High-precision prediction method of full-field dynamic displacement response under sparse observation condition

Abstract

The application discloses a high-precision prediction method for full-field dynamic displacement response under sparse observation conditions and relates to the technical field of structural health monitoring. The method comprises the following steps: extracting time-frequency features from sparse sensor observation data; generating a damage factor field through an E-Net network of space-time decoupling, and then constructing a time-varying elastic modulus field; mapping the elastic modulus field and external load into full-field displacement response by using a feature fusion operator network of a fusion feature modulation mechanism; introducing a variational physical information constraint in the training process, reducing the instability of high-order derivatives through weak form integration, and ensuring that the prediction result meets the structural dynamics equation. The estimation method of the application realizes high-precision inversion of the time-varying elastic modulus and high-fidelity prediction of the full-field displacement under the condition of sparse arrangement of sensors, has strong physical consistency, strong robustness and high calculation efficiency, and is suitable for real-time monitoring and safety evaluation of engineering structures.

Description

Technical Field

[0001] This invention relates to the field of structural health monitoring technology, and in particular to a high-precision prediction method for full-field dynamic displacement response under sparse observation conditions. Background Technology

[0002] The displacement response of a structural dynamics system is a core physical quantity characterizing the overall mechanical behavior of the structure. It directly reflects the deformation, vibration characteristics, and energy transfer mechanism of the structure under external loads. In recent years, with the development of technologies such as digital twin drones and imaging measurement, vision-based full-field displacement acquisition and reconstruction has received increasing attention. However, in engineering practice, continuous high-precision full-field measurement remains difficult to achieve due to limitations in sensor cost, field-of-view obstruction, and bandwidth. Therefore, how to reconstruct high-fidelity full-field displacement under sparse observation conditions is a core technical problem for establishing structural digital twins and realizing online security decision-making.

[0003] However, during the service of actual engineering structures, phenomena such as stiffness degradation, local damage propagation, contact degradation, and changes in material properties due to environmental variations occur. This causes the models initially based on static parameter assumptions to gradually deviate from observations. When the model struggles to capture these spatiotemporal changes, displacement predictions accumulate errors and rapidly lose the ability to identify key peak values ​​and damage details, weakening the credibility and early warning value of digital twins. In high-frequency or nonlinear structural dynamics scenarios, modal frequencies, energy transfer paths, and displacement field morphology are significantly affected by time-varying elastic modulus changes, profoundly highlighting the necessity of synchronously inverting the spatiotemporal field of the elastic modulus and predicting displacement.

[0004] Existing mainstream technologies have significant shortcomings in addressing the combined challenges of sparse observations and spatiotemporally variable elastic moduli. Traditional methods, such as the Finite Element Model Modification (FEMU) approach, represent the dynamic behavior of a structure using a parameterized finite element model. By minimizing the difference between the finite element model's predicted response and the monitored data, uncertain parameters in the model are corrected. Ogunniyi et al. proposed a method based on Modal Order Reduction and Local Eigenvalue Modification Procedure (LEMP), which can achieve near real-time model updates under high-frequency events and supports low-latency displacement estimation. However, this method relies on pre-existing modal information and reduced-order bases. When the elastic modulus undergoes a local abrupt change in space, it is difficult to recover high-resolution full-field displacement details using global or local modal correction. Similarly, while Bayesian FEMU can provide a large amount of uncertainty quantification and high-precision correction for displacement prediction, its iterative solution and multiple high-dimensional sensitivity calculations lead to low computational efficiency. On the other hand, purely data-driven methods such as deep learning and convolutional neural networks can achieve fast inference with large-scale training data, but due to the lack of physical constraints, they have shortcomings in extrapolation ability, noise robustness, and physical consistency. When damage causes the elastic modulus to exhibit high-frequency abrupt changes locally, purely data-driven models generally generate overly smooth displacement fields, lose sharp boundary information, and even induce non-physical responses. The PINN framework proposed by Raissi et al. incorporates PDE residuals into training, improving physical consistency and providing a cornerstone for joint learning of "data + physics". However, it still faces problems such as training instability, spectral bias, and difficulty in weight tuning when dealing with long-term accumulated errors of high-order spatial derivatives and balancing multiple loss terms. The hp-VPINN method proposed by Kharazmi et al. introduces the weak form of PDE into network training, transferring higher-order differentials to the test function, thereby reducing the order of automatic differentiation and enhancing robustness to noise. The Fourier feature mapping proposed by Tancik et al. can greatly enhance the fitting ability of MLP networks to high-frequency functions, thereby reducing spectral bias. However, Fourier features alone cannot replace physical constraints. In problems such as sparse observations or time-varying parameters, they may introduce pseudo-high-frequency oscillations and unreliable details in unobserved regions. Summary of the Invention

[0005] This invention provides a high-precision prediction method for full-field dynamic displacement response under sparse observation conditions. Under the condition of sparse sensor arrangement, it realizes high-precision inversion of time-varying elastic modulus and high-fidelity prediction of full-field displacement. It has strong physical consistency, strong robustness and high computational efficiency, and is suitable for real-time monitoring and safety assessment of engineering structures.

[0006] To solve the above-mentioned technical problems, the technical solution proposed by this invention is as follows:

[0007] A high-precision prediction method for full-field dynamic displacement response under sparse observation conditions includes the following steps:

[0008] Time-frequency features are extracted from displacement time-series data acquired by sparse sensors to obtain feature vectors characterizing the dynamic state of the system;

[0009] The feature vector is input into the spatiotemporally decoupled E-Net network to generate a damage factor field that satisfies causal constraints, and a time-varying elastic modulus field is constructed based on the damage factor field and the preset health state elastic modulus benchmark distribution.

[0010] The time-varying elastic modulus field and external load are input into the feature fusion operator network, and the feature fusion operator network outputs the predicted value of the full-field displacement response of the target structure.

[0011] The feature fusion operator network includes a training process, which constructs a phased training framework for physical-data collaboration. The framework divides the network optimization process into two stages: forward operator construction and full-field response reconstruction. By introducing physical information constraints based on variational weak forms, the predicted displacement response satisfies the structural dynamics control equations.

[0012] A further improvement to the above technical solution is as follows:

[0013] Preferably, the time-frequency feature extraction employs a frequency-domain-aware temporal feature encoder, comprising:

[0014] The displacement time-series signal of each sensor is subjected to short-time Fourier transform to obtain the corresponding time spectrum;

[0015] The time-frequency spectra of each sensor are combined into a multi-channel feature map, which is then input into a two-dimensional convolutional neural network for feature extraction.

[0016] The extracted features are interpolated along the time dimension to restore the original time resolution, and the feature vector is finally output.

[0017] Preferably, the E-Net network includes a spatial basis function network and a temporal coefficient network. The spatial basis function sub-network is used to learn a set of spatial impairment basis functions, and the temporal coefficient sub-network uses the feature vector... As input, predict the corresponding time coefficient increment. The damage factor field Generated by the following formula to embed hard causal constraints:

[0018]

[0019] in, Let be the sigmoid function, with a fitting coefficient of 0.95 to limit the damage to the interval [0, 0.95]. s is a learnable parameter that controls the sensitivity of the sigmoid function. is the unconstrained cumulative damage field, and b is a learnable parameter that controls the center offset of the Sigmoid function.

[0020] Preferably, the feature fusion operator network adopts the DeepONet architecture and includes a feature linear modulation mechanism, including:

[0021] A dual-stream branch network is used to receive the time-varying elastic modulus field and the external load, and output a set of dynamic modulation parameters and a projection coefficient vector;

[0022] The trunk network is used to receive spatiotemporal coordinates and, in at least one hidden layer, to perform an affine transformation on the network activation values ​​using the dynamic modulation parameters, ultimately outputting a set of adaptive spatiotemporal response basis functions.

[0023] The full-field displacement response is reconstructed by performing an inner product between the coefficient vector output by the branch network and the basis function output by the trunk network.

[0024] Preferably, in the forward operator construction stage, the feature fusion operator network is trained to learn the mapping from the elastic modulus field to the displacement field; under the condition of having full-field displacement label data, the main objective is to minimize the error between the predicted displacement and the actual displacement, while gradually introducing variational physical information constraints based on weak forms to train the forward mapping capability of the feature fusion operator network.

[0025] Preferably, in the full-field response reconstruction stage, the parameters of the trained feature fusion operator network are frozen, and the frequency-domain-aware temporal feature encoder and the spatiotemporally decoupled damage factor field generation network are fine-tuned using only the observation data of sparse sensors and the physical residuals to form a joint loss.

[0026] Preferably, the variational physical information constraint is constructed based on the weak form of the Euler-Bernoulli beam equation, and the requirement for higher-order derivatives of the displacement field is reduced by performing weighted integration and integration by parts on the governing equation.

[0027] Preferably, during the training process of the forward operator construction or full-field response reconstruction stage, a dynamic weight fusion algorithm based on gradient variance is adopted to adaptively balance the weights between the data loss term and the physical residual loss term, using the ReLoBRaLo-GVW algorithm.

[0028] The high-precision prediction method for full-field dynamic displacement response under sparse observation conditions provided by this invention has the following advantages compared with existing technologies:

[0029] (1) The high-precision prediction method for full-field dynamic displacement response under sparse observation conditions of the present invention establishes the FusionDeepONet operator network architecture for multi-physics coupling problems. By using the hybrid dual-stream topology and feature modulation mechanism, the elastic modulus field and load field are fused at the deep feature level, which effectively alleviates the inductive bias conflict between spatially dominant and temporally dominant features and improves the model's ability to characterize elastic modulus mutation and high-frequency dynamic features.

[0030] (2) The high-precision prediction method for full-field dynamic displacement response under sparse observation conditions of the present invention designs an E-Net parameter generation network with hard causal constraints. By using spatiotemporal decoupling modeling and Softplus-CumSum structure, the monotonicity of damage evolution is guaranteed from the network level, non-physical oscillations and pseudo solutions are suppressed, and the physical consistency and stability of time-varying parameter inversion are enhanced.

[0031] (3) The high-precision prediction method for full-field dynamic displacement response under sparse observation conditions of the present invention establishes a phased physical driving training strategy of forward operator construction and sparse physical fine-tuning, and uses the physical control equation as a differentiable constraint to efficiently propagate sparse observation information to the unobserved region, thereby achieving physical self-consistent full-field prediction under sparse data conditions. Attached Figure Description

[0032] Figure 1 This is a schematic diagram of the physical model of the present invention.

[0033] Figure 2 This is the overall framework of the gray-box operator guided by physical information in this invention.

[0034] Figure 3 To verify the time-varying elastic modulus field inversion results based on sparse observations in the experiment (t=0.5).

[0035] Figure 4 To verify the time-varying elastic modulus field inversion results based on sparse observations (t=1.0) in the experiment.

[0036] Figure 5 To verify the reconstruction of the full-field displacement space under sparse observations in the experiment (t=0.8).

[0037] Figure 6 The response prediction diagram is used to verify the low-frequency slight damage condition in the experiment.

[0038] Figure 7 The response prediction diagram is used to verify the experimental results under high-frequency severe damage conditions. Detailed Implementation

[0039] The following provides a detailed description of specific embodiments of the present invention. It should be understood that the specific embodiments described herein are for illustrative and explanatory purposes only and are not intended to limit the scope of the invention.

[0040] like Figure 1 As shown, the present invention provides a high-precision prediction method for full-field dynamic displacement response under sparse observation conditions. This method first extracts and encodes time-frequency features from sparse observation data to obtain feature vectors reflecting the dynamic state of the system; then, it uses a spatiotemporally decoupled E-Net network to map the feature vectors into a damage factor field satisfying causal constraints. And construct the spacetime field of elastic modulus Finally, the generated elastic modulus field is used as input to the forward solver module, and FusionDeepONet quickly predicts the full-field displacement response. During the training process, variational physics constraints based on weak forms are introduced as a regularization method to reduce the numerical instability of higher-order derivatives while ensuring that the prediction results satisfy the structural dynamics control equations.

[0041] Step S1: Design the Variational Physical Information Neural Network (VPINN).

[0042] Traditional Physical Information Neural Networks (PINNs) suffer from noise amplification and gradient instability when dealing with high-order partial differential equations (PDEs) due to high-order automatic differentiation. To prevent these problems, this invention employs a variational physical information neural network (VPINN) architecture. Unlike the traditional approach based on pointwise minimization of residuals using strong forms, VPINN uses a weak form approach to relax the requirements on the continuity of approximate solutions and utilizes the smoothing properties of integral operators to improve the robustness of the model.

[0043] Output of a deep neural network within the VPINN framework The trial function is shaped into a solution to the governing equations, where x represents the spatial coordinates. This represents the trainable parameters of the network. A finite set of tightly supported test basis functions is selected. In the support domain Expanding the PDE with weighted integrals and integrals by parts, we transfer the higher-order spatial derivatives acting on the test function to the test function, the th The variational residual equations for the test functions are expressed as follows:

[0044]

[0045] in, The integral by parts acts on the trial function. With test function The reduced-order differential operator satisfies the order complement relation; The physical parameters to be inverted; For known source terms; The boundary terms are derived from integration by parts. This transformation significantly reduces the derivative order required by the neural network and effectively suppresses numerical noise introduced by the depth of the computational graph.

[0046] Based on the above integral form of variational physical information loss The integral term is composed of the mean squared errors of the integral residuals of all test functions, and is calculated using the Gauss-Legendre quadrature method.

[0047]

[0048] in, Representatives include network predictions With physical parameters The numerical integral approximation of the trial function term; Representative Source Item The numerical integral approximation, This represents the total number of test functions selected. The external load in the governing equations.

[0049] Step S2: Construct a physical model and define the problem.

[0050] S2-1, Time-varying Euler-Bernoulli governing equations:

[0051] To establish a physical model of time-varying structural damage, the estimation method of this invention constructs a system as follows: Figure 1 The slender cantilever beam dynamic system shown is defined in the spatial domain. and time domain Above, the beam axis is oriented in the coordinate system shown. Extending along the axis, its lateral vibration displacement Then along The direction of the axis, the shaded area in the figure marks the preset time-varying damage area, while the sparse displacement sensors arranged along the beam span simulate the limited observation points in actual monitoring, providing data support for subsequent monitoring.

[0052] Assuming the beam satisfies the Euler-Bernoulli assumption, meaning its cross-section remains planar and perpendicular to the neutral axis after deformation, and considering viscous damping and time-varying elastic modulus distribution, its lateral displacement... The dynamic evolution is governed by the following fourth-order partial differential equation:

[0053]

[0054] in, The density of the material; It is the cross-sectional area; It is the viscous damping coefficient; The moment of inertia of the cross section; External loads applied to the structure; This is a time-varying elastic modulus field. For a cantilever beam with its fixed end at x=0 and its free end at x=L, where L is the total length of the beam, the system must satisfy the following boundary conditions: the displacement and rotation at the fixed end are zero, and the bending moment and shear force at the free end are also zero. That is:

[0055]

[0056]

[0057] The initial condition is usually set to a static state:

[0058]

[0059] In structural health monitoring, direct inversion This could potentially lead to physically uninterpretable solutions. To embed physical priors, this estimation method employs a gray-box modeling strategy. Assuming that the degradation of structural stiffness is relative to the initial healthy state, the elastic modulus field is parameterized as follows:

[0060]

[0061] in, It is a known baseline distribution of the elastic modulus of a healthy state; It is the damage factor field. Under this definition, the inversion problem is transformed into reconstructing a scalar field from sparse observation data. .

[0062] To prevent numerical optimization difficulties caused by differences in the magnitudes of different physical quantities, the governing equations are dimensionless. Dimensionless variables are defined as follows:

[0063]

[0064] in, These are the characteristic scales for length, time, displacement, elastic modulus, and load, respectively. After substituting them, the network input and output are normalized to the interval [-1,1] or [0,1], improving the convergence stability of the AdamW optimizer.

[0065] S2-2, variational weak form:

[0066] Following VPINN's general framework, this estimation method derives a specific weak form of the Euler-Bernoulli beam equation, selecting a test function defined on a local support domain. By performing a weighted integration of the governing equations and then using integration by parts to transfer the higher-order spatial derivatives of the elastic modulus term to the test function, the following integral residual form is obtained:

[0067]

[0068] in, This represents the weak-form integral residual of the governing equations within the support domain of the test function.

[0069] In the weak form, the displacement field The order of the high spatial derivative is reduced from fourth to second, which reduces the smoothness requirement of the network. Furthermore, the low-pass filtering characteristics of the integral operator are used to effectively suppress observation noise and improve the robustness of the algorithm.

[0070] Step S3: Design a gray-box prediction framework that integrates physical mechanisms.

[0071] This invention designs a gray-box prediction framework that integrates physical mechanisms, the overall process of which is as follows: Figure 2 As shown, this method achieves rapid and high-precision prediction of the full-field displacement response of a structure by utilizing limited sparse observation data. First, time-frequency features are extracted and encoded from the displacement responses acquired by a limited number of sensors to obtain feature vectors reflecting the dynamic state of the system. Subsequently, the feature vectors are mapped to a damage factor field that satisfies causal constraints through the spatiotemporally decoupled E-Net network. Furthermore, an elastic modulus spacetime field satisfying physical constraints is constructed. Finally, this parameter field is used as input to the operator-level forward model, and FusionDeepONet quickly predicts the full-field displacement response. .

[0072] This design enables the model to perceive changes in structural physical parameters like traditional finite element methods, significantly improving computational efficiency. The VPINN mechanism is introduced during the training phase, combining the physical residuals of the weakly formed structural dynamics equations with observation errors to form a loss function, thereby ensuring the physical consistency of the prediction results.

[0073] Specifically, the following steps are included:

[0074] S3-1, Construct a frequency-domain-aware temporal feature encoder:

[0075] Under sparse sensor observation conditions, the state evolution information of a structural dynamic system can only be indirectly reflected through a limited temporal response. To extract features that characterize the change of structural dynamic state over time from sparse displacements, this invention designs a frequency-aware convolutional coding network (FACE). Unlike traditional time-domain processing methods based on recurrent neural networks, this temporal feature encoder uses short-time Fourier transform (STFT) to explicitly transform the dynamic response into a time-frequency spectrum and selects a two-dimensional convolutional neural network (2D-CNN) to extract high-dimensional frequency domain features, thereby effectively overcoming the spectral bias problem commonly found in deep learning.

[0076] The model's input is the time-series observation matrix. This matrix was acquired by sparse sensors, where For batch size, For time steps, For the number of sensors. Taking into account the structural elastic modulus... The degradation of the signal inevitably leads to a drift in the system's natural frequency and a change in the modal energy distribution. Directly processing the time-domain waveform generally makes it difficult to capture these subtle high-frequency changes. Therefore, this invention first uses Short-Time Fourier Transform (STFT) to transform the one-dimensional vibration signal into a two-dimensional time spectrum. For the first... Signal from a sensor Its STFT transformation is defined as:

[0077]

[0078] in, The signal represents the time. With frequency Complex spectral density at; For The sliding window function centered on the signal is used to extract the signal within a local time period to achieve localized analysis of non-stationary signals. This is a Fourier transform kernel function used to extract the signal at angular frequencies. The frequency domain components at that location.

[0079] The complex spectrum obtained by STFT transformation contains amplitude and phase information. This invention performs modulo operation on the complex spectrum to obtain the amplitude spectrum, and stacks the amplitude spectra of consecutive time steps to construct a two-dimensional time spectrum describing the energy distribution. Subsequently, the generated time spectrum is treated as a multi-channel image input to a two-dimensional convolutional neural network. Here, the frequency axis is mapped to the spatial vertical axis of the image, and the sliding of the convolution kernel on the time-frequency plane allows it to treat low-frequency and high-frequency regions equally, thereby keenly capturing frequency bending and energy redistribution patterns caused by changes in the elastic modulus. To address the temporal downsampling problem caused by convolutional pooling operations, this invention introduces a temporal interpolation layer after the feature extraction layer. Linear interpolation restores the compressed feature map to its original temporal resolution, and the final output shape is... eigenvectors, where Represents the set of real numbers. Indicates the sample batch size. Indicates the feature dimension of the hidden layer. This represents the hidden state. This eigenvector is strictly aligned with the physical solver at the time step and serves as the conditional input for E-Net to generate the damage factor field.

[0080] S3-2, Constructing a spatiotemporal decoupling damage generator:

[0081] This invention utilizes a generative network (E-Net) to map feature vectors extracted from the FACE module and located in the abstract feature space back to a time-varying damage factor field with explicit physical meaning, thereby generating an elastic modulus field. Traditional fully convolutional networks typically face the problem of a large number of parameters and difficulty in capturing long-term dependencies when processing high-dimensional spatiotemporal data. To address this, this invention designs a spatiotemporally decoupled generative architecture based on the idea of ​​variable separation. This architecture assumes that the high-dimensional damage field can be approximated as a linear combination of a set of time coefficients and spatial basis functions, meaning that the spatiotemporal features of the entire field can be efficiently reconstructed by low-rank matrix operations.

[0082]

[0083] in, To reconstruct the full-field spatiotemporal feature matrix, The spatial damage basis function matrix, The time coefficient matrix, The order of the decomposition is... The number of discrete points in space. This represents the number of time steps.

[0084] Based on this assumption, E-Net was designed with a two-branch structure:

[0085] (I) Spatial Basis Function Network: The purpose of this branch is to obtain spatial basis function networks from spatial coordinates. Learn a set of orthogonal damage basis functions that can effectively describe the damage field. To meet the computational requirements of the Euler-Bernoulli beam equation for higher-order spatial derivatives, the network uses the SiLU activation function, utilizing its... The smoothing property prevents the problem of ReLU second derivative truncation, where It is infinitely differentiable; simultaneously, layer normalization is introduced to apply implicit Lipschitz continuity constraints, which effectively suppresses high-frequency oscillation artifacts commonly found in implicit neural representations, ensuring the generation of spatial damage bases. Physical priors that align with the continuous and gradual changes in material properties.

[0086] (II) The time coefficient network uses the feature vector output by the FACE module As a conditional input, a mapping is established using a multilayer fully connected network (MLP). It transforms the frequency domain features implicit in space into the damage evolution intensity in physical space, and outputs the corresponding time coefficient increment. ,in Representing the eigenvector Dimensions.

[0087] The evolution of structural damage exhibits extremely strict physical monotonicity, meaning that the damage is irreversible. Existing physical information neural networks generally use penalty terms to curb non-monotonic behavior, but during the training process, gradient conflicts with data loss are very likely to occur, leading to training instability.

[0088] This invention designs a hard causal constraint to strictly guarantee the monotonically increasing damage. Instead of directly predicting the damage value, the network predicts non-negative damage increments. By utilizing the Softplus activation function and a cumulative summation operation along the time dimension, an unconstrained cumulative damage field is established. :

[0089]

[0090] in, Let k be the spatial damage basis function. For the first Each mode in The increment of the time coefficient at any given moment.

[0091] This operation is mathematically consistent. At the underlying logic level, a strict physical constraint on the irreversibility of damage is implemented. Finally, numerical smoothing and boundary constraints are applied to the accumulated results to obtain the final damage factor field. ,for:

[0092]

[0093] Among them, the Sigmoid function A matching coefficient of 0.95 strictly limits the damage to the interval [0, 0.95], preventing the finite element stiffness matrix singularity problem caused by the complete decay of stiffness to zero. s is a learnable parameter controlling the sensitivity of the Sigmoid function, and b is a learnable parameter controlling the center offset of the Sigmoid function. A gating mechanism at time t=0 is also introduced to ensure that the initial state satisfies the damage-free assumption, constructing a time-varying elastic modulus field. .

[0094] S3-3, Design of Feature Fusion Operator Network:

[0095] To address the complex conditions of spatiotemporal evolution of the elastic modulus of structures during service, and to achieve rapid and high-fidelity prediction of the dynamic displacement response across the entire field, this invention constructs a feature fusion operator network (FusionDeepONet). The core task of this module is to learn and establish a spatiotemporally varying elastic modulus field. With external load To full field displacement response Nonlinear operator mapping .

[0096] In structural dynamics problems, the elastic modulus field Local perturbations can generate complex global nonlinear effects through the governing equations. The standard DeepONet architecture performs well in the field of operator learning, but its physical parameters are simply linearly multiplied by the output of the branch network and the trunk network in the last layer, making it difficult to capture such deep physical coupling effects. Therefore, the method of this invention introduces a deep feature fusion mechanism, which no longer treats physical parameters as static inputs but as adjustment signals, deeply involved in the generation process of the displacement field.

[0097] Furthermore, to address the space-dominated elastic modulus field With time-driven external loads To address the inductive bias conflict during feature extraction, the branch network is designed as a hybrid two-stream architecture. The time-varying elastic modulus field reconstructed by E-Net is received by the modulus branch, which utilizes multi-layer 2D convolutions to extract the spatial topology of the elastic modulus distribution and local damage intensity. The load branch uses multi-scale 1D convolutions to capture the temporal fluctuation characteristics of the excitation signal. After concatenation and fusion, the two feature streams are no longer directly output as static features. Instead, they are mapped to two sets of dynamic parameters using a hypernetwork mechanism: one set is the projection coefficient vector used for final displacement prediction. The other set is used to adjust the modulation parameters of the trunk network.

[0098] The trunk network is responsible for providing the spatiotemporal response basis functions for the displacement solution. To inject the constraints of physical parameters into the generation process of the basis functions, the method described in this invention introduces a characteristic linear modulation (FiLM) mechanism into each layer of the trunk network. Let the trunk network be... The input features of the layer are Its forward propagation process follows the affine transformation formula:

[0099]

[0100] in, ReLU is a non-linear activation function. The scaling factor is calculated by the branch network based on the current physical input; Let be the weight matrix of the j-th layer of the trunk network itself; Let be the bias vector of the j-th layer of the trunk network itself; The translation factor is calculated by the branch network based on the current physical input; This represents the Hadamard Product, which is an element-wise multiplication.

[0101] This type of transformation, which has clear physical interpretability, utilizes a scaling factor generated by a branch network. This simulation demonstrates the dynamic adjustment of local response sensitivity to changes in elastic modulus, enabling the network to adaptively amplify or suppress modal activation in specific regions based on the degree of damage. The overall shift in the response baseline is determined by a translation factor. Simulations were performed on external loads.

[0102] The FiLM modulation process is recursively expanded in each layer of the trunk network. Physical parameters are not treated as one-time initial conditions but are continuously injected into the feature transformation process. Dynamic modulation and correction are performed on the spatiotemporal coordinate feature representation at different network depths. With increasing network depth, this cumulative effect arises from the layer-by-layer injection of physical constraints. When the signal propagates to the last layer of the trunk network, its output characteristics are not purely geometric coordinate mappings but rather an adaptive field highly coupled with the current physical state. This invention transforms the output of the trunk network at its end... A dimensional vector is directly defined as a set of basis functions for the spatiotemporal response. Unlike the fixed basis functions in traditional DeepONet, the specific waveforms of this set of response basis functions are dynamically reconstructed according to changes in the stiffness field and load, providing various components with physical sensing capabilities for subsequent displacement field reconstruction.

[0103] Based on operator approximation theory, the full-field displacement response This can be represented as the inner product of the response basis function and the projection coefficients. The method of this invention combines the coefficient vector b output by the branch network with the basis function output by the trunk network. Perform dot product operation:

[0104]

[0105] in, For the i-th projection coefficient, Let be the basis function of the i-th spatiotemporal response.

[0106] By leveraging this explicit inner product reconstruction mechanism, the feature fusion operator network not only achieves rapid end-to-end mapping from the physical parameter space to the full-field response space, but also effectively overcomes the interpretability defects of traditional black-box models, ensuring high fidelity and physical consistency of prediction results under time-varying elastic modulus perturbation.

[0107] Step S4: Determine the loss function and training strategy.

[0108] To address the spatiotemporal evolution of structural elastic modulus under complex working conditions, and to achieve accurate prediction of full-field displacement response using only sparse sensor observations, this invention designs a physical-data collaborative phased training framework. The framework divides the network optimization process into two stages: forward operator construction (Stage F) and full-field response reconstruction (Stage I). By gradually introducing physical constraints, the model is guided to recover high-fidelity full-field displacement that satisfies the dynamic equations from sparse signals.

[0109] S4-1, Phased Training Framework:

[0110] In the forward operator construction phase, the goal is to train the feature fusion operator network to learn the operator mapping from the time-varying parameter field to the full-field displacement response field. To mitigate the numerical instability introduced by higher-order physical constraints in the early stages of training, this invention employs a course learning strategy, which minimizes data consistency loss in the first 20% of training. This guides the network to first establish a stable data-driven representation. The physical loss is then gradually introduced in the latter 80% of the training. Its weight is increased from 0 to 0.5 through dynamic warm-up to correct for non-physical deviations, while the fixed support boundary conditions and initial conditions are strictly constrained by the mean square error (MSE). The total loss function at this stage... for:

[0111]

[0112] in, This represents the mean square error of the total displacement. The physical loss weight coefficients are dynamically adjusted with each training round. To determine the total number of spatiotemporal sampling points in the training dataset, The full-field displacement response predicted by the operator network. This corresponds to the actual displacement response. and Strong constraints are applied to the boundary and initial state, respectively, and are constructed using the mean square error form; physical loss Based on the weak form construction of Euler-Bernoulli beam theory, the requirement for output smoothness is reduced by integrating by parts twice:

[0113]

[0114] in, and Representing the predicted displacements With test function The second derivative with respect to spatial coordinates; The density of the material; It is the cross-sectional area; The moment of inertia of the cross section; , where is the damping coefficient. After training, the feature fusion operator network will serve as a frozen, differentiable physical surrogate model to support prediction optimization in subsequent stages.

[0115] To address the ill-posedness of the inverse problem caused by the sparse distribution of sensors, a progressive strategy of fully supervised pre-training (Stage I-a) followed by physical consistency fine-tuning (Stage I-b) is employed in the full-field response reconstruction stage (Stage I). In Stage I-a, supervised learning of the feature extractor (E-Net) and parameter generator (FACE) is conducted using limited ground-value data, initially establishing a nonlinear mapping relationship from sparse temporal observations to the time-varying elastic modulus field. To ensure the smoothness of high-frequency damage details and the physical field, a composite loss function incorporating deep damage weighting, frequency domain consistency, and Sobolev regularization is introduced in this stage. The total loss function is... for:

[0116]

[0117] in, This represents the total number of spatiotemporal sampling points in the training data of the elastic modulus field. For the predicted values ​​of the time-varying elastic modulus field generated by the network, This corresponds to the true value of the elastic modulus field. To predict the first spatial gradient of the elastic modulus field, In response to Apply greater weight to areas of high damage; represents the weighting coefficients of the Sobolev regularization term, used to constrain the first-order spatial gradient of the predicted elastic modulus field to eliminate non-physical checkerboard artifacts.

[0118] Stage I-b employs a training mechanism that freezes the physical operator and fine-tunes the state generator. In this stage, the weights of the forward operator FusionDeepONet are completely frozen, treating it as a differentiable physical surrogate model embedded in the computation graph. During training, only the observation errors from sparse sensors are utilized. With physical residual A loss function is constructed, and the error gradient is backpropagated through the frozen FusionDeepONet via a backpropagation mechanism, updating only the upstream FACE and E-Net parameters. This mechanism forces the generative network to continuously correct the underlying elastic modulus field. This allows it to accurately fit the displacement data of sparse measurement points after mapping with the help of physical operators, while ensuring that the overall response satisfies the constraints of the governing equations. Under this physically driven implicit state correction, physical laws actually act as unsupervised regularization constraints, transmitting observation information from discrete sensor positions to the unobserved regions of the entire field, achieving high-precision, physically self-consistent full-field displacement prediction under sparse data conditions. Its total loss function... for:

[0119]

[0120] in, It is a set of spatiotemporal coordinates of sparse sensors; To dynamically balance the weights, For the first The spatial location of each sensor For sparse sensors in location The actual displacement observations collected at the location.

[0121] S4-2, Adaptive Optimization and Sampling Strategy:

[0122] To significantly enhance the convergence and accuracy of the prediction model under complex conditions, this invention specifically introduces a dynamic weight fusion and adaptive sampling mechanism. Firstly, to resolve the gradient conflict caused by the difference in convergence rates between observed data loss and physical constraint loss, the ReLoBRaLo-GVW dynamic weight algorithm is employed. This algorithm improves upon the traditional linear weighting model by introducing gradient variance (GVW) as a stability penalty term based on the relative loss reduction rate. The fusion weight of the i-th loss term in the t-th iteration is... Defined as:

[0123]

[0124] in, The suggested weights are calculated based on the relative decline rate. For gradient variance, To prevent numerical stability constants with a denominator of zero. Final weights. Depend on It is generated after normalization. The core advantage of this mechanism lies in utilizing the denominator term. It automatically detects and significantly suppresses unstable tasks with severe gradient oscillations, thereby effectively smoothing the optimization trajectory, preventing a single task from dominating network updates, and ensuring gradient balance and robust convergence in multi-objective optimization.

[0125] Meanwhile, to address the high-frequency oscillation of the displacement field in the damaged region caused by changes in the elastic modulus, this invention presents a gradient-guided stateless adaptive sampling strategy. Unlike traditional momentum methods that maintain historical states, this strategy utilizes the spatial gradient modulus of the currently inferred elastic modulus field. Constructing the dynamic probability density function :

[0126]

[0127] in, Represents the currently predicted elastic modulus field E( The spatial gradient L2 norm; is the focusing coefficient, used to adjust the sampling preference intensity for high-gradient regions; 1 is the uniform sampling baseline term. Instantaneous resampling is performed accordingly in each iteration.

[0128] Based on the probability density constructed from the material parameter gradient, weighted sampling is performed on high-importance regions at the discrete grid level to achieve coarse-grained localization at the grid level, and a uniform perturbation is applied within the selected grid. Generate fine-grained training coordinates. This mechanism causes physical collocations to adaptively focus on regions of drastic system state changes, enabling the network to more precisely describe local high-frequency dynamic characteristics through physical constraints, thus greatly improving the detail fidelity of the overall displacement prediction.

[0129] Experimental verification:

[0130] The performance of the method of this invention was quantitatively evaluated on a synthetic dataset to verify its effectiveness in time-varying parameter inversion and full-field displacement prediction under sparse observations.

[0131] All experiments were performed on a computing platform equipped with an NVIDIA RTX 5070 GPU, implemented using the PyTorch deep learning framework. To optimize memory efficiency and accelerate convergence, mixed-precision training, gradient checkpointing, and gradient accumulation techniques were combined during training. The experimental dataset was constructed from a one-dimensional Euler-Bernoulli beam synthetic dynamics simulation, covering random damage fields at different depths and locations. Model training strictly followed a phased strategy of forward operator construction (Stage F) – full-field response reconstruction (Stage I). For network hyperparameter configuration, the FusionDeepONet trunk network hidden layer dimension was set to 512, and the number of basis functions was P=64; the optimizer used was AdamW, with cosine annealing used to adjust the learning rate and weight decay. To verify the effectiveness of the framework in structural health monitoring, detailed numerical experimental results of time-varying elastic modulus cantilever beams will be analyzed from two dimensions: physical parameter inversion and full-field displacement response prediction.

[0132] (1) Spatiotemporal inversion of time-varying elastic modulus field

[0133] Figure 3 and Figure 4 The model demonstrates the time-varying elastic modulus across the entire field under conditions of only sparse sensor observations. The inversion performance. Select... and Quantitative analysis of cross-sections at two representative time points reveals a high degree of consistency between the E-Net's predicted curves and the actual values. During the mid-stage of damage, the true damage factor field is The model predicts the value. The absolute deviation is As the damage evolves to In the severe stage, the true damage factor field is The model predicts the value. Deviation control within This indicates that the model accurately captures the elastic modulus from... decay to The model exhibits high robustness in full-field reconstruction, with a mean root square error (RMSE) of [value missing] over the entire time period. The stability of this error distribution strongly validates the effectiveness of hard causal modules and physical prior constraints. The model performance does not degrade with the increase of system nonlinearity, and can provide a reliable diagnostic basis for the health monitoring of the structure throughout its entire life cycle.

[0134] (2) Full-field dynamic response prediction based on sparse observations

[0135] To evaluate the applicability of the proposed framework in actual engineering monitoring scenarios, the following experimental setup was conducted: 12 non-uniformly distributed displacement sensors were placed on the beam, and 1% random Gaussian noise was superimposed on all observation data to simulate the real field measurement environment. Figure 5 The model was demonstrated in The full-field displacement reconstruction result at time s. As can be seen from the figure, although the input observation points deviate from the true physical trajectory due to noise, the prediction curve output by the network framework does not mechanically overfit these errors. With the embedded partial differential equation (PDE) constraints, the network automatically filters out high-frequency random disturbances in the measurement data and accurately predicts the displacement response that conforms to physical laws in the vast blind zone where no sensors are deployed.

[0136] To further evaluate the generalization ability of the proposed method in the time dimension, a systematic analysis was conducted on the full-time displacement response prediction performance of unobserved nodes under different working conditions. Two representative harmonic excitation conditions were selected as test cases: an external load amplitude of 13N, a dominant frequency of 7.0Hz, and a maximum damage level of 12.3%; and an external load amplitude of 26N, a dominant frequency of 13.0Hz, and a maximum damage level of 22.4%, to test the model's adaptability when both the excitation frequency and damage level change significantly. Figure 6 and Figure 7 Two working conditions are given. A comparison of the displacement time history prediction results with the actual response was conducted. It can be observed that in both low-frequency and high-frequency scenarios, the model can accurately reconstruct the overall vibration pattern of the structure. Especially in the more complex high-frequency severe damage sample, facing the severe oscillations caused by the coupling of high-frequency excitation and stiffness degradation, the network's prediction results maintained good consistency with the actual response throughout the entire time domain, stably tracking the main vibration peaks and troughs without significant phase drift or frequency mismatch. Quantitative results show that the root mean square error (RMSE) of the prediction is 0.2464 mm under low-frequency mild damage conditions, while it is 0.2039 mm under high-frequency severe damage conditions, with the relative errors under both conditions controlled within 4%.

[0137] The above results show that the trained operator model can stably learn the intrinsic mapping relationship between time-varying elastic modulus and dynamic response under different frequency characteristics and different damage evolution levels, and achieve high-precision reconstruction of dynamic displacement response in unobserved areas under sparse observation conditions, thus providing important support for building a structural digital twin model with reliable prediction capabilities.

[0138] Although slight numerical oscillations remain in the damage edge region in the parameter inversion results, the corresponding displacement prediction results still maintain good smoothness and high fidelity, without significant noise amplification. From a structural dynamics perspective, the displacement response is determined by the integral relationship of the stiffness field through the governing equations. The integral operation naturally acts as a low-pass filter for high-frequency fluctuations in the parameter field during propagation. Furthermore, FusionDeepONet learns operator-level mappings from the parameter field to the solution space, rather than point-to-point functional relationships. Under the combined effect of physical and data constraints, the predicted displacement field is implicitly constrained within the physical solution space defined by the structural dynamics equations, thereby suppressing the amplification of local parameter errors to the response layer. These characteristics ensure that the model can still stably output reliable displacement prediction results even with a certain degree of parameter uncertainty.

[0139] Experiments have demonstrated that the estimation method of this invention can simultaneously achieve high-precision parameter inversion and high-fidelity displacement prediction in the dynamics of cantilever beams with time-varying elastic modulus, and possesses excellent robustness, showcasing its engineering application potential as a core modeling tool for structural digital twins and online health monitoring.

[0140] The above embodiments are merely preferred examples of the present invention and are not intended to limit the present invention in any way. Although the present invention has been disclosed above with reference to preferred embodiments, it is not intended to limit the present invention. Therefore, any simple modifications, equivalent changes, and alterations made to the above embodiments based on the technical essence of the present invention without departing from the scope of the present invention should fall within the protection scope of the present invention.

Claims

1. A high-precision prediction method for full-field dynamic displacement response under sparse observation conditions, characterized in that, Includes the following steps: Time-frequency features are extracted from displacement time-series data acquired by sparse sensors to obtain feature vectors characterizing the dynamic state of the system; The feature vector is input into the spatiotemporally decoupled E-Net network to generate a damage factor field that satisfies causal constraints, and a time-varying elastic modulus field is constructed based on the damage factor field and the preset health state elastic modulus benchmark distribution. The time-varying elastic modulus field and external load are input into the feature fusion operator network, and the feature fusion operator network outputs the predicted value of the full-field displacement response of the target structure. The feature fusion operator network includes a training process, which constructs a phased training framework for physical-data collaboration. The framework divides the network optimization process into two stages: forward operator construction and full-field response reconstruction. By introducing physical information constraints based on variational weak forms, the predicted displacement response satisfies the structural dynamics control equations. In the forward operator construction phase, the trained feature fusion operator network learns the mapping from the elastic modulus field to the displacement field; Given full-field displacement label data, the main objective is to minimize the error between the predicted displacement and the actual displacement. At the same time, variational physical information constraints based on weak form are gradually introduced to train the forward mapping capability of the feature fusion operator network. In the full-field response reconstruction stage, the parameters of the trained feature fusion operator network are frozen, and the joint loss is constructed using only the observation data of sparse sensors and physical residuals to fine-tune the frequency-domain-aware temporal feature encoder and the spatiotemporally decoupled damage factor field generation network. The variational physical information constraint is constructed based on the weak form of the Euler-Bernoulli beam equation, and the requirement for higher-order derivatives of the displacement field is reduced by performing weighted integration and integration by parts on the governing equation.

2. The high-precision prediction method for full-field dynamic displacement response under sparse observation conditions according to claim 1, characterized in that, The time-frequency feature extraction employs a frequency-domain-aware temporal feature encoder, including: The displacement time-series signal of each sensor is subjected to short-time Fourier transform to obtain the corresponding time spectrum; The time-frequency spectra of each sensor are combined into a multi-channel feature map, which is then input into a two-dimensional convolutional neural network for feature extraction. The extracted features are interpolated along the time dimension to restore the original time resolution, and the feature vector is finally output.

3. The high-precision prediction method for full-field dynamic displacement response under sparse observation conditions according to claim 2, characterized in that, The E-Net network comprises a spatial basis function network and a temporal coefficient network. The spatial basis function sub-network is used to learn a set of spatially impaired basis functions, and the temporal coefficient network uses the feature vector... As input, predict the corresponding time coefficient increment. The damage factor field Generated by the following formula to embed hard causal constraints: ； in, The Sigmoid function, with a coordination coefficient of 0.95, restricts the damage to the interval [0, 0.95]. Learnable parameters for controlling the sensitivity of the Sigmoid function. For unconstrained cumulative damage field, Learnable parameters that control the center offset of the Sigmoid function.

4. The high-precision prediction method for full-field dynamic displacement response under sparse observation conditions according to claim 2, characterized in that, The feature fusion operator network adopts the DeepONet architecture and includes a feature linear modulation mechanism, including: A dual-stream branch network is used to receive the time-varying elastic modulus field and the external load, and output a set of dynamic modulation parameters and a projection coefficient vector; The trunk network is used to receive spatiotemporal coordinates and, in at least one hidden layer, to perform an affine transformation on the network activation values ​​using the dynamic modulation parameters, ultimately outputting a set of adaptive spatiotemporal response basis functions. The full-field displacement response is reconstructed by performing an inner product between the coefficient vector output by the branch network and the basis function output by the trunk network.

5. The high-precision prediction method for full-field dynamic displacement response under sparse observation conditions according to claim 1, characterized in that, During the training process of the forward operator construction or full-field response reconstruction stage, a dynamic weight fusion algorithm based on gradient variance is adopted to adaptively balance the weights between the data loss term and the physical residual loss term, using the ReLoBRaLo-GVW algorithm.

Images