High-dimensional reliability analysis method combining physical information neural network and distance sampling

Abstract

The invention relates to the technical field of high-dimensional reliability analysis, in particular to a high-dimensional reliability analysis method combining a physical information neural network and distance sampling, which comprises the following steps of: measuring a response value of a measured solid under a to-be-measured condition by adopting a sensor, and taking the to-be-measured condition and the corresponding response value as a transfer learning sample; obtaining a plurality of transfer learning samples; generating a plurality of configuration points, wherein the configuration points record space-time positions and randomly generated attributes of the tested solid; carrying out physical information learning based on the configuration points, and carrying out transfer learning training based on transfer learning samples to obtain a deep neural network reliability analysis model; based on a deep neural network reliability analysis model, calculating the reliability of the high-dimensional reliability assessment configuration point as a reliability analysis result; the method can improve the generalization performance of the physical information neural network under a small amount of sample data.

Description

Technical Field

[0001] The present invention relates to the technical field of high-dimensional reliability analysis, and in particular to a high-dimensional reliability analysis method combining physical information neural network and distance sampling. Background Art

[0002] Due to the limitations of measurement technology and cognitive ability, uncertainty is widely present in practical engineering problems. In order to ensure the reliability and safety of products, it is necessary to properly quantify and control the uncertainty in production, namely reliability analysis. Reliability analysis methods include methods based on most likely points (such as first-order and second-order reliability methods), surrogate model methods, expansion methods and approximate integral methods. Among them, surrogate models are widely used in various engineering problems due to their simplicity and efficiency. Surrogate models are easy-to-evaluate models that approximately describe the relationship between system input and output. Different surrogate model technologies, such as support vector machines (SVM), Kriging, artificial neural networks (ANN) and response surface methods (RSM) have been developed. Surrogate models are usually constructed based on given training data, and then combined with Monte Carlo simulation to solve the reliability of the system. Solving reliability analysis problems based on surrogate model methods has also been fully explored in recent years.

[0003] On the other hand, due to the recognition of the physical laws governing the dynamic evolution of control systems, machine learning techniques have also been developed to directly learn the dynamics of the system without using data provided by traditional numerical solvers such as finite element models. One of the most popular recent works is the physical information neural network, which is widely used in fields such as fluid mechanics, robotic control, and high energy physics. It constructs a loss function using the residuals and boundary conditions of partial differential equations. The model is then trained by optimizing the loss function so that the prediction results satisfy the physical laws. In this case, the neural network itself can be considered as a physical information proxy model to provide solutions to the control equations. Physical information neural networks have also emerged in the field of reliability analysis, especially when the control equations of the system are expressed in the form of differential equations. There is still much research to be done on solving reliability analysis problems based on physical information neural networks. Summary of the invention

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

[0005] The present invention provides a high-dimensional reliability analysis method combining physical information neural network and distance sampling, comprising the following steps:

[0006] Step S1, using a sensor to measure the response value of the solid under the test conditions, taking the test conditions and the corresponding response values ​​as transfer learning samples, and obtaining multiple transfer learning samples; the test conditions include time-space position and 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, generating a plurality of configuration points, wherein the configuration points record the time-space position and the randomly generated properties of the solid being measured; 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, performing physical information learning based on the configuration points, and performing 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, the reliability of the high-dimensional reliability assessment configuration point is calculated as a result of the reliability analysis.

[0010] Preferably, the step S2 specifically includes:

[0011] Latin hypercube sampling is used to generate n that can evenly fill the variable space. pde PDE residual collocation point P pde ;

[0012] Determine n ic Initial condition configuration point P ic and n bc Boundary condition configuration point P bc , and obtain the response value U at the initial condition configuration point and the boundary condition configuration point based on the prior information of the initial condition and boundary condition ic and U bc .

[0013] Preferably, the step S3 specifically includes:

[0014] Step S3-1, initializing and training the physical information neural network;

[0015] Step S3-2, initializing 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 to transform the high-dimensional reliability assessment configuration points into feature space to obtain sample points;

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

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

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

[0020] Step S3-1-2, determining the loss function that violates the partial differential equation, the initial condition and the boundary condition, and iteratively training the physical information neural network with the goal of minimizing the loss function to obtain the optimal parameters of the physical information neural network;

[0021] The step S3-1-2 specifically includes:

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

[0023]

[0024] Among them, θ represents the parameters of the physical information neural network, t, s, and x are time variables, space variables, and input random variables representing the properties of the measured solid, respectively. Represents the loss function of the physical information neural network, MSE pde , MSE ic , MSE bc are the loss functions of partial differential equations, initial conditions and boundary conditions respectively, i is the configuration point index, represents the partial derivative of the predicted output response of the ith configuration point with respect to t, represents the predicted output response of the ith configuration point, represents the nonlinear differential operator that parameterizes the random variable; λ represents the parameter of the nonlinear differential operator, |·| 2 represents the square of the absolute value, u ic (t i ,s i ,x i ) represents the predicted output response of the initial condition of the i-th configuration point, u bc (t i ,s i ,x i ) represents the predicted output response of the boundary condition of the i-th configuration point;

[0025] Will Minimize, the expression is:

[0026]

[0027] in, It means finding the value of parameter θ so that the function value is minimized. * is the optimal parameter of the physical information neural network.

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

[0029] Step S3-2-1, setting the initial values ​​of the parameters in the deep neural network as the optimal parameters of the physical information neural network;

[0030] Step S3-2-2, setting 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] The step S3-3 specifically includes:

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

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

[0034] Step S3-4-1: Use the historical iteration deep neural network model to obtain the sample point Z mcs Find candidate experimental sites in

[0035] Step S3-4-2, downsampling the candidate experimental points based on distance to obtain newly added experimental points;

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

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

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

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

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

[0041]

[0042] in, represents the set of candidate experimental points for the τth iteration, τ is the number of iterations, and z is the sample point Z mcs The elements in is the transfer learning deep neural network for sample point Z mcs The predicted output response is and are two thresholds set in advance, and their expressions are:

[0043]

[0044] in, and are the thresholds of the predicted output response at the τth iteration, and are the thresholds of the predicted output response at the τ-1th iteration, Indicates calculation of all sample points Z mcs The minimum value of the corresponding function value, is the predicted output response of the τ-1th iteration of the transfer learning deep neural network.

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

[0046] All points in the feature space are regarded as the initial point cloud, and two sets are divided in the initial point cloud, called experimental point set D e and the candidate experimental point set Z c ;

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

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

[0049] Among them, min(·) means calculating the minimum value, ||·|| means calculating the modulus of the vector, z j , z k are the jth and ith 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 largest distance is added as a new point and put into the experimental point set D e , expressed as:

[0051] z * = arg max(dmin )

[0052] Among them, 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, the step S3-4-4 specifically includes:

[0055] Determine the loss function of the transfer learning deep neural network, which can be expressed as:

[0056]

[0057] in, represents the loss function of the transfer learning deep neural network, represents the i-th configuration point (t0, s0, x i )’s predicted output response; θ t The network parameters of the deep neural network for transfer learning include the weights W that are allowed to be updated t and bias b t ;

[0058] The loss function of the transfer learning deep neural network is used as the minimization optimization target, which is expressed as:

[0059]

[0060] in, The optimal parameter θ for transfer learning deep neural network * , It means to find the parameter θ t The value of makes the function value minimum; finally, the updated transfer learning deep neural network model parameters are obtained.

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

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

[0063]

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

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

[0066]

[0067] When the reliability difference e Pr Less than the threshold e cr , the iterative process stops and the training of the transfer learning deep neural network model is completed, otherwise it returns to step S3-4-1.

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

[0069] (1) In order to solve the problem of high computational cost faced by data-driven proxy models in solving high-dimensional reliability problems, the present invention extends the physical information neural network to high-dimensional reliability analysis and establishes a model for high-dimensional reliability analysis using the physical information neural network.

[0070] (2) In order to overcome the generalization problem of physical information neural network constructed with limited configuration points when dealing with high-dimensional reliability analysis problems, the present invention adopts transfer learning technology. In this case, transfer learning is data-driven. Unlike pure data-driven neural networks, this method retains useful physical information by freezing parameters, and only a small amount of training data is needed to complete the update of the neural network.

[0071] (3) The present invention proposes a distance-based downsampling strategy. On the one hand, it ensures the sparse positioning of the 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 will not be located near the existing samples, because they may provide less information for improving the accuracy of the deep neural network model. BRIEF DESCRIPTION OF THE DRAWINGS

[0072] The drawings are only for the purpose of illustrating particular embodiments and are not to be construed as limiting the invention.

[0073] Figure 1 A flow chart of a high-dimensional reliability analysis method combining physical information neural network and distance sampling provided by the present invention.

[0074] Figure 2 Schematic diagram of the physical information neural network architecture constructed with limited configuration points provided by the present invention.

[0075] Figure 3 Schematic diagram of transfer learning of the physical information neural network constructed with limited configuration points provided by the present invention.

[0076] Figure 4A schematic diagram of renaming layers in a deep neural network for transfer learning provided by the present invention.

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

[0078] Figure 6 A detailed flow chart of the high-dimensional reliability analysis method provided by the present invention. DETAILED DESCRIPTION

[0079] In order to more clearly understand the above-mentioned purpose, features and advantages of the present invention, the present invention is further described in detail below in conjunction with the accompanying drawings and specific embodiments. It should be noted that, in the absence of conflict, the embodiments of the present invention and the features in the embodiments can be combined with each other. In addition, the present invention can also be implemented in other ways different from those described herein, and therefore, the protection scope of the present invention is not limited by 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] Among them, P{·} is the probability measure, f X (x) is the joint probability density function (PDF) of X, R n is an n-dimensional variable space. In high-dimensional reliability analysis, formula (1) is usually a multidimensional integral problem, and it is difficult to directly obtain a closed-form solution. In order to solve the multidimensional integral problem with high precision, a sampling-based method has been proposed, such as Monte Carlo simulation. According to the Monte Carlo simulation method, formula (1) can be written as:

[0083]

[0084] Among them, E[g] represents the expectation; Ω F ={x:G(x)<0} is the fault set. In formula (2) It is called the indicator function and is defined as:

[0085]

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

[0087]

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

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

[0090] In order to illustrate the effectiveness of the method proposed by the present invention, the above technical solution of the present invention is described in detail below through a specific embodiment. Figure 1 , Figure 6 As shown, a high-dimensional reliability analysis method combining physical information neural network and distance sampling is disclosed, comprising the following steps:

[0091] Step S1, using a sensor to measure the response value of the solid under test conditions, taking the test conditions and the corresponding response values ​​as transfer learning samples, and obtaining multiple transfer learning samples; the test conditions include time-space positions and properties of the solid under test, and the properties of the solid under test include material properties, geometric dimensions and external loads.

[0092] The transfer learning samples may include input random variables x, time variables t, space variables s, and corresponding partial differential equation response values ​​in the field of solid mechanics. The random variables x may include material properties, geometric dimensions, and external loads.

[0093] In some embodiments, the material properties include elastic modulus, yield strength, density and thermal conductivity of the solid material, etc., which are usually obtained through experimental measurements, literature data or data provided by the manufacturer. For example, the elastic modulus of the solid material can be measured by a tensile test. Geometric dimensions include the length, width, height, thickness or three-dimensional mesh model of the object, etc., 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, etc., 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 may be physical quantities such as stress and strain.

[0095] Step S2: Generate multiple configuration points, wherein the configuration points record time information, space 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] Configuration points refer to data points used to evaluate the prediction results of neural networks when solving partial differential equations. The present invention uses Latin hypercube sampling (LHS) to generate configuration points that can evenly fill the variable space. The time variable t and the space variable s use uniform distribution to generate configuration points. For the random variable x, the uniform distribution is also used to generate configuration points. For the random variable x that does not obey the uniform distribution, it is converted into a uniform distribution U[μ-5σ,μ+5σ] and a configuration point is generated. Among them, μ and σ represent the mean and standard deviation of the random variable respectively. This method generates a total of n total Configuration points, including n pde PDE residual collocation point P 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 They represent the time variable, space variable and input random variable of the i-th configuration point respectively. The response values ​​of the initial conditions and boundary conditions at the configuration point can be calculated and represented 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 conditions and boundary conditions at specific input random variables x, time variables t, and space variables s can be obtained through experimental measurement, numerical calculation, or known data. For example, if the initial conditions and boundary conditions can be directly measured in the experiment, then these values ​​can be obtained through experimental equipment (such as sensors, measuring instruments), such as temperature sensors that can measure the initial temperature distribution. For conditions that cannot be measured directly, they can be obtained through numerical simulation or theoretical calculation, 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 by prior knowledge. The initial conditions are descriptions of the state of the system at the initial time, such as displacement fields, etc., which can be obtained by experimental measurement or known initial states. Boundary conditions can be constraints or conditions imposed on the boundaries of the system, such as fixed ends, free ends, which can be obtained by experimental measurement, design specifications or theoretical assumptions. For example, the displacement of the fixed end is zero, which can be determined by design specifications.

[0105] In order to speed up the training of the physical information neural network for high-dimensional reliability analysis, the present invention uses a limited number of configuration points to construct the physical information neural network, that is, n total Not a very large value.

[0106] At the same time, generate the configuration point P for high-dimensional reliability evaluation mcs The present invention mainly targets a specific spatiotemporal high-dimensional reliability analysis problem, and therefore sets the configuration point of the high-dimensional reliability evaluation as the time variable t=t0 and the space variable s=s0.

[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 the physical information neural network and performing transfer learning based on distance downsampling, which is introduced in detail below.

[0110] Step S3-1, initialize the physical information neural network.

[0111] The physical information neural network architecture constructed using limited configuration points is as follows 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, the learning rate, etc. Use the deep neural network model to approximate the hidden solution, that is, to predict the output response, which can be expressed as:

[0112]

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

[0114] Secondly, the 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:

[0115]

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

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

[0118]

[0119] Among them, θ represents the parameters of the physical information neural network, Represents the loss function of the physical information neural network.

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

[0121]

[0122] in, It 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, setting 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 weights and biases, and have the same architecture, such as the number of hidden layers, the number of neurons, the learning rate, etc.

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

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

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

[0128] Step S3-2-2, freeze the parameters corresponding to all layers above the penultimate layer or the second-to-last layer in the deep neural network, and do not update them anymore, and only update the parameters corresponding to the last layer or the last two layers in the deep neural network. It can be expressed as:

[0129]

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

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

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

[0133] Step S3-4: by iterative optimization, adopt a distance-based downsampling method to obtain new experimental points from the sample points, update the transfer learning deep neural network based on the new experimental points, and finally obtain a deep neural network reliability analysis model. This step specifically includes:

[0134] Step S3-4-1: Use the deep neural network model of the previous iteration to obtain the sample point Z mcs Find candidate experimental points in .

[0135] Based on the previously established deep neural network model, iteratively extracts samples from the feature space Z mcs In each iteration, the deep neural network model is updated to predict P mcs The limit state function output is then used to obtain the limit state function from the sample point Z using the deep neural network model of the previous iteration. mcs Find candidate experimental points in . The set of candidate experimental points is defined as:

[0136]

[0137] in, represents the set of candidate experimental points for the τth iteration, is the transfer learning deep neural network for sample point Z mcs The predicted output response is and are two thresholds set in advance, and their definitions are:

[0138]

[0139] in, and are the thresholds of the predicted output response at the τth iteration, and are the thresholds of the predicted output response at the τ-1th iteration, Indicates calculation of all sample points Z mcs The minimum value of the corresponding function value, is the predicted output response of the τ-1th iteration of the transfer learning deep neural network.

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

[0141] Since the feature space is obtained by nonlinear mapping of the variable space, the distribution information of the feature space is difficult to obtain, which means that the traditional method of selecting experimental points based on probability in the random variable space cannot be used to select experimental points in the feature space. Therefore, the sampling method in the feature space can be discussed from a geometric perspective.

[0142] The distance-based downsampling strategy provided by the present invention is as follows: Figure 5 As shown in Figure 2, first, all points in the feature space are regarded as the initial point cloud, and two sets are divided in the initial point cloud, called experimental point sets D e and the candidate experimental point set Z c , expressed 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] Among them, n e and n c are 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 number of initial point clouds, z j , z k are the j-th and i-th experimental point and candidate experimental point respectively.

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

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

[0148] Among them, ‖·‖ represents the modulus of the calculated vector.

[0149] Again, in the candidate experimental point set Z c Find a set of experimental points D eThe point with the maximum distance is the new point, and the new point is added to the experimental point set D e , expressed as:

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

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

[0152] Finally, repeat the above distance-based downsampling process to obtain multiple new experimental points Z e , expressed as:

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

[0154] Among them, z l is the lth newly added experimental point, and m is 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. Among them, the transfer learning sample points with true response value labels can be used as experimental points selected when constructing the training set, and the corresponding true response value labels can be used as output values ​​corresponding to the training set data.

[0157] The experimental points of each iteration are combined Update to the union of experimental points selected in all previous iterations:

[0158]

[0159] in, represents the union of experimental points in the τth iteration, Represents the set of experimental points in the τth iteration.

[0160] Get the configuration point corresponding to the union of the iterative experimental points of the current iteration in the variable space

[0161]

[0162] Among them, n t is the total number of experimental points in the experiment point collection.

[0163] Using the current iteration of the deep neural network to get The predicted output response of each configuration point in 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 is the configuration point (t0, s0, x i )’s predicted output response.

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

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

[0168]

[0169] in, represents the loss function of the transfer learning deep neural network, represents the i-th configuration point (t0, s0, x i )’s predicted output response. t The network parameters of the deep neural network for transfer learning include the weights W that are allowed to be updated t and bias b t ;

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

[0171]

[0172] in, The optimal parameter θ for transfer learning deep neural network * , It means to find the parameter θ t The value of makes the function value minimum. Finally, the updated transfer learning deep neural network model parameters are obtained.

[0173] Step S3-4-5, check the convergence of the transfer learning deep neural network model, and complete the training of the transfer learning deep neural network model when the convergence conditions are met, otherwise return to step S3-4-1.

[0174] In this step, P is first calculated based on the 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 is estimated using two consecutive iterations, e Pr To check convergence:

[0176]

[0177] in, is the reliability of the τth iteration, is the reliability of the τth iteration. Reliability P r Available through P r =1-P f Calculated.

[0178] Return to step S3-4-1 until the reliability difference e Pr Less than the threshold e cr (e.g. 0.005), the iterative 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, the reliability of the high-dimensional reliability assessment configuration point is calculated as a result of the reliability analysis.

[0180] In this step, similarly, P is calculated based on the deep neural network reliability analysis model. mcs The predicted output response is then used to evaluate the failure probability P f , through P r =1-P f The reliability P is calculated r .

[0181] The embodiment of the present invention discloses a numerical example of using the reliability analysis method of the present invention in a scenario where a solid structure is affected by a load.

[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, displacement, etc. over time and space. In this embodiment, in order to verify the ability of the method of the present invention to solve the 25-dimensional reliability analysis problem, the following partial differential equation representing the influence of the load on the strain u of the solid in the time t∈[0,2] range 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, 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] It means that at time t=0, the initial value of the state variable strain u is 0.

[0188] The limit state function is expressed as:

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

[0190] In the method provided by the present invention, first, in order to construct a physical information neural network, Latin hypercube sampling is used to generate 2100 collocation points, of which 1000 collocation points are used for the residual of the partial differential equation and 100 collocation points are used for the initial conditions. For the time variable t, the uniform distribution U[0,2] can be used to achieve the generation; for the random variable Since it follows the normal distribution N(1,0.2 2 ), and the converted uniform distribution U[0,2] is used to generate it. At the same time, in order to predict the reliability of the system, it is necessary to generate 10 5 configuration points, for time variable t = 1, for random variable According to the normal distribution N(1,0.2 2 ) to achieve generation.

[0191] Secondly, construct a physical information neural network. Use the generated 1000 partial differential variance residual configuration points and 100 initial condition configuration points to construct a physical information neural network. It is worth noting that for a 25-dimensional space, 1100 configuration points are obviously not fully and evenly filled, which means that for partial differential equations of solid structures affected by loads, the physical information neural network constructed with 1100 configuration points will face serious generalization problems. The specific details of constructing the physical information neural network are shown in Section 3.1. The parameters of the physical information neural network model during training are shown in Table 1.

[0192] Table 1

[0193]

[0194] Thirdly, a deep neural network for transfer learning is constructed. In order to perform data-driven transfer learning on the physical information neural network constructed with limited configuration points, the parameters of the constructed physical information neural network are loaded onto the 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, which means that there is only 1 migration layer. When the number of frozen layers is determined, the feature extractor consisting of the input layer, the frozen layer, and the last layer of the frozen layer maps the configuration points for reliability assessment in the variable space to the feature space and obtains the corresponding sample points.

[0195] Finally, the deep neural network is updated and the reliability is estimated. In each iteration, the proposed point cloud downsampling method is used to sample 1 sample point in the feature space, which means that M = 1. Usually, M does not need to be too large. On the one hand, useful physical information has been retained in the frozen layer. On the other hand, the frozen layer reduces the number of layers that need to be updated in the entire neural network, that is, it reduces the 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 and realize transfer learning. The iterative update process stops after 9 iterations, and the obtained reliability approximation is 0.9322, and the relative error of the benchmark result 0.9323 is negligible.

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

[0197] Table 2

[0198]

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

[0200] Although the specific embodiments of the present invention depict various actions or steps in a specific order, this should be understood as requiring such actions or steps to be performed in the specific order shown or in a sequential order, or requiring all illustrated actions or steps to be performed to obtain the desired results. Under certain circumstances, multitasking and parallel processing may be advantageous. Similarly, although several specific implementation details are included in the above discussion, these should not be interpreted as limiting the scope of the present disclosure. Certain features described in the context of a separate embodiment can also be implemented in a single implementation in combination. On the contrary, the various features described in the context of a single implementation can also be implemented in multiple implementations individually or in any suitable sub-combination. The above is only a preferred specific embodiment of the present invention, but the protection scope of the present invention is not limited thereto. Any changes or replacements that can be easily thought of by any technician familiar with the technical field within the technical scope disclosed by the present invention should be covered within the protection scope of the present invention.

[0201] The above description is only a preferred specific implementation manner of the present invention, but the protection scope of the present invention is not limited thereto. Any changes or substitutions that can be easily conceived by any technician familiar with the technical field within the technical scope disclosed by the present invention should be covered within the protection scope of the present invention.

Claims

1. A high-dimensional reliability analysis method combining physical information neural network and distance sampling, characterized in that: The following steps are involved: Step S1, using a sensor to measure the response value of the solid under the test conditions, taking the test conditions and the corresponding response values ​​as transfer learning samples, and obtaining multiple transfer learning samples; the test conditions include time-space position and properties of the solid under test, and the properties of the solid under test include material properties, geometric dimensions and external loads; Step S2, generating a plurality of configuration points, wherein the configuration points record the time-space position and the randomly generated properties of the solid being measured; the configuration points include residual configuration points, initial condition configuration points, boundary condition configuration points and high-dimensional reliability evaluation configuration points; Step S3, performing physical information learning based on the configuration points, and performing 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, the reliability of the high-dimensional reliability assessment configuration point is calculated as a result of the reliability analysis.

2. The high-dimensional reliability analysis method combining physical information neural network and distance sampling according to claim 1 is characterized in that: The step S2 specifically includes: Latin hypercube sampling is used to generate n that can evenly fill the variable space. pde PDE residual collocation point P pde ; Determine n ic Initial condition configuration point P ic and n bc Boundary condition configuration point P bc , and obtain the response value U at the initial condition configuration point and the boundary condition configuration point based on the prior information of the initial condition and boundary condition ic and U bc .

3. The high-dimensional reliability analysis method combining physical information neural network and distance sampling according to claim 2 is characterized in that: The step S3 specifically includes: Step S3-1, initializing and training the physical information neural network; Step S3-2, initializing the transfer learning deep neural network based on the trained physical information neural network; Step S3-3, using the transfer learning deep neural network to transform the high-dimensional reliability assessment configuration points into feature space to obtain sample points; Step S3-4: by means of iterative optimization, a distance-based downsampling method is used to obtain new experimental points from the sample points, and the transfer learning deep neural network is updated based on the new experimental points to finally obtain a deep neural network reliability analysis model.

4. The high-dimensional reliability analysis method combining physical information neural network and distance sampling according to claim 3 is characterized in that: The step S3-1 specifically includes: Step S3-1-1, initializing the parameters and hyperparameters of the physical information neural network; Step S3-1-2, determining the loss function that violates the partial differential equation, the initial condition and the boundary condition, and iteratively training the physical information neural network with the goal of minimizing the loss function to obtain the optimal parameters of the physical information neural network; The step S3-1-2 specifically includes: The loss function of the physical information neural network is determined as: Among them, θ represents the parameters of the physical information neural network, t, s, and x are time variables, space variables, and input random variables representing the properties of the measured solid, respectively. Represents the loss function of the physical information neural network, MSE pde , MSE ic , MSE bc are the loss functions of partial differential equations, initial conditions and boundary conditions respectively, i is the configuration point index, represents the partial derivative of the predicted output response of the ith configuration point with respect to t, represents the predicted output response of the ith configuration point, represents the nonlinear differential operator that parameterizes the random variable; λ represents the parameter of the nonlinear differential operator, | 2 represents the square of the absolute value, u ic (t i ,s i ,x i ) represents the predicted output response of the initial condition of the i-th configuration point, u bc (t i ,s i ,x i ) represents the predicted output response of the boundary condition of the i-th configuration point; Will Minimize, the expression is: in, It means finding the value of parameter θ so that the function value is minimized. * is the optimal parameter of the physical information neural network.

5. The high-dimensional reliability analysis method combining physical information neural network and distance sampling according to claim 4 is characterized in that: The step S3-2 specifically includes: Step S3-2-1, setting the initial values ​​of the parameters in the deep neural network as the optimal parameters of the physical information neural network; Step S3-2-2, setting 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; The step S3-3 specifically includes: The configuration point P of the high-dimensional reliability evaluation mcs After calculation by the feature extractor, the corresponding sample point Z in the feature space is obtained mcs ; The transfer learning sample is calculated by the feature extractor to obtain the corresponding transfer learning sample points in the feature space.

6. The high-dimensional reliability analysis method combining physical information neural network and distance sampling according to claim 5 is characterized in that: The step S3-4 specifically includes: Step S3-4-1: Use the historical iteration deep neural network model to obtain the sample point Z mcs Find candidate experimental sites in Step S3-4-2, downsampling the candidate experimental points based on distance to obtain newly added experimental points; Step S3-4-3, constructing and updating the training set using the newly added experimental points, and updating the union of experimental points in each iteration to the union of experimental points selected in all previous iterations; Step S3-4-4: Update the parameters of the transfer learning deep neural network model based on the updated training set; Step S3-4-5, check the convergence of the transfer learning deep neural network model, and complete the training of the transfer learning deep neural network model when the convergence condition is met, otherwise return to step S3-4-1 until the convergence condition is met.

7. The high-dimensional reliability analysis method combining physical information neural network and distance sampling according to claim 6 is characterized in that: The step S3-4-1 specifically includes: The candidate experimental points are obtained using the following formula: in, represents the set of candidate experimental points for the τth iteration, τ is the number of iterations, and z is the sample point Z mcs The elements in is the transfer learning deep neural network for sample point Z mcs The predicted output response is and are two thresholds set in advance, and their expressions are: in, and are the thresholds of the predicted output response at the τth iteration, and are the thresholds of the predicted output response at the τ-1th iteration, Indicates calculation of all sample points Z mcs The minimum value of the corresponding function value, is the predicted output response of the τ-1th iteration of the transfer learning deep neural network.

8. The high-dimensional reliability analysis method combining physical information neural network and distance sampling according to claim 7 is characterized in that: The step S3-4-2 specifically includes: All points in the feature space are regarded as the initial point cloud, and two sets are divided in the initial point cloud, called experimental point set D e and the candidate experimental point set Z c ; Calculate the candidate experimental point set Z c All points in the experimental point set D e The distance d min : d min =min(‖z k -With j ‖) Among them, min(·) means calculating the minimum value, ‖·‖ means calculating the modulus of the vector, and z j , z k are the jth and ith experimental point and candidate experimental point respectively; In the candidate experimental point set Z c Find a set of experimental points D e The point with the largest distance is added as a new point and put into the experimental point set D e , expressed as: z * =argmax(d min ) Among them, argmax(·) means finding the parameter value that maximizes the function value; Repeat the distance-based downsampling process multiple times to obtain multiple new experimental points Z e .

9. The high-dimensional reliability analysis method combining physical information neural network and distance sampling according to claim 8 is characterized in that: The step S3-4-4 specifically includes: Determine the loss function of the transfer learning deep neural network, which can be expressed as: in, represents the loss function of the transfer learning deep neural network, represents the i-th configuration point (t0, s0, x i )’s predicted output response; θ t The network parameters of the deep neural network for transfer learning include the weights W that are allowed to be updated t and bias b t ; The loss function of the transfer learning deep neural network is used as the minimization optimization target, which is expressed as: in, The optimal parameter θ for transfer learning deep neural network * , It means to find the parameter θ t The value of makes the function value minimum; finally, the updated transfer learning deep neural network model parameters are obtained.

10. The high-dimensional reliability analysis method combining physical information neural network and distance sampling according to claim 9 is characterized in that: The step S3-4-5 specifically includes: Calculate P based on transfer learning deep neural network model mcs The predicted output response is then used to estimate the probability of failure using the following formula: Among them, E[g] represents the expectation; Ω F ={x:G(x)<0} is the fault set; reliability P r Available through P r =1-P f Calculated; The difference in reliability estimated using two consecutive iterations is Pr To check convergence: When the reliability difference e Pr Less than the threshold e cr , the iterative process stops and the training of the transfer learning deep neural network model is completed, otherwise it returns to step S3-4-1.

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