Method for predicting mechanical properties of flexible structure

Abstract

The invention provides a method for predicting mechanical properties of a flexible structure, which comprises the following steps of: analyzing functional requirements of the structure, uniformly sampling in a load range of the flexible structure by adopting test design, obtaining a load set comprising a plurality of groups of load working conditions, and establishing a mechanical property data set of the flexible structure under the plurality of working conditions, constructing a coordinate feature matrix and a load feature matrix as input samples, and designing a multi-branch topological structure neural network based on physical information; introducing a stress weighting mechanism, establishing a physical-oriented composite loss function, adjusting neural network parameters based on the minimum loss function, establishing a Gaussian process regression model, predicting the residual error of the neural network model, and compensating the model to obtain a neural network prediction model after residual error compensation. According to the method, the calculation speed of solving the mechanical properties of the flexible mechanism can be effectively increased, the related data of the mechanical properties of the flexible structure can be given with relatively high precision, and the problem of low calculation precision of a traditional finite element method for the flexible mechanism is solved.

Description

Technical Field

[0001] This invention relates to a method for predicting the mechanical properties of flexible structures, belonging to the field of flexible structure mechanics technology. Background Technology

[0002] Flexible structures are widely used in key engineering fields such as aerospace, high-end equipment, and biomedicine. In complex service environments, they often endure the coupling of multiple forces and moments, accompanied by complex effects such as material nonlinearity, geometric nonlinearity, and constraint nonlinearity. Accurate assessment of their mechanical response directly affects the reliability and safety of the equipment. Currently, the engineering field commonly uses numerical simulation methods based on finite element analysis to predict mechanical properties. While this method can achieve high-precision analysis, the computational cost increases significantly when facing high-dimensional operating conditions. It not only consumes a large amount of hardware resources but also has a lengthy analysis cycle, severely restricting the efficiency of rapid review and evaluation and multi-scheme trade-off decision-making during the design phase.

[0003] To address the computational efficiency issues in high-dimensional working conditions, two main approaches have been developed, but both have significant limitations: one is model reduction methods, such as the Krylov subspace projection method. Although it can compress the model dimension by constructing a low-dimensional subspace and maintain good mechanical response reproduction in specific frequency bands, the prediction accuracy of high-frequency mechanical properties drops significantly when facing strongly nonlinear contact scenarios. The other is degree-of-freedom aggregation methods, which can achieve rapid degree-of-freedom reduction by aggregating similar dynamic properties, but are prone to accumulating errors due to modal coupling effects caused by strong nonlinearity, making it difficult to guarantee the stability of degree-of-freedom reduction. The other is traditional condensation methods, which rely on engineering experience to select the master degrees of freedom, and are prone to missing key information for complex flexible structures, resulting in poor accuracy controllability. Secondly, there are data-driven proxy model methods, such as neural networks and support vector machines. Although these methods can achieve rapid prediction by fitting the mapping relationship between input parameters and mechanical response, and have shown some practicality in some scenarios, they are highly dependent on the scale and coverage of training data. As the parameter dimension increases, the sample requirement grows exponentially, and their extrapolation ability outside the training data range is extremely poor. More importantly, they are detached from the structural physics governing equations and cannot explain the intrinsic mechanism of mechanical response. When faced with scenarios with dual nonlinear coupling of materials and geometry, they are prone to producing prediction results that violate physical laws.

[0004] Existing methods struggle to balance computational efficiency, prediction accuracy, and nonlinear adaptability, failing to meet the practical needs of predicting the mechanical properties of flexible structures under high-dimensional conditions. Therefore, there is an urgent need to design a neural network model based on physical information. By integrating the structural physics equations into the neural network construction process, this model combines the physical consistency of mechanistic modeling with the efficiency of data-driven approaches. It overcomes the data dependence bottleneck and lack of physical meaning inherent in traditional methods, while simultaneously improving adaptability to nonlinear effects, ultimately achieving efficient and high-precision prediction of the mechanical properties of flexible structures. Summary of the Invention

[0005] In view of this, the present invention provides a method for predicting the mechanical properties of flexible structures, which can effectively improve the calculation speed and accuracy of the mechanical properties of flexible structures under high-dimensional working conditions.

[0006] This invention provides a method for predicting the mechanical properties of flexible structures, comprising:

[0007] Step S1: Analyze the structural functional requirements, determine the load range of the flexible structure according to the design specifications, and use experimental design to uniformly sample within the load space to obtain a load set containing multiple load conditions.

[0008] Step S2: Based on the load sets of multiple load conditions, obtain the mechanical property dataset of the flexible machine structure under multiple load conditions. The mechanical property dataset includes the structural spatial coordinates, the loads applied, and the stresses applied.

[0009] Step S3: Based on the mechanical property dataset under multiple working conditions, construct the coordinate feature matrix and load feature matrix as input samples, and design a multi-branch topology neural network based on physical information.

[0010] Step S4 introduces a stress weighting mechanism, establishes a physics-oriented composite loss function, adjusts the neural network parameters based on minimizing the loss function, and obtains the trained physical information neural network model.

[0011] Step S5: Build a Gaussian process regression model, predict the residuals of the neural network model, and compensate the model to obtain the neural network prediction model after residual compensation.

[0012] In the above method, step S1 includes:

[0013] Analyze the structural functional requirements, set the load vector as a multidimensional variable containing multiple components of force and moment, and determine the value range and coupling constraints of each component according to the design specifications.

[0014] An experimental design method based on a greedy algorithm is used to uniformly sample in the load space and remove samples that do not meet the constraints or are highly redundant.

[0015] The sampling results are subjected to boundary coverage and correlation tests and evaluations, and the load set is stored in the form of a data file to obtain a load set containing multiple load conditions.

[0016] In the above method, step S2 includes:

[0017] Based on the flexible structure design, spatial geometric parameters are determined and material properties are assigned to the structure. Adaptive meshes are divided into regions to generate a finite element model of the flexible connection structure.

[0018] Boundary conditions are set at the nodes at both ends of the structure, with one end fixed and the selected load applied to a local rigid region at the other end.

[0019] Based on the load set of multiple load conditions, the boundary condition parameters are dynamically updated, and nonlinear mechanical analysis is performed to obtain a dataset of mechanical properties of flexible structures under multiple load conditions, including spatial coordinates, loads, and stresses.

[0020] In the above method, step S3 includes:

[0021] Transform spatial coordinates to generate feature vectors, forming a coordinate feature matrix. The feature vectors include basic features, first-order Fourier features, and interactive features.

[0022] The mean and standard deviation of the six-dimensional load are calculated based on the load set of multiple load conditions. The data in the mechanical property dataset under multiple load conditions are normalized to form a load feature matrix.

[0023] Using the coordinate feature matrix and load feature matrix as input, a multi-branch topology neural network is designed, in which the backbone network is used to extract the spatial features of the flexible structure, and the three output heads predict the quadratic coefficient, linear coefficient and constant term of the stress function, respectively.

[0024] In the above method, step S4 includes:

[0025] Different weights are assigned to the prediction errors of different stress regions, and a composite loss function is constructed using the weighted mean square error data term, physical constraint term, and regularization term.

[0026] By combining learning rate scheduling and early stopping mechanism to iteratively optimize the loss function, and using an adaptive optimization algorithm to adjust the network weights for training, a trained physical information neural network model is obtained.

[0027] In the above method, step S5 includes:

[0028] The residuals are calculated within the normalization domain, and a residual input vector is constructed. The residual input vector includes geometric features, normalized loads, and their physical interaction terms.

[0029] Cross-validation was used to determine the kernel function parameters, and a Gaussian process regression model was established. The mean of its residuals was linearly superimposed with the output of the neural network to obtain the neural network prediction model after residual compensation. Attached Figure Description

[0030] To more clearly illustrate the technical solutions of the embodiments of the present invention, the drawings used in the embodiments will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without any creative effort or labor.

[0031] Figure 1 This is a flowchart illustrating a method for predicting the mechanical properties of a flexible structure provided in an embodiment of the present invention.

[0032] Figure 2 This is a schematic diagram of a neural network for rapidly predicting the mechanical properties of flexible structures based on physical information, as described in an embodiment of the present invention.

[0033] Figure 3 This is a schematic diagram illustrating the fitting degree of the neural network based on physical information to predict the mechanical properties of flexible structures in an embodiment of the present invention. Specific Implementation

[0034] To better understand the technical solution of the present invention, the embodiments of the present invention will be described in detail below with reference to the accompanying drawings.

[0035] It should be understood that the described embodiments are merely some, not all, of the embodiments of the present invention. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention without inventive effort are within the scope of protection of the present invention.

[0036] This invention provides a method for predicting the mechanical properties of flexible structures, such as... Figure 1 As shown, this is a flowchart illustrating the method for predicting the mechanical properties of a flexible structure provided in an embodiment of the present invention. The method includes the following steps:

[0037] Step 101: Analyze the structural functional requirements, determine the load range of the structure according to the design specifications, and use experimental design to uniformly sample within the load space to obtain a load set containing multiple load conditions.

[0038] (1.1) Analyze the structural functional requirements, set the load vector as a multidimensional variable containing multiple components of force and moment, and determine the value range and coupling constraints of each component according to the design specifications, specifically including:

[0039] To formalize the load description, the six-component load vector is denoted as...

[0040]

[0041] in For load vectors; Forces in three directions; These are the torques in the X, Y, and Z directions; This indicates transpose.

[0042] Based on design specifications, fatigue conditions, and service scenarios, the closed intervals of each component are determined. and coupling constraints Forming a feasible region:

[0043]

[0044] in Let be the feasible region in the load space, where all elements satisfy component boundaries and all coupling constraints. for The magnitude of the load component, ; for The minimum value of the load component; for The maximum value of the component load.

[0045] (1.2) An experimental design method based on a greedy algorithm is used to uniformly sample the load space and remove samples that do not meet the constraints or are highly redundant. Specifically, this includes:

[0046] Generate initial Latin hypercube samples:

[0047]

[0048] in This is the initial set of feasible samples; For the first Six-dimensional loads under various working conditions; This represents the initial number of samples.

[0049] To improve spatial coverage, a greedy augmentation method based on the maximin criterion is adopted: defining the distance between samples. Iterative selection in the candidate set

[0050]

[0051] in For candidate sets; This is a diagonal matrix of weights for each dimension; It is the Euclidean norm; This is the new sample with the largest minimum neighbor distance in the current set.

[0052] (1.3) Perform boundary coverage and correlation tests and evaluations on the sampling results, and store the load set in the form of a data file to obtain a load set containing multiple load conditions, specifically including:

[0053]

[0054] in For the final inclusion The load set of the working conditions.

[0055] Step 102: Based on the load set of multiple load conditions, obtain the mechanical property dataset of the flexible machine structure under multiple conditions. The mechanical property dataset includes the structural spatial coordinates, the loads and stresses.

[0056] (2.1) Based on the flexible structure design, determine its spatial geometric parameters and assign material properties to the structure, divide the region into adaptive meshes, and generate a finite element model of the flexible connection structure;

[0057] (2.2) Set boundary conditions at the nodes at both ends of the structure, with one end fixed and a local rigid region established at the other end to apply the selected boundary conditions. Working load of each working condition Load vector Each condition corresponds to a specific working condition;

[0058] (2.3) Based on the load set of multiple load conditions, the boundary condition parameters are dynamically updated, and nonlinear mechanical analysis is performed to obtain the mechanical property dataset of the flexible structure under multiple load conditions, including spatial coordinates, applied loads, and applied stresses, specifically including:

[0059] MATLAB calls APDL loop traversal The node coordinates are obtained by performing a nonlinear solution. With equivalent stress For each set of working condition nodes Perform quality control and unit consistency; then implement sampling integration at the node level according to the following steps:

[0060] (2.3.1) First, constrain the sample size using the total number of nodes and the upper limit of the project to determine the target sampling size for a single working condition:

[0061]

[0062] in For working condition nodes The number of nodes; This represents the sampling ratio; This represents the upper limit of the single-condition sample size. This indicates rounding up to the nearest integer.

[0063] (2.3.2) To prioritize coverage of potential risk areas, high-stress nodes are first screened based on stress amplitude:

[0064] ,

[0065] in For nodal stress; ; It is the stability constant; To normalize the stress, It is a high-stress set; This is the quantile threshold.

[0066] (2.3.3) To take into account regions with drastic changes, the local gradient is approximated by nearest neighbor difference, and nodes with high gradients are selected:

[0067] , ,

[0068] in For nodes of Near-collar assembly; The distance is Euclidean. For local gradient intensity, , ; Its normalized value; , They are respectively the first / The spatial coordinate vector of each node; It is a set of high gradients; This is the quantile threshold; , It is the stability constant.

[0069] (2.3.4) To balance risk and full coverage within a limited budget, a seed set is first constructed, and then points are added using the spatial uniformity criterion:

[0070] , ,

[0071] in The seed set is selected according to joint priority; its size does not exceed [number]. ; This represents the percentage of seeds. This represents the number of samples that still need to be added. For the candidate set; Used as the base number.

[0072] (2.3.5) Then, to maximize the minimum distance, uniform spatial point filling is achieved, and this process is repeated:

[0073] , , (repeat Second-rate)

[0074] in The currently selected collection; To The candidate point that is furthest away.

[0075] (2.3.6) Finally, a dataset of mechanical properties of flexible structures under multiple working conditions is formed, including the sample set of a single working condition:

[0076]

[0077] (2.3.7) Summarize all working conditions to obtain a dataset of mechanical properties of flexible structures under multiple working conditions:

[0078]

[0079] in The coordinates of the nodes; The equivalent stress at the corresponding node; For the first Each operating condition load; The final set of node indices ( ).

[0080] Step 103: Based on the mechanical property dataset under multiple working conditions, construct the coordinate feature matrix and load feature matrix as input samples, and design a multi-branch topology neural network based on physical information.

[0081] (3.1) Transform spatial coordinates to generate feature vectors, forming a coordinate feature matrix. The feature vectors include basic features, first-order Fourier features, and interaction features, specifically including:

[0082] Will Mapped to cylindrical coordinates Generate feature vectors

[0083]

[0084] in Radial distance; Circumferential angle; Z-axis coordinate; ∈ represents the geometric eigenvector.

[0085] To avoid training instability caused by different units of measurement, the first... Feature vectors of each node Standardize the process:

[0086]

[0087] in The mean / standard deviation of the geometric features on the training set; "" indicates the standardized quantity.

[0088] Operating conditions have If there are selected nodes, the coordinate matrix is:

[0089]

[0090] in For the first The transpose of the eigenvectors of each node.

[0091] (3.2) Calculate the six-dimensional load mean based on the load set of multiple load conditions with standard deviation Loads in the mechanical property data set under multiple working conditions Normalization is performed:

[0092]

[0093] in The mean / standard deviation of the load components under operating conditions; "" indicates the standardized quantity.

[0094] As can be seen from (2.3), the working condition is... have Selected nodes, total number of samples Then the load matrix is:

[0095]

[0096] in For length is A vector of all 1s is used to copy the same load condition to all sample points of that load condition.

[0097] (3.3) Using the coordinate feature matrix and load feature matrix as input, design a multi-branch topology neural network, the specific structure of which is as follows: Figure 2 As shown. The backbone network is used to extract the spatial features of the flexible structure, and the three output heads predict the quadratic coefficients, linear coefficients, and constant term of the stress function, respectively, including:

[0098] The main trunk extracts representations from coordinate features:

[0099]

[0100] in The first Layer weights and biases; For activation functions; The hidden representation of layers 1-4 of the backbone network; To ultimately hide the representation, the current Equivalent to .

[0101] The three output heads corresponding to the parameters of "quadratic term", "linear term", and "constant term" are respectively

[0102]

[0103] To ensure the (semi)positive definiteness of the quadratic term Q, the following formula is used:

[0104] , ,

[0105]

[0106] Where softplus is the activation function; The "secondary head" hidden variable output by the backbone network; This is the stability constant term.

[0107] The prediction scalar for the standardized domain objective is then:

[0108] ,

[0109] in For the prediction scalar of the standard domain objective; It is the bias function; These are the location-dependent linear coefficients; It is a positive definite / semi-positive definite quadratic coefficient matrix; is the numerical stability constant.

[0110] Considering the non-negativity and long-tailed distribution of stress, a monotonically invertible logarithmic transformation is performed on the target:

[0111]

[0112]

[0113] in Original stress; For the dataset The mean and standard deviation; Physical domain prediction.

[0114] Step 104: Introduce a stress-weighted mechanism, establish a physics-oriented composite loss function, adjust the neural network parameters based on minimizing the loss function, and obtain the trained physical information neural network model.

[0115] (4.1) Different weights are assigned to the prediction errors of different stress regions, and a composite loss function is constructed using the weighted mean square error data term, physical constraint term, and regularization term, specifically including:

[0116] For the first For each sample, let the standardized error be... Define smoothing weights based on high quantile thresholds.

[0117] ,

[0118] in For the first The weights of each sample; Control the weight strength; The high quantile threshold for the training set; This is the Sigmoid function.

[0119] Define data items as The physical constraint terms include the residuals of the governing equations. Boundary condition deviation Regularization Therefore, the total loss is:

[0120]

[0121]

[0122] in The number of samples; From network parameters The generated field quantity; External force; For the physical residual evaluation point set, As the base; For the corresponding operators of the governing equations; For boundary operators; Give values ​​to the boundaries; For the set of boundary points; , , Weights for each item; for Layer weight matrix; It is the Frobenius norm; All trainable parameters of the network; These are the optimal parameters.

[0123] (4.2) Combining learning rate scheduling and early stopping mechanism, the loss function is iteratively optimized, and an adaptive optimization algorithm is used to adjust the network weights for training, resulting in a trained physical information neural network model, specifically including:

[0124] The physical domain uses access indicators for monitoring and early shutdown:

[0125]

[0126] in , These are the actual and predicted stresses, respectively. The true mean; RMSE / MAE / R 2 Each measure error scale, absolute deviation, and explanatory power.

[0127] Train the following model based on loss function minimization and pass rate:

[0128]

[0129] Decoded to the physical domain

[0130]

[0131] in pass It is trained by an adaptive optimizer and determined by early stopping.

[0132] Step 105: Build a Gaussian process regression model, predict the residuals of the neural network model, and compensate the model to obtain the neural network prediction model after residual compensation.

[0133] (5.1) Calculate the residuals within the normalized domain and construct the residual input vector. The residual input vector includes geometric features, normalized loads, and their physical interaction terms, specifically including:

[0134] Calculate residuals in the normalization domain Construct the input vector:

[0135]

[0136] in For the first Standardized residuals of each sample; This is the input vector for the Gaussian model; Geometric features; For standardized loads; except All others are physical interaction items.

[0137] (5.2) Cross-validation is used to determine the kernel function parameters, a Gaussian process regression model is established, and the residual mean is linearly superimposed with the neural network output to obtain the neural network prediction model after residual compensation, specifically including:

[0138] To ensure adaptive correlation across all dimensions, the ARD–Matérn–5 / 2 kernel is selected:

[0139] ,

[0140] in For the nuclear amplitude; It is an anisotropic length-scaled diagonal matrix; selected through cross-validation. ; For input dimensions; To be according to Scaled distance; , The residual input vectors are at different points.

[0141] The residual mean of the Gaussian process and the neural network output are processed in the same standardized domain:

[0142]

[0143]

[0144] in The kernel matrix of the training samples; This represents the kernel vector between the test points and the training samples; For noise variance; For unit array; To train the residual vector; and denoted as the posterior mean and variance of the test points, respectively.

[0145] Will The output of the neural network is linearly superimposed on the output of the neural network in the normalization domain:

[0146]

[0147] in For the compensated standardized domain prediction; This is the original prediction from the neural network.

[0148] Decoding back to the physical domain, we solve for the stress in the neural network prediction model after residual compensation:

[0149]

[0150] in , The mean and standard deviation of the dataset; This is for predicting the physical domain stress after residual compensation.

[0151] To verify the effectiveness of the method of this invention, a validation set of 17,547 samples was used as the validation object. The stress results predicted by the neural network under nine-dimensional working conditions were compared with the finite element analysis results. Figure 3As shown, compared to traditional methods that take several hours to predict, the mechanical property prediction method for flexible structures provided by this invention takes only seconds for a single prediction, indicating that the neural network has a faster prediction speed. Stress linear regression determination coefficient R0 2 The value is 0.9065, R 2 The slopes of both the regression and regression lines are close to 1, indicating a high degree of consistency between the neural network predictions and the finite element simulation results. The relative error distribution also satisfies... The principle, that is, the relative error distribution is in The probability of success is 0.6826, and the relative error distribution is as follows: The probability of success is 0.9544, and the relative error distribution is as follows: The probability of the result being correct is 0.9974, indicating that the prediction results of the neural network are relatively accurate.

[0152] According to the embodiments of the present invention, a method for predicting the mechanical properties of flexible structures can be implemented to quickly solve the mechanical properties of flexible structures. By using experimental design to uniformly sample within the load space, a dataset of the mechanical properties of the structure under multiple load conditions is obtained. Based on this, physical interaction feature terms are constructed, a stress weighting mechanism is introduced, and a physics-guided composite loss function is established to train a multi-branch topological neural network model based on physical information. Simultaneously, a Gaussian process regression model is built to predict residuals to compensate for the neural network model, thereby quickly obtaining relevant data on the mechanical properties of flexible structures. This invention effectively balances efficiency and accuracy in solving the mechanical properties of flexible structures under high-dimensional conditions, providing high-precision and fast data on the mechanical properties of flexible structures in a short time, thus overcoming the problem that traditional finite element methods cannot simultaneously balance computational speed and accuracy when dealing with flexible structures under multi-dimensional conditions.

[0153] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

[0154] The contents not described in detail in this specification are common knowledge to those skilled in the art.

Claims

1. A method for predicting the mechanical properties of flexible structures, characterized in that, The method includes: Step S1: Analyze the structural functional requirements, determine the load range of the flexible structure according to the design specifications, and use experimental design to uniformly sample within the load space to obtain a load set containing multiple load conditions. Step S2: Based on the load sets of multiple load conditions, obtain the mechanical property dataset of the flexible machine structure under multiple load conditions. The mechanical property dataset includes the structural spatial coordinates, the loads applied, and the stresses applied. Step S3: Based on the mechanical property dataset under multiple working conditions, construct the coordinate feature matrix and load feature matrix as input samples, and design a multi-branch topology neural network based on physical information. Step S4 introduces a stress weighting mechanism, establishes a physics-oriented composite loss function, adjusts the neural network parameters based on minimizing the loss function, and obtains the trained physical information neural network model. Step S5: Build a Gaussian process regression model, predict the residuals of the neural network model, and compensate the model to obtain the neural network prediction model after residual compensation.

2. The method according to claim 1, characterized in that, Step S1 includes: Analyze the structural functional requirements, set the load vector as a multidimensional variable containing multiple components of force and moment, and determine the value range and coupling constraints of each component according to the design specifications. An experimental design method based on a greedy algorithm is used to uniformly sample in the load space and remove samples that do not meet the constraints or are highly redundant. The sampling results are subjected to boundary coverage and correlation tests and evaluations, and the load set is stored in the form of a data file to obtain a load set containing multiple load conditions.

3. The method according to claim 1, characterized in that, Step S2 includes: Based on the flexible structure design, spatial geometric parameters are determined and material properties are assigned to the structure. Adaptive meshes are divided into regions to generate a finite element model of the flexible connection structure. Boundary conditions are set at the nodes at both ends of the structure, with one end fixed and the selected load applied to a local rigid region at the other end. Based on the load set of multiple load conditions, the boundary condition parameters are dynamically updated, and nonlinear mechanical analysis is performed to obtain a dataset of mechanical properties of flexible structures under multiple load conditions, including spatial coordinates, loads, and stresses.

4. The method according to claim 1, characterized in that, Step S3 includes: Transform spatial coordinates to generate feature vectors, forming a coordinate feature matrix. The feature vectors include basic features, first-order Fourier features, and interactive features. The mean and standard deviation of the six-dimensional load are calculated based on the load set of multiple load conditions. The data in the mechanical property dataset under multiple load conditions are normalized to form a load feature matrix. Using the coordinate feature matrix and load feature matrix as input, a multi-branch topology neural network is designed, in which the backbone network is used to extract the spatial features of the flexible structure, and the three output heads predict the quadratic coefficient, linear coefficient and constant term of the stress function, respectively.

5. The method according to claim 1, characterized in that, Step S4 includes: Different weights are assigned to the prediction errors of different stress regions, and a composite loss function is constructed using the weighted mean square error data term, physical constraint term, and regularization term. By combining learning rate scheduling and early stopping mechanism to iteratively optimize the loss function, and using an adaptive optimization algorithm to adjust the network weights for training, a trained physical information neural network model is obtained.

6. The method according to claim 1, characterized in that, Step S5 includes: The residuals are calculated within the normalization domain, and a residual input vector is constructed. The residual input vector includes geometric features, normalized loads, and their physical interaction terms. Cross-validation was used to determine the kernel function parameters, and a Gaussian process regression model was established. The mean of its residuals was linearly superimposed with the output of the neural network to obtain the neural network prediction model after residual compensation.

Images