A long-span arch bridge damage diagnosis method based on differential equation embedding

Abstract

The application relates to a long-span arch bridge damage diagnosis method based on differential equation embedding, and a physical information driven inverse problem operator learning framework is constructed; the framework constructs a dual-channel operator network architecture integrating node coordinates and structure displacement, realizes nonlinear operator reconstruction from limited sparse data to global continuous displacement field, and further introduces dimensionless and discrete node mapping strategies to overcome the numerical ill-posedness of the inverse problem. By decoupling the local stiffness into independent learnable parameters attached to the spatial grid, the global physical residual constraint is accurately constructed. Finally, a multi-source composite loss function of data, boundary conditions and physical residual is constructed, the network weight and discrete stiffness parameter are synchronously updated in the back propagation through joint optimization, the stiffness space distribution meeting the physical consistency is finally output, the accurate positioning and quantitative evaluation of damage are realized, and the effectiveness and practicability of the method are verified through numerical simulation and scale test.

Description

Technical Field

[0001] This invention belongs to the field of bridge structural health monitoring technology, and relates to a damage diagnosis method for long-span arch bridges based on differential equation embedding. In particular, it relates to a non-uniform damage identification method for long-span arch bridges that integrates discrete differential equations and DeepONet, and constructs a physical information-driven inverse problem operator learning framework. Background Technology

[0002] Long-span arch bridges, with their superior spanning capacity and mechanical properties, are a core component of transportation infrastructure in the mountainous areas of western my country. However, with increasing service life, structures are prone to local stiffness degradation and damage accumulation under the coupled effects of cyclic loads and the environment. Accurate damage identification has become a core requirement for ensuring operational safety. Current mainstream damage identification technologies have significant bottlenecks: methods based on dynamic and static response indices are either insufficiently sensitive to minor local damage or rely on complex analytical derivations, making them difficult to adapt to the nonlinear behavior of long-span arch bridges; methods based on finite element model updates have extremely high computational costs, are prone to getting trapped in local optima in inverse problem solving, exhibit significant ill-posedness in multi-damage scenarios, and often result in false damage misjudgments; while purely data-driven methods have strong feature extraction capabilities, they are "black box" models lacking physical interpretability and highly dependent on massive amounts of labeled data, making them difficult to adapt to scenarios where damage samples are scarce in actual engineering projects.

[0003] In recent years, Physical Information Neural Networks (PINNs) have provided a new approach to integrating mechanical mechanisms with data-driven approaches by embedding governing equations as regularization constraints into the network. However, existing parameter inversion methods for arch bridges still have significant limitations. Most methods construct physical constraints based on continuous differential equations, assuming that the structural stiffness is continuous and smooth throughout the entire field. This approach can only achieve overall stiffness inversion and cannot capture the stiffness abrupt changes and discontinuous features caused by local damage in actual engineering, easily leading to blurred damage boundaries and missed detection of minor damage. Some methods implicitly encode stiffness parameters in the network weights without establishing an explicit mapping with spatial nodes. When facing multi-parameter inversion scenarios, ill-posedness is not effectively suppressed, resulting in insufficient quantization accuracy and stability. In addition, existing methods have poor adaptability to low-quality monitoring data. Under actual service scenarios such as noise interference, sparse measuring points, and sensor failures, the identification error will increase significantly, making it difficult to meet the needs of engineering applications.

[0004] In summary, existing damage identification methods for long-span arch bridges suffer from three core shortcomings: a mismatch between continuous physical constraints and the discontinuous characteristics of local damage, hindering the accurate localization and quantification of inconsistent damage; unresolved ill-posedness in the inversion process, leading to spurious damage, large error fluctuations, and low computational efficiency; and insufficient robustness to low-quality monitoring data in actual service environments, limiting their engineering applicability. Therefore, it is urgent to overcome the compatibility issues of existing technologies and construct a new damage identification scheme for long-span arch bridges that integrates mechanical mechanisms and the advantages of deep learning, enabling accurate identification of local inconsistent damage and adaptability to low-quality data scenarios in actual engineering projects. Summary of the Invention

[0005] In view of this, in order to solve the problems of poor physical interpretability of traditional deep learning models and weak generalization ability due to reliance on massive labeled data in existing long-span arch bridge damage identification methods, this invention provides a long-span arch bridge damage diagnosis method based on differential equation embedding. The arch bridge control differential equation is used as a strong prior knowledge explicit embedding loss function, and the continuous physical response field is reconstructed by relying on the dual-channel architecture of operator network. The ill-posedness problem of inversion solution is effectively overcome by physical-data dual-drive.

[0006] To achieve the above objectives, the present invention provides the following technical solution:

[0007] A damage diagnosis method for long-span arch bridges based on differential equation embedding includes the following steps:

[0008] S1. Based on the DeepONet dual-branch architecture based on operator approximation theory, the displacement monitoring response at the structural measurement points is used as the input of the branch network (Branch Net), and the continuous spatial coordinates of the arch bridge are used as the input of the trunk network (Trunk Net). The global continuous displacement field is output through dot product aggregation. The automatic differentiation technique is used to solve the derivatives of the displacement field with respect to the spatial coordinates.

[0009] S2. Based on the geometric nonlinear differential equation of the long-span arch bridge, a dimensionless strategy is introduced to reconstruct the physical control equation, eliminating the numerical solution instability caused by the large difference in stiffness parameters. At the same time, based on the node discretization method, the solution domain of the long-span arch bridge is discretized into N spatial nodes. The derivatives obtained in step S1 are mapped and paired one by one with the dimensionless node stiffness coefficients discretized in step S2, and substituted into the reconstructed discrete form differential equation to construct physical residual terms based on each node and embed them into the model.

[0010] S3. Construct a multi-objective joint loss function that includes data loss, boundary condition loss and physical residual loss. Through the gradient descent algorithm, update the stiffness parameters of the deep neural network weights and discrete nodes synchronously during the backpropagation calculation of the network. Finally, output the spatial distribution of the stiffness coefficients of the entire bridge to complete the accurate location and quantitative assessment of the damage.

[0011] S4. Based on the spatial distribution of stiffness coefficients obtained from step S3, locate the stiffness abrupt change region to determine the damage location, quantify the stiffness reduction magnitude to determine the damage degree; verify the effectiveness and practicality of the method through numerical simulation and scaled-down tests.

[0012] A computer-readable storage medium storing a computer program that, when executed by a processor, implements the steps of the above-described method for diagnosing damage to long-span arch bridges based on differential equation embedding.

[0013] An electronic device includes a memory, a processor, and a computer program stored in the memory. When the processor executes the program, it implements the steps of the above-described method for diagnosing damage to long-span arch bridges based on differential equation embedding.

[0014] The beneficial effects of this invention are as follows:

[0015] 1. The damage diagnosis method for long-span arch bridges based on differential equation embedding disclosed in this invention firstly reconstructs the discrete form of the differential equations using a node-based discretization method, thereby transforming the stiffness coefficients of each node into independent optimizable parameters. Then, relying on the DeepONet dual-network structure, the displacement field is reconstructed, and the derivatives of each order are solved and mapped one-to-one with the discrete node stiffness coefficients to construct a model based on the physical residuals of each node. Secondly, a composite loss function is used for joint optimization to accurately invert the stiffness coefficients of each node, achieving precise identification of the location and extent of local damage. Finally, the effectiveness and practicality of this method are verified through numerical simulations and scaled-down tests. The results show that the proposed method strictly controls the stiffness quantification error within 2.32% under single-damage conditions; under complex conditions with multiple concurrent damages, the vast majority of identification errors remain below 4%. Compared with traditional numerical methods, the error reduction is generally 70%, and spurious damage is significantly reduced, demonstrating excellent accuracy and numerical stability, and enabling efficient and high-precision local damage location and degree quantification in bridges.

[0016] 2. The damage diagnosis method for long-span arch bridges based on differential equation embedding disclosed in this invention uses DeepONet as the baseline architecture. It embeds the geometrically nonlinear differential equations characterizing the mechanical behavior of the arch bridge as physical prior constraints, deeply into the global optimization process of the operator network. This architecture not only endows the deep model with rigorous physical interpretability and effectively reduces the algorithm's dependence on large-scale, high-quality labeled data, but also significantly improves its solution efficiency under sparse monitoring data by avoiding the complex high-dimensional matrix iteration and reanalysis of traditional methods. Compared with traditional numerical methods, this method improves computational efficiency by more than 74% under various working conditions.

[0017] 3. The damage diagnosis method for long-span arch bridges based on differential equation embedding disclosed in this invention addresses the bottleneck of traditional numerical methods, which are prone to gradient divergence and getting trapped in local multivalues ​​when dealing with high-dimensional nonlinear objective functions. This study constructs a multi-source objective collaborative optimization framework jointly driven by physical residuals, boundary conditions, and limited observation data. Based on this, a local parameter mapping strategy combining dimensionless mapping and node discretization effectively suppresses numerical ill-conditioned oscillations that easily occur in solving high-order differential equations. This strategy effectively improves feature crosstalk under complex working conditions, enabling the model to achieve high-resolution spatial decoupling and high-fidelity quantitative assessment even when facing multi-point concurrent damage.

[0018] 4. The damage diagnosis method for long-span arch bridges based on differential equation embedding disclosed in this invention exhibits excellent robustness under harsh measurement environments. Even under various signal-to-noise ratio conditions, extremely sparse measurement points, and severe sensor malfunctions, it still maintains an error of less than 5% in identifying the location and extent of damage and reconstructs a high-fidelity global continuous displacement field, demonstrating strong noise resistance and robustness, effectively overcoming the dependence on high-quality labeled data. Through scaled-down model experiments, the proposed method not only accurately inverts the local damage level of the real structure in its initial state but also accurately tracks the local stiffness degradation trend of the structure under multi-point graded progressive loading without relying on real-time iteration of a high-dimensional finite element model. The identified damage evolution trajectory highly matches the actual process of eventual instability and failure.

[0019] Other advantages, objectives, and features of the invention will be set forth in part in the description which follows, and in part will be apparent to those skilled in the art from the following examination, or may be learned from practice of the invention. The objectives and other advantages of the invention can be realized and obtained through the following description. Attached Figure Description

[0020] To make the objectives, technical solutions, and advantages of the present invention clearer, the preferred embodiments of the present invention will be described in detail below with reference to the accompanying drawings, wherein:

[0021] Figure 1 This is a flowchart illustrating the model flow framework of this invention that integrates physical constraints and DeepONet.

[0022] Figure 2 This is a schematic diagram illustrating the derivation of the geometric nonlinear analysis model equations for the arch bridge of the present invention;

[0023] Figure 3 This is a numerical model diagram of the arch bridge of the present invention;

[0024] Figure 4 This is a loss curve during the pre-training process of the PI-DeepONet model of this invention;

[0025] Figure 5 This is a comparison chart of the stiffness identification performance of the PI-DeepONet model of this invention and traditional numerical methods under single-damage conditions; wherein, Figure 5 (a) Damage at L / 8, Figure 5 (b) shows the damage at L / 4. Figure 5 (c) Damage at 3L / 8, Figure 5 (d) represents the damage at L / 2;

[0026] Figure 6 The graph shows a comparison of the relative errors of PI-DeepONet and traditional numerical methods under various single-damage conditions.

[0027] Figure 7 This is a comparison chart of the stiffness recognition performance of the PI-DeepONet model of this invention and traditional numerical methods under multiple damage conditions; wherein, Figure 7 (a) Damage at L / 8 and L / 4, Figure 7 (b) Damage at L / 8 and L / 2, Figure 7 (c) Damage at L / 4 and 5L / 8, Figure 7 (d) Damage at L / 8, L / 2, and 3L / 4;

[0028] Figure 8 The graph shows a comparison of the relative errors of PI-DeepONet and traditional numerical methods under various working conditions with multiple damages.

[0029] Figure 9 A comparison chart of noise levels during verification under low data quality conditions;

[0030] Figure 10 This is a comparison chart of displacement field data reconstruction results under different noise levels during verification under low data quality conditions. Figure 10 (a) is operating condition 3-30. Figure 10 (b) is operating condition 3-31. Figure 10 (c) is operating condition 3-32. Figure 10 (d) refers to operating condition 3-33;

[0031] Figure 11 This is a comparison chart of the stiffness coefficient inversion results of the PI-DeepONet model under different noise levels during low data quality verification. Figure 11 (a) is operating condition 3-30. Figure 11 (b) is operating condition 3-31. Figure 11 (c) is operating condition 3-32. Figure 11 (d) refers to operating condition 3-33;

[0032] Figure 12This is a comparison of the stiffness coefficient inversion results of the PI-DeepONet model of this invention under different sensor numbers during low data quality verification. Figure 12 (a) is operating condition 3-34. Figure 12 (b) is operating condition 3-35. Figure 12 (c) is operating condition 3-36. Figure 12 (d) refers to operating condition 3-37. Figure 12 (e) refers to operating condition 3-38;

[0033] Figure 13 This is a comparison chart of displacement field data reconstruction results under different numbers of sensor failures during low data quality verification. Figure 13 (a) is operating condition 3-39. Figure 13 (b) is for operating condition 3-40. Figure 13 (c) is operating condition 3-41. Figure 13 (d) refers to operating condition 3-42;

[0034] Figure 14 A comparison of the stiffness coefficient inversion results of the PI-DeepONet model of this invention under different numbers of sensor failures during low data quality verification is shown in the figure. Figure 14 (a) is operating condition 3-39. Figure 14 (b) is for operating condition 3-40. Figure 14 (c) is operating condition 3-41. Figure 14 (d) refers to operating condition 3-42;

[0035] Figure 15 This is a diagram showing the arrangement of the test arch ribs in the indoor scaled-down test.

[0036] Figure 16 This is a comparison chart of loading curves at different observation sections during indoor scale-down tests, where: Figure 16 (a) is the initial damage load-time curve. Figure 16 (b) is the initial damage displacement-time curve. Figure 16 (c) shows the load-time curve of the entire bridge under load. Figure 16 (d) is the full-bridge loading displacement-time curve;

[0037] Figure 17 These are diagrams illustrating the damage at each stage during the indoor scale-down test verification, including: Figure 17 (a) shows the initial damage crack. Figure 17 (b) is the final destruction;

[0038] Figure 18 The diagram shows the identification results of single-damage conditions in the indoor scale-down test.

[0039] Figure 19This is a diagram showing the identification results of multiple damage conditions during indoor scale-down testing; where: Figure 19 (a) Under load level 12 Figure 19 (b) is under a load of level 16. Detailed Implementation

[0040] The following specific examples illustrate the implementation of the present invention. Those skilled in the art can easily understand other advantages and effects of the present invention from the content disclosed in this specification. The present invention can also be implemented or applied through other different specific embodiments, and various details in this specification can also be modified or changed based on different viewpoints and applications without departing from the spirit of the present invention.

[0041] like Figure 1 The damage diagnosis method for long-span arch bridges based on differential equation embedding, as shown, includes the following steps:

[0042] S1. Based on the DeepONet dual-branch architecture based on operator approximation theory, the displacement monitoring response at the structural measurement points is used as the input of the branch network (Branch Net), and the continuous spatial coordinates of the arch bridge are used as the input of the trunk network (Trunk Net). The global continuous displacement field is output through dot product aggregation. The automatic differentiation technique is used to solve the derivatives of the displacement field with respect to the spatial coordinates.

[0043] Specifically, settings To characterize the mapping operator corresponding to the behavior of the bridge system, its input function is the displacement u at the structural measurement point extracted from the high-dimensional hidden layer state features. Let be the corresponding continuous spatial response function. For Arbitrary spatial coordinates of the arch bridge in the continuous space of the domain Used to construct continuously differentiable spatial basis, output For the physical response at this point, adaptive tensor aggregation is performed on these two sets of features at the network end. To achieve effective decoupling and reconstruction of system input features and spatial dimensions, this model adopts a dual-channel architecture of Branch Net and Trunk Net. The model effectively overcomes the limitations of traditional point-to-point mapping and realizes nonlinear operator reconstruction from finite sensor observation data to a full-bridge high-fidelity continuous physical response field.

[0044] The Branch Net's role is to map the discretized input function space to the feature space. In real-world scenarios, the input function u typically cannot be directly input in analytical form; instead, it is represented as a vector of observations at m discrete sensor locations. Branch Net, as a deep feature extractor, aims to map the high-dimensional discrete observation vector to a low-dimensional variable space and output a feature vector. .

[0045] The purpose of a Trunk Net is to provide a continuous set of basis functions for the output domain. This network is independent of the external excitation input u, receiving only continuous spatial coordinates within the domain defined by the arch bridge structure. As input, the core function of Trunk Net is to adaptively construct a set of continuous and differentiable basis functions within its domain through multiple nonlinear transformations, and output feature vectors. .

[0046] Dot Product: At the end of the dual-channel architecture, the network uses the Dot Product to adaptively aggregate the discrete state features extracted by the Branch Net with the continuous spatial basis constructed by the Trunk Net, thereby reconstructing the physical field response in the continuous domain.

[0047] (1)

[0048] S2. Based on the geometric nonlinear differential equations of long-span arch bridges, a dimensionless strategy is introduced to reconstruct the physical control equations, eliminating numerical instability caused by large differences in stiffness parameters. Simultaneously, based on a node-based discretization method, the solution domain of the long-span arch bridge is discretized into N spatial nodes. The derivatives obtained in step S1 are mapped and paired one-to-one with the dimensionless node stiffness coefficients discretized in step S2, and substituted into the reconstructed discrete differential equations to construct physical residual terms based on each node and embed them into the model. To overcome the numerical ill-posedness caused by the large magnitude when solving the inverse problem, the model introduces a dimensionless and discrete node mapping strategy, explicitly defining the local stiffness to be inverted as an independent dimensionless parameter bound to the spatial nodes, thereby accurately calculating the spatial physical residuals at each node.

[0049] The aforementioned dual-channel operator architecture achieves high-fidelity reconstruction of discrete observation data into a continuous spatial displacement field, but it exhibits low sensitivity to local damage characteristics. The mapping paradigm, which relies solely on data fitting, lacks the support of underlying mechanical laws, making it difficult to accurately locate and quantitatively assess local damage. Therefore, we consider introducing a physical information neural network architecture, explicitly embedding the differential equations controlling the deformation and force balance of the arch bridge into the optimization of the operator network, thereby transforming damage identification into a solution process constrained by mechanical laws.

[0050] Specifically, based on traditional deflection theory, an additional bending moment caused by the reaction force at the arch abutment was considered, and a geometrical nonlinear analysis model for arch bridges was established. Based on this, for bridges such as... Figure 2 For any hingeless arch bridge, neglecting the influence of temperature conditions, the geometrically nonlinear differential governing equation can be expressed as:

[0051] (2)

[0052] (3)

[0053] In the formula, and These represent the horizontal and vertical displacements at point x, respectively. E is the horizontal angle; E is the elastic modulus of the material; A is the cross-sectional area; M is the bending moment; I is the moment of inertia of the cross section.

[0054] For a typical hingeless arch, the arch feet at both ends are fixed. Therefore, the vertical displacement and rotation of the arch feet are zero, that is:

[0055] (4)

[0056] In the formula, l represents the calculated span.

[0057] The aforementioned geometric nonlinear governing equations establish a nonlinear mapping relationship between the structural mechanical response and the cross-sectional stiffness. When local damage occurs in a structure, its physical essence manifests as a degradation of local stiffness. Therefore, the residuals of the aforementioned differential governing equations can be introduced into the training of a neural network as physical constraint loss terms. By solving the inverse problem, the stiffness distribution parameters of the entire structure can be retrieved, thereby achieving the goal of accurate identification of structural damage.

[0058] However, the bending stiffness EI and tensile stiffness EA of the arch bridge in Equation (2) are on the order of magnitude. Directly solving them can easily induce gradient explosion and convergence stagnation, resulting in numerical ill-posed problems that seriously affect the convergence performance of the model. To address this, this study introduces a dimensionless strategy. Specifically, instead of directly using the stiffness parameters as the parameters to be inverted, dimensionless coefficients are introduced through Equation (5), transforming the original large-scale inversion problem into a problem of identifying elastic modulus coefficients within the range of [0,1], thereby significantly improving the convergence efficiency and stability of the algorithm.

[0059] (5)

[0060] Substituting equation (5) into equation (2), we get:

[0061] (6)

[0062] To reduce dimensionless coefficients Multiply both terms on the left and right sides of equation (6) by have to:

[0063] (7)

[0064] Furthermore, considering that the stiffness parameter is a global parameter, it is impossible to identify local damage.

[0065] Therefore, the continuous solution domain of the governing equations is discretized into N spatial nodes. Discretized into: a network set consisting of N nodes Unlike the implicit expression of physical parameters through network weights, this method uses the stiffness coefficient field to be identified. Explicitly defined as a set of independently optimizable scalar parameters. To ensure the coefficients always remain positive, an exponential space is introduced. For any measurement point... Its corresponding local stiffness is expressed as:

[0066] (8)

[0067] (9)

[0068] in The baseline elastic modulus; It corresponds to the spatial location. Learnable dimensionless parameters; Represents the exponential function This ensures that the dimensionless coefficients always remain non-negative.

[0069] S3. Construct a multi-objective joint loss function that includes data loss, boundary condition loss and physical residual loss. Through the gradient descent algorithm, update the stiffness parameters of the deep neural network weights and discrete nodes synchronously during the backpropagation calculation of the network. Finally, output the spatial distribution of the stiffness coefficients of the entire bridge to complete the accurate location and quantitative assessment of the damage.

[0070] Specifically, to incorporate the aforementioned discretized local stiffness parameters into network training to drive model updates, a specific tensor injection and residual evaluation mechanism needs to be constructed. This involves defining a discrete mapping operator. : Make each spatial coordinate The index k in the parameter vector is uniquely corresponding. When the network is at the configuration point... When solving the residuals of the governing equations, the dimensionless coefficient term The value is directly injected into the governing equations from the k-th component through an exponential transformation and in the form of a zero-order tensor. At this point, the physical residual... exist The local form at a point can be expressed as about Functions of its derivatives:

[0071] (10)

[0072] (11)

[0073] In the formula, These are the weights of the neural network.

[0074] Therefore, the physical residual loss function of this method can be expressed as:

[0075] (12)

[0076] To ensure that the neural network conforms to the physical constraints of the boundary region, the boundary condition loss function is:

[0077] (13)

[0078] In addition, the displacement monitoring data of the arch bridge is used to add a data loss function to the network as follows:

[0079] (14)

[0080] Therefore, for the damage identification problem, the overall optimization objective is defined as a weighted combination of multi-source errors, encompassing data-driven observation errors, boundary conditions, and the residuals of the governing equations related to physical constraints. The overall loss function can be expressed as:

[0081] (15)

[0082] S4. Based on the spatial distribution of stiffness coefficients obtained from step S3, locate the stiffness abrupt change region to determine the damage location, quantify the stiffness reduction magnitude to determine the damage degree; verify the effectiveness and practicality of the method through numerical simulation and scaled-down tests.

[0083] Numerical simulation verification of the embodiment

[0084] (1) Overview of the numerical model; the numerical simulation takes the design requirements of a certain upper-bearing concrete arch bridge as a reference, such as Figure 3 As shown, the main arch axis adopts a catenary, with a calculated span l = 600 m, rise f = 125 m, rise-to-span ratio f / l = 1 / 4.8, arch axis coefficient m = 1.9, and main arch ring cross-sectional area A = 18.6375 m². 2 The moment of inertia of the cross section is I = 232.024 m. 4 The elastic modulus E = 4.075 × 10 10 N / m 2 .

[0085] (2) Working condition settings: The structural damage is simulated by the element stiffness reduction method, that is, by proportionally reducing the elastic modulus of the elements near a specific location to characterize different degrees of local damage. To comprehensively evaluate the effectiveness of the identification method, two types of damage were considered, as shown in Table 1: single damage and multiple damage. First, for single damage scenarios, four key sections distributed along the span direction (L / 8, L / 4, 3L / 8, and 1L / 2) were selected, and damage levels of 5%, 10%, 15%, and 20% were set respectively to examine the model's ability to identify damage from slight to significant at different spatial locations. Second, considering the complexity of concurrent multi-source damage, multi-point concurrent damage scenarios were constructed, covering damage combinations of adjacent locations (L / 8 and L / 4), large span intervals (L / 8 and L / 2), and asymmetric distributions (L / 4 and 5L / 8). Furthermore, more complex scenarios of concurrent damage at different levels at L / 8, L / 2, and 3L / 4 were introduced to verify the identification accuracy of the method under multiple damage interference and damage levels (5%~20%).

[0086] Table 1. Settings for each operating condition

[0087]

[0088] (3) Model training configuration; The numerical computing platform is configured with an Intel Core i5-13490F (2.5 GHz) processor and an NVIDIA RTX 4060Ti GPU. The algorithm program is written in Python 3.7 and implemented in the PyTorch 1.13.1 deep learning framework.

[0089] Given that the geometric control equations contain higher-order differential terms, Tanh is chosen as the activation function to ensure the smoothness and continuous differentiability of the network's output response field in the calculation of second-order and higher derivatives. For optimization, the Adam optimization algorithm is employed, with a total of 50,000 training epochs and an initial learning rate of 0.001. To prevent the model from getting stuck in local optima or experiencing gradient oscillations in the later stages of training, an exponentially decaying learning rate scheduling strategy is introduced, setting the learning rate to 95% of the previous stage every 1000 epochs.

[0090] In the joint optimization of PINNs, the reasonable allocation of weights for each loss term is crucial for the model to accurately capture the laws of mechanics. Therefore, before formal training, a magnitude assessment experiment is first conducted, temporarily setting all loss weight coefficients to 1. This is achieved through... Figure 4 Observations of the loss curves during pre-training revealed a significant imbalance in the magnitudes of the loss terms due to the fundamental difference between the physical dimensions and numerical scales of the physical constraints. Therefore, the training hyperparameters were set as shown in Table 2.

[0091] Table 2 Hyperparameter settings for model training

[0092]

[0093] (4) Single damage identification; Figure 5 The performance of PI-DeepONet (blue) and traditional numerical methods (red) in identifying multi-scale damage at key cross sections of a full bridge was compared and evaluated. Figure 5 (a) Damage at L / 8, Figure 5 (b) shows the damage at L / 4. Figure 5 (c) Damage at 3L / 8, Figure 5 (d) Damage at L / 2. Overall, the proposed PI-DeepONet demonstrates significant advantages in damage localization resolution and quantization accuracy. 1) In terms of spatial localization accuracy, PI-DeepONet exhibits excellent local feature capture capabilities. The minimum points of its stiffness coefficient curve are precisely locked near the preset damage location, without significant phase shift or peak misalignment. Furthermore, the curve converges rapidly after deviating from the damage center. In contrast, traditional methods show significant localization shifts at L / 8, and multiple false damages affecting damage identification occur in various conditions, making it difficult to accurately define the damage range. 2) Regarding the quantification of damage severity, whether it is 5% early minor damage or 20% severe damage, the extreme values ​​identified by this model are highly consistent with the preset stiffness reduction rate. In contrast, traditional methods have very limited quantization capabilities, and the quantification results of damage severity under various conditions are unsatisfactory. Especially at the mid-span L / 2, the stiffness coefficient calculated by traditional methods far exceeds the preset damage severity. 3) In the non-damage region, PI-DeepONet effectively suppresses numerical divergence caused by ill-posedness in the inverse problem by leveraging operator approximation theory and regularization constraints of physical equations. Its non-damage region curve closely follows the baseline, and the global oscillation amplitude is strictly controlled within 2%. However, traditional methods are accompanied by high-frequency and large-amplitude numerical oscillations across the entire domain, which seriously obscures the objective identification of real damage.

[0094] To further quantify and evaluate the overall performance of the model, the minimum stiffness near the damage location was selected as the final identification value. Table 3 details the quantitative identification values, relative errors, and computation time for each working condition. A detailed analysis of Table 3 leads to the following conclusions: 1) From the perspective of relative error, PI-DeepONet exhibits extremely high quantitative accuracy. Its absolute relative error is strictly controlled within 2.5% for all working conditions, and the error for more than half of the conditions is below 1.0%. This indicates that the algorithm has a stable and high-precision fitting ability for stiffness reduction at different locations and to varying degrees. In contrast, the quantitative accuracy of traditional numerical methods exhibits severe instability. When damage occurs at L / 8, the traditional method is not sensitive enough to identifying the degree of damage, with an average error of 8.82%, indicating a significant underestimation of the damage degree; while when damage occurs at L / 2, the average error is as high as -14.8%, indicating a serious overestimation of the damage degree. The traditional method exhibits large error fluctuations, proving that it is prone to getting trapped in local optima and divergent distortion when dealing with nonlinear inverse problems. 2) In terms of computational efficiency, PI-DeepONet demonstrates an overwhelming advantage. Traditional numerical methods, relying on frequent high-dimensional matrix inversions and physics reanalysis, have single-case processing times ranging from 1027.15 seconds to 2098.17 seconds. PI-DeepONet, through its operator mapping mechanism, successfully avoids the cumbersome physics iteration process, significantly reducing its processing time to between 189.53 seconds and 535.74 seconds. Overall, PI-DeepONet reduces the average processing time by 74.24% compared to traditional methods.

[0095] Table 3 Results of Single Damage Condition Identification

[0096]

[0097] Figure 6 This image compares the error accuracy of PI-DeepONet and traditional numerical methods for identifying multi-scale damage at key cross-sections of a full bridge. Figure 6 As can be seen, PI-DeepONet achieves a significant leap in accuracy under most operating conditions. Statistical data shows that the error reduction rate of this method is generally above 70%. Particularly noteworthy is that in the regions where numerical methods perform worst, namely operating conditions 3-1 to 3-3 and 3-13 to 3-16, the error reduction of this model is close to or exceeds 90%, reaching a maximum of 99%. This result fully demonstrates that the PI-DeepONet model not only effectively overcomes the shortcomings of traditional numerical methods in terms of divergence and large errors in specific sensitive areas, but also achieves high-precision and high-stability identification of structural damage across the entire domain, showcasing the enormous potential and superiority of the physics-based deep learning framework in solving complex inverse problems.

[0098] (5) Multi-damage identification; Figure 7 The performance of PI-DeepONet (blue) and traditional methods (red) in stiffness identification under multi-damage conditions is demonstrated. Figure 7 (a) Damage at L / 8 and L / 4, Figure 7 (b) Damage at L / 8 and L / 2, Figure 7 (c) Damage at L / 4 and 5L / 8, Figure 7 (d) represents the damage at L / 8, L / 2, and 3L / 4. The results show that damage at multiple locations significantly increases the solution dimensionality and coupling complexity of the nonlinear inverse problem, further validating the comprehensive performance of the method.

[0099] Under multi-damage conditions, PI-DeepONet demonstrates superior spatial feature decoupling capabilities. The minimum values ​​of its recognition curve accurately pinpoint the corresponding damage locations, and adjacent damage intervals are clearly defined without spatial dimensional damage fusion. In contrast, traditional methods exhibit significant lateral drift in positioning at L / 8 and L / 4 due to the cumulative influence of multiple damage sources. Furthermore, PI-DeepONet successfully shields against coupling interference from multi-point concurrency, achieving high-precision recognition of the true stiffness reduction rate from 5% to 20% under various conditions without significant attenuation or amplification. In comparison, traditional methods experience a dramatic increase in distortion when handling multi-point inverse problems, and the identified damage level is generally greater than the actual damage under the cumulative influence of multiple damages.

[0100] As shown in Table 4, the PI-DeepONet model exhibits extremely high quantization accuracy and stability. The results in Table 4 demonstrate that under all operating conditions, the relative error of this model is strictly controlled within 3% for the vast majority of cases. Even under complex operating conditions involving three different damage gradients, its maximum error is only 2.48%, accurately achieving independent reconstruction of different spatial locations and damage degrees. In contrast, traditional methods are prone to falling into local optima traps during multivariate optimization, failing to effectively isolate and quantify multiple interfering damage signals, resulting in a maximum error as high as -16.83%. Furthermore, as the number of damage nodes to be inverted increases, the processing time of PI-DeepONet does not show a significant increase, remaining stable between 342.04 seconds and 509.81 seconds. Traditional methods, limited by cumbersome physical domain iteration and high-dimensional search, have a single-condition processing time as high as 1437.71 seconds to 1838.83 seconds. Statistical analysis shows that the average processing time of PI-DeepONet per operating condition is reduced by 73.94% compared to traditional methods.

[0101] Table 4. Results of Multi-Damage Condition Identification

[0102]

[0103] observe Figure 8It can be observed that under multiple damage conditions, compared with traditional numerical methods, the error reduction rate of this model is generally stable at over 70% in most damage conditions, demonstrating excellent accuracy improvement capability. Although the error reduction rate shows negative fluctuations of -147% and -93% at L / 4 of condition 3-20 and 5L / 8 of condition 3-26, respectively, combined with the relative error histogram... Figure 8 As can be seen, the absolute identification errors of both methods are now within a relatively small range, and do not substantially affect the overall accuracy evaluation of the model. In summary, the PI-DeepONet model achieves significant improvements in damage localization capability and damage quantification accuracy compared to traditional methods, regardless of whether the damage is single or multi-damage conditions, effectively overcoming the ill-posedness of the inverse problem.

[0104] Implementation Examples: Low Data Quality Condition Verification

[0105] (1) Recognition performance under noise influence; Considering that sensor signal acquisition is often affected by noise in service environment, in order to examine the recognition performance of this method under noise conditions, this section sets up four progressive noise conditions based on the data of condition 3-8. The signal-to-noise ratio is set to 20 dB (corresponding to condition 3-30), 15 dB (corresponding to condition 3-31), 10 dB (corresponding to condition 3-32) and 5 dB (corresponding to condition 3-33), respectively. The lower the signal-to-noise ratio, the higher the level of noise introduced.

[0106] The introduction of noise adopts the following method: Add, for example, noise level data. Figure 9 As shown. Wherein:

[0107]

[0108]

[0109] In the formula, Signal-to-noise ratio; For displacement data with added noise, This is noise data; for , Gaussian white noise; n is the number of displacement data measurement points.

[0110] Combination Figure 10As shown in Table 5, PI-DeepONet exhibits excellent denoising and reconstruction performance under multi-scale environmental noise interference. Although the contaminated input signal exhibits severe high-frequency oscillations and distortion as the signal-to-noise ratio decreases, the reconstructed displacement curve output by PI-DeepONet consistently matches the original true value. Even in a 5dB noisy environment, the determination coefficient of the reconstructed displacement field remains at a high value of 0.9512, and both MAE and RMSE are tightly controlled within 10. -4 The noise level is relatively low. This excellent noise resistance relies on the physical regularization term forcing the mapping of the network space to follow the laws of mechanics, which fundamentally blocks the overfitting tendency of the data-driven model to high-frequency non-physical noise, thus giving it strong robustness in harsh engineering monitoring environments.

[0111] Table 5. Multi-index evaluation of displacement field reconstruction performance

[0112]

[0113] After verifying the model's ability to reconstruct noisy displacement fields with high fidelity, Figure 11 Table 6 further reveals the model's recognition performance under high-frequency environmental noise. PI-DeepONet exhibits strong noise robustness in spatial localization; even under extreme interference of 5 dB, its minimum point remains precisely located near L / 4 without significant phase drift, fully validating the efficient decoupling ability of physical constraints on spatial features in complex environments. However, limited by heavily polluted input data, the model's rigidity quantification accuracy shows some sensitivity to extreme noise: under light to moderate noise conditions of 15 dB and above, its quantification error is strictly controlled within 5%; but when the signal-to-noise ratio deteriorates to 10 dB and below, noise excites local oscillations in the non-damaged area, leading to an overestimation of the damage level at extreme values, such as an error of -26.15% in condition 5-4. Overall, however, the results show that this model still possesses highly reliable damage recognition performance under harsh monitoring environments.

[0114] Table 6. Damage identification results for each signal-to-noise ratio.

[0115]

[0116] (2) Recognition performance under incomplete spatial data

[0117] 1) The impact of the number of sensors

[0118] The number of sensors deployed also has a significant impact on structural health monitoring. The sparsity of the measuring point density directly determines the completeness of the monitoring data in the spatial domain and also affects the system's economics. Therefore, this section systematically analyzes the impact of different numbers of sensors on the model's recognition performance based on working condition 3-8. As shown in Table 7, five working conditions were designed with the number of sensors decreasing from 60 to 20.

[0119] Table 7 Operating Conditions with Different Number of Sensors

[0120]

[0121] The results for each working condition are as follows Figure 12 As shown in Table 8, PI-DeepONet exhibits strong robustness in sparse data scenarios. Even in the extreme case where the number of sensors is reduced to 20, its local stiffness inversion error remains within 3.31%, fully validating that introducing physical constraints can effectively compensate for the information loss caused by insufficient discrete observation data. Furthermore, as the density of measurement points decreases, the model only induces spurious damage in the non-damage region due to spatial resolution limitations when the number of sensors drops to 30 and 20. In summary, this framework successfully overcomes the ill-posedness of the inverse problem under sparse data by deeply integrating physical priors and data-driven approaches, significantly reducing the hardware dependence of high-precision damage identification on traditional high-density sensor networks.

[0122] Table 8. Identification Results for Each Working Condition

[0123]

[0124] 2) Impact of sensor failure

[0125] In practical engineering applications, sensors operate in complex environments for extended periods, inevitably leading to data acquisition failures that weaken the continuous reliability of structural health monitoring systems. Therefore, this study systematically investigates the identification performance under different sensor failure conditions based on operating conditions 3-8. The experiment used an initial set of 61 sensors as a baseline and employed a gradient random loss strategy to construct operating conditions: first, data from four sensors were randomly removed; then, based on the existing failure nodes, the number of failures was further increased, successively forming simulated scenarios with 8, 16, and 32 failure nodes. As shown in Table 9, with the increase in the number of faulty sensors, the sensor failure rate significantly increased from 6.56% to 52.46%. This gradient operating condition design aims to verify the robustness of physical constraints in damage identification and data reconstruction under harsh failure environments.

[0126] Table 9 Sensor Fault Condition Table

[0127]

[0128] Under the aforementioned gradient random failure condition Figure 13 This demonstrates the reconstruction performance of the PI-DeepONet model on missing data. As the number of fault sensors increases progressively from 4 to 32, the discrete spatial data input to the model exhibits high sparsity and discontinuity. However, the reconstructed displacement values ​​at all fault nodes remain accurately anchored to the true displacement response curves. This result verifies that embedding partial differential equations as physical constraints into a deep operator network allows the model to move beyond simply fitting discrete data and learn the mechanical laws governing the global deformation of the arch bridge from the ground up.

[0129] Combination Figure 14 Table 10, showing the stiffness inversion curves and identification results, further reveals the impact of sensor failures on the model's damage identification performance. Regarding damage quantization accuracy, PI-DeepONet demonstrates excellent fault tolerance and robustness: even under the extreme data loss condition of 32 sensor failures, the model's quantitative inversion error only slightly increases from 2.62% to 4.44%, remaining strictly controlled within the 5% engineering tolerance limit. This fully demonstrates that the global prior constraints of the physical equations can effectively solve the quantization bias caused by large-scale data loss. In terms of spatial positioning characteristics, except for the significant spurious damage generated near 400m due to severe sparsity of the long-span physical supervision signal under the 52.64% severe failure rate condition, no significant local distortion occurred under other failure rate gradients, and the spatial positioning of the main damage remained accurate throughout. In summary, this model can still maintain high-fidelity quantification and reliable positioning in damage identification even under harsh conditions of large-area hardware failure.

[0130] Table 10 Recognition Results

[0131]

[0132] Indoor scale-down test verification of the example

[0133] To verify the practicality of the research method, experiments were conducted on damaged reinforced concrete arch ribs. This experiment was conducted on a specific upper-bearing, stiffened-frame concrete arch bridge. Figure 15 The test was conducted at a 1:100 scale. Detailed dimensions of the arch rib are shown in Table 11. Static loading was performed using a counterweight mass block combined with a lever system. To comprehensively capture the spatial mechanical behavior of the structure, seven key observation sections (L / 8, L / 4, 3L / 8, L / 2, 5L / 8, 3L / 4, and 7L / 8) were evenly distributed along the longitudinal axis of the arch rib, and displacement gauges, loading point pressure gauges, and strain gauges were deployed. Figure 16 This is a comparison chart of multi-point loading curves. Figure 16 (a) is the initial damage load-time curve. Figure 16(b) is the initial damage displacement-time curve. Figure 16 (c) is the load-time curve of the entire bridge under load. Figure 16 (d) is the displacement-time curve of the entire bridge under load.

[0134] Table 11 Dimensions of the Test Arch

[0135]

[0136] To simulate the progressive damage evolution process from localized cracking to global failure during the service life of an arch bridge, a single-point static load was first applied at section L / 3 as shown in the figure, until the concrete at sections L / 3 and L / 4 showed signs of cracking. Figure 17 The obvious crack in (a) was then unloaded to a zero state to simulate local initial damage. Subsequently, while maintaining the aforementioned initial damage state, single-point progressive loading tests were performed sequentially on nodes L / 6, L / 3, and L / 2 of the left half-span, ensuring that the applied load did not exceed the load applied to the initial damage. Finally, progressively increasing loads based on the equivalent force similarity ratio of the actual bridge columns were simultaneously added to all five loading points of the entire bridge. A steady-state period was set after each loading level to eliminate transient dynamic disturbances until structural damage occurred. Figure 17 The final destruction in (b).

[0137] Therefore, the detailed working conditions of this experiment are shown in Table 12. Based on the initial damage state and loading mode, the experimental working conditions are divided into two main categories: one is the single-damage working condition with loading at different spatial locations; the other is the multi-damage working condition with simultaneous loading at five points. Considering that the complete instability failure of the arch rib observed in the experiment occurred at the 20th load level, in order to effectively reveal the evolution law of structural damage stages and effectively avoid the interference of highly nonlinear and abrupt data in the near-failure stage, the analysis of the multi-damage working conditions specifically selected the 12th and 16th load states in the early and middle stages of failure as the research objects.

[0138] Table 12 Test Condition Settings

[0139]

[0140] The identification results of single-damage conditions are as follows: Figure 18 As shown in the figure. The results indicate that this method can accurately pinpoint the local stiffness degradation region. The significant abrupt change in stiffness reduction occurs within the range of 1.5 m to 2.25 m, which is consistent with the actual preset local damage location in the experiment. In the non-damaged section, the stiffness coefficients identified under the three loading conditions remain between 0.85 and 1.0, which, considering the influence of factors such as noise, is consistent with the condition that the structure has not suffered significant damage.

[0141] Figure 19This study demonstrates the spatial distribution and evolution of the arch rib section stiffness coefficient under multi-point loading with increasing load level. As shown in the figure, when the load reaches level 12, the structure exhibits a significant increase in stiffness degradation in the pre-defined initial damage zone of the left half-span, with the section stiffness coefficient further reduced compared to the single-damage case. Simultaneously, significant stiffness attenuation begins to appear near the 3.75 m and 5.25 m measuring points in the right half-span, indicating the emergence of new local damage under complex stress conditions. With further increases in load level, the damage evolution of the entire bridge shows an accelerated expansion trend: the initial damage in the left half-span continues to worsen, while the stiffness degradation in the newly formed damage zone of the right half-span is particularly pronounced, showing a trend of concentration towards the 5.25 m section, where the stiffness reduction coefficient drops sharply. Combined with actual experimental observations, when the load reaches level 20, the specimen completely loses its bearing capacity and fails completely near the 7L / 8 section. The damage evolution trajectory revealed by this method is highly consistent with the actual physical process of the structure's eventual failure in the experiment, fully verifying the accuracy and engineering applicability of this method in identifying damage under complex stress conditions.

[0142] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and are not intended to limit it. Although the present invention has been described in detail with reference to preferred embodiments, those skilled in the art should understand that modifications or equivalent substitutions can be made to the technical solutions of the present invention without departing from the spirit and scope of the present invention, and all such modifications or substitutions should be covered within the scope of the claims of the present invention.

Claims

1. A damage diagnosis method for long-span arch bridges based on differential equation embedding, characterized in that, Includes the following steps: S1. Based on the DeepONet dual-branch architecture based on operator approximation theory, the displacement monitoring response at the structural measurement point is used as the input of the branch network, and the continuous spatial coordinates of the arch bridge are used as the input of the backbone network. The continuous displacement field of the whole domain is output by dot product aggregation. The automatic differentiation technology is used to solve the derivatives of the displacement field with respect to the spatial coordinates. S2. Based on the geometric nonlinear differential equation of the long-span arch bridge, a dimensionless strategy is introduced to reconstruct the physical control equation, eliminating the numerical solution instability caused by the large difference in stiffness parameters. At the same time, based on the node discretization method, the solution domain of the long-span arch bridge is discretized into N spatial nodes. The derivatives obtained in step S1 are mapped and paired one by one with the dimensionless node stiffness coefficients discretized in step S2, and substituted into the reconstructed discrete form differential equation to construct physical residual terms based on each node and embed them into the model. Based on traditional deflection theory, and considering the additional bending moment caused by the arch foot reaction, the geometrically nonlinear differential governing equations are established as follows: （2） （3） In the formula, and These represent the horizontal and vertical displacements at point x, respectively. Here is the equation for the arch axis; E is the horizontal angle; E is the elastic modulus of the material; A is the cross-sectional area; M is the bending moment; I is the moment of inertia of the cross section. For a common hingeless arch, the arch feet at both ends are fixed, therefore the vertical displacement and rotation of the arch feet are zero, that is: （4） In the formula, To calculate the span; By introducing dimensionless coefficients through equation (5), the original large-scale inversion problem is transformed into the problem of identifying elastic modulus coefficients within the range of [0,1]. （5） Substituting equation (5) into equation (2), we get: （6） To reduce dimensionless coefficients Multiply both terms on the left and right sides of equation (6) by have to: （7） Obtain stiffness parameters suitable for global damage identification; Discretize the continuous solution domain of the governing equation (7) into: N spatial nodes Discretized into a network set consisting of N nodes The stiffness coefficient field to be identified Explicitly defined as a set of independently optimizable scalar parameters for any measurement point Its corresponding local stiffness is expressed as: （8） （9） in, The baseline elastic modulus; It corresponds to the spatial location. Learnable dimensionless parameters; Represents the exponential function ; S3. Construct a multi-objective joint loss function that includes data loss, boundary condition loss and physical residual loss. Through the gradient descent algorithm, update the stiffness parameters of the deep neural network weights and discrete nodes synchronously during the backpropagation calculation of the network. Finally, output the spatial distribution of the stiffness coefficients of the entire bridge to complete the accurate location and quantitative assessment of the damage. Step S3 incorporates the discretized local stiffness parameters into the network training to drive model updates. This requires constructing a specific tensor injection and residual evaluation mechanism: defining the discrete mapping operator. : Make each spatial coordinate The index k in the parameter vector corresponds uniquely to the network at the configuration point. When solving the residuals of the governing equations, the dimensionless coefficient term The value is directly injected into the governing equations from the k-th component through an exponential transformation and in the form of a zero-order tensor. At this point, the physical residual... exist The local form of the location is expressed as about Functions of its derivatives: （10） （11） In the formula, These are the weights of the neural network; S4. Based on the spatial distribution of stiffness coefficients obtained from step S3, locate the stiffness abrupt change region to determine the damage location, quantify the stiffness reduction magnitude to determine the damage degree; verify the effectiveness and practicality of the method through numerical simulation and scaled-down tests.

2. The damage diagnosis method for long-span arch bridges based on differential equation embedding as described in claim 1, characterized in that, In step S1, set To characterize the mapping operator corresponding to the behavior of the bridge system, its input function is the displacement u at the structural measurement point extracted from the high-dimensional hidden layer state features. For the corresponding continuous spatial response function; Arbitrary spatial coordinates of the arch bridge in the continuous space of the domain Used to construct continuously differentiable spatial basis, output For the physical response at that point, adaptive tensor aggregation is performed on these two sets of features at the network end.

3. The damage diagnosis method for long-span arch bridges based on differential equation embedding as described in claim 2, characterized in that, The physical field response reconstructed in the continuous domain in step S1 is: (1) In the formula, For spatial coordinates, The displacement at the measuring point, In a branch network, the output feature vector maps a high-dimensional discrete observation vector to a low-dimensional variable space. The output feature vectors of the basis functions are adaptively constructed through multi-layer nonlinear transformations in the backbone network.

4. The damage diagnosis method for long-span arch bridges based on differential equation embedding as described in claim 1, characterized in that, The physical residual loss function for step S3 is expressed as: （12） The boundary condition loss function is expressed as: （13） The data loss function added to the network using displacement monitoring data from the arch bridge is as follows: （14） For the damage identification problem, the overall optimization objective is defined as a weighted combination of multi-source errors, encompassing data-driven observation errors, boundary conditions, and the residuals of the governing equations related to physical constraints. The overall loss function is expressed as: （15）。 5. The damage diagnosis method for long-span arch bridges based on differential equation embedding as described in claim 1, characterized in that, Step S4 verifies the method through numerical simulation and scaled-down tests. The stiffness quantification error under single-damage conditions is controlled within 2.32%, and the identification error under multiple-damage concurrent conditions is mostly maintained below 4%. Compared with traditional numerical methods, the error reduction is generally more than 70%. It is suitable for non-uniform damage identification of long-span arch bridges. Through the independent inversion of discrete node parameters, the degree of stiffness degradation at different locations of the structure can be accurately quantified.

6. A computer-readable storage medium storing a computer program, characterized in that, When the program is executed by the processor, it implements the steps of the damage diagnosis method for long-span arch bridges based on differential equation embedding as described in any one of claims 1 to 5.

7. An electronic device comprising a memory, a processor, and a computer program stored in the memory, characterized in that, When the processor executes the program, it implements the steps of the damage diagnosis method for long-span arch bridges based on differential equation embedding as described in any one of claims 1 to 5.

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