Reliability design optimization method combining transfer learning and physical information neural network

Abstract

This invention belongs to the field of reliability-based design optimization technology, specifically a reliability design optimization method combining transfer learning and physical information neural networks. It includes constructing a physical information neural network model using training data from the source domain and evaluating the reliability of the source domain. Utilizing layer freezing and fine-tuning techniques from transfer learning, the physical information neural network model constructed using the source domain training data is updated to evaluate the reliability of target domains with different statistical properties, thus saving computational overhead. A first-order scoring function is used to perform stochastic sensitivity analysis on the failure probability to ensure the normal progress of design optimization. Finally, the estimated reliability, objective function value, and corresponding sensitivity information are provided to the optimizer to obtain the corresponding updated design point. The design process is repeated iteratively until the optimal design is achieved. This invention accelerates the training speed of the physical information neural network model and improves its accuracy.

Description

Technical Field

[0001] This invention belongs to the field of reliability-based design optimization technology, and particularly relates to the design optimization of constraint functions expressed in the form of ordinary differential equations or partial differential equations in complex mechanical structures or systems. Specifically, it is a reliability design optimization method that combines transfer learning and physical information neural networks. Background Technology

[0002] Reliability-Based Design Optimization (RBDO) fully considers the impact of uncertainties on constraints and objectives during the optimization process, resulting in an optimized solution that satisfies the target reliability, thus achieving a good balance between constraint reliability and target optimization. Generally, the mathematical formula for a reliability-based design optimization problem can be expressed as:

[0003]

[0004] In the formula, the random input X is determined by X rv and X rp It consists of two components, namely X = [X rv ,X rp ] T , where X rv =[X1,X2,…,X nd ] T X represents a vector composed of random design variables; rp =[X nd+1 ,X nd+1 ,…,X nr ] T Let d = [d1, d2, ..., dn] represent a vector of random parameters. nd ] T The design vector is a vector X consisting of random design variables. rv The mean; d L and d U These represent the lower and upper boundaries of the design vector, respectively. Let be the target failure probability of the j-th constraint function; nc, nd, and nr represent the number of constraint functions, the number of random design variables, and the number of random inputs, respectively. It's easy to see that to obtain the precise optimal design point, it's necessary to accurately evaluate the failure probability of each constraint function. In other words, this means performing reliability analysis on each constraint function. The failure probability in reliability analysis is defined as:

[0005]

[0006] In the formula, ψ is a vector of distribution parameters, typically including the mean and standard deviation of the random input X; P{·} is a probability measure; f X(x; ψ) denotes the joint probability density function (PDF) of X. Equation (2) is usually a multidimensional integral problem, and its closed-form solution is difficult to obtain directly. In order to solve multidimensional integral problems with high accuracy, sampling-based methods can be used, such as Monte Carlo simulation. In this case, equation (2) can be expressed as:

[0007]

[0008] Where E[·] is the expectation operator; Ω F ={x:G(x)<0} is the invalid set. In equation (3), it is called the indicator function, and is defined as follows:

[0009]

[0010] Due to the use of X rv The mean value d is used as the design vector. When calculating the failure probability in equation (3), d can be simply used to replace the distribution parameter vector ψ.

[0011] The core of the constraint function G(X) in many optimization design 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:

[0012]

[0013] Where u(t,s,X) represents the latent (hidden) solution of the PDE, where 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; Therefore, in this case, the constraint function G(X) can be expressed by the latent (hidden) solution u and the threshold u0 as:

[0014] G(X)=g(u)-u0 (6)

[0015] The mathematical formula for RBDO can be reformulated as:

[0016]

[0017] The solution process of the RBDO method typically involves two nested optimization layers. The outer layer optimizes the design variables, while the inner layer performs reliability analysis; that is, each design point in the optimization process requires a reliability assessment. Since constraint functions in practical engineering are usually computationally expensive numerical models, this nesting relationship makes RBDO computationally inefficient. To improve the computational efficiency of RBDO, three mainstream methods have been developed: the dual-loop method, the decoupled loop method, and the single-loop method. The dual-loop method nests two different levels of optimization processes. The outer layer optimizes the design variables to generate candidate design points, while the inner layer performs reliability analysis. In the decoupled loop method, the RBDO problem is transformed into a series of deterministic optimization problems. The main idea of ​​the single-loop method is to replace probabilistic constraints with optimality conditions and solve the RBDO problem within a single loop.

[0018] In RBDO (Reliability-Based Problem Solving), performing accurate and efficient reliability analysis is crucial, as it affects the accuracy of the optimal solution and computational cost. To this end, numerous reliability analysis methods have been proposed, including sampling methods (such as Monte Carlo simulation), 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, with their simplicity and efficiency, are widely used in various engineering problems. A surrogate model is an easily evaluated model that approximately describes the relationship between the system's 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 simulation to solve for the system's reliability. Solving RBDO problems based on surrogate model methods has also been extensively explored in recent years.

[0019] On the other hand, with the recognition 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 (PINNs) are among the most popular recent developments, widely applied in fluid mechanics, robot 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. PINNs are also showing promise in reliability analysis, especially when the system's control equations are expressed in differential equation form. Solving RBDO problems using PINN-based methods remains to be studied.

[0020] In the process of realizing this invention, the inventors discovered at least the following problems in the prior art:

[0021] (1) Although traditional surrogate model methods have been extensively explored in the field of RBDO, their essence is to use data-driven surrogate models to solve RBDO problems, and the quality and prediction accuracy of the model directly depend on the training data. Although various strategies are used to minimize the size of the database, the original constraint functions still need to be evaluated. Especially when the constraint functions are expressed in the form of differential equations, it is necessary to solve the differential equations repeatedly. Usually, methods such as finite element method (FE) and finite difference method (FD) are used. However, using these methods to solve complex problems can be very time-consuming.

[0022] (2) Physical information neural networks (PINs) have not been fully explored in the field of RBDO (Restricted Baseline Design of Dots). However, as with data-driven surrogate models in solving RBDO problems, these methods can be directly and simply integrated into the sampling-based RBDO framework. However, this simple integration is inefficient, mainly because the generalization ability of the PSN models trained by these methods is limited. This means that in solving RBDO problems, the PSN model must be rebuilt in each iteration to accurately assess the reliability of the current design point. Although PSNs do not require evaluation of constraint functions expressed as differential equations (i.e., "simulation-free"), this does not mean that PSNs are computationally inexpensive. In fact, training PSNs is also time-consuming. Therefore, rebuilding the PSN model in each iteration in solving RBDO problems also faces significant computational costs. Summary of the Invention

[0023] To address the aforementioned problems in the existing technology, this invention proposes a reliability design optimization method combining transfer learning and physical information neural networks, primarily solving the computational accuracy and efficiency issues of the RBDO problem expressed by partial differential equations. This mainly includes: (1) the problem of obtaining sample data for reliability analysis due to expensive numerical models; and (2) the generalization problem of physical information neural networks caused by covariate shifts.

[0024] The complete technical solution of this invention includes:

[0025] The sample space and its distribution based on the initial design point are called the source domain, and the sample space and its distribution based on the current design point are called the target domain. First, a physical information neural network model is constructed using training data from the source domain, and the reliability of the source domain is evaluated. Second, to evaluate the reliability of the target domain, which has different statistical properties, layer freezing and fine-tuning techniques from transfer learning are used to update the physical information neural network model constructed using the training data from the source domain, instead of reconstructing a physical information neural network model using the training data from the target domain, thus saving computational overhead. Third, to ensure the normal progress of design optimization, a stochastic sensitivity analysis of the failure probability is performed using a first-order scoring function. Finally, the estimated reliability, objective function value, and corresponding sensitivity information are provided to the optimizer to obtain the updated design point. The iterative design process will be repeated continuously until the optimal design is achieved.

[0026] The complete technical solution of this invention includes:

[0027] A reliability design optimization method combining transfer learning and physical information neural networks includes the following steps:

[0028] (1) Define the initial design point of the RBDO;

[0029] (2) Generate source domain placement points, wherein the source domain is the sample space and its distribution based on the initial design points described in step (1), and source domain placement points are generated using Latin hypercube sampling; generate placement points for constructing the physical information neural network model. Simultaneously generate configuration points for source domain reliability assessment.

[0030] (3) Construct a physical information neural network model for source domain reliability analysis;

[0031] (4) Source domain reliability and sensitivity analysis: The constructed physical information neural network model is used to predict the location point. The response is analyzed, and a reliability analysis is performed on the constraints; at the same time, a first-order scoring function is used to perform a stochastic sensitivity analysis.

[0032] (5) Update the design point: The updated design point is obtained by providing the optimizer with the estimated failure probability, objective function value and corresponding sensitivity information;

[0033] (6) Generate target domain placement points: The target domain is the sample space and its distribution based on the updated design points. Latin hypercube sampling is used to generate target domain placement points; placement points are generated for transfer learning. Simultaneously generate configuration points for target domain reliability assessment.

[0034] (7) Transfer learning is performed on the physical information neural network model used for source domain reliability analysis; the transfer learning includes first freezing the layers and then fine-tuning to update the physical information neural network model;

[0035] (8) Target domain reliability and sensitivity analysis: Predicting placement points using an updated physical information neural network model. The response is analyzed, and a reliability analysis is performed on the constraints; at the same time, a first-order scoring function is used to perform a stochastic sensitivity analysis.

[0036] (9) Check convergence condition: If the convergence condition is met, stop the iteration process and output the optimal design; otherwise, return to step (5) and repeat the iteration process until the convergence condition is met.

[0037] Furthermore, in step (1), the initial design point of RBDO is the optimal solution under deterministic optimization.

[0038] Furthermore, in step (2), the configuration points for constructing the physical information neural network model include initial condition configuration points. Boundary condition configuration points and the residual collocation point of partial differential equations

[0039] Furthermore, in step (3), the process of constructing the physical information neural network model for source domain reliability analysis includes: first, initializing a deep neural network model; then, using an automatic differentiation mechanism to satisfy the initial conditions, boundary conditions, and partial differential equations; and finally, minimizing the training data. The loss of physical information.

[0040] Furthermore, in step (6), the configuration points used for transfer learning Including initial condition configuration points Boundary condition configuration points and the residual collocation point of partial differential equations

[0041] Furthermore, in step (6), the layer freezing includes freezing a portion of the layers of the physical information neural network model used for source domain reliability analysis, and then freezing the training data. Update the parameters of the remaining layers.

[0042] Furthermore, in step (6), the fine-tuning includes unfreezing the entire model or a portion of the model obtained above, and then processing the training data. The model was then retrained at a lower learning rate.

[0043] Furthermore, the lower learning rate is 0.0001.

[0044] Furthermore, the system structure is optimized using the aforementioned reliability design optimization method.

[0045] Furthermore, the system includes force-bearing components.

[0046] Furthermore, the system can be a cylindrical rod, an undamped single-degree-of-freedom system, or a thin plate subjected to uneven loads.

[0047] The advantages of this invention over the prior art are:

[0048] (1) This invention uses a physical information neural network for reliability analysis, which efficiently solves the reliability design optimization problem of complex engineering structures expressed in the form of ordinary differential equations or partial differential equations. The physical information neural network incorporates physical equations as constraints into the neural network, which not only makes the fitted results more in line with physical laws, but also avoids the simulation of traditional numerical solution methods such as finite element analysis, and realizes efficient uncertainty quantification.

[0049] (2) This invention uses a first-order scoring function to analyze the sensitivity of failure probability. By providing the optimizer with the estimated reliability, objective function value, and corresponding sensitivity information, the corresponding updated design point can be obtained.

[0050] (3) This invention introduces transfer learning technology, which efficiently solves the generalization problem caused by covariate shift when the physical information neural network method deals with design optimization problems. The layer freezing technique in transfer learning, on the one hand, preserves the useful physical information in the physical information neural network model constructed from the source domain; on the other hand, it reduces the gradients that need to be calculated to update the physical information neural network model, thus accelerating the training speed of the physical information neural network model. The fine-tuning technique in transfer learning further improves the accuracy of the physical information neural network updated using the layer freezing technique. Attached Figure Description

[0051] Figure 1 A flowchart illustrating a simulation-free reliability design optimization method that combines transfer learning and physical information neural network models.

[0052] Figure 2 This is the architecture of a physical information neural network.

[0053] Figure 3 This is a schematic diagram of the layer freezing technology.

[0054] Figure 4 This is a schematic diagram of the fine-tuning technique.

[0055] Figure 5 This is a schematic diagram of the structure of the simulation-free reliability design optimization method combining transfer learning and physical information neural network models as described in this invention.

[0056] Figure 6 This is a schematic diagram of the cylindrical rod in Example 1. Detailed Implementation

[0057] The present invention will now be described in detail with reference to embodiments and accompanying drawings. However, it should be understood that the embodiments and drawings are for illustrative purposes only and do not constitute any limitation on the scope of protection of the present invention. All reasonable modifications and combinations included within the inventive spirit of the present invention fall within the scope of protection of the present invention.

[0058] Figure 1 and Figure 5 The flowchart and structural diagram of an embodiment of the simulation-free reliability design optimization method combining transfer learning and physical information neural networks described in this invention are shown in the figure. The method includes the following steps:

[0059] S1 defines the initial design point.

[0060] In general, the initial design point for reliability-based optimization design is the optimal solution under deterministic optimization.

[0061] S2 generates source domain configuration points.

[0062] The sample space and its distribution based on the initial design points are called the source domain, and Latin hypercube sampling is used to generate the locus of points in the source domain. These locus of points are then used to construct the physical information neural network model. Including initial condition configuration points Boundary condition configuration points and the residual collocation point of partial differential equations Simultaneously generate configuration points for source domain reliability assessment.

[0063]

[0064] This invention employs Latin hypercube sampling to generate locust points. For the time variable t and the spatial variable s, a uniform distribution is used to generate locust points. For the random variable X, the source domain D is used... S The distribution is used to generate [the data]. This method generates a total of [number] [data].

[0065] Configuration points include The residual collocation points of a partial differential equation Initial condition configuration points Boundary condition configuration points in:

[0066]

[0067] The values ​​of the initial and boundary conditions at the configuration point can be calculated and are expressed as follows: in:

[0068]

[0069] Simultaneously generate configuration points for source domain reliability assessment. This invention primarily addresses specific spatiotemporal reliability analysis and reliability-based design optimization problems. This means that for time variables t = t0 and space variables s = s0, where:

[0070]

[0071] S3 constructs a physical information neural network model for source domain reliability analysis, as shown in the model below. Figure 2 As shown.

[0072] S301 uses a deep neural network model to represent the approximate solution. The deep neural network model requires initialization, including parameters and hyperparameters such as the number of hidden layers and neurons. The approximate hidden solution using the deep neural network model can be represented as:

[0073]

[0074] S302 uses an automatic differentiation mechanism to satisfy initial conditions, boundary conditions, and partial differential equations. The loss function for violating partial differential equations, initial conditions, and boundary conditions can be expressed as follows:

[0075]

[0076] Therefore, the loss function of the physical information neural network model used for source domain reliability analysis can be expressed as:

[0077]

[0078] S303 minimizes the loss function. To calculate the parameter θ... S Including W S and b S Minimizing the loss function in equation (18) can be expressed as:

[0079]

[0080] S4 source domain reliability and sensitivity analysis, including:

[0081] S401 source domain reliability analysis. Once the network is trained, it can be used to predict partial differential equations at placement points. The solution is obtained. This yields the probability distribution function of the estimated response, used for source domain reliability analysis, where the placement points... Based on the estimated response, it is divided into two categories: failure and safety, represented as follows:

[0082]

[0083] in, This represents an approximate solution obtained from a physical information neural network model. Then, based on the number of failure samples and the placement points... The ratio of their magnitudes, used to approximate reliability, is expressed as:

[0084]

[0085] S402 source domain sensitivity analysis. To find the optimal solution, the failure probability P... F (d) Sensitivity information is necessary. The failure probability for the i-th design variable d i Find the partial derivatives:

[0086]

[0087] According to Leibniz's differential rule, the integral operator and the differential operator are interchangeable, and equation (22) can be written as:

[0088]

[0089] because Not d i The function of d. In equation (23), the logarithmic function of the joint PDF is related to d. i The partial derivative of is called d i The first-order scoring function can be expressed as:

[0090]

[0091] Therefore, equation (23) can be reformulated as:

[0092]

[0093] The sensitivity can be directly approximated based on the identified Monte Carlo failure samples as follows:

[0094]

[0095] Where, n mcs Let be the size of the Monte Carlo sample population. Equation (26) can effectively approximate the sensitivity information of the failure probability to the design variables without incurring additional computational costs. Therefore, the sensitivity in the source domain can be expressed as follows:

[0096]

[0097] S5 updated design points.

[0098] By providing the optimizer with the estimated failure probability, objective function value, and corresponding sensitivity information, the updated design point can be obtained.

[0099] S6 generates target domain configuration points.

[0100] The sample space and its distribution based on the updated design points are called the target domain, and Latin hypercube sampling is used to generate the source domain locust points. For the time variable t and the spatial variable s, a uniform distribution is used to generate the locust points. For the random variable X, the locust points are generated according to the target domain D. S The distribution is used to generate [the data]. This method generates a total of [number] [data]. Configuration points include The residual collocation points of a partial differential equation Initial condition configuration points Boundary condition configuration points in:

[0101]

[0102]

[0103] The values ​​of the initial and boundary conditions at the configuration point can be calculated and are expressed as follows: in:

[0104]

[0105] Simultaneously generate configuration points for target domain reliability assessment. in:

[0106]

[0107] S7 performs transfer learning on the physical information neural network model used for source domain reliability analysis.

[0108] S701 layer frozen

[0109] The initial values ​​of the parameters in the physical information neural network model used for target domain reliability analysis are set to the parameter values ​​in the physical information neural network model used for source domain reliability analysis. This means that the physical information neural network models used for target domain reliability analysis and source domain reliability analysis have the same architecture. Subsequently, in the training data... The model is trained by updating the parameters of the last few layers in the physical information neural network model, rather than updating the parameters of all layers. For example... Figure 3 As shown, it can be represented as:

[0110]

[0111] in, This represents the weights and biases that are allowed to be updated during the layer freezing process. The loss function that violates the partial differential equations, initial conditions, and boundary conditions can be expressed as follows:

[0112]

[0113] Therefore, the loss function of the physical information neural network model used for target domain reliability analysis can be expressed as:

[0114]

[0115] To calculate parameters include Minimizing the loss function in equation (38) can be expressed as:

[0116]

[0117] Layer freezing is implemented using a custom function. This function first uses the `model.parameters()` method to return an iterator to iterate over all the model's parameters. Then, the `requires_grad` attribute is used to set whether each parameter requires a gradient. Layer freezing, on the one hand, preserves useful physical information in the physical information neural network model built from the source domain. On the other hand, it reduces the gradients that need to be calculated to update the physical information neural network model, thus accelerating its training.

[0118] S702 fine-tuning.

[0119] Fine-tuning involves unfreezing the entire model (or a portion of the model) obtained above, and then processing it on the training data. The model is then retrained at a lower learning rate (e.g., 0.0001), such as... Figure 4 As shown, it can be represented as:

[0120]

[0121] in, Let represent the weights and biases that allow for fine-tuning, respectively. The training data for the fine-tuning process and the layer freezing process are consistent. Therefore, the loss function for violating partial differential equations, initial conditions, and boundary conditions during the fine-tuning process can also be expressed by equations (35), (36), and (37). The loss function of the physical information neural network model used for target domain reliability analysis can be expressed as:

[0122]

[0123] To calculate parameters include Minimizing the loss function in equation (41) can be expressed as:

[0124]

[0125] Typically, the number of fine-tuning iterations is a small value to prevent overfitting. Fine-tuning further improves the model's prediction accuracy.

[0126] S8 target domain reliability analysis and random sensitivity analysis.

[0127] S801 target domain reliability analysis. Once the network is trained, it can be used to predict partial differential equations at placement points. The solution is obtained. This yields the probability distribution function of the estimated response, used for reliability analysis in the target domain, where the placement points... Based on the estimated response, it is divided into two categories: failure and safety, represented as follows:

[0128]

[0129] in, This represents an approximate solution obtained from a physical information neural network model. Then, based on the number of failure samples and the placement points... The ratio of their magnitudes, used to approximate reliability, is expressed as:

[0130]

[0131] S802 Target Domain Sensitivity Analysis. Similar to sensitivity analysis in the source domain, sensitivity in the target domain can be expressed as:

[0132]

[0133] S9 checks the convergence condition. If convergence is achieved, the iteration process stops and the optimal design is output; otherwise, the iteration process returns to S5.

[0134] The method proposed in this invention achieves high-precision and high-efficiency optimization of reliability design for engineering structures. The high precision of the proposed method is attributed to two main reasons: First, the layer-freezing technique in transfer learning overcomes the generalization problem caused by statistical differences in solving the RBDO problem using a physical information neural network model. Second, based on the physical information neural network model updated using the layer-freezing technique, the model is retrained using fine-tuning techniques in transfer learning, further improving the model's prediction accuracy. The high efficiency of the proposed method is attributed to two main reasons: First, compared to directly solving the RBDO problem using a data-driven neural network model, the proposed method fully utilizes prior physical information, avoiding the use of traditional, expensive numerical simulation models. Second, compared to directly solving the RBDO problem using a physical information neural network model, the proposed method retains useful physical information from the previous physical information neural network model through the layer-freezing technique, eliminating the need to retrain the physical information neural network model in each optimization process, thus greatly reducing the training overhead of the physical information neural network model.

[0135] Example 1

[0136] Taking a cylindrical straight rod as an example, the invention will be further explained as follows: Figure 6 As shown, the cylindrical rod has a length l = 5, is fixed on the left side, and bears a load F on the right side. Young's modulus E, cross-sectional area A, and load F are considered random variables, with cross-sectional area A being a random design variable and the others being non-design variables considered as random parameters. The distribution information of all random variables is shown in Table 1.

[0137] Table 1 Distribution information of random variables in Example 1

[0138] random variable mean Standard deviation distributed F 100 10 normal distribution A <![CDATA[d1]]> 0.1 normal distribution E 206 15 normal distribution

[0139] The optimization objective is to minimize the mass of the cylindrical rod, which is equivalent to minimizing its cross-sectional area. The maximum allowable displacement of the rod is u. max =1.5. The RBDO problem is represented as:

[0140]

[0141] in, u0 = 1.5. In this embodiment, the initial point is set to d0 = 1.500, and it is assumed that the standard deviation of the random design variables remains constant. The method proposed in this invention is used to solve the RBDO problem.

[0142] In this embodiment, the sample space and its distribution based on the initial design point d0 are referred to as the source domain. To perform reliability analysis on the source domain, firstly, Latin hypercube sampling was used to generate 2100 collocation points, including 2000 partial differential equation residual collocation points and 100 boundary condition collocation points, which were used to train the physical information neural network model. This means... Simultaneously generate 10 5 One configuration point is used for reliability assessment, which means... Secondly, a physical information neural network model, denoted as SD-PINN, is constructed. The network structure contains three hidden layers, each with 50 neurons. Except for the last layer, the activation function of the remaining layers is tanh. Thirdly, the constructed physical information neural network model is used to evaluate the failure probability of the source domain. Finally, a first-order scoring function is used to perform stochastic sensitivity analysis on the failure probability. By providing the optimizer with the estimated failure probability, the objective function value, and the corresponding sensitivity information, the updated design point d1 = 1.9146 can be obtained.

[0143] The sample space and its distribution based on the updated design point d1 are referred to as the target domain. To perform reliability analysis on target domains with different distribution attributes, firstly, Latin hypercube sampling was used to generate 2100 placement points, including 2000 partial differential equation residual placement points and 100 boundary condition placement points, which were used to update the physical information neural network model. This means... Simultaneously generate 10 5 One configuration point is used for target domain reliability assessment, which means... Secondly, layer freezing and fine-tuning techniques are used to update the physical information neural network model, denoted as FL-PINN and FT-PINN respectively. In this problem, the last two layers are allowed to be updated. Furthermore, for efficiency comparison, the physical information neural network model is directly constructed using the training data of the target domain, denoted as TD-PINN. Next, the updated physical information neural network model is used to evaluate the failure probability of the target domain. Finally, a first-order scoring function is used to perform stochastic sensitivity analysis on the failure probability. By providing the optimizer with the estimated failure probability, the objective function value, and the corresponding sensitivity information, the updated design point d2 = 1.9872 can be obtained. The iterative design process will be repeated until an optimal design point d2 is obtained. opt =2.0077.

[0144] The above-described embodiments are merely some implementation methods of this application. For those skilled in the art, various modifications and improvements can be made without departing from the inventive concept of this application, and these all fall within the protection scope of this application.

Claims

1. A reliability design optimization method combining transfer learning and physical information neural networks, characterized in that, The reliability design optimization method is used to optimize the system structure, which is a cylindrical rod with a length l = 5. The left side is fixed, and the right side bears a load F. The Young's modulus E, cross-sectional area A, and load F are random variables. The cross-sectional area A is a random design variable, while the Young's modulus E and load F are non-design variables. The optimization objective is to minimize the mass of the cylindrical rod, i.e., minimize the cross-sectional area. The specific steps include: (1) Define the initial design point of the RBDO; (2) Generate source domain placement points, wherein the source domain is the sample space and its distribution based on the initial design points described in step (1), and source domain placement points are generated using Latin hypercube sampling; generate placement points for constructing the physical information neural network model. Simultaneously generate configuration points for source domain reliability assessment. (3) Construct a physical information neural network model for source domain reliability analysis; (4) Source domain reliability and sensitivity analysis: The constructed physical information neural network model is used to predict the location point. The response is analyzed, and a reliability analysis is performed on the constraints; at the same time, a first-order scoring function is used to perform a stochastic sensitivity analysis. (5) Update the design point: The updated design point is obtained by providing the optimizer with the estimated failure probability, objective function value and corresponding sensitivity information; (6) Generate target domain placement points: The target domain is a sample space and its distribution based on the updated design points. Latin hypercube sampling is used to generate target domain placement points. Generate configuration points for transfer learning Simultaneously generate configuration points for target domain reliability assessment. (7) Transfer learning is performed on the physical information neural network model used for source domain reliability analysis; the transfer learning includes first applying layer freezing, followed by fine-tuning, to update the physical information neural network model; specifically including: S701 layer frozen The initial values ​​of the parameters in the physical information neural network model used for target domain reliability analysis are set to the parameter values ​​in the physical information neural network model used for source domain reliability analysis. Subsequently, at the configuration points used for transfer learning... The above trains the physical information neural network model used for target domain reliability analysis, and updates the parameters of the last few layers in the physical information neural network model, as shown below: in, and The weights and biases that are allowed to be updated during the layer freezing process; The loss functions that violate partial differential equations, initial conditions, and boundary conditions are expressed as follows: The loss function of the physical information neural network model used for target domain reliability analysis is expressed as: To calculate parameters include and Minimizing the loss function in equation (5) is expressed as: Layer freezing is implemented using a custom function. The custom function first uses the model.parameters() method to return an iterator, which is used to traverse all the parameters of the model. Then, it uses the requires_grad attribute to set whether each parameter needs a gradient. S702 fine-tuning Fine-tuning involves unfreezing the entire model or a portion of the model obtained above, and then processing it on the training data. The model is retrained with a learning rate of 0.0001, as shown below: in, and Let represent the weights and biases that allow fine-tuning, respectively. The training data for the fine-tuning process and the layer freezing process are consistent. The loss function for violating partial differential equations, initial conditions, and boundary conditions during the fine-tuning process is expressed by equations (2), (3), and (4). The loss function of the physical information neural network model used for target domain reliability analysis is expressed as: Calculation parameters include and Minimizing the loss function in equation (8) is expressed as: (8) Target domain reliability and sensitivity analysis: Predicting placement points using an updated physical information neural network model. The response is analyzed, and a reliability analysis is performed on the constraints; at the same time, a first-order scoring function is used to perform a stochastic sensitivity analysis. (9) Check convergence condition: If the convergence condition is met, stop the iteration process and output the optimal design; otherwise, return to step (5) and repeat the iteration process until the convergence condition is met.

2. The reliability design optimization method combining transfer learning and physical information neural networks according to claim 1, characterized in that, In step (1), the initial design point of RBDO is the optimal solution under deterministic optimization.

3. The reliability design optimization method combining transfer learning and physical information neural networks according to claim 2, characterized in that, In step (2), the configuration points for constructing the physical information neural network model include initial condition configuration points. Boundary condition configuration points and the residual collocation point of partial differential equations 4. The reliability design optimization method combining transfer learning and physical information neural networks according to claim 3, characterized in that, In step (3), the process of constructing the physical information neural network model for source domain reliability analysis includes: first, initializing a deep neural network model; then, using an automatic differentiation mechanism to satisfy the initial conditions, boundary conditions, and partial differential equations; and finally, minimizing the training data. The loss of physical information.

5. The reliability design optimization method combining transfer learning and physical information neural networks according to claim 4, characterized in that, In step (6), the configuration points used for transfer learning Including initial condition configuration points Boundary condition configuration points and the residual collocation point of partial differential equations

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