A high-dimensional reliability analysis method combining physical information neural networks and distance sampling

Abstract

This invention relates to the field of high-dimensional reliability analysis technology, specifically to a high-dimensional reliability analysis method combining physical information neural networks and distance sampling. The method includes: measuring the response value of a solid under test using a sensor under test conditions; using the test conditions and corresponding response values ​​as transfer learning samples to obtain multiple transfer learning samples; generating multiple configuration points, each configuration point recording its spatiotemporal location and randomly generated solid attributes; performing physical information learning based on the configuration points and transfer learning training based on the transfer learning samples to obtain a deep neural network reliability analysis model; and calculating the reliability of the high-dimensional reliability assessment configuration points based on the deep neural network reliability analysis model, using this as the result of the reliability analysis. This invention can improve the generalization performance of physical information neural networks with limited sample data.

Description

Technical Field

[0001] This invention relates to the field of high-dimensional reliability analysis technology, specifically to a high-dimensional reliability analysis method that combines physical information neural networks and distance sampling. Background Technology

[0002] Due to limitations in measurement technology and cognitive abilities, uncertainty is prevalent in practical engineering problems. To ensure product reliability and safety, it is necessary to appropriately quantify and control uncertainties in production, i.e., reliability analysis. Reliability analysis methods include most probable point-based methods (such as first-order and second-order reliability methods), surrogate model methods, expansion methods, and approximate integration methods. Among these, surrogate models are widely used in various engineering problems due to their simplicity and efficiency. A surrogate model is an easily evaluated model that approximately describes the relationship between system inputs and outputs. Different surrogate model techniques, such as Support Vector Machines (SVM), Kriging, Artificial Neural Networks (ANN), and Response Surface Methods (RSM), have been developed. Surrogate models are typically constructed based on given training data, and then combined with Monte Carlo simulations to solve for the system's reliability. Solving reliability analysis problems based on surrogate model methods has also been extensively explored in recent years.

[0003] On the other hand, due to the understanding of the physical laws governing the dynamic evolution of control systems, machine learning techniques have also developed, enabling the direct learning of system dynamics without the need for data provided by traditional numerical solvers (such as finite element models). Physical information neural networks (PINs) are among the most popular recent developments, widely applied in fluid mechanics, robotics control, and high-energy physics. They utilize the residuals and boundary conditions of partial differential equations to construct loss functions. The model is then trained by optimizing the loss function to ensure that the predictions satisfy physical laws. In this context, the neural network itself can be considered a physical information proxy model, providing solutions to the control equations. PINs have also begun to emerge in reliability analysis, especially when the system's control equations are expressed in differential equation form. However, methods based on PINs for solving reliability analysis problems still require further investigation. Summary of the Invention

[0004] In view of the above problems, the present invention provides a high-dimensional reliability analysis method that combines physical information neural networks and distance sampling, which solves the technical problems of difficulty in obtaining sample data for reliability analysis and poor generalization performance of physical information neural networks in the prior art.

[0005] This invention provides a high-dimensional reliability analysis method that combines physical information neural networks and distance sampling, comprising the following steps:

[0006] Step S1: Use a sensor to measure the response value of the solid under test under the test conditions, and use the test conditions and the corresponding response values ​​as transfer learning samples to obtain multiple transfer learning samples; the test conditions include spatiotemporal location and the properties of the solid under test, and the properties of the solid under test include material properties, geometric dimensions and external loads;

[0007] Step S2: Generate multiple configuration points, which record the spatiotemporal location and randomly generated properties of the solid under test; the configuration points include residual configuration points, initial condition configuration points, boundary condition configuration points, and high-dimensional reliability evaluation configuration points.

[0008] Step S3: Perform physical information learning based on the configuration points, and perform transfer learning training based on the transfer learning samples to obtain a deep neural network reliability analysis model;

[0009] Step S4: Based on the deep neural network reliability analysis model, calculate the reliability of the high-dimensional reliability assessment configuration point as the result of the reliability analysis.

[0010] Preferably, step S2 specifically includes:

[0011] Latin hypercube sampling is used to generate n that can uniformly fill the variable space. pde The residual collocation point P of the partial differential equation pde ;

[0012] Determine n ic Initial condition configuration point P ic and n bc Boundary condition configuration point P bc Based on prior information about the initial and boundary conditions, the response values ​​U at the initial and boundary condition configuration points are obtained. ic and U bc .

[0013] Preferably, step S3 specifically includes:

[0014] Step S3-1: Initialize and train the physical information neural network;

[0015] Step S3-2: Initialize the transfer learning deep neural network based on the trained physical information neural network;

[0016] Step S3-3: Using the transfer learning deep neural network, the high-dimensional reliability evaluation configuration points are transformed into the feature space to obtain sample points;

[0017] Step S3-4: Through iterative optimization, a distance-based downsampling method is used to obtain new experimental points from the sample points. The transfer learning deep neural network is updated based on the new experimental points to finally obtain the deep neural network reliability analysis model.

[0018] Preferably, step S3-1 specifically includes:

[0019] Step S3-1-1: Initialize the parameters and hyperparameters of the physical information neural network;

[0020] Step S3-1-2: Determine the loss function that violates the partial differential equation, initial conditions, and boundary conditions. With the goal of minimizing the loss function, iteratively train the physical information neural network to obtain the optimal parameters of the physical information neural network.

[0021] Step S3-1-2 specifically includes:

[0022] The loss function of the physical information neural network is determined as follows:

[0023]

[0024] Where θ represents the parameters of the physical information neural network, and t, s, and x are the time variable, spatial variable, and input random variable representing the properties of the solid being measured, respectively. The loss function of a physical information neural network, MSE pde MSE ic MSE bc , respectively, are the loss functions for the partial differential equation, initial conditions, and boundary conditions, where i is the index of the placement point. Let represent the partial derivative of the predicted output response at the i-th configuration point with respect to t. This represents the predicted output response of the i-th configuration point. Let |·| denote the nonlinear differential operator that parameterizes the random variable; λ denotes the parameter of the nonlinear differential operator, |·| 2 u represents the square of the absolute value. ic (t i ,s i ,x i ) represents the predicted output response of the initial conditions at the i-th configuration point, u bc (t i ,s i ,x i ) represents the predicted output response of the boundary conditions at the i-th configuration point;

[0025] Will Minimize, the expression is:

[0026]

[0027] in, This means finding the value of parameter θ that minimizes the function value. * These are the optimal parameters for the physical information neural network.

[0028] Preferably, step S3-2 specifically includes:

[0029] Step S3-2-1: Set the initial values ​​of the parameters in the deep neural network to the optimal parameters of the physical information neural network;

[0030] Step S3-2-2: Set the deep neural network to update only the weights and bias parameters corresponding to the last layer or the last two layers in the network;

[0031] Step S3-3 specifically includes:

[0032] The configuration point P of the high-dimensional reliability assessment mcs After calculation by the feature extractor, the corresponding sample point Z in the feature space is obtained. mcs The transfer learning samples are processed by a feature extractor to obtain the corresponding transfer learning sample points in the feature space.

[0033] Preferably, step S3-4 specifically includes:

[0034] Step S3-4-1: Utilize a deep neural network model based on historical iterations from sample point Z mcs Searching for candidate experimental sites;

[0035] Step S3-4-2: Perform distance-based downsampling on the candidate experimental points to obtain new experimental points;

[0036] Step S3-4-3: Construct and update the training set using the newly added experimental points, and update the union of experimental points in each iteration to the union of experimental points selected in all previous iterations;

[0037] Step S3-4-4: Based on the updated training set, update the parameters of the transfer learning deep neural network model;

[0038] Step S3-4-5: Check the convergence of the transfer learning deep neural network model. If the convergence condition is met, complete the training of the transfer learning deep neural network model; otherwise, return to step S3-4-1 until the convergence condition is met.

[0039] Preferably, step S3-4-1 specifically includes:

[0040] Candidate experimental points are obtained using the following formula:

[0041]

[0042] in, Z represents the set of candidate experimental points in the τ-th iteration, where τ is the iteration number and z is the sample point Z. mcs The elements in It is a transfer learning deep neural network for sample point Z mcs The predicted output response, and These are two pre-set thresholds, and their expressions are:

[0043]

[0044] in, and Let be the threshold values ​​for the predicted output response in the τ-th iteration. and These are the thresholds for the predicted output response in the (τ-1)th iteration. This indicates the calculation of Z for all sample points. mcs The minimum value of the corresponding function. It is the predicted output response of the transfer learning deep neural network in the (τ-1)th iteration.

[0045] Preferably, step S3-4-2 specifically includes:

[0046] Consider all points in the feature space as the initial point cloud, and divide the initial point cloud into two sets, called the experimental point sets D. e and candidate experimental point set Z c ;

[0047] Calculate the candidate experimental point set Z c All points in the experimental point set D e distance d min :

[0048] d min =min(||z k -z j ||)

[0049] Where min(·) represents calculating the minimum value, ||·|| represents calculating the magnitude of the vector, and z j , z k These are the j-th and i-th experimental points and candidate experimental points, respectively;

[0050] In the candidate experimental point set Z c Find a set of experimental points D e The point with the maximum distance is selected as the new point and added to the experimental point set D. e , is represented as:

[0051] z * =arg max(dmin )

[0052] Here, arg max(·) means finding the parameter value that maximizes the function value;

[0053] Repeat the distance-based downsampling process multiple times to obtain multiple new experimental points Z. e .

[0054] Preferably, step S3-4-4 specifically includes:

[0055] The loss function for the transfer learning deep neural network is determined, and the loss function of the transfer learning deep neural network can be expressed as:

[0056]

[0057] in, This represents the loss function of a deep neural network for transfer learning. Represents the i-th configuration point (t0, s0, x) i The predicted output response of θ; t For transfer learning of deep neural network parameters, including the allowable update weights W t and bias b t ;

[0058] The loss function of the transfer learning deep neural network is minimized as the optimization objective, expressed as:

[0059]

[0060] in, To find the optimal parameters θ for transfer learning deep neural networks * , This indicates that the parameter θ has been found. t The value of is found to minimize the function value; finally, the updated parameters of the transfer learning deep neural network model are obtained.

[0061] Preferably, step S3-4-5 specifically includes:

[0062] P is calculated based on a deep neural network model of transfer learning. mcs The predicted output response is then used to evaluate the failure probability using the following formula:

[0063]

[0064] Where E[g] represents the expectation; Ω F ={x:G(x)<0} is the fault set; reliability P r Through P r =1-P f Calculated;

[0065] The difference in reliability e is estimated using two consecutive iterations. Pr To test convergence:

[0066]

[0067] When the reliability difference e Pr Less than threshold e cr If the iteration stops and the training of the transfer learning deep neural network model is completed, then return to step S3-4-1.

[0068] Compared with the prior art, the present invention has at least the following beneficial effects:

[0069] (1) To address the problem of excessive computational cost in solving high-dimensional reliability problems using data-driven proxy models, this invention extends physical information neural networks to high-dimensional reliability analysis and establishes a model for high-dimensional reliability analysis using physical information neural networks.

[0070] (2) To overcome the generalization problem of physical information neural networks constructed with finite configuration points when dealing with high-dimensional reliability analysis problems, this invention employs transfer learning technology. In this case, transfer learning is data-driven. Unlike purely data-driven neural networks, this method preserves useful physical information by freezing parameters, requiring only a small amount of training data to update the neural network.

[0071] (3) This invention proposes a distance-based downsampling strategy. On the one hand, it ensures sparse localization of additional sample points, which can effectively capture the shape of the feature space, thereby improving the accuracy of the reliability approximation. On the other hand, it ensures that the newly added samples are not located near existing samples, as they may provide less information to improve the accuracy of the deep neural network model. Attached Figure Description

[0072] The accompanying drawings are for illustrative purposes only and are not intended to limit the scope of the invention.

[0073] Figure 1 The flowchart shows the high-dimensional reliability analysis method that combines physical information neural networks and distance sampling provided by this invention.

[0074] Figure 2 A schematic diagram of the physical information neural network architecture constructed using finite configuration points provided by this invention.

[0075] Figure 3 A schematic diagram of transfer learning for a physical information neural network constructed using finite configuration points provided by this invention.

[0076] Figure 4This is a schematic diagram illustrating the renaming of layers in a deep neural network used for transfer learning, as provided by the present invention.

[0077] Figure 5 This is a schematic diagram of the distance-based downsampling strategy provided by the present invention.

[0078] Figure 6 A detailed flowchart of the high-dimensional reliability analysis method provided by this invention. Detailed Implementation

[0079] To better understand the above-described objectives, features, and advantages of the present invention, the invention will be further described in detail below with reference to the accompanying drawings and specific embodiments. It should be noted that, unless otherwise specified, the embodiments of the present invention and the features thereof can be combined with each other. Furthermore, the present invention can be implemented in other ways different from those described herein; therefore, the scope of protection of the present invention is not limited to the specific embodiments disclosed below.

[0080] In reliability analysis, the limit state function G(X) is generally defined based on the failure mechanism, where X represents a vector of random input variables. The failure probability can be defined as:

[0081]

[0082] Where P{·} is the probability measure, f X (x) is the joint probability density function (PDF) of X, R n Let n be the variable space. In high-dimensional reliability analysis, equation (1) is usually a multidimensional integral problem, and it is difficult to obtain a closed-form solution directly. In order to solve multidimensional integral problems with high accuracy, sampling-based methods have been proposed, such as Monte Carlo simulation. According to the Monte Carlo simulation method, equation (1) can be written as:

[0083]

[0084] Where E[g] represents the expectation; Ω F ={x:G(x)<0} is the fault set. In equation (2) This is called the index function, and it is defined as:

[0085]

[0086] The core of the limit state function G(X) for many problems in science and engineering is driven by physical laws and expressed in the form of linear or nonlinear ordinary differential equations / partial differential equations. For spacetime phenomena, these equations can be expressed as:

[0087]

[0088] Where u(t,s,X) represents the latent (hidden) solution of the PDE, t represents the time variable, and s represents the space variable. t It is the partial derivative of u(s,t,X) with respect to t; Ω represents a nonlinear differential operator parameterized by a random variable X; a subset of Since the space is a multidimensional real space, the function G(X) can be expressed by the latent (hidden) solution u and the threshold u0 as follows:

[0089] G(X)=g(u)-u0 (5)

[0090] To illustrate the effectiveness of the method proposed in this invention, the following detailed description of the above technical solution is provided through a specific embodiment, such as... Figure 1 , Figure 6 As shown, a high-dimensional reliability analysis method combining physical information neural networks and distance sampling is disclosed, including the following steps:

[0091] Step S1: Use a sensor to measure the response value of the solid under test under the test conditions, and use the test conditions and the corresponding response values ​​as transfer learning samples to obtain multiple transfer learning samples; the test conditions include spatiotemporal location and the properties of the solid under test, and the properties of the solid under test include material properties, geometric dimensions and external load.

[0092] Transfer learning samples can include input random variable x, time variable t, spatial variable s, and the corresponding partial differential equation response values ​​in the field of solid mechanics. The random variable x can include material properties, geometric dimensions, and external loads, etc.

[0093] In some embodiments, the material properties include the elastic modulus, yield strength, density, and thermal conductivity of solid materials, which are typically obtained through experimental measurements, literature data, or data provided by the manufacturer. For example, the elastic modulus of a solid material can be measured through a tensile test. Geometric dimensions include the length, width, height, thickness, or shape of an object in a three-dimensional mesh model, which can be obtained through design drawings, measuring tools (such as calipers, laser rangefinders), or CAD models. External loads include forces, pressures, and temperature changes applied to the structure, which can be obtained through experimental application and measurement, sensor data, standard specifications, or theoretical calculations.

[0094] In some embodiments, for partial differential equations in the field of solid mechanics, the response value can be physical quantities such as stress and strain.

[0095] Step S2: Generate multiple configuration points, each containing time information, spatial information, and random configuration variables; the configuration points include residual configuration points, initial condition configuration points, boundary condition configuration points, and high-dimensional reliability evaluation configuration points.

[0096] Locating points are data points used to evaluate the prediction results of a neural network when solving partial differential equations. This invention uses Latin hypercube sampling (LHS) to generate locating points that uniformly fill the variable space. Time variable t and spatial variable s are generated using a uniform distribution. For the random variable x, a uniform distribution is also used to generate locating points. For random variables x that do not follow a uniform distribution, they are transformed into a uniform distribution U[μ-5σ,μ+5σ], and locating points are generated accordingly. Here, μ and σ represent the mean and standard deviation of the random variable, respectively. This method generates a total of n locating points. total There are n configuration points. pde The residual collocation point P of the partial differential equation pde n ic Initial condition configuration point P ic and n bc Boundary condition configuration point P bc ,in:

[0097] P pde ={(t i ,s i ,x i |i = 1, 2, K, n pde} (6)

[0098] P ic ={(t i ,s i ,x i |i = 1, 2, K, n ic} (7)

[0099] P bc ={(t i ,s i ,x i |i = 1, 2, K, n bc} (8)

[0100] Where i is the configuration point index, t i ,s i ,x i Let U represent the time variable, spatial variable, and input random variable at the i-th placement point, respectively. The response values ​​of the initial conditions and boundary conditions at the placement point can be calculated and denoted as U. ic and U bc ,in:

[0101] U ic ={(t i,s i ,x i |i = 1, 2, K, n ic} (9)

[0102] U bc ={(t i ,s i ,x i |i = 1, 2, K, n bc} (10)

[0103] In some embodiments, the response values ​​corresponding to the initial and boundary conditions at specific input random variables x, time variables t, and spatial variables s can be obtained through experimental measurements, numerical calculations, or known data. For example, if the initial and boundary conditions can be directly measured in an experiment, these values ​​can be obtained through experimental equipment (such as sensors or measuring instruments), such as a temperature sensor that can measure the initial temperature distribution. For conditions that cannot be directly measured, they can be obtained through numerical simulations or theoretical calculations, such as calculating the stress distribution on the boundary through finite element analysis (FEA).

[0104] In some embodiments, the initial conditions and boundary conditions may be known or determined through prior knowledge. Initial conditions are a description of the system state at the initial time, such as the displacement field, and can be obtained through experimental measurement or a known initial state. Boundary conditions can be constraints or conditions imposed on the system boundaries, such as fixed ends or free ends, and can be obtained through experimental measurement, design specifications, or theoretical assumptions. For example, the zero displacement at the fixed end can be determined through design specifications.

[0105] To accelerate the training speed of the physical information neural network used for high-dimensional reliability analysis, this invention uses a finite number of placement points to construct the physical information neural network, that is, n total It's not a very large value.

[0106] Simultaneously, configuration point P is generated for high-dimensional reliability assessment. mcs This invention primarily addresses a specific spatiotemporal high-dimensional reliability analysis problem; therefore, the configuration points for high-dimensional reliability assessment are set as the time variable t = t0 and the spatial variable s = s0. Wherein:

[0107] P mcs ={(t0,s0,x i |i = 1, 2, K, n mcs} (11)

[0108] Step S3: Perform physical information learning based on the configuration points, and perform transfer learning training based on the transfer learning samples to obtain a deep neural network reliability analysis model.

[0109] This step includes training a physical information neural network and performing transfer learning through distance-based downsampling, which will be described in detail below.

[0110] Step S3-1: Initialize the physical information neural network.

[0111] Physical information neural network architecture built using finite configuration points, such as Figure 2 As shown. First, initialize the deep neural network, including parameters and hyperparameters, such as the number of hidden layers, the number of neurons, and the learning rate. Using the deep neural network model to approximate the hidden solution, i.e., predict the output response, can be expressed as:

[0112]

[0113] in, DNN(g) represents the predicted output response, and DNN(g) represents a deep neural network. Let W represent the activation function, and b represent the weights and biases, respectively.

[0114] Secondly, an automatic differentiation mechanism is used to satisfy the initial conditions, boundary conditions, and partial differential equations. The loss functions for violating the partial differential equations, initial conditions, and boundary conditions can be expressed as follows:

[0115]

[0116] Among them, MSE pde MSE ic MSE bc These are the loss functions for the partial differential equation, initial conditions, and boundary conditions, respectively. Let represent the partial derivative of the predicted output response at the i-th configuration point with respect to t. This represents the predicted output response of the i-th configuration point. Denotes a nonlinear differential operator that parameterizes a random variable; λ represents the parameter of the nonlinear differential operator, | 2 u represents the square of the absolute value. ic (t i ,s i ,x i ) represents the predicted output response of the initial conditions at the i-th configuration point, u bc (t i ,s i ,x i ) represents the predicted output response of the boundary conditions at the i-th configuration point.

[0117] In summary, the loss function of the physical information neural network used for high-dimensional reliability analysis can be expressed as:

[0118]

[0119] Where θ represents the parameters of the physical information neural network. This represents the loss function of a physical information neural network.

[0120] Finally, the loss function is minimized to calculate the optimal parameters θ of the physical information neural network. * Minimizing the loss function in equation (16) can be expressed as:

[0121]

[0122] in, This means finding the value of parameter θ that minimizes the function value.

[0123] Step S3-2: Initialize the transfer learning deep neural network based on the trained physical information neural network.

[0124] Step S3-2-1: Set the initial values ​​of the parameters in the deep neural network to the optimal parameters of the physical information neural network; the initial deep neural network and the physical information neural network have the same parameters, such as the same weights and biases, and have the same architecture, such as the same number of hidden layers, the same number of neurons, and the same learning rate.

[0125] like Figure 3 As shown, a deep neural network can be represented as:

[0126] dnn(t,s,x)=DNN(t,s,x)(18)

[0127] Here, dnn(·) represents a deep neural network used for transfer learning.

[0128] Step S3-2-2: Freeze the parameters corresponding to all layers above the penultimate or penultimate layer in the deep neural network, and do not update them again. Only update the parameters corresponding to the last layer or the last two layers in the deep neural network. This can be expressed as:

[0129]

[0130] in, W represents the predicted output response. t and b t These represent the weights and biases that are allowed to be updated, respectively.

[0131] Step S3-3: Using the transfer learning deep neural network, the configuration points used for high-dimensional reliability assessment are transformed into the feature space to obtain sample points.

[0132] The deep neural network model used for transfer learning can be simplified into three parts: the input layer, the frozen layer, and the transfer layer. Figure 4As shown. The placement point P in the variable space used for high-dimensional reliability assessment. mcs After passing through the feature extractor (i.e., the frozen layer), the corresponding sample point Z can be obtained in the feature space. mcs The transfer learning samples are processed by a feature extractor to obtain corresponding transfer learning sample points. These transfer learning sample points have real labels and can be used to construct subsequent training sets.

[0133] Step S3-4: Through iterative optimization, a distance-based downsampling method is used to obtain new experimental points from the sample points. Based on these new experimental points, the transfer learning deep neural network is updated, ultimately yielding the deep neural network reliability analysis model. This step specifically includes:

[0134] Step S3-4-1: Using the deep neural network model from the previous iteration, start from sample point Z. mcs We searched for candidate experimental sites.

[0135] Based on the previously established deep neural network model, iteratively analyze the feature space samples Z. mcs Select experimental points. In each iteration, update the deep neural network model to predict P. mcs The limit state function is output. Then, the deep neural network model from the previous iteration is used to extract the sample point Z. mcs We search for candidate experimental points. The set of candidate experimental points is defined as follows:

[0136]

[0137] in, Let represent the set of candidate experimental points in the τ-th iteration. It is a transfer learning deep neural network for sample point Z mcs The predicted output response, and These are two pre-set thresholds, defined as follows:

[0138]

[0139] in, and Let be the threshold values ​​for the predicted output response in the τ-th iteration. and These are the thresholds for the predicted output response in the (τ-1)th iteration. This indicates the calculation of Z for all sample points. mcs The minimum value of the corresponding function. It is the predicted output response of the transfer learning deep neural network in the (τ-1)th iteration.

[0140] Step S3-4-2: Perform distance-based downsampling on the candidate experimental points to obtain new experimental points.

[0141] Since the feature space is obtained by nonlinearly mapping the variable space, its distribution information is difficult to obtain. This means that traditional methods for selecting experimental points based on probability in the random variable space cannot be used to select experimental points in the feature space. Therefore, sampling methods in the feature space can be explored from a geometric perspective.

[0142] The distance-based downsampling strategy provided by this invention is as follows: Figure 5 As shown, firstly, all points in the feature space are considered as the initial point cloud, and two sets are divided in the initial point cloud, called the experimental point set D. e and candidate experimental point set Z c , is represented as:

[0143] D e ={z j |j=1,2,K,n e} (twenty three)

[0144] Z c ={z k |k=1,2,K,n c} (twenty four)

[0145] Where, n e and n c Let n be the number of experimental point sets and the number of candidate experimental point sets, respectively, and satisfy n e ≤N, n c ≤N, N is the initial number of point clouds, z j , z k These are the j-th and i-th experimental points and candidate experimental points, respectively.

[0146] Secondly, calculate the candidate experimental point set Z. c All points in the experimental point set D e distance d min The distance from a point to the set is defined as the minimum distance from the point to all points in the set, expressed as:

[0147] d min =min(||z k -z j ||) (25)

[0148] Where, ‖·‖ represents the calculation of the magnitude of the vector.

[0149] Secondly, in the candidate experimental point set Z c Find a set of experimental points D eThe point with the maximum distance is the newly added point, and it is added to the experimental point set D. e , is represented as:

[0150] z * =arg max(d min (26)

[0151] Here, arg max(·) means finding the parameter value that maximizes the function value.

[0152] Finally, the distance-based downsampling process was repeated to obtain multiple new experimental points Z. e , is represented as:

[0153] Z e ={z l |l=1,2,K,m} (27)

[0154] Among them, z l Let m be the l-th newly added experimental point, and m be the total number of newly added experimental points.

[0155] Step S3-4-3: Construct and update the training set using the newly added experimental points.

[0156] In this step, new experimental points are added to construct the training set required for training the transfer learning deep neural network. Specifically, transfer learning sample points with labels of real response values ​​can be used as experimental points when constructing the training set, and the corresponding real response value labels can be used as the output values ​​of the training set data.

[0157] Union of the experimental points in each iteration Updated to the union of experimental points selected from all previous iterations:

[0158]

[0159] in, Let represent the union of experimental points in the τth iteration. Let represent the set of experimental points in the τth iteration.

[0160] Obtain the placement point in the variable space corresponding to the union of the iterative experiment points of the current iteration.

[0161]

[0162] Where, n t The total number of experimental points and the total number of experimental points.

[0163] Using the current iteration of the deep neural network to obtain The predicted output response of each configuration point is used to obtain the training set D. t :

[0164] D t ={((t0,s0,x i ),u i i = 1, 2, K, n t} (30)

[0165] Among them, u i The configuration point (t0, s0, x) corresponding to the i-th experimental point i The predicted output response.

[0166] Step S3-4-4: Based on the updated training set, update the parameters of the transfer learning deep neural network model.

[0167] Based on the updated training set, the network model is updated using the loss function of the transfer learning deep neural network. The loss function of the transfer learning deep neural network can be expressed as:

[0168]

[0169] in, This represents the loss function of a deep neural network for transfer learning. Represents the i-th configuration point (t0, s0, x) i The predicted output response of θ. t For transfer learning of deep neural network parameters, including the allowable update weights W t and bias b t ;

[0170] Minimizing the loss function in equation (31) can be expressed as:

[0171]

[0172] in, To find the optimal parameters θ for transfer learning deep neural networks * , This indicates that the parameter θ has been found. t The goal is to find the value that minimizes the function value. This leads to the updated parameters of the transfer learning deep neural network model.

[0173] Step S3-4-5: Check the convergence of the transfer learning deep neural network model. If the convergence condition is met, complete the training of the transfer learning deep neural network model; otherwise, return to step S3-4-1.

[0174] In this step, P is first calculated based on a transfer learning deep neural network model. mcs The predicted output response is obtained, and the failure probability is evaluated by formula (2).

[0175] Then, the difference in reliability e is estimated using two consecutive iterations. Pr To test convergence:

[0176]

[0177] in, Let be the reliability of the τth iteration. Let P be the reliability of the τth iteration. r Through P r =1-P f Calculated.

[0178] Return to step S3-4-1 until the reliability difference e Pr Less than threshold e cr (For example, 0.005), the iteration process stops, and the training of the transfer learning deep neural network model is completed.

[0179] Step S4: Based on the deep neural network reliability analysis model, calculate the reliability of the high-dimensional reliability assessment configuration point as the result of the reliability analysis.

[0180] In this step, similarly, P is calculated based on a deep neural network reliability analysis model. mcs The predicted output response is obtained, and the failure probability P is evaluated using formula (2). f Through P r =1-P f The reliability P is calculated. r .

[0181] The embodiments of the present invention disclose numerical examples of reliability analysis using the present invention in scenarios where solid structures are affected by loads.

[0182] Taking solid mechanics as an example, partial differential equations in solid mechanics are usually used to describe the changes of physical quantities such as stress, strain, and displacement with time and space. In this embodiment, in order to verify the ability of the method of the present invention to solve 25-dimensional reliability analysis problems, the following partial differential equation representing the influence of load on the strain u of a solid in the time range t∈[0,2] is analyzed:

[0183]

[0184] in, is a random variable that follows a normal distribution with a mean of 1 and a standard deviation of 0.2, where t represents time, d is the number of random variables, and σ represents the standard deviation of the random variable.

[0185] The boundary conditions are:

[0186] u(0,X)=0

[0187] This indicates that the initial value of the state variable strain u is 0 at time t = 0.

[0188] The limit state function is expressed as:

[0189] G(t,X)=u-1.5

[0190] In the method provided by this invention, firstly, to construct a physical information neural network, 2100 placement points are generated using Latin hypercube sampling, of which 1000 placement points are used for the residuals of the partial differential equations, and 100 placement points are used for the initial conditions. For the time variable t, a uniform distribution U[0,2] can be used for generation; for the random variable... Since it follows a normal distribution N(1, 0.2) 2 The transformed uniform distribution U[0,2] is used to generate the system. Simultaneously, to predict system reliability, 10... 5 There are several configuration points, for the time variable t=1, for the random variable... According to the normal distribution N(1, 0.2) 2 This is used to generate the data.

[0191] Secondly, a physical information neural network is constructed. The network is built using 1000 generated partial differential variance residual locus points and 100 initial condition locus points. It is worth noting that for a 25-dimensional space, 1100 locus points are clearly insufficient for sufficiently uniform filling. This means that for partial differential equations relating solid structures to loads, the physical information neural network constructed with 1100 locus points will face serious generalization problems. Specific details of constructing the physical information neural network are presented in Section 3.1. The parameters during the training process of the physical information neural network model are shown in Table 1.

[0192] Table 1

[0193]

[0194] Next, a deep neural network for transfer learning is constructed. To perform data-driven transfer learning on a physical information neural network built with finite placement points, the parameters of the constructed physical information neural network are loaded onto a deep neural network with the same structure, and the number of frozen layers in the deep neural network is determined. In this problem, the number of frozen layers is set to 4, meaning there is only 1 transfer layer. Once the number of frozen layers is determined, a feature extractor consisting of the input layer, the frozen layer, and the last layer of the frozen layer maps the placement points used for reliability assessment in the variable space to the feature space, obtaining the corresponding sample points.

[0195] Finally, the deep neural network is updated and its reliability is estimated. In each iteration, the proposed point cloud downsampling method is used, sampling one sample point in the feature space, which means M=1. Typically, M does not need to be too large. On the one hand, the frozen layer already retains useful physical information. On the other hand, the frozen layer reduces the number of layers that need to be updated in the entire neural network, thus reducing the number of parameters that need to be trained. Therefore, in this case, only a small amount of training data is needed to update the deep neural network, achieving transfer learning. The iterative update process stops after 9 iterations, and the relative error between the obtained reliability approximation of 0.9322 and the benchmark result of 0.9323 is negligible.

[0196] The reliability analysis results of the example provided by this invention are shown in Table 2:

[0197] Table 2

[0198]

[0199] The method provided by this invention effectively utilizes the physical information contained in the problem, and the deviation of the result (0.0107%) has higher accuracy and efficiency compared with the purely data-driven method (0.2360%).

[0200] While the specific embodiments of the present invention depict actions or steps in a particular order, this should be understood as requiring such actions or steps to be performed in the shown specific order or sequential order, or requiring all illustrated actions or steps to be performed to achieve the desired result. In certain environments, multitasking and parallel processing may be advantageous. Similarly, although several specific implementation details are included in the above discussion, these should not be construed as limiting the scope of this disclosure. Certain features described in the context of individual embodiments may also be implemented in combination in a single implementation. Conversely, various features described in the context of a single implementation may also be implemented individually or in any suitable sub-combination in multiple implementations. The above descriptions are merely preferred embodiments of the present invention, but the scope of protection of the present invention is not limited thereto. Any variations or substitutions that can be easily conceived by those skilled in the art within the technical scope disclosed in the present invention should be included within the scope of protection of the present invention.

[0201] The above description is only a preferred embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any changes or substitutions that can be easily conceived by those skilled in the art within the scope of the technology disclosed in the present invention should be included within the scope of protection of the present invention.

Claims

1. A high-dimensional reliability analysis method combining physical information neural networks and distance sampling, characterized in that, Includes the following steps: Step S1: Use a sensor to measure the response value of the solid under test under the test conditions, and use the test conditions and the corresponding response values ​​as transfer learning samples to obtain multiple transfer learning samples; the test conditions include spatiotemporal location and the properties of the solid under test, and the properties of the solid under test include material properties, geometric dimensions and external loads; Step S2: Generate multiple configuration points, which record the spatiotemporal location and randomly generated properties of the solid under test; the configuration points include residual configuration points, initial condition configuration points, boundary condition configuration points, and high-dimensional reliability evaluation configuration points. Step S3: Perform physical information learning based on the configuration points, and perform transfer learning training based on the transfer learning samples to obtain a deep neural network reliability analysis model; Step S4: Based on the deep neural network reliability analysis model, calculate the reliability of the high-dimensional reliability assessment configuration point as the result of the reliability analysis; Step S2 specifically includes: Latin hypercube sampling is used to generate variables that can uniformly fill the variable space. The residual collocation points of a partial differential equation ; Sure Initial condition configuration points as well as Boundary condition configuration points Based on prior information about the initial and boundary conditions, the response values ​​at the initial and boundary condition configuration points are obtained. and ; Step S3 specifically includes: Step S3-1: Initialize and train the physical information neural network; Step S3-2: Initialize the transfer learning deep neural network based on the trained physical information neural network; Step S3-3: Using the transfer learning deep neural network, the high-dimensional reliability evaluation configuration points are transformed into the feature space to obtain sample points; Step S3-4: Through iterative optimization, a distance-based downsampling method is used to obtain new experimental points from the sample points. The transfer learning deep neural network is updated based on the new experimental points to finally obtain the deep neural network reliability analysis model. Step S3-1 specifically includes: Step S3-1-1: Initialize the parameters and hyperparameters of the physical information neural network; Step S3-1-2: Determine the loss function that violates the partial differential equation, initial conditions, and boundary conditions. With the goal of minimizing the loss function, iteratively train the physical information neural network to obtain the optimal parameters of the physical information neural network. Step S3-1-2 specifically includes: The loss function of the physical information neural network is determined as follows: in, The parameters representing the physical information neural network, These are time variables, spatial variables, and input random variables representing the properties of the solid being measured, respectively. The loss function of a neural network representing physical information. , , , respectively, are the loss functions for the partial differential equation, initial conditions, and boundary conditions, where i is the index of the placement point. This represents the predicted output response pair for the i-th configuration point. The partial derivatives, This represents the predicted output response of the i-th configuration point. This represents a nonlinear differential operator that parameterizes a random variable. The parameters represent the nonlinear differential operators. Represents the square of the absolute value. This represents the predicted output response of the initial conditions at the i-th configuration point. This represents the predicted output response of the boundary conditions at the i-th configuration point; Will Minimize, the expression is: in, Indicates that the parameter has been found. The value of that minimizes the function value. These are the optimal parameters for the physical information neural network.

2. The high-dimensional reliability analysis method combining physical information neural networks and distance sampling according to claim 1, characterized in that, Step S3-2 specifically includes: Step S3-2-1: Set the initial values ​​of the parameters in the deep neural network to the optimal parameters of the physical information neural network; Step S3-2-2: Set the deep neural network to update only the weights and bias parameters corresponding to the last layer or the last two layers in the network; Step S3-3 specifically includes: The configuration points of the high-dimensional reliability assessment After calculation by the feature extractor, the corresponding sample points in the feature space are obtained. The transfer learning samples are processed by a feature extractor to obtain the corresponding transfer learning sample points in the feature space.

3. The high-dimensional reliability analysis method combining physical information neural networks and distance sampling according to claim 2, characterized in that, Step S3-4 specifically includes: Step S3-4-1: Utilize a deep neural network model based on historical iterations from sample points Searching for candidate experimental sites; Step S3-4-2: Perform distance-based downsampling on the candidate experimental points to obtain new experimental points; Step S3-4-3: Construct and update the training set using the newly added experimental points, and update the union of experimental points in each iteration to the union of experimental points selected in all previous iterations; Step S3-4-4: Based on the updated training set, update the parameters of the transfer learning deep neural network model; Step S3-4-5: Check the convergence of the transfer learning deep neural network model. If the convergence condition is met, complete the training of the transfer learning deep neural network model; otherwise, return to step S3-4-1 until the convergence condition is met.

4. The high-dimensional reliability analysis method combining physical information neural networks and distance sampling according to claim 3, characterized in that, Step S3-4-1 specifically includes: Candidate experimental points are obtained using the following formula: in, Indicates the first The set of candidate experimental points for each iteration For the number of iterations, For sample points The elements in It is a transfer learning deep neural network for sample points The predicted output response, and These are two pre-set thresholds, and their expressions are: in, and The first The threshold for the predicted output response in the next iteration. and The first The threshold for the predicted output response in the next iteration. This indicates the calculation of all sample points. The minimum value of the corresponding function. It is the first deep neural network to transfer learning. The predicted output response of the next iteration.

5. The high-dimensional reliability analysis method combining physical information neural networks and distance sampling according to claim 4, characterized in that, Step S3-4-2 specifically includes: Consider all points in the feature space as the initial point cloud, and divide the initial point cloud into two sets, called the experimental point sets. and candidate experimental point set ; Calculate the candidate experimental point set All point-to-experiment set distance : in, This indicates calculating the minimum value. This indicates the calculation of the magnitude of a vector. , These are the j-th and i-th experimental points and candidate experimental points, respectively; In the candidate experimental point set Find a set of experimental points The point with the maximum distance is selected as the new point and added to the experimental point set. , is represented as: in, This means finding parameter values ​​that maximize the function value; Repeat the distance-based downsampling process multiple times to obtain several new experimental points. .

6. The high-dimensional reliability analysis method combining physical information neural networks and distance sampling according to claim 5, characterized in that, Step S3-4-4 specifically includes: The loss function for the transfer learning deep neural network is determined, and the loss function of the transfer learning deep neural network can be expressed as: in, This represents the loss function of a deep neural network for transfer learning. Represents the i-th configuration point The predicted output response; To transfer learn the network parameters of a deep neural network, including the weights that are allowed to be updated. and bias ; The loss function of the transfer learning deep neural network is minimized as the optimization objective, expressed as: in, Optimal parameters for transfer learning deep neural networks , Indicates that the parameter has been found. The value of is found to minimize the function value; finally, the updated parameters of the transfer learning deep neural network model are obtained.

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