with respect to the hyperparameters θ and Λ, and solve to obtain... and hyperparameters Step 5: According to get and Then according to and Obtain the posterior variational distribution Will Substituting into F(q(f),q(g)), the optimal posterior variational distribution is obtained by solving. Obtain new main beam temperature data Obtain the latent function value log-variance function value of noise Will and Substitute p(f) into each * |f) and p(g * In |g), we get and Will and Substituting into the undetermined heteroscedastic Gaussian regression model, we obtain the final predicted distribution. according to For y * Make predictions.
2. The method for predicting bridge temperature-induced strain under the influence of non-constant noise according to claim 1, characterized in that: In step two, p(f,g) is determined according to equation (2.1): In the formula: f is the sample latent function of y; g is the noise log-variance function of y; p(f,g) is the joint prior distribution of f and g; p(f) is the prior distribution of f; p(g) is the prior distribution of g; x i x j Let be the i-th and j-th main beam temperature data in the training set, i≠j; 0 is the zero mean vector of f; μ01 is the mean vector of g; f is based on x i x j The covariance matrix between them; For g based on x i x j The covariance matrix between them; Wherein, p(f) is determined according to equation (2.2): Wherein, p(g) is determined according to equation (2.4): In the formula: μ0 is the prior mean of g(x), which is an undetermined hyperparameter; 1 r Represents an r×1 vector of all 1s; Wherein, the covariance matrix and Both adopt the form of a squared exponential kernel function and are determined according to equations (2.6) and (2.7) respectively: In the formula: θ=[θ f ,θ g [μ0] is the set of hyperparameters. and l is The hyperparameters are denoted as θ. f , and l is The hyperparameters are denoted as θ. g ; l、 These represent the function variance, length dimension, and noise variance, respectively; ||x i -x j ||2 is x i x j The n-dimensional Euclidean distance between them.
3. The method for predicting bridge temperature-induced strain under the influence of non-constant noise according to claim 2, characterized in that: In step two, within the framework of heteroscedastic Gaussian processes, based on p(f,g) and p(y|f,g), p(f,g|y) is obtained according to equation (2.8): In the formula: p(f,g|y) is the true joint posterior distribution of f and g given y; p(f,g) is the joint prior distribution of f and g; p(y|f,g) is the likelihood function of y; p(y) is the marginal likelihood function of y; Where p(y|f,g) follows a multivariate normal distribution and is determined according to equation (2.9): In the formula: y represents the temperature-induced strain of the main beam in the training set. I r Let r×r be the identity matrix. For I r The transpose of y; R is the noise variance matrix of y; Wherein, R is determined according to formula (2.10): R=diag(exp(g(x1),exp(g(x2),...,exp(g(x r )))(2.10) In the formula: diag represents the transformation of a vector into a diagonal matrix, whose diagonal elements are exp(g(x1)), exp(g(x2)), ..., exp(g(x...)). r )), all off-diagonal elements are 0.
4. The method for predicting bridge temperature-induced strain under the influence of non-constant noise according to claim 3, characterized in that: In step two, based on p(f,g|y) and p(y) * |f * ,g * ), p(f * |f), p(g * |g), p(y) * |y) is obtained according to equation (2.14): p(y * |y)=∫p(y * |f * ,g * )·p(f * |f)·p(g * |g)·p(f,g|y)dfdgdf * dg * (2.14) In the formula: p(y * |y) represents the temperature-induced strain y of the main beam in the test concentration. * conditional distribution; f * For y * The latent function; g * For y * The noise log-variance function; p(y * |f * ,g * ) is a given f * and g * Next y * The conditional distribution of p(f); * |f) is f under a given f * The conditional distribution of p(g); * |g) represents g given g. * The conditional distribution.
5. The method for predicting bridge temperature-induced strain under the influence of non-constant noise according to claim 4, characterized in that: In step three, F(q(f),q(g)) is determined according to equation (3.1): F(q(f),q(g))=logp(y)-KL(q(f,g)||p(f,g|y)) (3.1) In the formula: F(q(f),q(g)) is the variational free energy bound; logp(y) is the log marginal likelihood function; KL(q(f,g)||p(f,g|y)) is the KL divergence between the joint posterior variational distributions q(f,g) and p(f,g|y); Wherein, q(f,g) is determined according to equation (3.2): q(f,g)=q(f)q(g)(3.2) In the formula: q(f) is the posterior variational distribution of f; q(g) is the posterior variational distribution of g; q(f) and q(g) are independent of each other and both follow a multivariate Gaussian distribution; Where p(y) is obtained according to equation (3.3): p(y)=∫∫p(y|f,g)p(f)p(g)dfdg (3.3) In the formula: p(y) is the marginal likelihood function of y; The KL divergence between q(f,g) and p(f,g|y) is obtained according to equation (3.4): In the formula: KL(q(f)q(g)||p(f,g|y)) is the KL divergence between q(f,g) and p(f,g|y).
6. The method for predicting bridge temperature-induced strain under the influence of non-constant noise according to claim 5, characterized in that: In step three, q * (f) Determined according to formula (3.5): In the formula: q * (f) represents the optimal posterior variational distribution; F(q(f), q(g)) is the maximum value of q(f); Z(q(g)) is the normalization constant, ensuring that q * (f) The integral is 1; q(g) is the posterior variational distribution of g; logp(y|f,g) is the log-conditional likelihood function of y given f and g; F(q(g)) is obtained according to equation (3.6): F(q(g))=logZ(q(g))-KL(q(g)||p(g)) (3.6) In the formula: F(q(g)) is the marginal boundary; KL(q(g)||p(g)) is the KL divergence between q(g) and p(g); p(g) is the prior distribution of g.
7. The method for predicting bridge temperature-induced strain under the influence of non-constant noise according to claim 6, characterized in that: In step four, F(μ,Σ) is obtained according to equation (4.4): In the formula: μ is the mean vector of q(g); Σ is the covariance matrix of q(g); R' is the noise variance diagonal matrix; tr(Σ) is the trace of the covariance matrix Σ.
8. The method for predicting bridge temperature-induced strain under the influence of non-constant noise according to claim 7, characterized in that: In step four, μ(Λ) and ∑(Λ) are determined according to equations (4.5) and (4.6), respectively; In the formula: Λ is a positive semi-definite diagonal matrix; for An invertible matrix.
9. The method for predicting bridge temperature-induced strain under the influence of non-constant noise according to claim 8, characterized in that: In step five, Determine according to formula (5.2): In the formula: α is the weight vector, The results obtained according to (5.3) and (5.4) respectively are as follows: In the formula: and They are respectively and The posterior variational distribution; They are respectively The predicted mean and predicted variance; They are respectively The predicted mean and predicted variance; in, Determine according to equations (5.5) and (5.6) respectively: In the formula: For f and Covariance matrix between; for The transpose of the matrix; for and The covariance value; in, Determine according to equations (5.7) and (5.8) respectively: In the formula: For g and Covariance matrix between; for The transpose of the matrix; for and The covariance value.
10. The method for predicting bridge temperature-induced strain under the influence of non-constant noise according to claim 9, characterized in that: In step five, Determine according to formulas (5.9) and (5.10): Right now In the formula: Main beam temperature-induced strain y * The final predicted distribution; Therefore, the main beam temperature response y * The predicted mean and variance are obtained according to equation (5.11): In the formula: Based on the training set data D and the newly acquired main beam temperature data Temperature-induced strain y of lower main beam * The predicted mean; Based on the training set data D and the newly acquired main beam temperature data Temperature-induced strain y of lower main beam * The prediction variance.