Structural damage detection of shear frame model using parallel physics-informed neural network

Abstract

Conventional numerical and data-driven methods for structural damage identification often struggle with problems such as poor generalization, sensitivity to noise, and limited accuracy in quantifying and localizing damage, especially under real-world uncertainties. This study introduces a novel parallel physics-informed neural network framework that addresses these limitations by integrating physical constraints directly into the learning process and employing a nondimensionalized formulation of the governing equations to improve training stability and convergence. The method is applied to simultaneously identify the location and severity of damage in numerical and experimental shear frame model using measured acceleration responses from the structure. In the numerical case, different damage scenarios are analyzed under white Gaussian noise and real earthquake excitations, across various noise levels. The model demonstrates high accuracy in identifying damaged locations even under high noise conditions, and outperforms traditional response surface methods by reducing false positive cases. Experimental validation using open-source shear frame data further confirms the model’s effectiveness. Despite the presence of modeling uncertainty and measurement, the model accurately identifies damage locations and estimates their severity in agreement with experimental observations. Overall, the model proves its efficacy in predicting damage locations and assessing the severity levels even under uncertainty and noise levels.

Graphical abstract

Keywords

structural health monitoring / damage detection / parallel physics-informed neural network / deep learning

1 Introduction

Structural health monitoring (SHM) and damage identification aim to detect, locate, and quantify damage in existing engineering structures such as buildings, bridges, and tunnels using actual measured dynamic and/or static structural responses. These techniques contribute to enhancing structural safety, optimizing maintenance strategies, and assessing the remaining service life of infrastructure systems [1]. Structural damage typically alters material and geometric properties, leading to changes in dynamic characteristics such as mass, stiffness, and damping [2,3]. Consequently, vibration-based damage identification methods are widely employed to determine the presence, location, and severity of damage. Traditionally, this is achieved by comparing measured structural responses in both damaged and undamaged states [1,4]. Over the past several decades, a wide range of methodologies have been developed in this field, which can generally be categorized into two main groups: physics-based models and data-driven models [5].

Physics based models have a long development history in structural damage detection primarily due to their strong physical interpretability and well-established theoretical basis. Common approaches in this category include metaheuristic algorithm-based methods [6–8], sensitivity-based methods [9–11], wavelet-based methods [12–14], and response surface-based approaches [15]. These methods typically rely on the availability of an initial accurate finite element model (FEM) of the target structure, along with measured structural response data under present conditions. By leveraging the initial FEM in conjunction with optimization algorithms, these approaches aim to identify both the location and severity of damage by comparing changes in dynamic characteristics such as natural frequencies, modal damping, and mode shapes between undamaged and damaged states. Despite their extensive development and successful application in many scenarios, these physics-based methods present certain limitations. Most notably, their performance heavily depends on the accuracy of the FEM, which is often compromised by modeling errors (e.g., in stiffness, material density, or boundary conditions) and measurement noise. Such uncertainties can significantly impact the reliability of model updating and the accuracy of damage identification results [1,16,17].

On the other hand, data-driven models identify damage location and severity by extracting damage sensitive features directly from the measured vibration signals [18–20]. These models do not require a FEM and rely solely on structural response data. In recent years, deep learning (DL) models have emerged as state-of-the-art data-driven methods in the field of SHM, finding successful applications in damage identification [21–23], crack detection [24,25], and structural condition assessment [26,27]. DL models offer the advantages of automatically and effectively extracting highly nonlinear and nonstationary features relevant to structural damage [28]. However, their performance depends heavily on the availability of extensive labeled data that include both undamaged and damaged conditions [29]. For large scale structures such as bridges or high-rise buildings, acquiring damage state data can be prohibitively time-consuming, labor-intensive, and costly [30,31] making the lack of labeled damage data one of the most significant challenges in practical SHM applications. Furthermore, these models are typically regarded as black boxes, offering limited insight into the physical behavior of the system, thereby reducing scientific interpretability. In addition, they may exhibit poor generalization ability in the scenarios with unseen data patterns [32].

Given that both physics-based methods and data-driven methods possess distinct limitations and complementary strengths, a combined approach that leverages the advantages of both would be particularly beneficial for SHM. Physics-informed neural networks (PINNs) have emerged as a promising hybrid framework that integrates the physical interpretability of physics-based methods with the powerful feature learning capabilities of driven models [33,34]. This physics-guided learning paradigm enables the model to not only fit observed data but also adhere to the governing physical laws of the system. PINNs are mesh-free scientific computing approaches capable of solving a wide range of equations, including complex nonlinear partial differential equations (PDEs) [35,36]. These models approximate the solution of such equations by training neural networks (NNs) to minimize a composite loss function that can be formulated either using the strong form of the governing equations or their weak form, depending on the problem to be solved [37–39]. While the strong form directly penalizes the residuals of the PDEs at collocation points, weak form formulations provide greater flexibility in solving problems involving complex boundaries, discontinuities, or irregular geometries. Several weak-form-based variants have been developed to extend the applicability of PINNs. For instance, the deep energy method and its extension deep complementary energy method leverage energy principles to construct loss functions, enabling accurate solutions of boundary value problems and elasticity [40,41]. Boundary-integral type neural networks reduce dimensionality through boundary integral equations while naturally incorporating boundary conditions [42], whereas conservative energy method with neural networks and subdomains extends energy-based formulations to heterogeneous and complex geometries via subdomain networks [43]. Such approaches have been successfully applied to frictionless contact problems under large deformation [44], where energy-based PINNs deliver robust nonlinear solutions, and to elastodynamics without labeled data [45], where hybrid outputs improve accuracy and stability. More recently, operator-learning-based methods such as the variational physics-informed neural operator have improved scalability and convergence for large-scale PDEs [46]. In parallel, PINN-based dual weighted residual framework enables efficient error estimation for nonlinear PDEs [47], while peridynamic and phase-field models [48–50] capture crack initiation, branching, and thermal effects with improved multiscale fidelity. Apart from this, novel frameworks such as radial point interpolation method empowered with neural network solvers and kolmogorov–arnold-informed neural networks have enhanced robustness in nonlinear solid mechanics while improving computational efficiency through alternative architectures [51,52]. Furthermore, transfer learning in PINNs [53] demonstrates that strategies like low-rank adaptation can significantly accelerate adaptation to new boundary conditions, materials, and geometries, positioning transfer learning as a key enabler for large pre-trained PDE models.

A key advantage of PINNs lies in their ability to address both forward and inverse problems. In forward problems, PINNs predict the evolution of physical states given initial conditions and known parameters, whereas in inverse problems, they estimate unknown system parameters or boundary conditions from sparse or noisy measurements, all while ensuring consistency with physical laws [35,36]. As SHM encompasses both forward simulations and inverse identification tasks, PINNs provide a unified and physics-consistent approach well suited to addressing these challenges [5,54]. In recent years, there has been a growing interest in the application of PINNs to a wide range of SHM tasks, including dynamic response simulation, dynamic system identification, and structural damage detection. In these applications, the structural dynamic governing equations are embedded into the loss function to enforce physical consistency during training. For example, Yu et al. [32] demonstrated the use of PINNs for simulating the dynamic responses of both linear multi-degree-of-freedom (MDOF) and nonlinear single-degree-of-freedom systems. Similarly, Zhang et al. [55] proposed a physics-guided convolutional neural network (CNN) for predicting the seismic response of buildings, illustrating the integration of physical constraints into data-driven models. Another study was conducted by Kapoor et al. [56], employing the concept of nondimensionalization (ND) to solve both forward and inverse problems related to complex beam dynamics. They proved that while the original dimensional form of the Euler–Bernoulli beam equation led to high loss values and poor prediction accuracy, the nondimensionalized formulation significantly improved model convergence and predictive performance. In the context of structural system identification, several recent studies have demonstrated the effectiveness of PINNs. Lai et al. [57] proposed a physics-informed neural ordinary differential equation framework for both free and forced vibration analysis of MDOF systems with nonlinearities. Another study formulated the loss function based on a multivariate nonlinear regression expression to enhance structural parameter identification [58]. Liu and Meidani [59] proposed PIDynNet for system identification of MDOF systems exhibiting various types of nonlinearity. Zhang et al. [60] addressed the inverse problem of continuous beams using a parallel PINNs approach. Furthermore, both parallel and sequential PINNs frameworks were employed for parameter and state estimation in linear and nonlinear systems [61]. Their findings indicated that the parallel PINNs offered higher accuracy with reduced computational cost, making it suitable for application to more complex systems.

Damage detection studies have been conducted on a numerical simply supported beam and an experimental three-storey frame, where structural damage was simulated through a reduction in stiffness [62]. A physics-guided deep neural network (PGDNN) framework using a CNN model was proposed for damage identification in both numerical suspension bridge models and an experimental frame, considering modeling uncertainties and environmental noise [63]. Expanding on this concept, Yin et al. [16] applied a similar PGDNN-based framework for damage localization in bridge structures employing the VGG-16 model. The proposed approach achieved superior prediction accuracy compared to conventional methods. Notably, these damage identification frameworks employ the forward PINNs approach and are primarily focused on damage localization tasks. Another notable approach for damage identification was proposed by Mai et al. [64], termed as damage-informed NN. The proposed framework demonstrates superior performance in damage identification, as well as faster convergence compared to conventional metaheuristic algorithms. More recently, PINNs have been applied to damage identification in both numerical and experimental studies involving simply supported beam structures [5], showing the capability to detect complex damage scenarios which are not explicitly included in the training data. In the context of composite materials, PINNs have also been utilized for fatigue damage detection in carbon fiber reinforced polymer materials [65]. Furthermore, Wang et al. [17] introduced a physics-guided residual NN for structural damage identification, demonstrating accurate results even with limited training data. All the aforementioned methods perform damage localization and quantification using the forward PINNs approach, primarily relying on vibration-based information. In contrast, Zhou and Xu [66] introduced a method based on inverse PINNs for damage identification in a cantilever beam using flexural guided wavefield data. Extending this idea, the same author later applied the methodology to both numerical and experimental plate structures [54]. This inverse approach demonstrated robustness in damage identification, even in the presence of significant measurement noise. These methods are based on inverse approach for damage identification but uses the concept of flexural guided wave instead of vibration signals.

Most of these aforementioned studies primarily focused on damage localization, with only a limited number addressing both localization and severity quantification. Furthermore, the majority of these works adopt a forward PINNs approach for damage identification, which constrains their ability to infer unknown system parameters directly from measured data. In contrast, inverse PINNs frameworks have demonstrated strong potential and superior performance in structural simulation and system identification tasks, especially under conditions of limited or noisy data. Additionally, recent advancements such as parallel P-PINNs and ND techniques have shown significant advantages in simulation accuracy and parameter identification. P-PINNs offer greater architectural flexibility by allowing distinct NNs models to be assigned to each degree of freedom (DOFs), which is particularly beneficial for complex and coupled PDE-based systems. Meanwhile, ND enhances model generalization and numerical stability by reducing the dependency on specific physical scales. Despite these promising developments, there remains a clear gap in leveraging the combined strengths of P-PINNs and ND for inverse damage identification, particularly in applications involving both numerical and experimental structural data.

To address this gap, the present study introduces a novel inverse P-PINNs framework enhanced with ND techniques to simultaneously predict both the location and severity of damage in shear building structures. The proposed approach is validated on numerical and experimental shear building models, which have not been addressed in previous studies, making our contribution distinct. Numerical validation is performed under varying excitation conditions including Gaussian white noise and El Centro earthquake data, with various noise levels in measured acceleration data. Experimental validation is done using open-source acceleration data measured from a three dimensional (3D) shear building model, while employing a simplified two dimensional (2D) dynamics equation in the loss formulation. Furthermore, the stability and reliability of the proposed method are demonstrated through multiple trials with different random initializations of trainable NN parameters and damage indices. Overall, the key novelty of our work lies in bridging inverse P-PINN architectures and ND into a unified framework that advances the state-of-the-art for reliable and scalable structural damage identification. The rest of the paper is organized as follows. Section 2 describes the methodology, including the structural dynamics formulation, ND, P-PINNs, and the formulation of the loss function. Sections 3 and 4 present the evaluation of the proposed approach in the numerical and experimental studies, respectively. Section 5 discusses about the comparison of the proposed method with conventional algorithm. Finally, Section 6 summarizes the conclusions of the study.

2 Methodology

Structural damage identification aims to develop robust models capable of accurately capturing the complex nonlinear relationships between a structure’s dynamic responses (e.g., vibration characteristics) and its material and geometric properties. DL models have proven effective in learning such nonlinear mappings and are therefore widely adopted for structural damage detection. However, their reliance on purely data-driven training makes them susceptible to problems such as limited data availability and measurement noise. To overcome these challenges, recent research has focused on incorporating physical laws specifically, the governing equations of structural dynamics into the training process. This physics-informed approach enhances model performance by constraining the solution space to physically feasible regions. Among various advancements in this direction, the integration of ND techniques and P-PINNs has demonstrated significant promise for structural simulation and system identification. These concepts further enhance the model’s generalizability, numerical stability, and computational efficiency. Building upon this foundation, the proposed approach combines the strengths of ND and P-PINNs for structural damage detection, where the governing equations of motion represent the underlying physics, and acceleration time history data serve as the measurement input. The proposed framework is presented in subsequent subsections. First, the structural dynamic equations of motion and damage index formulation which serve as physics constraints linking structural states to dynamic responses are briefly introduced. Then, the concept of ND is discussed in the context of the dynamic equation of motion for MDOF system, followed by a detailed explanation of the P-PINNs architecture and loss function formulation.

2.1 Structural dynamics formulation

In structural engineering, the dynamic behavior of a structure is governed by the second-order linear differential equation given by Eq. (1):

(1)\mathit{M}\ddot{\mathit{u}}\left(t\right)+\mathit{C}\dot{\mathit{u}}\left(t\right)+\mathit{K}\mathit{u}\left(t\right)=\mathit{f}(t),

where M, C, K represents mass, damping, and stiffness matrix of the structure, respectively. The vectors \ddot{\mathit{u}}\left(t\right), \dot{\mathit{u}}\left(t\right), \mathit{u}\left(t\right) denote the structural acceleration, velocity, and displacement responses at time t while \mathit{f}(t) is the external excitation vector applied to the corresponding DOFs. Equation (1) forms the basis for dynamic analysis of structures and can be solved numerically using time integration methods such as the Newmark-β method. Structural damage identification is usually based on the assumptions that damage will cause changes in the structural parameters such as stiffness, mass and damping, which in turn alter the structure’s dynamic response characteristics. In this study, damage is modeled as a reduction in stiffness at specific storey levels of a shear building. To quantify damage, a set of scalar variables {\alpha }_i\in (0,1) in introduced, where i = 1,2,…,n, representing the stiffness reduction at each storey level. The damage index {\alpha }_i is defined for each floor, with i = 1 corresponding to the top floor and i = n corresponding to the ground floor of an n-storey shear building. The damaged stiffness at each storey is then expressed as

(2)k_{\mathrm{d},i}=(1-{\alpha }_i)k_{\mathrm{u},i},

where k_{\mathrm{u},i} denotes the stiffness of the (n + 1 − i)th storey in the undamaged state, and k_{\mathrm{d},i} denotes the corresponding stiffness in the damaged state.

2.2 Nondimensionalization

Data normalization is a widely adopted preprocessing step in DL to ensure that input features have comparable magnitudes and ranges, thereby promoting faster convergence and enhancing training stability. In PINNs, ND serves a similar purpose by ensuring that all output variables remain within a reasonable and interpretable range. ND is both a mathematical and physical technique used to simplify and analyze complex systems by converting governing equations and parameters into their dimensionless forms [56]. Once the independent variables are appropriately scaled, all dependent variables can subsequently be expressed in terms of these dimensionless quantities. This approach is particularly beneficial in real-world applications, where variables often differ significantly in units and scale, thereby improving numerical stability and generalization in the learning process. The ND formulation of MDOF system is described as follows. Consider a damaged MDOF system with dynamic equation of motion as given by Eq. (3):

(3)\mathit{M}\ddot{\mathit{u}}(t)+\mathit{C}\dot{\mathit{u}}(t)+{\mathit{K}}_{\mathrm{d}}\mathit{u}(t)=\mathit{f}(t).

The damaged stiffness matrix {\mathit{K}}_{\mathrm{d}} is formulated using the damaged stiffness value as computed by Eq. (2) for each floor levels. To dimensionalize this equation, following changes should be made for the fundamental variable, i.e., time:

(4)\mathit{u}(t)=u\times {\mathit{u}}^{\text{nd}}(\tilde{t})\mathrm{a}\mathrm{n}\mathrm{d}t=T_0\tilde{t}\Rightarrow \frac{\mathrm{\partial }\tilde{t}}{\mathrm{\partial }t}=\frac{1}{T_0}.

The first order derivative is computed as

(5)\frac{\mathrm{\partial }\mathit{u}}{\mathrm{\partial }t}=u\frac{\mathrm{\partial }{\mathit{u}}^{\text{nd}}}{\mathrm{\partial }t}=u\frac{\mathrm{\partial }{\mathit{u}}^{\text{nd}}}{\mathrm{\partial }\tilde{t}}\frac{\mathrm{\partial }\tilde{t}}{\mathrm{\partial }t}=\frac{u}{T_0}\frac{\mathrm{\partial }{\mathit{u}}^{\text{nd}}}{\mathrm{\partial }\tilde{t}}.

Similarly, the second order derivative is computed as

(6)\begin{array}{rl}\frac{{\mathrm{\partial }}^2\mathit{u}}{\mathrm{\partial }{t}^2}=& \frac{\mathrm{\partial }}{\mathrm{\partial }t}\left(\frac{u}{T_0}\frac{\mathrm{\partial }{\mathit{u}}^{\text{nd}}}{\mathrm{\partial }\tilde{t}}\right)=\frac{u}{T_0}\frac{\mathrm{\partial }}{\mathrm{\partial }t}\left(\frac{\mathrm{\partial }{\mathit{u}}^{\text{nd}}}{\mathrm{\partial }\tilde{t}}\right)\\ =& \frac{u}{T_0}\frac{\mathrm{\partial }}{\mathrm{\partial }\tilde{t}}\frac{\mathrm{\partial }\tilde{t}}{\mathrm{\partial }t}\left(\frac{\mathrm{\partial }{\mathit{u}}^{\text{nd}}}{\mathrm{\partial }\tilde{t}}\right)=\frac{u}{T_0^2}\frac{{\mathrm{\partial }}^2{\mathit{u}}^{\text{nd}}}{\mathrm{\partial }{\tilde{t}}^2}.\end{array}

Since acceleration data are used to formulate the loss function, so

(7)\frac{u}{T_0^2}={\ddot{u}}_{max}\Rightarrow u={\ddot{u}}_{max}\times T_0^2.

Substituting these in above equation,

(8)\mathit{M}\frac{u}{T_0^2}\frac{{\mathrm{\partial }}^2{\mathit{u}}^{\text{nd}}}{\mathrm{\partial }{\tilde{t}}^2}+\mathit{C}\frac{u}{T_0}\frac{\mathrm{\partial }{\mathit{u}}^{\mathrm{n}\mathrm{d}}}{\mathrm{\partial }\tilde{t}}+{\mathit{K}}_{\mathrm{d}}u{\mathit{u}}^{\text{nd}}(\tilde{t})=\mathit{f}(T_0\tilde{t}).

To simplify, assuming T_0=\sqrt{{\displaystyle \frac{m_{max}}{k_{max}^{\mathrm{a}}}}}

(9)\mathit{M}\frac{u\times k_{max}^{\mathrm{a}}}{m_{max}}\frac{{\mathrm{\partial }}^2{\mathit{u}}^{\text{nd}}}{\mathrm{\partial }{\tilde{t}}^2}+\mathit{C}\frac{u}{\sqrt{m_{max}/k_{max}^{\mathrm{a}}}}\frac{\mathrm{\partial }{\mathit{u}}^{\text{nd}}}{\mathrm{\partial }\tilde{t}}+{\mathit{K}}_{\mathrm{d}}u{\mathit{u}}^{\text{nd}}(\tilde{t})=\mathit{f}(T_0\tilde{t}).

Dividing by u\times k_{max}^{\mathrm{a}},

(10)\begin{array}{rl}& \frac{\mathit{M}}{m_{max}}\frac{{\mathrm{\partial }}^2{\mathit{u}}^{\text{nd}}}{\mathrm{\partial }{\tilde{t}}^2}+\frac{\mathit{C}}{\sqrt{k_{max}^{\mathrm{a}}\cdot m_{max}}}\frac{\mathrm{\partial }{\mathit{u}}^{\text{nd}}}{\mathrm{\partial }\tilde{t}}+\frac{{\mathit{K}}_{\mathrm{d}}}{k_{max}^{\mathrm{a}}}{\mathit{u}}^{\text{nd}}(\tilde{t})\\ =& \frac{1}{\left(u\times k_{max}^{\mathrm{a}}\right)}\mathit{f}(T_0\overline{t}),\end{array}

(11){\mathit{M}}^{\text{nd}}{\ddot{\mathit{u}}}^{\text{nd}}\left(\tilde{t}\right)+{\mathit{C}}^{\text{nd}}{\dot{\mathit{u}}}^{\text{nd}}\left(\tilde{t}\right)+{\mathit{K}}^{\text{nd}}{\mathit{u}}^{\text{nd}}\left(\tilde{t}\right)={\mathit{f}}^{\text{nd}}\left(\tilde{t}\right),

where

\begin{array}{rl}& {\mathit{M}}^{\text{nd}}=\frac{\mathit{M}}{m_{max}},{\mathit{C}}^{\text{nd}}=\frac{\mathit{C}}{\sqrt{k_{max}^{\mathrm{a}}\cdot m_{max}}},\\ & {\mathit{K}}^{\text{nd}}=\frac{{\mathit{K}}_{\mathrm{d}}}{k_{max}^{\mathrm{a}}},{\mathit{f}}^{\text{nd}}(\tilde{t})=\frac{1}{\left(u\times k_{max}^{\mathrm{a}}\right)}\mathit{f}(T_0\overline{t}),\end{array}

where m_{max} denotes the maximum value in the mass matrix M and {\ddot{u}}_{max} represents maximum value of acceleration across all DOFs. k_{max}^{\mathrm{a}} refers to the maximum value in the stiffness matrix of the undamaged system, where the stiffness matrix is computed using an assumed approximate value of stiffness for each storey. Equation (11) expresses the dimensionless form of Eq. (3). In this formulation, {\mathit{M}}^{\text{nd}}, {\mathit{C}}^{\text{nd}}, {\mathit{K}}^{\text{nd}} denote the non-dimensionalized mass, damping and stiffness matrix, respectively. Similarly, {\ddot{\mathit{u}}}^{\text{nd}}\left(\tilde{t}\right), {\dot{\mathit{u}}}^{\text{nd}}\left(\tilde{t}\right), {\mathit{u}}^{\text{nd}}\left(\tilde{t}\right), {\mathit{f}}^{\text{nd}}(\tilde{t}) represent dimensionless acceleration, velocity, displacement and force vector, respectively, which is a function of dimensionless time.

2.3 Parallel physics-informed neural networks and formulation of loss function

A schematic representation of the P-PINNs framework is illustrated in Fig. 1. The framework consists of key components which includes actual input data, dimensionless inputs, a set of parallel NNs, automatic differentiation mechanisms and formulation of loss function. Each NNs is designed to predict the displacement corresponding to a specific DOFs. For ith DOFs, a fully connected feed forward NN {\hat{u}}_i({\tilde{t}}_i,{\theta }_i) is constructed to approximate the solution {\hat{u}}_i({\tilde{t}}_i), using dimensionless time data as input, as depicted in Fig. 1. Assuming the ith NNs consists of the L_i hidden layers, with the 0th layer representing the input layer and {(L}_i+1)th layer denoting the output layer, the hidden variables of the kth hidden layer are denoted by {\mathit{\text{z}}}_i^k. The architecture of the ith NNs can thus be expressed as

(12)\begin{array}{rl}& \mathrm{I}\mathrm{n}\mathrm{p}\mathrm{u}\mathrm{t}\mathrm{l}\mathrm{a}\mathrm{y}\mathrm{e}\mathrm{r}:{\mathit{\text{z}}}_i^0=x_i,\\ & \mathrm{H}\mathrm{i}\mathrm{d}\mathrm{d}\mathrm{e}\mathrm{n}\mathrm{l}\mathrm{a}\mathrm{y}\mathrm{e}\mathrm{r}:{\mathit{\text{z}}}_i^k=\sigma \left(w_i^k{\mathit{\text{z}}}_i^{k-1}+b_i^k\right),1\le k\le L_i,\\ & \mathrm{O}\mathrm{u}\mathrm{t}\mathrm{p}\mathrm{u}\mathrm{t}\mathrm{l}\mathrm{a}\mathrm{y}\mathrm{e}\mathrm{r}:{\mathit{\text{z}}}_i^{L_i+1}=w_i^{L_i+1}{\mathit{\text{z}}}_i^{L_i}+b_i^{L_i+1},\end{array}

where w_i^k and b_i^k denote the weight matrix and bias vector, respectively, of the kth layer of the ith NNs, respectively and \sigma represents a nonlinear activation function. Each NNs is allowed to have a distinct architecture. The complete set of trainable parameters, comprising all weights and biases across the networks, is collectively denotes by \theta in this study.

The NNs output for each DOFs is passed through automatic differentiation [67] to compute higher derivatives of predicted displacements with respect to input using the chain rule of differentiation. These derivatives are then used to formulate the loss function, which comprises two components: the residual loss and the measurement loss. The residual loss is derived from the dimensionless form of the governing equation. For a MDOF system, the residual loss is computed using Eq. (13) where {\ddot{\hat{u}}}^{\text{nd}}\left({\tilde{t}}_j\right), {\dot{\hat{u}}}^{\text{nd}}\left({\tilde{t}}_j\right), {\hat{u}}^{\text{nd}}\left({\tilde{t}}_j\right) denote the predicted dimensionless acceleration, velocity and displacement vector at the jth time instant, respectively. {\hat{\mathit{K}}}^{\text{nd}} represents predicted dimensionless stiffness matrix which is parameterized by the damage indices \alpha and the initial storey stiffness value. The trainable parameters of the NNs and the damage indices are represented by \theta and \alpha, respectively. The measurement loss, expressed in Eq. (14), is computed as the mean squared error (MSE) between the predicted and actual acceleration values across all DOFs. Here n and N_{\mathrm{p}} represents total number of DOFs and measurement points, respectively. It should be noted that the residual loss is computed for same number of points as in measurement data. {\hat{\ddot{u}}}_{d,j} and {\ddot{u}}_{d,j}^{\text{nd}} represents jth predicted and dimensionless acceleration corresponding to dth DOFs. The total loss is the weighted sum of residual loss and measurement data loss. In this study, the weight coefficient for each loss term is considered unity. Finally, the total loss function given by Eq. (15) is optimized using suitable optimizer to obtain the optimum value of NNs parameters and unknown damage indices for each storey. The pseudo code for developed P-PINNs algorithm for damage identification of shear frame model is outlined in Algorithm 1.

(13)\begin{array}{rl}{\mathcal{L}}_{\text{residual}}\left(\tilde{t};\theta ,\alpha \right)=& \frac{1}{N_{\mathrm{p}}}\sum _{j=1}^{N_{\mathrm{p}}}||{\mathit{M}}^{\text{nd}}{\ddot{\hat{\mathit{u}}}}^{\text{nd}}\left({\tilde{t}}_j\right)+{\mathit{C}}^{\text{nd}}{\dot{\hat{\mathit{u}}}}^{\text{nd}}\left({\tilde{t}}_j\right)\\ & +{\hat{\mathit{K}}}^{\text{nd}}{\hat{\mathit{u}}}^{\text{nd}}\left({\tilde{t}}_j\right)-{\mathit{f}}^{\text{nd}}\left({\tilde{t}}_j\right)|{|}^2,\end{array}

(14){\mathcal{L}}_{\text{MD}}\left(\tilde{t};\theta \right)=\sum _{d=1}^n\left(\frac{1}{N_{\mathrm{p}}}\sum _{j=1}^{N_{\mathrm{p}}}{\left|\left|{\hat{\ddot{u}}}_d\left({\tilde{t}}_j\right)-{\ddot{u}}_d^{\text{nd}}\left({\tilde{t}}_j\right)\right|\right|}^2\right),

(15){\mathcal{L}}_{\text{total}}\left(\tilde{t};\theta ,\alpha \right)={\mathcal{L}}_{\text{residual}}\left(\tilde{t};\theta ,\alpha \right)+{\mathcal{L}}_{\text{MD}}\left(\tilde{t};\theta \right).

3 Numerical examples

The proposed method is validated on a six-storey shear frame model as illustrated in Fig. 2. The mass and stiffness values for the structure are adopted from Ref. [15]. Following the precedent in the literature, damping is neglected in the model. The dynamic equation of motion for the undamaged structure, corresponding to each DOFs, is presented in Eq. (16), while its dimensionless form is given in Eq. (17). For the damaged condition, the dimensionless equation is described by Eq. (18), and by incorporating the damage index, the modified equation is expressed in Eq. (19). The residual loss for jth time instant of ith DOF is computed using Eq. (20). It is important to note that the matrix form of the governing equations inherently satisfies the boundary conditions; therefore, while formulating the loss function for each DOFs, these boundary conditions must be appropriately enforced. The measurement data loss term for jth time instant of ith DOFs is defined by Eq. (21). The MSE is calculated individually for each DOF, and these values are summed to form the total loss function.

(16)m_i{\ddot{u}}_i+\left(k_{\mathrm{u},i-1}+k_{\mathrm{u},i}\right)u_i-k_{\mathrm{u},i}{u}_{i+1}-k_{\mathrm{u},i-1}{u}_{i-1}=f_i(t),

(17)m_i^{\text{nd}}{\ddot{u}}_i+\left(k_{\mathrm{u},i-1}^{\text{nd}}+k_{\mathrm{u},i}^{\text{nd}}\right)u_i-k_{\mathrm{u},i}^{\text{nd}}{u}_{i+1}-k_{\mathrm{u},i-1}^{\text{nd}}{u}_{i-1}=f_i^{\text{nd}}(\tilde{t}),

(18)m_i^{\text{nd}}{\ddot{u}}_i+\left(k_{\mathrm{d},i-1}^{\text{nd}}+k_{\mathrm{d},i}^{\text{nd}}\right)u_i-k_{\mathrm{d},i}^{\text{nd}}{u}_{i+1}-k_{\mathrm{d},i-1}^{\text{nd}}{u}_{i-1}=f_i^{\text{nd}}(\tilde{t}),

(19)\begin{array}{rl}& m_i^{\text{nd}}{\ddot{u}}_i+\left[\left(1-{\alpha }_{i-1}\right)k_{\mathrm{u},i-1}^{\text{nd}}+\left(1-{\alpha }_i\right)k_{\mathrm{u},i}^{\text{nd}}\right]u_i\\ -& \left(1-{\alpha }_i\right)k_{\mathrm{u},i}^{\text{nd}}{u}_{i+1}-\left(1-{\alpha }_{i-1}\right)k_{\mathrm{u},i-1}^{\text{nd}}{u}_{i-1}=f_i^{\text{nd}}(\tilde{t}),\end{array}

(20)\begin{array}{rl}& m_i^{\text{nd}}{\ddot{\hat{u}}}_i({\tilde{t}}_j)+\left[\left(1-{\hat{\alpha }}_{i-1}\right)k_{\mathrm{u},i-1}^{\text{nd}}+\left(1-{\hat{\alpha }}_i\right)k_{\mathrm{u},i}^{\text{nd}}\right]{\hat{u}}_i({\tilde{t}}_j)\\ -& \left(1-{\hat{\alpha }}_i\right)k_{\mathrm{u},i}^{\text{nd}}{\hat{u}}_{i+1}({\tilde{t}}_j)-\left(1-{\hat{\alpha }}_{i-1}\right)k_{\mathrm{u},i-1}^{\text{nd}}{\hat{u}}_{i-1}({\tilde{t}}_j)-f_i^{\text{nd}}({\tilde{t}}_j)=0,\end{array}

(21){\hat{\ddot{u}}}_i\left({\tilde{t}}_j\right)-{\ddot{u}}_i^{\text{nd}}\left({\tilde{t}}_j\right)=0.

Three damage scenarios are considered in this study: single-storey damage (Case 1), two-storey damage (Case 2), and damage across all storeys (Case 3), as outlined in Table 1. Cases 1 and 2 are adopted from existing literature to facilitate direct comparison with the proposed method, while Case 3 is introduced to evaluate the effectiveness of the proposed approach under more challenging conditions. The proposed method is validated under two types of base excitation signals: white Gaussian noise excitation and the El Centro earthquake record. Acceleration response data are generated using the Newmark-β method. To simulate realistic measurement conditions, different levels of noise (10, 20, and 30 dB) are added to the acceleration data. In contrast, prior studies have typically focused only on Cases 1 and 2 and considered noise levels of 30, 35, and 40 dB. The inclusion of lower noise levels and a more complex damage case (Case 3) in this study is intended to demonstrate the robustness and superiority of the proposed method. Initially, Case 1 is analyzed to compare the performance of the proposed approach against the parallel PINNs method using actual value of parameter, i.e., without ND. After validating the method’s accuracy, it is further applied to the remaining damage scenarios. The predicted results obtained using the proposed method are also compared with those reported in the literature to ensure consistency and reliability.

A consistent NNs architecture is employed across all damage cases. Each network consists of five hidden layers with 100 neurons per layer, utilizing a sine activation function [68]. The networks are trained using the Adam optimizer with a learning rate scheduling strategy based on exponential decay, applying a decay rate of 0.9 every 1000 steps. Training was conducted for 40000 epochs which was determined through multiple trial of experiments to ensure optimal performance. The total number of trainable parameters in the network is 244212, comprising 244206 parameters from the NNs and six trainable damage indices corresponding to the six storeys of the shear building model. Notably, the damage index for each storey is treated as a trainable parameter for every damage case, enabling the algorithm to simultaneously identify both the location and severity of structural damage. To assess the stability and robustness of the proposed method, multiple training runs (ten trials) are performed with random initialization of both NNs and damage parameters. The final predicted results presented in this study represent the mean values obtained across these ten trials. All the present codes are executed on the Param Shakti high-performance computing supercomputer using 1 CPU node. The system configuration includes an Intel® Xeon® Gold SKL G-6148 processor with 40 cores per node operating at 2.40 GHz, a 64-bit Linux operating system, and 192 GB of RAM.

3.1 White Gaussian noise excitation

First of all, white Gaussian noise is applied as the base excitation to the six-storey shear frame model. White noise excitation is widely used due to its ability to excite a wide range of vibration modes simultaneously. The zero mean Gaussian white noise of peak ground acceleration of 0.01g is generated with sampling time and frequency of 15 s and 500 Hz, respectively. In the existing literature, the structure is excited to similar signal to compute frequency and mode shapes which is further used for damage identification using response surface methods (RSM) method [15]. In this study, acceleration responses of all floors are obtained by solving the dynamic equilibrium equations using Newmark’s β method with appropriate time-stepping parameters. To simulate realistic measurement conditions, three noise levels having signal-to-noise ratios, namely 10, 20, and 30 dB are added to the computed acceleration responses. This allows a thorough evaluation of the robustness and accuracy of the proposed method under increasing levels of measurement noise.

Algorithm 1 Pseudo code for P-PINNs with ND method for damage identification in shear frame model

1: Initialize.

2: Compute/Measure the actual response, i.e., acceleration time history corresponding to each DOFs.

3: Nondimensionalize the time and acceleration response as discussed in Subsection 2.2.

4: Divide the structural systems to n substructures corresponding to each DOFs.

5: Create a NN {\hat{u}}_i(t_i,{\theta }_i) for each DOFs.

6: Set the hyperparameters of each NN (number of layers, number of neurons, learning rate, and activation function).

7: Set the initial value of each NN parameter ({\theta }_i) and damage indices ({\alpha }_i).

8: Use automatic differentiation to compute the higher derivatives of NN output.

9: Set N_{max} as the maximum number of epochs using the Adam optimizer.

10: While Convergence not reached do

11: for k = 1 to N_{max} do

12: for d = 1 to n do

13: for j = 1 to N_{\mathrm{p}} do

14: Compute all the loss terms, i.e., {\mathcal{L}}_{\text{residual}} and {\mathcal{L}}_{\text{MD}} using Eqs. (13) and (14).

15: end for

16: Compute the total loss {\mathcal{L}}_{\text{total}}\left(\tilde{t};\theta ,\alpha \right) using Eq. (15).

17: Update the parameters (\theta ,{\alpha }_i) by minimizing the total loss function.

18: end for

19: end while

20: Deployment:

21: The estimated damage indices are {\alpha }_i.

First of all, a single NN combined with the concept of ND is employed to predict the Case 3 damage scenarios for the six-storey shear building model. To ensure a fair comparison, the same hyperparameters discussed earlier are used. The convergence of the total loss and damage indices is analyzed to evaluate performance. Figure 3(a) shows the evolution of the total loss function. Despite training for 100000 epochs, the loss curve flattens prematurely and fails to decrease sufficiently, indicating poor convergence. Figure 3(b) illustrates the convergence of predicted damage indices. The results show that the predictions fail to converge and instead saturate around incorrect values, leading to significant errors. This behavior demonstrates the limitations of employing a single NN for multi-storey systems, as the complex dynamics of multiple degrees of freedom cannot be effectively captured by a shared representation. Therefore, the use of parallel NNs, with one network assigned to each storey becomes essential to enable independent learning for each DOF and achieve accurate convergence to the true damage indices.

Secondly, the parallel PINNs framework is employed for the damage identification using the actual values of the structural parameters for case 1 damage scenarios. The performance of the parallel PINN model during training is illustrated in Fig. 4(a), which presents the evolution of the total loss with respect to the number of epochs. Figure 4(b) depicts the convergence behavior of the predicted damage index over the same training period. As observed from Fig. 4(a), the total loss for each trial decreases initially but stabilizes around a value of approximately 3.5 × 10⁹ at nearly 40000 epochs. Beyond this point, the loss remains nearly constant even when training is extended to 100000 epochs. This constant value in the loss function suggests that the optimization algorithm has become trapped in a local minimum, thereby limiting its ability to further minimize the loss and improve prediction accuracy. The stagnation in training is further supported by Fig. 4(b), which shows that the predicted damage index also fails to converge to the expected values, indicating poor learning performance. This indicates that the model, even after extended training fails to accurately identify the damage scenario likely due to poor convergence behavior. One possible reason for this poor convergence behavior is the inherently complex loss landscape associated with the governing differential equations, particularly when high coefficient values are used in the model. As discussed by Krishnapriyan et al. [69], such high parameter values can lead to a highly non-convex and stiff loss surface, making it difficult for gradient-based optimizers to navigate effectively and locate the global minimum.

To address the challenges associated with the complex loss landscape arising from high coefficient values in the governing differential equations, the proposed method incorporates a ND strategy. By rescaling the coefficients to be of order unity, the stiffness of the optimization landscape is significantly reduced, thereby facilitating smoother convergence during training. The improved performance of the proposed method can be seen in the training results for the same damage case, i.e., Case 1. As shown in Fig. 5(a), which illustrates the convergence of the total loss across multiple trials along with their mean value, the mean total loss value at the end of 40000 epochs is approximately 8.15 × 10⁻³. This represents a significant reduction compared to the earlier approach without ND. The observed fluctuations in the total loss, despite using full-batch training, can be attributed to a high learning rate which is mitigated in later epochs as the learning rate is decreased, leading to a more stable convergence. The consistent and rapid convergence of the loss function indicates that the network has successfully minimized the residuals of the governing equations and likely converged to the true solution. This assertion is further supported by Fig. 5(b), which displays the convergence of the predicted damage index at the damaged location across multiple trials. The figure includes both individual trial trajectories and their average, confirming that the predicted values are stable and closely aligned with the actual damage location. These results illustrate the stability of the proposed framework, as it consistently converges to true values despite varying initial conditions. Furthermore, the effectiveness of the proposed method is benchmarked against an RSM-based approach. As illustrated in Fig. 5(c), while the RSM method tends to produce false positive damage predictions i.e., nonzero damage indices at undamaged locations up to 1.5% the proposed method accurately identifies the damage location with minimal predictions at undamaged elements, close to zero. Minimizing false positive predictions in structural damage identification is critical to maintaining the reliability, efficiency, and cost-effectiveness of monitoring systems. Quantitatively, the RSM method overestimates the damage index at the true damage location by approximately 2.36%, whereas the proposed method underestimates it by only 2.15%, indicating a superior predictive accuracy. The robustness of the proposed approach is further evaluated under various noise levels. As shown in Fig. 5(d), the prediction error remains below 2% across different noise intensities, and the false positive rates are consistently low. Table 2 presents the mean and standard deviation of various damage cases and noise levels. The minimal difference of predicted values across various noise levels indicates the model’s strong ability to localize and quantify single damage accurately. The associated standard deviation is low across all noise levels (between ±0.0013 and ±0.0024), highlighting the method’s stability and low variance in prediction, even when measurement noise is introduced. This robustness to noise highlights a key advantage of proposed frameworks over data-driven models. The inclusion of the physics-based loss component acts as an inherent regularizer, guiding the network to learn solutions that comply with the governing physical laws. As a result, even in the presence of noisy data, the model effectively filters out noise by rejecting physically inconsistent solutions. This demonstrates the capability of the proposed framework to provide accurate and reliable damage identification, even in the presence of measurement noise, outperforming RSM in both precision and robustness.

The effectiveness of the proposed framework is further validated through case 2, which involves multiple damage scenarios. As shown in Fig. 6(a), the mean loss at the end of training is 5.64 × 10⁻³. Compared to case 1 damage scenarios, the variability in total loss across individual trials is higher, reflecting the increased complexity of identifying multiple damage locations. This variability can be observed in the spread of the loss curves. Despite this complexity, Fig. 6(b) demonstrates that the damage indices from all trials converge closely to the ground truth, with the average trend aligning well with the true damage values. This consistent convergence across multiple runs indicates the robustness of the proposed framework to different initializations. Quantitatively, the RSM method underestimates the damage at 5th storey by 3.13% and overestimates the damage at the 3rd storey by 0.82%. In contrast, the proposed approach achieves significantly lower errors of 1.55% and 0.31%, respectively. Furthermore, as shown in Fig. 6(c), the RSM method exhibits false positive identifications of up to 1.5% at undamaged locations, while the proposed framework demonstrates negligible false positives, enhancing its reliability for practical applications. The model’s performance under noisy conditions further supports its robustness. As shown in Fig. 6(d), the proposed framework maintains high accuracy with error levels below 1.65%, even when trained with data corrupted by 10 dB noise. The standard deviations for both locations remain low (mostly below ±0.0006), except a slight increase at higher noise levels, indicating only modest increases in prediction variability.

To further evaluate the effectiveness and generalizability of the proposed framework, a more challenging scenario (Case 3) is considered, where damage is simultaneously introduced at all storeys of the structure. The mean total loss at the end of training is 2.20 × 10⁻³, which confirms effective model convergence. Figure 7(a) illustrates the total loss curves for multiple independent trials, along with the average trend. Despite the increased number of damaged locations, all trials exhibit stable and consistent convergence behavior, validating the robustness of the training process. Figure 7(b) shows the convergence of the mean predicted damage indices at different storeys. Each predicted index converges closely to its corresponding true value, with an average error of only 1.95% in the noise-free case. Under 10 dB measurement noise, the average error increases slightly to 2.47%, which demonstrates that the framework maintains high accuracy even in the presence of noise. Figure 7(c) presents bar plots of the predicted damage indices for different noise levels. The result shows close agreement with the actual damage values across all storeys, highlighting the reliability of the framework under noisy conditions. Larger damage magnitudes (e.g., α₅ and α₆) display slightly higher standard deviations at 10 dB noise (e.g., ±0.0092 for α₅), which reflects an increase in prediction uncertainty as both damage magnitude and noise level grow. Nevertheless, these deviations remain within practically acceptable limits for SHM. Overall, Fig. 7 demonstrates that the proposed approach can accurately identify multiple damage locations, preserve robustness under noisy conditions, and ensure negligible false-positive detection rates.

3.2 El Centro earthquake excitation

While the proposed framework demonstrates high accuracy and robustness under white Gaussian noise excitation, it is essential to validate its performance using real-time earthquake data to ensure its practical applicability. Earthquake records present complex, non-stationary, and nonlinear loading conditions that are more representative of actual structural demands. Validation with such data allows assessment of the model’s generalization ability, its response under realistic and extreme loading scenarios, and its reliability in capturing true damage patterns. This step is critical for transitioning from simulation-based validation to real-world deployment, thereby enhancing the trust and acceptance of the proposed method in SHM and seismic damage assessment applications. The El Centro earthquake ground motion, recorded during the 1940 Imperial Valley earthquake in California, is one of the most widely used seismic inputs for structural dynamic analysis and earthquake engineering studies. It represents a real-world scenario involving transient, non-stationary ground acceleration with complex frequency content. In this study, the north–south component of the El Centro earthquake acceleration time history is applied as the base excitation to the six-storey shear building model. The acceleration response of the structure is computed using the Newmark-β method with a time step of 0.005 s and a total duration of 15 s. As with the previous case, noise is added to the simulated acceleration data at 10, 20, and 30 dB noise levels to mimic sensor inaccuracies and assess the robustness of the proposed approach. Similar damage scenarios as discussed in Table 1 are considered in this case too.

For Case 1 damage scenario, Fig. 8(a) presents the total loss curves for individual trials along with the average performance. These curves exhibit greater variability compared to those obtained under white Gaussian noise conditions for the same damage scenario. The final loss value achieved at the end of training is 3.07 × 10⁻⁵, which is approximately two orders of magnitude lower than the corresponding value in the white Gaussian noise case, indicating significantly better convergence behavior. As shown in Fig. 8(b), the convergence of the damage index curves across all 10 trials is stable, with each trial closely approaching the true damage index value. The damage index at the actual damage location is predicted with a relative error of approximately 1% under the no-noise condition. However, this error increases to about 5.50% at a noise level of 10 dB, as illustrated in Fig. 8(c), highlighting the moderate influence of stronger noise on prediction accuracy. Despite this, the standard deviation remains consistently low (±0.0004) across all noise levels as reported in Table 3. This consistency in the standard deviation reflects the model’s robustness and stability in predicting single-location damage even under relatively high noise intensities. Furthermore, the model maintains excellent performance in avoiding false positives as the predicted damage indices at undamaged locations remain negligibly small across all noise levels.

For Case 2 damage scenarios, the mean total loss at the end of training is 8.35 × 10⁻⁶, as shown in Fig. 9(a). The total loss curves for individual trials consistently exhibit a downward trend, indicating effective learning behavior of the proposed method across multiple damage scenarios. As shown in Fig. 9(b), both individual trials and the average convergence of damage indices at various locations closely match the true damage values, demonstrating reliable and consistent training outcomes. The maximum error in predicting the damage index under the no-noise condition occurs at the second damage location ({\alpha }_2) and is approximately 3.91%. This error increases to around 7% at a noise level of 10 dB, reflecting a moderate sensitivity of the model to measurement noise, particularly when identifying smaller damage magnitudes as seen in Fig. 9(c). The standard deviation for {\alpha }_2 increases from ±0.0002 (no noise) to ±0.0006 (10 dB), suggesting a rise in uncertainty with increasing noise intensity. In contrast, predictions for {\alpha }_4, which corresponds to a larger damage value, remain highly stable across all noise levels. The mean predicted values consistently lie in the range of 0.200–0.201, with standard deviations remaining below ±0.0005.

For Case 3 scenarios, the total loss curves for individual trials along with the average performance are presented in Fig. 10(a), with the mean loss value at the end of training recorded as 1.94 × 10⁻⁵. Figure 10(b) illustrates the convergence behavior of the predicted damage indices at all locations, where it can be observed that the model successfully converges to the true damage values with high accuracy and minimal variance. The model demonstrates strong predictive performance across different damage magnitudes, especially under low and moderate noise conditions as shown in Fig. 10(c). The average prediction error in the absence of noise is 1.82%, which increases to 6.45% when 10 dB noise is introduced. Notably, the damage index {\alpha }_6 consistently maintains a predicted mean in the range of 0.300–0.301, with extremely low standard deviations (as small as ±2.63 × 10^–5), even under high noise levels, showcasing the model’s robustness for identifying significant damage. However, some sensitivity to noise is evident at certain locations. For example, {\alpha }_3 exhibits a drop in its mean predicted value from 0.2025 (no noise) to 0.1856 (10 dB), indicating that moderate noise can impact the estimation accuracy of mid-range damage levels. The highest prediction error is observed at {\alpha }_1, which corresponds to the smallest damage magnitude. The error reaches 6.74% in the no-noise scenario and further increases to 9.55% at 10 dB noise, as depicted in Fig. 10(c). This highlights that smaller damage magnitudes are more susceptible to the effects of measurement noise. This increased sensitivity can be attributed to the lower signal-to-noise ratio associated with subtle structural responses caused by small damages, making it more difficult for the NNs to distinguish these effects from background noise. Additionally, for small actual values, even minor absolute deviations result in disproportionately large relative percentage errors, further amplifying the perceived inaccuracy. Despite these challenges, the standard deviations across all damage indices remain consistently low, affirming the model’s overall stability and reliability in predicting damage, even under noisy conditions. Compared to the white Gaussian noise case, although the error under strong noise is slightly higher, the overall predictive performance remains highly reliable and stable.

4 Experimental validation

To further evaluate the generalization capability and practical applicability of the proposed method, validation is carried out using an open-source experimental data set obtained from a laboratory scale shear building model. The experimental validation introduces various uncertainties and practical limitations that are not present in synthetic numerical models. These include modeling errors such as assuming a 2D representation for a 3D shear building, rigid floor diaphragms, idealized rigid joints, neglecting damping, and inaccuracies in mass and stiffness estimates as well as sensor-related issues like electrical interference, sensor drift, misalignment, and mounting imperfections, all of which degrade signal quality. Despite these challenges, the robustness of the proposed method is assessed by comparing its predictions against experimentally known damage states. Validating the model on such data ensures that it is not simply overfitting to synthetic simulations but is also capable of interpreting complex and noisy real-world measurements with high fidelity. For this validation, a benchmark data set from Ref. [15] is employed. The test structure consists of a six-storey, three-dimensional miniature shear building model (approximately 1:20 scale), constructed from steel. Free vibration testing is conducted by applying initial displacements to the structure, enabling the capture of dynamic response data. According to the referenced literature, each storey has a height of 187.5 mm. The beams and columns have cross-sectional dimensions of 6 mm × 6 mm and 7.8 mm × 7.8 mm, respectively. To simulate structural damage, the columns between the third and fourth floors are replaced with thinner columns having a cross-section of 6.5 mm × 6.5 mm, corresponding to an estimated 51.7% reduction in stiffness at that location. Acceleration responses are recorded at each floor level for both undamaged and damaged states, with a sampling interval of 0.00024 s. The total mass at each floor level, as reported in the original study, is provided in Table 4. This setup offers a challenging and realistic testing environment, allowing for the evaluation of the model’s robustness in detecting localized damage under conditions representative of real structural systems.

To replicate the workflow used in the numerical studies, a subset of experimental data corresponding to both undamaged and damaged states is utilized as input to the proposed framework. The model is trained using acceleration responses from both structural conditions. Damage parameters, including location and severity, are predicted solely based on the measured acceleration data. These predicted results are then compared against the actual experimental damage configuration to evaluate accuracy. All input quantities, including acceleration response, mass, and stiffness, are converted into a dimensionless form to maintain consistency with the numerical study. The model is first validated on undamaged acceleration data to predict the damage index across all storeys. The same set of hyperparameters as those used in the numerical studies is employed to maintain methodological consistency. Furthermore, to assess the stability of the approach against random initializations of NNs and damage parameters, the algorithm is executed ten times. In each trial, the initial damage index for each storey is assigned a random value, allowing the model to simultaneously learn and predict both the location and severity of damage. This multi-trial approach ensures that the proposed method is not only capable of identifying damage under realistic conditions but is also stable and repeatable across independent runs.

For the undamaged case, the total loss curves for individual trials and the average trend, as shown in Fig. 11(a), exhibit a steadily decreasing behavior and eventually stabilize, indicating consistent convergence. The mean total loss at the end of training is approximately 1.20 × 10⁻³. Minor fluctuations observed in loss curve are attributed to a relatively high initial learning rate. As training progresses and the learning rate is adaptively decreased, these fluctuations diminish, leading to improved stability. Figure 11(b) illustrates the convergence of damage indices at various storeys for the undamaged case, all of which converge to near-zero values, clearly indicating the absence of structural damage. Table 5 presents the mean and standard deviation of the predicted results for experimental data. The predictions for the undamaged structure are remarkably consistent, with mean values ranging between 0.0005 and 0.0007 and standard deviations as low as ±0.0002 to ±0.0001 (e.g., α₁ = 0.0006 ± 0.0002, α₃ = 0.0005 ± 0.0001). These values are close to zero and exhibit minimal variability, highlighting the model’s robustness, precision, and resilience to experimental noise. Overall, the results confirm that the proposed framework reliably identifies the healthy condition of the structure, even in the presence of uncertainties inherent in real-world experimental data.

For the damaged case, the mean total loss at the end of training is approximately 1.49 × 10⁻³, as illustrated in Fig. 12(a), where both the individual trial curves and the mean loss curve consistently converge to this value. Figure 12(b) demonstrates the convergence of the predicted damage index at the actual damaged location across multiple trials, all of which converge to similar values indicating the consistency and effectiveness of the proposed method. Furthermore, Fig. 12(c) shows the convergence behavior of the mean damage index for each storey, where the values stabilize near zero at undamaged storeys and converge to nonzero values at the damaged location. This highlights the method’s robustness and reliability in avoiding false positives. Figure 12(d) compares the predicted damage indices from the proposed method and the RSM. Although RSM predicts the damaged location with a lower error of 1.08%, it also incorrectly identifies damage at several undamaged locations with false positive predictions reaching up to 21%, thereby reducing its reliability. In contrast, the proposed method identifies the damaged location with a slightly higher error of 14%, but it exhibits no false positive predictions, making it a more trustworthy approach for practical use. Although the accuracy in estimating the exact severity is somewhat lower, the method consistently pinpoints the correct damage location with low variability, demonstrating high confidence and robustness in real-world conditions. The proposes framework demonstrates excellent accuracy and precision in identifying undamaged and damaged locations, outperforming RSM in avoiding false positives despite various modeling uncertainties and measurement noise in the data.

5 Comparison with conventional algorithm and the proposed method

The proposed method is further evaluated against a conventional algorithm, the unscented kalman filter (UKF), to assess its performance for systems with a higher number of storeys and under longer input durations. In both cases, white Gaussian noise excitation is applied. For a fair comparison, the hyperparameters of the proposed model are kept the same as those discussed in Subsection 3.1. Additionally, 30 dB noise and Case 3 damage scenario is considered.

5.1 Validation for higher number of storeys

The study is validated on shear building models with 3, 6, 10, 12, 15, 20, 25, and 30 storeys. Table 6 presents the average percentage errors of damage indices using the UKF method, the proposed approach, and the relative improvement of the proposed method over UKF. The predicted results from UKF show very high error percentages (ranging from 80% to 96%), while the proposed method consistently achieves errors below 10% after training for 50k epochs. This leads to a substantial relative improvement, exceeding 88% in all cases and reaching up to 99.84% for the 3-storey structure. Overall, the proposed method demonstrates significantly higher accuracy and robustness compared to UKF across increasing numbers of storeys. Although the proposed method requires higher computational effort, it provides significant advantages in robustness, accuracy, and reliability, while effectively minimizing false positives, making it a valuable tool for complex damage detection problems.

5.2 Validation with longer time duration

To investigate this aspect, simulations were conducted on the six-storey shear building, where the excitation duration was varied from 15 to 100 s. The results were obtained using both the UKF and the proposed framework. Table 7 presents the average prediction errors of damage indices using UKF and the proposed method for different excitation durations. The UKF error increases sharply with duration, rising from about 30% at 15 s to more than 350% at 100 s. In contrast, the proposed method consistently maintains very low errors, all below 3.2%, regardless of duration. These findings highlight the limitations of step-by-step iterative methods for long-term predictions, whereas the proposed approach maintains a nearly constant percentage error across different durations. This demonstrates that the proposed approach is more robust and effective for long-duration problems, as it optimizes all time states simultaneously instead of propagating errors through stepwise updates.

6 Conclusions

This study presents a novel parallel P-PINNs framework for structural damage identification in shear building models using acceleration data. By incorporating a nondimensionalized form of the governing equations into the loss function, the method achieves efficient training and stable convergence. Under Gaussian noise excitation, the proposed framework achieves significantly lower prediction errors and improved convergence compared to existing methods. The model achieves near-zero predictions at undamaged locations and accurate identification at damaged storeys (Table 2). It outperforms traditional RSM, particularly in reducing false positives and maintaining robustness across varying noise levels (Figs. 5(c) and 6(c)). Moreover, the prediction variance is very small which confirms the method’s reliability and repeatability. Under real earthquake excitation, the framework demonstrates strong generalization capability and practical relevance. While damage locations with higher severity are accurately identified across all noise levels, lower damage indices exhibit increased sensitivity due to reduced signal-to-noise ratios (Table 3). Nevertheless, the model avoids false positives and maintains high confidence in undamaged regions, reaffirming its robustness for seismic applications. Despite slightly higher errors under strong noise compared to Gaussian excitation, the overall predictive accuracy remains high and stable. The experimental validation further confirms the method’s applicability in real-world scenarios. The proposed framework reliably detects damage locations and provides damage severity estimates that align well with experimental observations, even in the presence of modeling uncertainty and measurement noise (Table 5). Although the accuracy of the proposed method in quantifying damage is slightly lower compared to RSM, it shows no false positive cases, unlike RSM, which exhibits significant false-positive damage estimations at undamaged locations (Fig. 12(d)). In summary, the proposed framework demonstrates superior performance over traditional methods like RSM, particularly by eliminating false positives and maintaining robustness under varying noise levels and real earthquake excitations. Its consistent accuracy across numerical and experimental cases highlights its potential for practical SHM under real-world uncertainties.

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