Full-field vibration reconstruction method and system based on sparse sampling and physical depth operator

Abstract

The application provides a full-field vibration reconstruction method and system based on sparse sampling and a physical depth operator, and belongs to the technical field of structural health monitoring, and comprises the following steps: constructing a full-field vibration displacement snapshot matrix; constructing a rigid body enhanced modal base containing overall translation, rigid body tilt and low-order elastic modal, and taking a sparse measurement point observation matrix as an object, and screening sparse measurement point coordinates in combination with an observability index; measuring sparse displacement data based on the sparse measurement point coordinates, inputting the sparse displacement data into a pre-trained physical information depth operator network after dimensionless, and obtaining a structure full-field vibration field through forward propagation calculation of the physical information depth operator network; wherein, the physical information depth operator network introduces constraints of measurement point confidence weighted data consistency, thin plate bending strain energy, boundary condition residual and curvature adaptive sampling in training. The application improves the stability and physical consistency of the reconstruction result.

Description

Technical Field

[0001] This invention belongs to the field of structural health monitoring technology, and in particular relates to a full-field vibration reconstruction method and system based on sparse sampling and physical depth operators. Background Technology

[0002] The statements in this section are merely background information related to the present invention and do not necessarily constitute prior art.

[0003] Typical engineering structures such as thin-walled structures of aerospace vehicles, plate and shell equipment components, and chemical pipeline systems are subjected to complex environmental loads, alternating dynamic loads, fluid pulsation excitations, and temperature and pressure fluctuations during service. This can cause vibration anomalies in critical structures. Therefore, it is necessary to conduct rapid and high-precision monitoring of the full-field vibration field of the above-mentioned critical structures to support structural health assessment, fault early warning, and digital twin model updates.

[0004] To overcome the mass load and on-site deployment limitations of contact measurements, non-contact scanning laser Doppler vibrometers are used in full-field vibration monitoring. However, when performing high-resolution measurements, this equipment requires point-by-point scanning, which is time-consuming and cannot meet the needs of rapid or near-real-time monitoring. Furthermore, it suffers from optical obstruction in complex environments. In recent years, field reconstruction methods based on deep learning have become an important direction for solving this problem. However, problems such as the lack of a unified representation of the low-frequency vibration characteristics of real structures, invalid or redundant measurement point acquisition, and insufficient training stability and prediction accuracy still exist, leading to reconstruction instability and physical distortion. Summary of the Invention

[0005] To overcome the shortcomings of the existing technologies, this invention proposes a full-field vibration reconstruction method and system based on sparse sampling and physical depth operators. This addresses the issues in vibration monitoring of thin-walled structures of aerospace vehicles, plate and shell equipment components, and chemical pipeline systems. These issues include the tendency of densely placed contact sensors to generate mass load effects, the high time cost of full-field point-by-point scanning with scanning laser Doppler vibrometers, and the instability and physical distortion caused by existing pure data-driven reconstruction methods, which suffer from problems such as lack of physical observability constraints on measurement point placement, insufficient generalization ability for unknown working conditions, susceptibility to non-physical distortion, and incomplete evaluation indicators. This invention aims to achieve rapid, high-precision, and physically consistent full-field vibration reconstruction with a limited number of laser measurement points.

[0006] To achieve the above objectives, one or more embodiments of the present invention provide the following technical solutions: In a first aspect, this invention discloses a full-field vibration reconstruction method based on sparse sampling and physical depth operators, comprising: The dominant natural modes of the structure are extracted by finite element simulation, and modal superposition calculation is performed by combining broadband random excitation to construct a full-field vibration displacement snapshot matrix; Based on the full-field vibration displacement snapshot matrix, a rigid body enhanced mode basis including global translation, rigid body tilt and low-order elastic modes is constructed, and the sparse measurement point observation matrix is ​​used as the object, and the coordinates of sparse measurement points are screened by combining observability index. Based on the sparse measurement point coordinates, sparse displacement data is measured. After the sparse displacement data is dimensionless, it is input into a pre-trained physical information deep operator network. The whole field vibration field of the structure is obtained through forward propagation calculation of the physical information deep operator network. The physical information deep operator network incorporates constraints such as measurement point confidence weighted data consistency, thin plate bending strain energy, boundary condition residuals, and curvature adaptive sampling during training.

[0007] Secondly, this invention discloses a full-field vibration reconstruction system based on sparse sampling and physical depth operators, comprising: The sample construction module is configured to: extract the dominant natural modes of the structure through finite element simulation, and perform modal superposition calculations in combination with broadband random excitation to construct a full-field vibration displacement snapshot matrix; The measuring point layout module is configured to: construct a rigid body enhanced mode basis containing overall translation, rigid body tilt and low-order elastic modes based on the full-field vibration displacement snapshot matrix, and select sparse measuring point coordinates by taking the sparse measuring point observation matrix as the object and combining observability index. The full-field reconstruction module is configured to: measure sparse displacement data based on the coordinates of the sparse measuring points, input the dimensionless sparse displacement data into a pre-trained physical information deep operator network, and obtain the full-field vibration field of the structure through forward propagation calculation of the physical information deep operator network. The physical information deep operator network incorporates constraints such as measurement point confidence weighted data consistency, thin plate bending strain energy, boundary condition residuals, and curvature adaptive sampling during training.

[0008] Thirdly, the present invention discloses an electronic device, including a memory and a processor, and computer instructions stored in the memory and running on the processor, wherein the computer instructions, when run by the processor, complete the steps of the above-mentioned full-field vibration reconstruction method based on sparse sampling and physical depth operators.

[0009] Fourthly, the present invention discloses a computer-readable storage medium for storing computer instructions, which, when executed by a processor, complete the steps of the above-mentioned full-field vibration reconstruction method based on sparse sampling and physical depth operators.

[0010] Compared with the prior art, the beneficial effects of the present invention are as follows: This invention provides a full-field vibration field reconstruction method based on modal observable sparse sampling and a physical information deep operator network. On one hand, a rigid-body reinforced modal basis is constructed, and sparse measurement points are placed by combining observability indices such as the rank of the observation matrix, condition number, minimum singular value, and modal noise amplification factor. This guides a scanning laser Doppler vibrometer to perform discrete measurements on only a small number of key points, thereby reducing scanning time and improving the ability to acquire key vibration information. On the other hand, a physical information deep operator network is constructed. Feature scale control, measurement point confidence, thin plate bending strain energy, boundary conditions, and curvature adaptive constraints are introduced into the network structure and loss function. A physical consistency evaluation index system is also established. This method can improve the accuracy of full-field vibration field reconstruction in environments with few measurement points and noise, suppress non-physical distortions under unknown working conditions, and improve the stability and physical consistency of the reconstruction results.

[0011] Advantages of additional aspects of the invention will be set forth in part in the description which follows, and in part will be obvious from the description, or may be learned by practice of the invention. Attached Figure Description

[0012] The accompanying drawings, which form part of this invention, are used to provide a further understanding of the invention. The illustrative embodiments of the invention and their descriptions are used to explain the invention and do not constitute an improper limitation of the invention.

[0013] Figure 1 This is a flowchart of the full-field vibration reconstruction method based on sparse sampling and physical depth operators as described in Embodiment 1 of the present invention.

[0014] Figure 2 This is a schematic diagram of the optimal sensor measurement points for modal observable sparse sampling as described in Embodiment 1 of the present invention.

[0015] Figure 3 This is a diagram of the overall architecture of the deep operator network described in Embodiment 1 of the present invention.

[0016] Figure 4 This is a schematic diagram of the micro-level architecture and physical constraint loop described in Embodiment 1 of the present invention.

[0017] Figure 5 This is a schematic diagram of the reconfiguration performance of the distributed external operating conditions described in Embodiment 1 of the present invention.

[0018] Figure 6 This is a graph showing the reconfiguration performance and dual-index evaluation of the distributed external operating conditions described in Embodiment 1 of the present invention.

[0019] Figure 7 This is a comparison chart of reconstruction errors under different measurement point placement strategies described in Embodiment 1 of the present invention. Detailed Implementation

[0020] It should be noted that the following detailed descriptions are exemplary and intended to provide further illustration of the invention. Unless otherwise specified, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this invention pertains.

[0021] It should be noted that the terminology used herein is for the purpose of describing particular implementations only and is not intended to limit the exemplary implementations of the present invention.

[0022] Where there is no conflict, the embodiments and features in the embodiments of the present invention can be combined with each other.

[0023] Thin-walled structures in aerospace vehicles, plate and shell equipment components, and chemical piping systems are typical engineering structures that endure long-term, complex environmental loads, alternating dynamic loads, fluid pulsation excitation, and temperature and pressure fluctuations during service. Critical areas such as thin plates, shells, pipe elbows, flange connection areas, and near supports and hangers are prone to problems such as localized vibration amplification, fatigue accumulation, loosening of connections, and even crack propagation. Especially for chemical piping systems, abnormal vibration can further induce sealing failures, leakage risks, and operational safety hazards. Therefore, rapid and high-precision monitoring of the full-field vibration field of these critical structures, and using this data to support structural health assessments, fault early warning, and digital twin model updates, is of significant engineering importance for ensuring the safe operation of high-end equipment and chemical process systems.

[0024] Traditional full-field vibration monitoring technology still faces significant bottlenecks in practical applications. First, online finite element simulation requires substantial computation, making it difficult to meet the rapid response requirements of digital twins and on-site monitoring. Second, densely arranging contact accelerometers on lightweight, thin-walled structures or the outer walls of chemical pipelines not only complicates construction but may also introduce localized mass load effects, altering the original dynamic characteristics of the measured structure and leading to distorted measurement results. Furthermore, the installation, maintenance, and long-term reliability of contact sensors are further limited in high-temperature, high-pressure, corrosive, or space-constrained chemical pipeline environments.

[0025] To overcome the mass load and on-site deployment limitations of contact measurements, non-contact scanning laser Doppler vibration meters have significant application value in vibration detection of thin-walled structures, rotating machinery components, and pipelines. However, when performing high-resolution full-field vibration measurements, this equipment typically requires controlling the laser beam for dense point-by-point spatial scanning, resulting in long measurement times and making it difficult to meet the needs of rapid on-site detection or near-real-time monitoring. Furthermore, complex equipment compartments and chemical pipeline environments may present problems such as optical obstruction, unstable reflection conditions, and limited measurable areas. Therefore, how to obtain key vibration information using only a small number of sparse laser measurement points and rapidly reconstruct the full-field vibration field of the structure based on this information is a pressing technical problem to be solved in the condition monitoring of aerospace structures, thin-walled equipment components, and chemical pipelines.

[0026] In recent years, field reconstruction methods based on deep learning have gradually become an important direction for solving the above problems. These methods construct neural networks to learn a high-dimensional nonlinear mapping between sparse measurement point responses and the entire vibration field. However, in practical engineering applications, existing purely data-driven models still suffer from theoretical and application shortcomings, such as a lack of physical observability constraints on measurement point placement, insufficient mechanical priors in the network training process, and difficulty in reflecting local physical consistency and blind zone errors in the evaluation of reconstruction results. These shortcomings manifest in the following three aspects: First, existing methods for measuring vibrations with few measurement points often rely on empirical, uniform, random, or purely mathematical decomposition methods for point placement, lacking a unified characterization of the base motion, rigid body tilt, and low-order elastic mode coupling characteristics in the low-frequency vibrations of real structures. Furthermore, existing methods typically fail to comprehensively evaluate the effectiveness of measurement point placement based on factors such as the rank of the observation matrix, condition number, minimum singular value, and noise amplification characteristics. This can lead to measurement points potentially falling into nodal regions, low-response regions, or acquiring redundant information, thus limiting modal observability and reconstruction stability under conditions of very few measurement points.

[0027] Second, existing deep learning-based full-field vibration reconstruction methods typically learn the mapping relationship from sparse measurement point data to the full-field displacement field directly. When the dimensionality of the network's latent features is high, the direct inner product fusion of the branch network and the backbone network can easily cause output scale fluctuations or feature variance shifts, affecting training stability and amplitude prediction accuracy. Furthermore, existing models often employ a single data fitting loss, failing to fully integrate measurement point confidence, thin plate bending strain energy, boundary conditions, curvature distribution, and multi-loss weight coordination mechanisms. This leads to non-physical distortions in models operating under conditions with very few measurement points, noisy measurements, and unknown disturbances.

[0028] Third, existing full-field vibration reconstruction methods often use single numerical error indicators such as mean square error and relative error for performance evaluation. These mainly reflect the overall amplitude deviation and are insufficient to characterize the rationality of the predicted displacement field in terms of local curvature, bending energy distribution, and mechanical consistency. Furthermore, existing evaluation methods typically do not separately examine the reconstruction error in blind zones far from sparse measurement points, making it difficult to promptly reveal local non-physical fluctuations in areas without measurement point coverage. This limits the accuracy of model performance comparison, parameter optimization, and reliability assessment for engineering applications.

[0029] In summary, this invention proposes a full-field vibration field reconstruction method that can accurately decouple modes to guide the optimal dimensionality reduction point selection of scanning laser Doppler vibration meters, and deeply integrate structural elasticity and mechanical physical constraints, thereby maintaining high robustness and high accuracy even with very few laser measurement points and unknown extreme conditions.

[0030] Example 1 In one or more embodiments, a full-field vibration reconstruction method based on sparse sampling and physical depth operators is disclosed. It mainly addresses the problems in vibration monitoring of aerospace thin-walled structures, plate and shell equipment components, and chemical pipeline systems, such as the limitation of the number of contact sensors deployed due to mass load effects, the high time cost of full-field point-by-point scanning with scanning laser Doppler vibrometers, and the insufficient generalization ability and easy generation of non-physical distortion of traditional pure data-driven models under conditions with few measurement points, noise, and unknown working conditions. It can be applied to digital twin and structural health monitoring technology of equipment structures, aerospace thin-walled structures, and chemical pipeline systems based on scanning laser Doppler vibrometers.

[0031] In this embodiment, the whole-field vibration field is mainly reconstructed based on the whole-field vibration displacement field of the structure; for thin plate and shell components, the whole-field vibration field can be the transverse vibration displacement field of the structural surface; for pipe or cylindrical shell structure, the whole-field vibration field can be the normal or radial vibration displacement field of the structural surface.

[0032] This method leverages the non-contact measurement characteristics of scanning laser Doppler vibrometers. It determines a small number of key measurement points through modal observability constraints and controls the laser beam for directional discrete measurements, replacing traditional full-field point-by-point scanning. The overall implementation process includes multi-condition data generation and dimensionless transformation, placement of sparse modal observable measurement points, construction of a physical information deep operator network, data-driven pre-training, and joint fine-tuning based on physical constraints. Figure 1 As shown, it includes the following steps: Step S1: Extract the dominant natural modes of the structure through finite element simulation, and perform modal superposition calculations by combining broadband random excitation to construct a full-field vibration displacement snapshot matrix. The full-field vibration displacement snapshot matrix It is used to store full-field discrete displacement response samples under different time steps or different excitation conditions; when training the physical information deep operator network, the sparse measurement point observation matrix is ​​extracted by sparse measurement point coordinates. And the corresponding discrete displacement vector of the whole field is used as the true label of the whole field. .

[0033] Step S101: Extract the dominant natural modes of the structure through finite element simulation. Specifically: establish a finite element model based on the geometric dimensions, material parameters and boundary connection conditions of the structure to be tested, perform modal analysis on the finite element model, and solve for the natural frequencies and corresponding natural mode shapes of the structure; select the first q dominant natural modes as the basis for subsequent modal superposition and reconstruction of the basis structure in order of natural frequencies from low to high, and in combination with the modal participation factor or modal effective mass contribution.

[0034] Step S102: The dominant natural modes of the structure are combined with broadband random excitation to perform modal superposition calculations and construct a full-field vibration displacement snapshot matrix.

[0035] The true physical full-field vibration field of the structure under broadband random excitation It can be generated by the linear superposition of the first q dominant modes: (1) Where q is the order of the dominant intrinsic mode participating in the modal superposition; The random mode coefficients of the i-th order mode at time t, and these random mode coefficients have the characteristic of decaying as the frequency increases; Let i be the i-th order intrinsic mode function. These are the physical dimensions of the thin-plate structure along the x and y directions. This represents the actual physical full-field vibration field of the structure.

[0036] The actual physical full-field vibration field of the structure Sampling is performed at discrete nodes across the entire field. For the m-th time step or the m-th excitation condition, the discrete displacement vector across the entire field is expressed as: (2) Where N is the number of discrete nodes in the entire field; Let be the discrete displacement vector of the whole field corresponding to the m-th sample.

[0037] Arrange the discrete displacement vectors of the whole field under m time steps or m excitation conditions in columns to obtain the whole field vibration displacement snapshot matrix: (3) in, is the full-field vibration displacement snapshot matrix, where m is the m-th snapshot sample.

[0038] (4) The full-field vibration displacement snapshot matrix constructed in the above manner is used for subsequent evaluation of the reconstructed basis representation ability, screening of sparse measurement points, and construction of training samples for physical information deep operator networks.

[0039] Step S2: Construct a rigid body enhanced mode basis containing global translation, rigid body tilt and low-order elastic modes based on the full-field vibration displacement snapshot matrix, and construct a sparse measurement point observation matrix based on the rigid body enhanced mode basis; and screen sparse measurement point coordinates with good observability and inversion stability by using observability indices such as observation matrix rank, null space dimension, minimum singular value, condition number and modal noise amplification factor.

[0040] Step S201: Construct a rigid body enhanced mode basis containing global translation, rigid body tilt and low-order elastic modes based on the full-field vibration displacement snapshot matrix.

[0041] Specifically, the full-field vibration displacement snapshot matrix obtained from step S1 In the above, the full-field discrete displacement vector corresponding to the m-th snapshot sample is: For simplicity, the following will use... Let represent any discrete displacement vector across the entire field, and we have: Let the reconstructed basis matrix be... Where N is the number of discrete nodes in the entire field; r is the number of reconstructed basis vectors. To avoid the influence of different basis vector magnitude scales on subsequent projection error evaluation and sparse measurement point arrangement, the reconstructed basis matrix is... Perform column space orthogonalization to satisfy: (5) in, express The conjugate transpose of the matrix. Represents the identity matrix.

[0042] Under this reconstructed basis, the full-field vibration response vector It can be decomposed into: (6) in, This represents the response components that the reconstructed basis can express. Represents the basis coefficient vector. This represents the basis residual vector that the reconstructed basis cannot represent. The basis coefficient vector and the basis residual vector are respectively: (7) (8) Let the sparse measurement point selection matrix be... Where K is the number of sparse measurement points, and Let the observed response vector at the sparse measuring points be... The measurement noise or fitting error vector is Then the sparse observation equation is: (9) Substituting the full-field response decomposition into the sparse observation equation, we get: (10) Therefore, the full-field reconstruction error under the condition of few measurement points is not only related to the reconstruction basis matrix of sparse measurement point pairs. This is related to the observation capability, and also to the basis residuals resulting from the inability of the reconstructed basis to represent the true response. Related. Let the sparse observation matrix be . ,when When the column rank is full, the full-field reconstruction error satisfies the following upper bound relationship: (11) in, This represents the reconstructed full-field vibration response vector. Represents a sparse observation matrix The smallest singular value.

[0043] This relationship indicates that the accuracy of sparse full-field reconstruction is controlled by two types of factors: one type is... The reconstruction basis expression error is characterized by another type of error caused by The stability of sparse observations is characterized. Therefore, the arrangement of measurement points in this step does not solely pursue the minimum condition number of the observation matrix, nor does it solely pursue the minimum projection error of the candidate basis. Instead, it uses a joint constraint of expressive power and sparse recoverability to determine the reconstruction basis and the sparse measurement point set.

[0044] Specifically, let the normalized planar coordinates of the discrete nodes across the entire structure be x and y. Considering the coupled components of global translation, rigid body tilting, and low-order elastic deformation in the low-frequency response of the real structure, this step constructs the original rigid body enhanced modal basis matrix containing global translation, rigid body tilting in two directions, and low-order elastic modes: (12) in, This represents the overall translational trend vector. and These represent the rigid body tilt tendency vectors in two directions, respectively. to This represents the first six low-order elastic mode vectors. The first three columns of this basis matrix describe the low-frequency overall motion and rigid body tilting tendency, while the last six columns describe the low-order elastic deformation of the structure. Therefore, the original rigid body enhanced mode basis matrix... The dimension is .

[0045] Subsequently, the original rigid body enhanced modal basis matrix was... By orthogonalizing the column space, we obtain the rigid body-enhanced modal basis matrix for numerical evaluation and measurement point placement. And satisfy: ;in, Used to characterize the physical composition and interpretability of rigid body reinforced modal bases. It is used for subsequent projection error evaluation, sparse observation matrix construction, and measurement point placement calculation. Orthogonalization does not change the reconstructed subspace spanned by the original rigid body enhanced modal basis, but only changes the coordinate representation in the basis space.

[0046] For candidate reconstructed basis matrix First, the projection normalization error is calculated to evaluate the ability of the candidate basis to represent the true full-field response. For the j-th full-field response sample... Its projection normalization error is defined as: (13) in, This refers to the projection normalization error; The average projection error across multiple response samples is: (14) in, denoted as the average projection error across multiple response samples; n represents the number of response samples. The smaller the value, the stronger the expressive power of the candidate reconstruction basis for the true full-field response, and the stronger the basis residual. The smaller.

[0047] Step S202: Using the sparse measurement point observation matrix as the object, and combining the observability index, select sparse measurement point coordinates with good observability and stability.

[0048] Based on the candidate basis meeting the expressive power requirements, a sparse observation matrix is ​​constructed, and the following observability and stability indices are calculated.

[0049] First, calculate the sparse observation matrix. rank: (15) And calculate its null dimension: (16) in, Let be the rank of the sparse observation matrix, used to determine whether the basis coefficients are observable at a finite number of measurement points. The zero-space dimension is used to determine whether there are unobservable basis coefficient degrees of freedom. When This indicates that there are unobservable directions under the current sparse measurement point set, and sparse inversion has the risk of underdeterminacy.

[0050] Therefore, this step will and This serves as the basic solvability threshold for sparse measurement point sets to enter subsequent stability evaluation; measurement point sets that do not meet this threshold are no longer compared with condition number and noise amplification factor, but are directly judged as not having complete modal observability.

[0051] After satisfying the basic solvability threshold, the sparse observation matrix is ​​calculated. The condition number: (17) in, and They represent The maximum and minimum singular values. Condition number. The larger the value, the closer the sparse observation matrix is ​​to being ill-conditioned, and the more susceptible the coefficient estimation is to the influence of basis residuals and measurement noise disturbances.

[0052] Furthermore, the modal noise amplification factor is calculated: (18) when When the column is full, the above formula can be written as: (19) in, express The pseudo-inverse matrix, Denotes the Frobenius norm. express The i-th singular value. MNAF( The larger the value, the easier it is to amplify measurement noise and baseline residuals when estimating coefficients under sparse measurement points.

[0053] In specific judgments, condition numbers and modal noise amplification factor It is not used as the first threshold for judging solvability, but rather when the set of measurement points already satisfies... and Based on this, it is used to further compare the inversion stability and noise resistance of different feasible measurement point sets.

[0054] First, for the candidate measurement point set that has passed the initial screening of basic solvability, calculate its corresponding sparse observation matrix. The minimum singular value and condition number. Let the minimum singular value threshold be... The condition number threshold is In one embodiment, take ,Pick The observation matrix of a candidate sparse measurement point set is considered to be ill-conditioned if any of the following conditions are met: (20) (twenty one) in, The smaller the value, the weaker the observable direction in the observation matrix, and the more sensitive the basis coefficient inversion is to basis residuals and measurement noise; The larger the value, the greater the difference in magnification across different basis directions, and the closer the observation matrix is ​​to being ill-conditioned. The principle behind setting this threshold is to limit the level of error amplification in sparse inversion, ensuring that the candidate sparse measurement point set has acceptable numerical stability while maintaining full column rank. Therefore, this type of measurement point set is not considered a preferred set, or its priority is reduced in the candidate ranking.

[0055] Secondly, among the candidate measurement points that meet the condition number requirements, the modal noise amplification factors are further compared. When two or more candidate test point sets all have rank and condition number representations that satisfy the above requirements, the preferred choice is... The smallest set of measurement points. The reason is that... It reflects the overall noise amplification level across all observable basis directions and, compared to a single condition number index, provides a more comprehensive evaluation of the sparse inversion process's sensitivity to measurement noise and basis residuals.

[0056] Therefore, this step adopts the following hierarchical decision-making process: First, based on... and Determine whether the set of measurement points satisfies the basic solvability; then... and Determine whether the measurement point set has good inversion stability; select the measurement point set with the smallest MNAF(C) as the optimal criterion to prevent noise amplification risk. The measurement point set selected according to the above hierarchical determination process is the optimal measurement point set, which will be used for the subsequent placement of key measurement points in the scanning laser Doppler vibrometer.

[0057] Step S203: Construct noisy normalized training samples based on the final sparse measurement point set.

[0058] In step S202, the final sparse measurement point set is determined. Afterwards, according to Construct the final sparse measurement point selection matrix The full-field vibration displacement snapshot matrix obtained from step S1. In the above, the full-field discrete displacement vector corresponding to the m-th snapshot sample is: Then the noise-free displacement response of the m-th snapshot sample at the final sparse measurement point is defined as: (twenty two) in, The noise-free displacement response of the m-th snapshot sample at the final K sparse measurement points; Select the matrix for the final sparse measurement points; K is the final number of sparse measurement points; N is the total number of discrete nodes in the field.

[0059] To simulate the noise impact of a scanning laser Doppler vibrometer during actual measurement, Gaussian measurement noise is injected into the unnoised displacement response at the final sparse measurement points. First, the mean value of the unnoised displacement response at the final sparse measurement points for the m-th snapshot sample is defined as: (twenty three) Define the standard deviation of the noise-free displacement response of the m-th snapshot sample at the final sparse measurement points as: (twenty four) in, For vectors The displacement response component corresponding to the kth sparse measuring point in the middle; for The mean; for The standard deviation.

[0060] The noisy sparse measurement point observation matrix data is defined as follows: (25) (26) in, This is the noisy sparse measurement point observation matrix data corresponding to the m-th snapshot sample; The noise level coefficient is taken as [value] in one embodiment. ; It is a K-dimensional standard normal random noise vector; It is a K-order identity matrix.

[0061] Simultaneously, the full-field discrete displacement vector corresponding to the m-th snapshot sample... As a true label for the entire event, namely: (27) To eliminate the impact of dimensional differences on the gradient backpropagation of the physical information deep operator network, the maximum absolute value of the full-field displacement corresponding to the full-field vibration displacement snapshot matrix in step S1 is used to normalize the noisy sparse measurement point observation matrix data and the full-field true labels. The maximum absolute value of the full-field displacement is defined as: (28) in, Full-field vibration displacement snapshot matrix The maximum absolute value of the full-field displacement across all snapshot samples; m is the number of snapshot samples; The infinite norm of a vector is the maximum absolute value of its elements.

[0062] The normalized sparse measurement point input and the true labels for the entire field are as follows: (29) (30) in, Input the sparse measurement points after normalization of the m-th snapshot sample; This represents the true label of the entire field after normalization for the m-th snapshot sample; the subscript n indicates normalization, not the number of samples. It consists of the labels corresponding to all snapshot samples. and This forms the training sample set for the subsequent deep operator network of physical information.

[0063] The following explains how to determine the final reconstruction basis for point selection and inversion. As shown in Table 1, this table does not simply compare the reconstruction errors of different basis bases, but rather screens candidate reconstruction basis bases according to the logic of prior recoverability evaluation, expressive power check, and posterior reconstruction verification. First, prior indices such as the rank / null space dimension, minimum singular value, condition number, and MNAF of the observation matrix are used to determine whether the candidate basis base has the ability to recover stable coefficients under finite measurement points. Second, the expressive power of the basis base to the true full-field response is checked by projective NRMSE. Finally, the optimal sparse reconstruction NRMSE is used to verify whether the aforementioned basis selection can be translated into an improvement in actual reconstruction performance.

[0064] Table 1. Joint evaluation results of the expressive power and sparsity recoverability of candidate reconstructive bases.

[0065] As can be seen from Table 1, the traditional purely elastic modal basis Although it has a good observation matrix condition number and a low risk of noise amplification, it is not good at expressing the true low-frequency response because it does not include global translation and rigid body tilt terms. Although it has the lowest projection error, indicating the strongest full-field representation capability, its basis dimension is greater than 10 measurement points. Under the current sparse measurement point conditions, there are unobservable directions, so it is not suitable as the principal reconstruction basis. compared to This further reduced the projection error, but significantly decreased the minimum singular value while significantly increasing the condition number and MNAF, indicating the introduction of a weakly observable direction, leading to a decrease in coefficient inversion stability at finite measurement points. In contrast, Introducing rigid body enhancement terms significantly improves the ability to represent the true low-frequency response while maintaining full column rank and a relatively stable observation matrix, achieving optimal results in the posterior sparse reconstruction error. Therefore, the final selection for modal observable sparse measurement point placement... As the target basis, point selection is performed on this basis to obtain a sensor layout consistent with the final reconstructed basis, such as... Figure 2 As shown, this is the spatial distribution of the optimal sensor measurement points for modal observable sparse sampling. In this embodiment, ten measurement points are shown.

[0066] This embodiment constructs a rigid-body reinforced modal basis that matches the true low-frequency response, and applies observability constraints to the sparse measurement point placement scheme based on this modal basis, thereby improving the ability to observe the low-frequency vibration characteristics of the structure under conditions of a small number of laser measurement points. Unlike methods that only rely on the numerical properties of the observation matrix for measurement point placement, this embodiment further incorporates the expressive power of the reconstructed basis to the true low-frequency response into the sparse measurement point placement process, thereby achieving a synergistic constraint between the expressive power of the reconstructed basis and the sparse recoverability of the finite number of measurement points.

[0067] This invention establishes a rigid body enhanced modal basis including overall translation, rigid body tilt, and low-order elastic modes, enabling the characterization of base motion, attitude changes, and elastic deformation features in low-frequency vibrations of real structures. Furthermore, using the observation matrix with sparse measurement points as the object, the method performs a priori evaluation of the measurement point placement effect by combining matrix rank, condition number, minimum singular value, and modal noise amplification factor. This method reduces the possibility of measurement points falling into nodal regions or acquiring redundant information, improves the observation capability of a small number of laser measurement points for key vibration features, thereby enhancing measurement efficiency, inversion stability, and the accuracy of full-field vibration field reconstruction.

[0068] Step S3: Measure sparse displacement data based on the coordinates of the sparse measuring points, and input the dimensionless sparse displacement data into a pre-trained physical information deep operator network. Calculate the full-field vibration field of the structure through the forward propagation of the physical information deep operator network. The physical information deep operator network incorporates constraints such as measurement point confidence weighted data consistency, thin plate bending strain energy, boundary condition residuals, and curvature adaptive sampling during training.

[0069] Step S301: Import the coordinates of sparse measurement points into the control system of the scanning laser Doppler vibrometer, and control the laser beam to only perform directional measurement on key measurement points, thereby reducing the time cost of full-field scanning.

[0070] It should be understood that in this embodiment, the rigid body enhanced modal basis and modal observability index in step S2 are mainly used for sparse measurement point placement, measurement point confidence construction, and network input design; the physical information deep operator network is used to predict the full-field displacement field based on the sparse measurement point response. B9 basis inversion can be used as a priori evaluation, baseline model, or network auxiliary constraint, rather than as the final output path competing with the physical information deep operator network.

[0071] Step S302: Construct a deep operator network with scaled dot products.

[0072] The physical information deep operator network is composed of a branch network and a backbone network in parallel. For example... Figure 3 As shown, the overall parallel architecture, feature extraction flow, and automatic differentiation physical constraint loop of the physical information deep operator network based on the principle of strain energy minimization are constructed. Further, as... Figure 4 As shown, the specific layer depth, node hidden dimension, activation function type, and forward propagation data flow of the multilayer perceptron within the branch network and backbone network are as follows, representing a micro-level architecture: The measurement point order is fixed and serves as the input order for the branch network. The branch network receives normalized sparse measurement point data through a multilayer perceptron. Output a high-dimensional latent feature vector with dimension p. : (31) in, Input the sparse measurement points after normalization of the m-th snapshot sample obtained in step S203; Let be the hidden feature vector of the branch network corresponding to the m-th snapshot sample, and , where p is the dimension of the common latent features output by the branch network and the backbone network; The number of layers in the branch network; Here is the weight matrix of the l-th layer of the branch network; This is the bias vector of the l-th layer of the branch network; The symbol represents the hyperbolic tangent activation function; the subscript b indicates a branch network; and the subscript l indicates the network layer number.

[0073] The backbone network is used to extract the spatial coordinate features of all discrete nodes in the field. Let the physical coordinates of the j-th discrete node in the field be... ,in Define the backbone network input coordinate vector of the j-th discrete node as: (32) Backbone network reception Output the spatial latent feature vector corresponding to the j-th discrete node in the entire field: (33) in, It is the spatial latent feature vector output by the backbone network at the j-th discrete node of the whole field. Number of layers in the backbone network; This is the weight matrix of the l-th layer of the backbone network; This is the bias vector of the l-th layer of the backbone network; the subscript t represents the backbone network, and the subscript l represents the network layer number.

[0074] To alleviate the problems of inner product variance shift and physical information depth operator network saturation caused by excessive feature dimension, this invention introduces a scaling dot product mechanism and global output bias at the end of the physical information depth operator network, outputting the normalized predicted displacement of the m-th snapshot sample at the j-th global discrete node: (34) in, For physical information deep operator networks in terms of parameters The normalized predicted displacement is given for the m-th snapshot sample and the j-th discrete node output of the whole field. Hidden feature vectors of branch networks The i-th component; The hidden feature vectors of the backbone network The i-th component; For global output bias; These are the trainable parameters for a physical information deep operator network.

[0075] In the reasoning stage, the same displacement scale parameters as in step S203 are used. Inverse normalization is performed on the normalized predicted displacement to obtain the predicted physical displacement value of the m-th snapshot sample at the j-th discrete node in the whole field: (35) in, The predicted physical displacement of the m-th snapshot sample at the j-th discrete node of the whole field; (36) in, This is the reconstructed full-field vibration displacement vector corresponding to the m-th snapshot sample; Let be the normalized full-field predicted displacement vector corresponding to the m-th snapshot sample.

[0076] Furthermore, a loss function is designed for the aforementioned physical information deep operator network, and a two-stage training strategy is adopted.

[0077] The first stage constructs a weighted data consistency loss function based on the confidence of modal observable measurement points. This loss function is used to constrain the prediction results of the physical information deep operator network at sparse measurement point locations to be consistent with the normalized sparse measurement point input. Let the final sparse measurement point set be... The discrete node number corresponding to the k-th sparse measurement point in the whole field is: The weighted data consistency loss function is then defined as: (37) in, The data consistency loss function is weighted by the confidence level of the measurement points. These are the trainable parameters for a deep operator network of physical information; For the final sparse measurement point set The discrete node number of the whole field corresponding to the kth sparse measurement point; Input the normalized sparse measurement points in step S203 The kth component ； For the physical information deep operator network at the m-th snapshot sample, the... Normalized predicted displacement output at each discrete node in the entire field; is the confidence weight of the k-th sparse measurement point.

[0078] In one embodiment, confidence weight The coefficient is determined by the observation contribution coefficient of the k-th sparse measurement point and the local signal-to-noise ratio coefficient. (38) in, The observation contribution coefficient corresponding to the kth sparse measurement point; Let be the local signal-to-noise ratio coefficient corresponding to the k-th sparse measurement point. If there is no prior local signal-to-noise ratio, then let . ; For the final sparse observation matrix The k-th row; Let represent the 2-norm of a vector. Based on the above definition, measurement points with higher observability contributions and higher local signal-to-noise ratios have higher weights in data consistency loss.

[0079] The first stage relies on sparse observation point allocation for data-driven pre-training, aiming to quickly fit the main mapping relationship between the sparse measurement point response and the full-field vibration field. To fully utilize the results of the modal observable sparse measurement point deployment, this invention does not use the traditional equal-weighted mean square error, but instead constructs a weighted data consistency loss based on the confidence of modal observable measurement points. This loss function comprehensively considers the observability contribution of each measurement point to the rigid body enhanced modal basis observation matrix and the local measurement signal-to-noise ratio, so that measurement points with higher observability contributions and better measurement quality have higher data fidelity weights in network training.

[0080] The second stage involves constructing a joint loss function based on physical constraints, including thin plate bending strain energy, boundary condition residuals, and curvature adaptive sampling.

[0081] To ensure that the dimensions of the physical constraint terms are consistent with those of the thin plate mechanical parameters, the normalized predicted displacement is first converted into a physical displacement prediction field: (39) in, Let be the predicted physical displacement field corresponding to the m-th snapshot sample.

[0082] Based on Kirchhoff's thin plate theory, and through the second-order automatic differentiation of the physical displacement prediction field with respect to spatial coordinates x and y, the bending strain energy density of a point-state thin plate is defined as: (40) in, Let represent the point-state bending strain energy density of the thin plate at position (x, y) for the m-th snapshot sample. D is the bending stiffness of the thin plate, and ν is Poisson's ratio. When the thickness of the thin plate is g and the elastic modulus is E, the bending stiffness D of the thin plate is: (41) To enhance the physical constraint strength in high curvature regions, an adaptive curvature sampling mechanism is introduced during the joint fine-tuning of physical constraints. Let the number of adaptively selected points in the h-th training iteration be... The spatial coordinates of the j-th adaptive collocation point are ( Then, the bending strain energy loss function of a thin plate under curvature adaptive sampling is defined as: (42) in, Let be the thin plate bending strain energy loss function under curvature adaptive sampling in the h-th training iteration; The number of points for curvature adaptive collocation; The curvature sampling weights are the adaptive collocation points in the h-th training iteration.

[0083] The curvature sampling weights are updated based on the local second derivative magnitudes of the local curvature magnitudes obtained in the current round or the most recent update: (43) in, The local curvature magnitude at the j-th adaptive collocation point in the h-th training iteration; To prevent extremely small positive numbers with a denominator of zero, the average curvature sampling weight of all adaptive collocation points is 1 after the above normalization process.

[0084] Boundary condition residuals are constructed based on the boundary support form of the structure under test. For thin plate structures, boundary condition residuals include at least one of the following: centrally connected region constraint residuals, fixed-support boundary residuals, simply supported boundary residuals, and free boundary residuals.

[0085] For a centrally connected thin-plate structure, let the set of boundary points of the central screw connection area or connecting block area be . Its constrained residual is defined as: (44) in, To provide a set of points connecting the boundaries of the central region, Assign points to the boundaries of the central connecting area. This represents the number of snapshot samples.

[0086] For a fixed-support thin-plate structure, let the set of collocation points at the fixed-support boundary be... Its boundary residual is defined as: (45) in, For the set of collocation points of the fixed boundary, The number of points to be assigned to the fixed boundary. This indicates the normal direction outside the boundary. This residual is used to constrain the displacement and normal rotation at the fixed boundary.

[0087] For a simply supported thin plate structure, let the set of collocation points of the simply supported boundary be... Its boundary residual is defined as: (46) in, For the set of collocation points of simply supported boundary, The number of points to be collocated on a simply supported boundary. This represents the bending moment residual at the j-th simply supported boundary point of the m-th snapshot sample. This residual is used to constrain the displacement and bending moment conditions at the simply supported boundary.

[0088] For a free-boundary thin-plate structure, let the set of collocation points at the free boundary be... Its boundary residual is defined as: (47) in, For the set of points on the free boundary, To determine the number of points on the free boundary, This represents the equivalent shear force residual at the j-th free boundary collocation point of the m-th snapshot sample. This residual is used to constrain the bending moment and equivalent shear force natural boundary conditions at the free boundary.

[0089] For pipe or cylindrical shell structures, cylindrical shell coordinates (s, Construct boundary condition residuals, where s represents the axial coordinate. Represents the circumferential coordinates. For the fixed support end of the pipeline, let the set of points at the fixed support end be... Its boundary residual is defined as: (48) in, For the fixed support points of the pipe or cylindrical shell, This corresponds to the number of matching points. This residual is used to constrain the radial displacement and axial rotation at the fixed support end of the pipeline.

[0090] For a simply supported end of a pipe, let the set of points for the simply supported end be... Its boundary residual is defined as: (49) in, For the simply supported end of a pipe or cylindrical shell, provide a set of connection points. For the corresponding number of points, This represents the axial bending moment residual. This residual is used to constrain the radial displacement and axial bending moment conditions at the simply supported ends of the pipe.

[0091] For the free end of the pipeline, let the set of points at the free end be... Its boundary residual is defined as: (50) in, For the set of points at the free ends of a pipe or cylindrical shell, For the corresponding number of points, This represents the axial equivalent shear force residual. This residual is used to constrain the axial bending moment and axial equivalent shear force natural boundary conditions at the free end of the pipe.

[0092] Based on the above boundary forms, the boundary condition residual loss function is defined as: (51) in, , , , , , and These are the weighting coefficients for the corresponding boundary residual terms.

[0093] In practical implementation, one or more of the above residual terms are selected based on the actual boundary support form of the structure under test. The weight coefficients of each boundary residual term are determined according to the actual boundary support form of the structure under test. Let the set of boundary residual term types be: (52) For any boundary residual term Q, its weighting coefficient is defined as follows: When this type of boundary or support form does not exist in the structure being tested, let The boundary residual term is not included in the boundary condition loss calculation; when this type of boundary or support exists in the structure under test, let The boundary residual term is used in the calculation of boundary condition loss.

[0094] In one embodiment, for boundary conditions that actually exist and are of equal importance, their corresponding weight coefficients are all set to 1; for boundary conditions that play a dominant role in structural constraints, such as the central connection area and fixed support ends, their corresponding weight coefficients can be appropriately increased. Thus, unused boundary terms are excluded by setting their weight coefficients to 0, while used boundary terms participate in the boundary condition residual loss through positive weight coefficients, thereby enabling the boundary condition residuals to be flexibly constructed according to different structural types and actual support forms.

[0095] Based on the above-mentioned thin plate bending strain energy loss and boundary condition residual loss, a joint physical loss function is constructed for the h-th training iteration: (53) in, Let h be the joint physical loss function in the h-th training iteration; Weighting of bending strain energy loss in thin plates; This represents the weight of the boundary condition residual loss.

[0096] Finally, a multi-task adaptive weight dynamic correction mechanism is introduced to dynamically adjust the weight coefficients in the overall objective function based on the relative gradient strengths of the data loss and each physical loss term during gradient backpropagation. The overall loss function in the h-th training iteration is defined as: (54) Right now: (55) To balance the gradient contributions of data loss and physical loss during backpropagation, the gradient intensity ratios of the thin plate bending strain energy loss and the boundary condition residual loss relative to the data loss are defined respectively: (56) (57) Based on the aforementioned gradient strength ratio, the weights for the two types of physical loss are dynamically updated: (58) (59) in, The gradient intensity ratio of data loss to the bending strain energy loss of the thin plate in the h-th training iteration; The gradient strength ratio of data loss to boundary condition residual loss in the h-th training iteration; This is the weighted adjustment rate; The preset gradient balance target value is set to 1 in one embodiment; To prevent smooth terms with a denominator of zero.

[0097] Through the aforementioned dynamic weight correction mechanism, the gradient contributions among data consistency loss, thin plate bending strain energy loss, and boundary condition residual loss can be coordinated during training, reducing gradient competition between physical constraint terms and data fitting terms, and improving the stability and physical consistency of network training in environments with few measurement points and noise.

[0098] In this embodiment, thin-plate elasticity physical constraints are introduced for joint fine-tuning in the second stage of training. Addressing the engineering challenge of traditional pure data-driven models easily generating non-physical high-frequency distortions in the blind zone, this invention innovatively constructs a joint loss function combining multi-source physical constraints and adaptive weight allocation. For pipe or cylindrical shell structures, the thin-plate bending strain energy constraint can be replaced with a bending-membrane coupled strain energy constraint based on cylindrical shell theory.

[0099] This invention constructs a physical information deep operator network composed of branch networks and a backbone network to extract sparse measurement point response features and spatial coordinate features, respectively. Scale fluctuations during high-dimensional feature fusion are controlled through a scaling dot product mechanism and global bias. Simultaneously, joint loss constraints are introduced, including measurement point confidence-weighted data consistency, minimization of thin plate bending strain energy, boundary condition residuals, curvature adaptive sampling, and multi-task weight adjustment, enabling the network to balance data fitting ability and thin plate mechanical consistency. This method improves training stability under conditions with very few measurement points and suppresses non-physical distortions under unknown working conditions, thereby enhancing the accuracy of full-field vibration field reconstruction.

[0100] Step S303: After model training is complete, the observation data at the optimal sparse measurement points are input into the physical information deep operator network, and the predicted results of the structural vibration displacement across the entire field are obtained through a single forward propagation. This method can improve reconstruction accuracy under conditions with few measurement points, noise, and unknown operating conditions, and suppress non-physical distortions in areas without measurement points.

[0101] Preferably, a dual-index evaluation system consisting of the relative error of local bending strain energy density and the distance-weighted blind zone relative error is used to evaluate the reconstruction results of the whole field vibration field, so as to avoid the problem that the mean square error alone is insufficient to reflect the physical consistency and blind zone reconstruction capability.

[0102] 1. Construct the relative error of local bending strain energy density.

[0103] This metric is used to evaluate the physical consistency of the predicted displacement field at the second-order curvature and bending energy levels. Let the actual physical displacement field corresponding to the m-th snapshot sample be... The predicted physical displacement field is The corresponding point-state thin plate bending strain energy densities are denoted as follows: and The relative error of local bending strain energy density is defined as: (60) in, This represents the relative error of local bending strain energy density; To prevent extremely small positive numbers with a denominator of zero, the smaller this index is, the closer the predicted field is to the real field in terms of local curvature and bending energy distribution, and the fewer non-physical high-frequency fluctuations and local distortions there are.

[0104] 2. Construct distance-weighted blind zone relative error.

[0105] This metric is used to evaluate the model's reconstruction capability in regions far from sparse measurement points. Let the spatial coordinate vector of the j-th discrete node in the entire field be... The spatial coordinate vector corresponding to the k-th sparse measurement point in the final sparse measurement point set is: The distance from the j-th discrete node to the nearest sparse measurement point is defined as: (61) Normalize the distance: (62) The distance-weighted blind zone relative error is defined as follows: (63) in, Let be the distance from the j-th discrete node in the whole field to the nearest sparse measurement point; Normalized distance weights ； and To prevent extremely small positive numbers with a denominator of zero ； This represents the distance-weighted blind zone relative error. A larger normalized distance indicates that the node is farther from the nearest sparse measurement point. The smaller the value, the smaller the reconstruction error of the model in areas without measurement point coverage.

[0106] The smaller this index, the smaller the reconstruction error of the model in areas without measurement point coverage, and the stronger the full-field reconstruction capability under conditions with few measurement points.

[0107] The two indicators mentioned above correspond to the two core evaluation objectives of this invention: the relative error of local bending strain energy density is used to verify the suppression effect of physical constraints on non-physical distortions; the relative error of distance-weighted blind zone is used to verify the reconstruction capability of the region without measuring points under the condition of sparse measuring point deployment based on modal observability. Through this dual-indicator system, the reconstruction effect of the whole-field vibration field can be evaluated from two aspects: physical consistency and generalization with few measuring points.

[0108] This invention establishes a physical consistency evaluation index system for full-field reconstruction results, introducing the relative error of local bending strain energy density and the distance-weighted blind zone relative error to evaluate the physical rationality of the predicted displacement field in terms of curvature and bending energy distribution, as well as the reconstruction accuracy in regions far from sparse measurement points. This evaluation system overcomes the problem that a single mean square error cannot reflect local non-physical distortions and errors in regions without measurement points, providing a more comprehensive quantitative basis for model performance comparison, parameter optimization, and reliability assessment in engineering applications.

[0109] The two-dimensional thermal distribution comparison results of the absolute error of the full-field reconstruction under extreme off-peak conditions, where the proposed physical information deep operator network and the pure data-driven model exceed historical peak values, are as follows: Figure 5 As shown, Figure 5 (a) in the figure shows the reference truth value, as shown in Figure 1. Figure 5 (b) shows the data-driven model prediction, such as Figure 5 (c) shows the prediction of the physical information constrained model, as shown in Figure 1. Figure 5 (d) in the figure shows the absolute error distribution of the data-driven model, as shown in Figure 1. Figure 5 Figure (e) shows the absolute error distribution of the physical information constraint model. Figure 5 It can be seen that the pure data-driven model produces significant non-physical local distortions within the observation blind zone without sensor coverage; while the physical information deep operator network, due to the introduction of elastic mechanical bending strain energy constraints, exhibits a flatter global error distribution and a significant decrease in numerical value under the same working conditions. The dual-index evaluation results under out-of-distribution working conditions are as follows: Figure 6 As shown, Figure 6 Figure (a) shows the improvement of the relative error index of local bending strain energy density, as shown in Figure (a). Figure 6 Figure (b) shows the improvement in the distance-weighted blind zone relative error index. Figure 6As can be seen, under out-of-distribution conditions exceeding historical peak values, compared to the pure data-driven model, this method significantly reduces both the relative error of local bending strain energy density and the relative error of distance-weighted blind zone. Specifically, the relative error of local bending strain energy density is reduced by approximately 59.29%, indicating that this method can effectively constrain the local curvature and bending energy distribution of the predicted displacement field, reducing non-physical high-frequency fluctuations and local distortions. The relative error of distance-weighted blind zone is reduced by approximately 42.06%, indicating that this method still possesses strong full-field inference and generalization capabilities in regions without measurement points, far from sparse measurement points. These results demonstrate that this method not only improves the numerical reconstruction accuracy under out-of-distribution conditions but also significantly enhances the physical consistency, blind zone reliability, and engineering robustness of the reconstruction results.

[0110] Furthermore, to verify the necessity of the dimension reduction and point selection strategy of this invention, a comparative verification of different sensor point placement strategies was conducted under the constraint of an equally sparse number of measurement points. The results are as follows: Figure 7 As shown, traditional uniform point placement strategies are prone to falling into the nodal blind zone of higher-order modes, leading to abnormally high local maximum reconstruction errors; random point placement strategies exhibit significant instability and extremely wide error distribution boundaries. In contrast, the modally observable sparse measurement point placement strategy adopted in this invention outperforms both uniform and random point selection strategies in terms of both average and maximum reconstruction errors, and shows smaller error fluctuations in multiple independent experiments, indicating that this measurement point placement strategy can improve the reconstruction accuracy and result stability under extremely sparse measurement point conditions.

[0111] Example 2 In one or more embodiments, a full-field vibration reconstruction system based on sparse sampling and physical depth operators is disclosed, specifically including: The sample construction module is configured to: extract the dominant natural modes of the structure through finite element simulation, and perform modal superposition calculations in combination with broadband random excitation to construct a full-field vibration displacement snapshot matrix; The measuring point layout module is configured to: construct a rigid body enhanced mode basis containing overall translation, rigid body tilt and low-order elastic modes based on the full-field vibration displacement snapshot matrix, and select sparse measuring point coordinates by taking the sparse measuring point observation matrix as the object and combining observability index. The full-field reconstruction module is configured to: measure sparse displacement data based on the coordinates of the sparse measuring points, input the dimensionless sparse displacement data into a pre-trained physical information deep operator network, and obtain the full-field vibration field of the structure through forward propagation calculation of the physical information deep operator network. The physical information deep operator network incorporates constraints such as measurement point confidence weighted data consistency, thin plate bending strain energy, boundary condition residuals, and curvature adaptive sampling during training.

[0112] Example 3 This embodiment provides an electronic device, including a memory and a processor, as well as computer instructions stored in the memory and running on the processor. When the computer instructions are executed by the processor, they complete the steps of the above-mentioned full-field vibration reconstruction method based on sparse sampling and physical depth operators.

[0113] Example 4 This embodiment provides a computer-readable storage medium for storing computer instructions. When the computer instructions are executed by a processor, they complete the steps of the above-described full-field vibration reconstruction method based on sparse sampling and physical depth operators.

[0114] This invention is described with reference to flowchart illustrations and / or block diagrams of methods and apparatus (systems) according to embodiments of the invention. It will be understood that each block of the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchart illustrations and / or block diagrams, can be implemented by computer instructions. These computer instructions can be provided to a processor of a general-purpose computer, special-purpose computer, embedded processor, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, generate instructions for implementing the process. Figure 1 One or more processes and / or boxes Figure 1 A device that provides the functions specified in one or more boxes.

[0115] These computer instructions may also be stored in a computer-readable storage medium that can direct a computer or other programmable data processing device to function in a particular manner, such that the instructions stored in the computer-readable storage medium produce an article of manufacture including instruction means, which are implemented in a process Figure 1 One or more processes and / or boxes Figure 1 The function specified in one or more boxes.

[0116] These computer instructions can also be loaded onto a computer or other programmable data processing equipment, whereby a series of operational steps are performed to produce a computer-implemented process, thereby providing instructions that execute on the computer or other programmable equipment for implementing the process. Figure 1 One or more processes and / or boxes Figure 1 The steps of the function specified in one or more boxes.

[0117] The descriptions of each embodiment in the above embodiments have different focuses. For parts not described in detail in a certain embodiment, please refer to the relevant descriptions in other embodiments.

[0118] The above description is merely a preferred embodiment of the present invention and is not intended to limit the invention. Various modifications and variations can be made to the present invention by those skilled in the art. 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.

Claims

1. A full-field vibration reconstruction method based on sparse sampling and physical depth operators, characterized in that, include: The dominant natural modes of the structure are extracted by finite element simulation, and modal superposition calculation is performed by combining broadband random excitation to construct a full-field vibration displacement snapshot matrix; Based on the full-field vibration displacement snapshot matrix, a rigid body enhanced mode basis including global translation, rigid body tilt and low-order elastic modes is constructed, and the sparse measurement point observation matrix is ​​used as the object, and the coordinates of sparse measurement points are screened by combining observability index. Based on the sparse measurement point coordinates, sparse displacement data is measured. After the sparse displacement data is dimensionless, it is input into a pre-trained physical information deep operator network. The whole field vibration field of the structure is obtained through forward propagation calculation of the physical information deep operator network. The physical information deep operator network incorporates constraints such as measurement point confidence weighted data consistency, thin plate bending strain energy, boundary condition residuals, and curvature adaptive sampling during training.

2. The full-field vibration reconstruction method based on sparse sampling and physical depth operator as described in claim 1, characterized in that, The process of selecting sparse measurement point coordinates based on the sparse measurement point observation matrix and observability indicators specifically includes: The rank of the sparse observation matrix and the null dimension are used to determine whether the set of measurement points satisfies the basic solvability. The minimum singular value and condition number of the sparse observation matrix are used to determine whether the set of measurement points has good inversion stability; Modal noise amplification factor is used as the preferred criterion for noise amplification risk. When multiple candidate measurement point sets simultaneously meet the above conditions, the measurement point set with the best overall performance is selected for the subsequent placement of key measurement points in the scanning laser Doppler vibration meter.

3. The full-field vibration reconstruction method based on sparse sampling and physical depth operators as described in claim 1, characterized in that, The physical information deep operator network is composed of a branch network and a backbone network in parallel. The branch network receives normalized sparse measurement point data through a multilayer perceptron and outputs a latent feature vector. The backbone network receives normalized spatial grid coordinates and outputs spatial basis function vectors. The hidden feature vector and spatial basis function vector are subjected to a scaling dot product mechanism and a global bias to output a normalized predicted displacement. The normalized predicted displacement is mapped by inverse normalization to obtain the physical full-field vibration field.

4. The full-field vibration reconstruction method based on sparse sampling and physical depth operator as described in claim 3, characterized in that, The normalized predicted displacement is output by employing a scaled dot product mechanism and a global bias: in, For physical information deep operator networks in terms of parameters The normalized predicted displacement is given for the m-th snapshot sample and the j-th discrete node output of the whole field. Hidden feature vectors of branch networks The i-th component; The hidden feature vectors of the backbone network The i-th component; For global output bias; denoted as the trainable parameters of the physical information deep operator network; p represents the common latent feature dimension of the branch network and the backbone network outputs.

5. The full-field vibration reconstruction method based on sparse sampling and physical depth operator as described in claim 1, characterized in that, The physical information deep operator network adopts a two-stage training strategy. The loss function includes a weighted data consistency loss function based on the confidence of modal observable measurement points and a physical constraint joint loss function based on adaptive sampling of thin plate bending strain energy and curvature. The physical constraint joint loss function based on thin plate bending strain energy and curvature adaptive sampling includes: constructing a joint physical loss function based on strain energy residuals and boundary condition residuals; constructing a thin plate bending strain energy loss function by converting normalized predicted displacement into a physical displacement prediction field and performing second-order automatic differentiation of the physical displacement prediction field with respect to spatial coordinates; and fine-tuning the thin plate bending strain energy loss function using a curvature adaptive sampling mechanism. A multi-task adaptive weight dynamic correction mechanism is adopted to dynamically adjust the weight coefficients in the overall objective function based on the relative convergence rates of the weighted data consistency loss function and each physical loss term during gradient backpropagation. Where h is the number of training iterations; These are the trainable parameters for a deep operator network of physical information; The data consistency loss function is weighted by the confidence level of the measurement points. Weighting of bending strain energy loss in thin plates; Thin plate bending strain energy loss function; Weights for boundary condition residual loss; This is the boundary condition residual loss function.

6. The full-field vibration reconstruction method based on sparse sampling and physical depth operator as described in claim 5, characterized in that, The weighted data consistency loss function based on the confidence of modal observable points is specifically as follows: in, The data consistency loss function is weighted by the confidence level of the measurement points. These are the trainable parameters for a deep operator network of physical information; For the final sparse measurement point set The discrete node number of the whole field corresponding to the kth sparse measurement point; Input for normalized sparse measurement points The kth component ； For the physical information deep operator network at the m-th snapshot sample, the... Normalized predicted displacement output at each discrete node in the entire field; represents the confidence weight of the k-th sparse measurement point; K is the final number of sparse measurement points. This represents the number of snapshot samples.

7. The full-field vibration reconstruction method based on sparse sampling and physical depth operator as described in claim 1, characterized in that, It also includes using a dual-index evaluation system consisting of the relative error of local bending strain energy density and the relative error of distance-weighted blind zone to evaluate the reconstruction results of the full-field vibration field; The relative error of the local bending strain energy density is used to evaluate the physical consistency of the predicted displacement field at the level of second-order curvature and bending energy. The distance-weighted blind zone relative error is used to evaluate the model's reconstruction capability in regions far from sparse measurement points.

8. A full-field vibration reconstruction system based on sparse sampling and physical depth operators, characterized in that, include: The sample construction module is configured to: extract the dominant natural modes of the structure through finite element simulation, and perform modal superposition calculations in combination with broadband random excitation to construct a full-field vibration displacement snapshot matrix; The measuring point layout module is configured to: construct a rigid body enhanced mode basis containing overall translation, rigid body tilt and low-order elastic modes based on the full-field vibration displacement snapshot matrix, and select sparse measuring point coordinates by taking the sparse measuring point observation matrix as the object and combining observability index. The full-field reconstruction module is configured to: measure sparse displacement data based on the coordinates of the sparse measuring points, input the dimensionless sparse displacement data into a pre-trained physical information deep operator network, and obtain the full-field vibration field of the structure through forward propagation calculation of the physical information deep operator network. The physical information deep operator network incorporates constraints such as measurement point confidence weighted data consistency, thin plate bending strain energy, boundary condition residuals, and curvature adaptive sampling during training.

9. An electronic device, characterized in that, It includes a memory and a processor, as well as computer instructions stored in the memory and running on the processor. When the processor executes the computer instructions, it performs the full-field vibration reconstruction method based on sparse sampling and physical depth operators as described in any one of claims 1-7.

10. A computer-readable storage medium, characterized in that, Used to store computer instructions, which, when executed by a processor, complete the full-field vibration reconstruction method based on sparse sampling and physical depth operators as described in any one of claims 1-7.

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