Interval type uncertainty model parameter correction method based on Riemannian manifold and Gaussian process model

Abstract

The invention discloses an interval type parameter uncertainty model correction method based on a Riemannian manifold and Gaussian process model, and belongs to the technical field of engineering parameter uncertainty quantification and model correction. According to the method, aiming at the defect that traditional interval analysis cannot represent parameter correlation, a convexly optimized minimum volume ellipsoid model is constructed, and a coupling relation between parameters is captured through a geometric learning framework; designing a Gaussian process regression agent model based on a logarithm Euclidean metric kernel function, and keeping symmetric positive definite matrix constraints by using a manifold kernel function; and providing a Riemann gradient optimization algorithm, and realizing parameter space unconstrained optimization through matrix logarithm mapping. The technical scheme comprises three core modules: an ellipsoid convex model parameterization module for realizing and explicit representation of parameter correlation, a manifold embedding agent model module for guaranteeing mathematical consistency of physical constraints, and a manifold gradient optimization module for improving high-dimensional parameter correction efficiency. According to the method, the problems that a traditional method depends on heuristic projection, the calculation efficiency is low, and constraint keeping is difficult are effectively solved, and a high-precision and interpretable uncertainty parameter correction tool is provided for a numerical model in engineering.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of engineering uncertainty quantification and model updating, and particularly relates to an interval-type uncertainty model parameter updating method based on a Riemannian manifold and a Gaussian process model. BACKGROUND

[0002] The existing uncertainty quantification method has three technical bottlenecks: first, the traditional interval analysis uses independent parameter intervals (such as confidence intervals and prediction intervals) to represent the parameters, which cannot capture the coupling relationship between the parameters, resulting in the accumulation of uncertainty propagation errors; second, the proxy model (such as polynomial regression and Kriging model) lacks geometric perception ability in the symmetric positive definite matrix space, and needs to be artificially intervened by truncating the eigenvalues to maintain the positive definiteness of the matrix, which destroys the physical consistency; third, the parameter optimization method based on random sampling (such as Metropolis-Hastings algorithm and transition Markov chain) has low computational efficiency in the manifold constraint space, and is difficult to handle high-dimensional engineering problems. The existing improvement schemes such as evidence theory and probability box method can handle part of the uncertainty, but have problems such as high computational complexity and strong conservatism. In particular, in the structure model updating involving material parameter correlation, the traditional ellipsoid model relies on empirical parameter adjustment and lacks mathematical rigor, and the black box model such as neural network has poor interpretability. Therefore, it is urgent to develop an uncertainty quantification framework that can maintain geometric constraints and efficiently handle parameter correlation to realize high-precision updating of engineering models. SUMMARY

[0003] The application discloses an interval-type uncertainty model parameter updating method based on a Riemannian manifold and a Gaussian process model, and belongs to the technical field of engineering parameter uncertainty quantification and model updating. The model uses a manifold kernel function to maintain the symmetric positive definite matrix constraint, thereby ensuring the mathematical consistency in the model updating process.

[0004] An interval-type uncertainty model parameter updating method based on a Riemannian manifold and a Gaussian process model, the model parameter updating method comprising the following steps:

[0005] Step 1: multiple repeated experiments are carried out on an engineering structure to obtain multiple groups of experimental result sample points, and an ellipsoid characteristic matrix representing the experimental results is fitted from the sample points;

[0006] Step 2: a finite element model of the engineering structure is established, the finite element model prediction results under M groups of different design parameters are calculated, and M groups of design parameter-prediction result sample pairs are formed;

[0007] Step 3: N samples are randomly selected from the M groups of design parameter-prediction result sample pairs, and an ellipsoid characteristic matrix of the prediction samples is fitted from the sample points by using a minimum volume ellipsoid method; the process is repeated M times to obtain M design parameter-prediction characteristic matrix samples.

[0008] Step 4: Transform the M design parameter-prediction feature matrix sample pairs using the logarithmic mapping formula, mapping them to the tangent space of the Riemannian manifold, and construct a Gaussian process regression model using the logarithmic-Euclidean kernel function;

[0009] Step 5: Search for the corrected parameter in the Gaussian process regression model, input the current value of the corrected parameter into the Gaussian process regression model, calculate its predicted value, calculate the gradient of the corrected parameter with respect to the loss function, and update the corrected parameter;

[0010] Step 6: Calculate the corrected parametric ellipsoid eigenma matrix using exponential mapping to inversely map the parameters from the tangent space back to the Riemannian manifold.

[0011] Furthermore, step two specifically involves:

[0012] Step two involves establishing a finite element model of the engineering structure, including mesh generation, material parameter settings, and boundary condition settings. Within the design parameter space of the finite element model, sample points are randomly selected using the Latin hypercube method, with a total of M independent samplings. The prediction results of the finite element model under M different design parameters are calculated, forming M sets of design parameter-prediction result sample pairs.

[0013] Furthermore, step three specifically involves:

[0014] Step 3: Randomly select N samples from the M sets of design parameter-prediction result sample pairs, where N≈0.5M, and use the minimum volume ellipsoid method to fit the sample points to obtain the feature matrix P of one corrected parameter. i And the ellipsoidal feature matrix (P) of a finite element model predicting a sample. pred ) i Repeat this process M times to obtain M sample pairs of design parameters and predicted feature matrices.

[0015] Furthermore, step four specifically involves:

[0016] Step four, apply the formula logarithmic mapping formula S i =log(P i ) and (S pred ) i =log((P) pred ) i The M parameters from step three are used to predict the feature matrix sample pairs. By performing a transformation and mapping it to the tangent space of the Riemannian manifold, we obtain...

[0017] use As the input and output of the Gaussian process model, a log-Euclidean kernel function is used to construct the Gaussian process regression model, and the bandwidth parameter and noise variance of the Gaussian process regression model are set.

[0018] Further, the step five is specifically:

[0019] Step five, according to the expert experience or randomly generating an initial search point S0 of the modified parameter in the Gaussian process regression model, setting the learning rate η, in the t, t = 1, 2,..., T max , in the iteration, the current value S t of the modified parameter is input into the Gaussian process regression model, the prediction value μ(S t ) is calculated, and the loss function

[0020] The gradient of the modified parameter to the loss function is calculated The gradient is symmetrized

[0021] The update formula of the modified parameter is When the loss change quantity of continuous 2 iterations satisfies , the iteration is stopped, and ε is a threshold value.

[0022] Further, in the step one, the minimum volume ellipsoid method is used to fit the experimental sample points.

[0023] Further, in the step two, the Latin hypercube method is used to randomly sample in the prior space of the modified parameter.

[0024] Further, in the step four, the log-Euclidean kernel function is used to construct the Gaussian process regression model.

[0025] Further, in the step four, the k-fold cross-validation is used to optimize the hyperparameters of the Gaussian process model.

[0026] Further, in the step five, the gradient of the modified parameter to the loss function needs to be symmetrized

[0027] Further, in the step six, the exponential mapping P = exp(S) needs to be used to calculate the modified parameter ellipsoid.

[0028] The technical scheme of the application comprises three core modules:

[0029] 1. Ellipsoid convex model parameterization module: this module fits the experimental sample points by the minimum volume ellipsoid method, realizes the explicit representation of parameter correlation, and captures the coupling relationship between parameters.

[0030] 2. Manifold embedding surrogate model module: the module adopts a log-Euclidean kernel function to construct a Gaussian process regression model, uses a manifold kernel function to maintain a symmetric positive definite matrix constraint, and guarantees the mathematical consistency of physical constraints.

[0031] 3. Manifold gradient optimization module: the module proposes a Riemannian gradient optimization algorithm, realizes unconstrained optimization of the parameter space through matrix logarithmic mapping, and improves the efficiency of high-dimensional parameter correction.

[0032] The technical scheme of the present application effectively solves the problems of traditional methods, such as dependence on heuristic projection, low computational efficiency, and difficulty in constraint maintenance, and provides a high-precision, interpretable uncertainty parameter correction tool for numerical models in engineering. This method not only improves the accuracy and efficiency of model correction, but also enhances the interpretability of the model, which has important theoretical and practical significance for engineering parameter uncertainty quantification and model correction. BRIEF DESCRIPTION OF DRAWINGS

[0033] Figure 1 is a flowchart of an interval-type uncertainty model parameter correction method based on Riemannian manifold and Gaussian process model according to the present application;

[0034] Figure 2 is a 3-degree-of-freedom vibration system based on the interval-type uncertainty model parameter correction method based on Riemannian manifold and Gaussian process model according to the present application;

[0035] Figure 3 is a schematic diagram of fitting sample points using the minimum ellipsoid method in the interval-type uncertainty model parameter correction method based on Riemannian manifold and Gaussian process model according to the present application;

[0036] Figure 4 is a schematic diagram of parameter changes during the iteration process of the interval-type uncertainty model parameter correction method based on Riemannian manifold and Gaussian process model according to the present application;

[0037] Figure 5 is a loss function convergence curve during the iteration process of the interval-type uncertainty model parameter correction method based on Riemannian manifold and Gaussian process model according to the present application. DETAILED DESCRIPTION

[0038] In order to facilitate the understanding of those skilled in the art, the present application will be further described below in conjunction with the embodiments and the drawings. The content mentioned in the embodiments is not a limitation of the present application.

[0039] The application provides an interval type uncertainty model parameter correction method based on a Riemannian manifold and a Gaussian process model, which aims to solve the deficiency of traditional interval analysis methods in representing parameter correlation, and explicitly represents the coupling relationship between parameters by constructing a convex optimization minimum volume ellipsoid model. The core of the application is to design a Gaussian process regression surrogate model based on a logarithmic Euclidean metric kernel function, which maintains the symmetric positive definite matrix constraint by using a manifold kernel function, thereby ensuring mathematical consistency in the model correction process. In addition, the application also proposes a Riemannian gradient optimization algorithm, which realizes unconstrained optimization of the parameter space through matrix logarithmic mapping, significantly improving the efficiency of high-dimensional parameter correction.

[0040] The modeling method comprises the following steps:

[0041] Step 1: experimental data generation:

[0042] 1.1 Perform repeated experiments on engineering structures such as bridges, buildings, and engineering machinery to obtain X sets of experimental result sample points, X>5. Use the minimum volume ellipsoid method to fit the experimental sample points to obtain the characteristic matrix P of the minimum volume ellipsoid of the experimental value obs .

[0043] Step 2: finite element modeling and sample point calculation:

[0044] 2.1 Establish a finite element model of the engineering structure, including meshing, material parameter setting, and boundary condition setting.

[0045] 2.2 Randomly extract sample points in the design parameter space of the finite element model using the Latin hypercube method, a total of M times, M>20.

[0046] 2.3 Calculate the finite element model prediction results under M sets of different design parameters to form M sets of design parameter-prediction result sample pairs.

[0047] Step 3: characteristic sample generation:

[0048] 3.1 Randomly extract N samples from the M sets of design parameter-prediction result sample pairs, where N≈0.5M, and use the minimum volume ellipsoid method to fit the sample points to obtain a characteristic matrix P of the corrected parameter i and an ellipsoid characteristic matrix (P pred ) of the finite element model prediction sample i .

[0049] 3.2 Repeat M times to obtain M sets of design parameter-prediction characteristic matrix sample pairs

[0050] Step 4: Gaussian process surrogate model construction:

[0051] 4.1 Logarithmic mapping of the sample space: The logarithmic mapping formula S is used. i =log(P i ) and (S pred ) i =log((p pred ) i )right Perform the transformation to obtain

[0052] 4.2 Model Training: For input and output, a logarithmic Euclidean kernel function is used. Construct a Gaussian process regression model and set the bandwidth parameter δ and noise variance σ. 2 During the modeling process, k-fold cross-validation was used to optimize the hyperparameters.

[0053] Step 5: Manifold constraint optimization and correction:

[0054] 5.1 Parameter initialization: Based on expert experience or by randomly generating an initial search point S0, set the learning rate η.

[0055] 5.2 Iterative Optimization: In the t-th iteration, the predicted value is calculated using a Gaussian process surrogate model. And calculate the loss function. Calculate the gradient of the corrected parameters with respect to the loss function. Gradient symmetry processing The correction formula for the parameter being corrected is:

[0056] 5.3 Termination condition: When the change in loss after two consecutive iterations satisfies... The iteration stops when ε is the threshold.

[0057] Step 6: Output the results:

[0058] Calculate the corrected parametric ellipsoid P = exp(S).

[0059] This embodiment utilizes a 3-DOF spring-mass vibration system. For example... Figure 1 The diagram shown is a flowchart illustrating a specific embodiment of the present invention. Figure 2 The model includes three lumped mass points and three springs, where the stiffness values ​​of the three springs are parameters to be corrected. The specific steps are as follows:

[0060] Step 1: Experimental Data Generation

[0061] 1.1 System Parameter Settings: The nominal values ​​of mass blocks m1, m2, and m3 are all 1000 kg, and the nominal values ​​of spring stiffness k1 to k6 are all 1000 N / m. Define the parameters to be corrected as k1 to k3, and their true ellipsoidal characteristic matrix is:

[0062]

[0063] 1.2 Statistical experimental observation data: 90 repeated frequency response function measurements are carried out on a 3-DOF vibration system, and the frequencies at the peak of the frequency response function are counted. The observed ellipsoid characteristic matrix is obtained by fitting using the minimum volume ellipsoid method: (such as Figure 3 )

[0064]

[0065] Step 2: Finite element modeling and sample point calculation:

[0066] 2.1 Establish the finite element model of the engineering structure, including meshing using BSUH elements and lumped mass elements, setting spring stiffness and lumped mass parameters, and setting fixed boundary conditions.

[0067] 2.2 In the modified parameter space k1~k3 of the finite element model, randomly sample points using the Latin hypercube method, and independently sample 20 times.

[0068] 2.3 Calculate the prediction results of the finite element model under 20 different design parameters to form 20 sets of design parameter-prediction sample pairs.

[0069] Step 3: Feature sample generation:

[0070] 3.1 Randomly select 10 samples from the 20 sets of design parameter-prediction result sample pairs, and fit the sample points using the minimum volume ellipsoid method to obtain a characteristic matrix P of the modified parameter i and an ellipsoid characteristic matrix (P pred ) i of the finite element model prediction sample.

[0071] 3.2 Repeat 20 times to obtain 20 sets of design parameter-prediction characteristic matrix sample pairs

[0072] Step 4: Gaussian process surrogate model construction

[0073] 4.1 Sample space logarithmic mapping: transform using the formula i log(P i ) and (S pred ) i = log((p pred ) i ) to obtain

[0074] 4.2 Model training: take as input and output, and use the logarithmic Euclidean kernel function Gaussian process regression model is constructed with bandwidth parameter δ = 0.01 and noise variance σ 2 = 0.01. The k = 5-fold cross-validation is used to optimize the hyperparameters during modeling.

[0075] Step 5: Manifold-constrained optimization correction

[0076] 5.1 Parameter initialization: an initial search point is generated according to expert experience or randomly

[0077]

[0078] 5.2 Iterative optimization: in the t-th iteration, the predicted value μ(S t ) is calculated by the Gaussian process surrogate model, and the loss function The gradient of the modified parameter on the loss function is calculated The gradient is symmetrized The correction formula of the modified parameter is The learning rate η = 0.1 is set.

[0079] 5.3 Termination condition: when t = 30, the change of the damage function satisfies Stop iteration, ε = 10 -5 .

[0080] Step 6: result output:

[0081] Calculate the parameter ellipsoid after correction

[0082]

[0083] The parameter space projection diagram (as shown in Figure 4 ) shows that the ellipsoid axis gradually aligns with the true distribution; the loss function descent curve (as shown in Figure 5 ) verifies the convergence of the algorithm.

[0084] This example proves that the loss function decreases from 48.84 to 3.12 × 10 -4 Before and after model correction. The model correction calculation time is only 0.8 minutes, which is 10 times faster than the traditional MCMC method. It provides a reusable technical example for the uncertainty quantification of mechanical vibration systems.

[0085] The embodiment discloses an interval type uncertainty model parameter correction method based on Riemannian manifold and Gaussian process model. The method firstly constructs an ellipsoid convex model of a parameter space through semi-definite programming optimization, and collects frequency response data of a vibration system to fit and generate an observation ellipsoid characteristic matrix. Then, Latin hypercube sampling is used to generate training samples, and a Gaussian process proxy model is constructed by using a logarithmic Euclidean kernel function, so that the geometric mapping of a symmetric positive definite matrix to a tangent space is realized. Subsequently, gradient optimization is implemented on the Riemannian manifold, the constrained optimization problem is converted into an unconstrained problem through matrix logarithmic transformation, and the symmetric gradient descent algorithm is combined to iteratively correct the ellipsoid parameters. Finally, the intersection-over-union ratio calculation, noise robustness test and dynamic characteristic verification are performed, the significant improvement effect of the corrected model in the parameter correlation representation accuracy, calculation efficiency and anti-interference ability is confirmed, and the uncertainty quantization closed loop is completed. The example results prove that the method is not only effective, but also can provide a high-precision and interpretable uncertainty parameter correction tool for a numerical model in engineering, significantly improves the accuracy and efficiency of model correction, and enhances the interpretability of the model.

[0086] The above is only the preferred embodiment of the present application, and it should be noted that those skilled in the art can make several improvements and adjustments without departing from the principles of the present application, and these improvements and adjustments should also be considered within the protection scope of the present application.

Claims

1. A method for interval-type uncertainty model parameter correction based on a Riemannian manifold and a Gaussian process model, characterized in that, The model parameter correction method comprises the following steps: Step one: a plurality of repeated experiments are carried out on the engineering structure, a plurality of experimental result sample points are obtained, and sample points are fitted to obtain an ellipsoid characteristic matrix representing the experimental results; Step two: a finite element model of the engineering structure is established, the finite element model prediction results under M groups of different design parameters are calculated, and M groups of design parameter-prediction result sample pairs are formed; Step three: N samples are randomly selected from the M groups of design parameter-prediction result sample pairs, and the minimum volume ellipsoid method is used to fit the sample points to obtain an ellipsoid characteristic matrix of the prediction samples; the process is repeated M times to obtain M design parameter-prediction characteristic matrix samples; Step four: the M groups of design parameter-prediction characteristic matrix samples are transformed by using a logarithmic mapping formula to map them to the tangent space of the Riemann manifold, and a Gaussian process regression model is constructed by using a log-Euclidean kernel function; Step five: the corrected parameter in the Gaussian process regression model is searched, the current value of the corrected parameter is input into the Gaussian process regression model, the prediction value is calculated, the gradient of the corrected parameter on the loss function is calculated, and the corrected parameter is updated; Step six: the parameter ellipsoid characteristic matrix after correction is calculated by using exponential mapping to inversely map the parameter from the tangent space back to the Riemann manifold.

2. The interval-type uncertainty model parameter correction method based on Riemannian manifold and Gaussian process model according to claim 1, characterized in that, The step two is specifically: Step two, establishing a finite element model of the engineering structure, including meshing, material parameter setting, and boundary condition setting; in the design parameter space of the finite element model, sample points are randomly selected by using the Latin hypercube method, and M times of independent sampling are performed; the finite element model prediction results under M groups of different design parameters are calculated, and M groups of design parameter-prediction result sample pairs are formed.

3. The interval-type uncertainty model parameter correction method based on Riemannian manifold and Gaussian process model according to claim 1, characterized in that, The step three is specifically: Step three, randomly select N samples from the M group of design parameter-predicted result sample pairs, where N≈0.5M, and use the minimum volume ellipsoid method to fit the sample points to obtain a characteristic matrix P of the modified parameters i and an ellipsoid characteristic matrix (P pred ) of the finite element model prediction sample i ; repeat M times to obtain M design parameter-prediction characteristic matrix sample pairs 4. The interval-type uncertainty model parameter correction method based on Riemannian manifold and Gaussian process model according to claim 1, characterized in that, The step four is specifically: Step four, apply the formula logarithmic mapping formula S i =log(P i ) and (S pred ) i =log((P) pred ) i The M parameters from step three are used to predict the feature matrix sample pairs. By performing a transformation and mapping it to the tangent space of the Riemannian manifold, we obtain... Utilizing As the input and output of the Gaussian process model, a log-Euclidean kernel function is used to construct a Gaussian process regression model, and a bandwidth parameter and a noise variance of the Gaussian process regression model are set.

5. The interval-type uncertainty model parameter correction method based on Riemannian manifold and Gaussian process model according to claim 1, characterized in that, The step five is specifically: Step five, according to the expert experience or randomly generate an initial search point S0 of the modified parameter in the Gaussian process regression model, set the learning rate η, in the t, t = 1, 2,..., T max , in the step iteration, input the current value S t of the modified parameter into the Gaussian process regression model, calculate the prediction value μ(S t ), and calculate the loss function The gradient of the loss function with respect to the modified parameter is calculated The gradient is symmetrized The update formula of the modified parameter is The iteration is stopped when the loss change amount of two consecutive iterations satisfies ε is a threshold value.

6. The method of claim 2 to 5, wherein, In the step one, the experimental sample points are fitted by using the minimum volume ellipsoid method.

7. The method of claim 1, wherein, The step four adopts logarithm-Euclidean kernel function A Gaussian process regression model is constructed.

8. The interval-type uncertainty model parameter correction method based on Riemannian manifold and Gaussian process model according to claim 1, characterized in that, The gradient of the loss function with respect to the modified parameters in the fifth step needs to be symmetrized 9. The interval-type uncertainty model parameter correction method based on Riemannian manifold and Gaussian process model according to claim 1, characterized in that, In the step six, the parameter ellipsoid after correction needs to be calculated by using exponential mapping P=exp(S).

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