# A parallel framework of physics-informed neural networks for model identification of linear and nonlinear systems

Nikhil Mahar, Subhamoy Sen, Laurent Mevel

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## To cite this version:

Nikhil Mahar, Subhamoy Sen, Laurent Mevel. A parallel framework of physics-informed neural networks for model identification of linear and nonlinear systems. Structures, 2025, 79, pp.109454. 10.1016/j.istruc.2025.109454. hal-05140778

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https://inria.hal.science/hal-05140778v1

Submitted on 2 Jul 2025

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# A Parallel Framework of Physics-Informed Neural Networks for Model Identification of Linear and Nonlinear Systems

Nikhil Mahar$^a$, Subhamoy Sen$^{a,1}$, Laurent Mevel$^b$

$^a$i4S Laboratory, Indian Institute of Technology Mandi, Mandi, HP, India

$^b$Univ. Gustave Eiffel, Inria, Cosys-SII, I4S, Campus de Beaulieu, Rennes, France

## Abstract

Recent advances in sensor and communication technologies have enabled widespread monitoring of complex structures using both model-based and data-driven methods for Structural Health Monitoring (SHM). While model-assisted approaches struggle with the complexity of real-world systems, purely data-driven methods offer simplicity and speed but lack physical interpretability. Physics-Informed Neural Networks (PINNs) offer a promising middle ground by integrating data with governing physical laws. However, most existing PINN applications focus on solving forward problems, with limited attention to inverse parameter estimation relevant to SHM.

This study presents a parallel PINN framework for efficient state and parameter estimation in complex structural systems. To address the computational burden of high-dimensional systems, the global dynamics are reformulated into a set of coupled lower-dimensional subproblems, one for each degree of freedom, without any physical partitioning of the domain. This algebraic decoupling retains structural continuity through shared global matrices, enabling localized learning while preserving global consistency. The approach is validated on both linear and nonlinear systems using harmonic and real earthquake excitations, demonstrating robustness under high noise, sparse data, and varying system complexity. An experimental study on a scaled shear frame further supports its practical relevance for real-world SHM.

**Keywords:** Parameter estimation, Coupled dynamical system, Parallel physics-informed neural networks.

## 1. Introduction

Within the realm of structural engineering, endeavors to model and predict structural behavior can be broadly classified into two primary approaches: data-driven and physics-driven methods. Data-driven methods, encompassing statistical techniques defined in time [1–4] and frequency domains [5–7], along with latent space tools like blind source separation (BSS) [8–11] have been extensively studied. These approaches

<footnote>*Corresponding author; E-mail address: subhamoy@iitmandi.ac.in</footnote>
typically rely on empirical data for model construction but often exhibit constraints when dealing with
non-linear systems. In contrast, physics-driven methods assume a predefined model structure, either in the
form of a parameterized model or a set of governing differential equations, which are subsequently updated
through optimization or Bayesian approaches [12–17].

In contrast to model-based approaches for parameter estimation, such as Bayesian filtering algorithms [18–20], model-independent techniques [8–11] generally rely on certain fault-sensitive statistical distances and may not leverage the known physics to inculcate inherent spatial correlation in the data into the estimation process. Conversely, the inclusion of a support model, while theoretically advantageous, can introduce significant computational burdens and compromise qualitative precision due to the sensitivity of statistical distances concerning the model, making them susceptible to errors.

The adoption of a comprehensive optimization-based approach instead of a simple hypothesis framework
may inadvertently slow down the overall process and reduce efficiency. Additionally, integrating data with
physical models for estimation can introduce significant computational and modeling constraints, potentially
resulting in suboptimal performance. However, a breakthrough is emerging in the realm of machine learning
(ML) and artificial intelligence (AI)-based algorithms.

Continuing in the domain of vibration-based structural health monitoring (SHM) and system identifi-
cation, Artificial Neural Networks (ANNs) have gained considerable attention in the past decade [21, 22].
Researchers have successfully utilized ANNs to ascertain various parameters of structural systems, including
mass, stiffness, damping coefficients, eigenvalues, and eigenmodes, utilizing acceleration and vibration data
as inputs [23–26]. Nevertheless, the accuracy, efficiency, and generalizability of such networks heavily depend
on the availability of extensive training data, limiting their use for sparsely measured systems. Moreover,
their “black box” nature poses challenges in interpreting the underlying physics of the system. These limita-
tions eventually impede their integration into real-life structural system identification, where data collection
is often expensive, time-consuming, and uncertain [27, 28]. Additionally, the fixed architecture of ANNs
may lead to errors when the system properties undergo changes, necessitating assumptions and a substantial
amount of input data capturing a wide spectrum of system variability.

To overcome inherent limitations in modeling complex physical systems, researchers have introduced
PINNs as a viable solution [29]. PINNs integrate the physics of the system, represented by governing equa-
tions, into the loss function, combining the strengths of ANNs with physical laws. This integration allows
for more precise modeling in situations with limited data and simplified system representations. PINNs have
showcased remarkable adaptability, as seen in innovations such as Conservative PINNs (cPINNs) and Vari-
ational PINNs (VPNNs), customized for specific problem domains [30, 31]. [32] introduced physics-guided
CNNs and physics-informed Long Short-Term Memory (LSTM) networks, showcasing the utility of PINNs in
predicting building responses to earthquake excitation. Similarly [33, 34] introduced an uncertainty-oriented,
physics-informed LSTM neural network and a hybrid model and data-driven framework for dynamic load
identification. These advancements underscore PINNs' versatility in diverse fields, spanning fluid mechanics and solid mechanics [35, 36]. Their proficiency in handling differential equations, commonly used to describe physical system behavior, further enhances their utility [37].

While PINNs have been extensively used for inverse problems, limited studies have explored their application in state-parameter estimation, specifically for inverse problems in structural dynamics. [38] proposed physics-informed neural ODEs to address linear and nonlinear structural identification problems in discrete systems. However, a significant limitation is their reliance on full state measurements at all relevant degrees of freedom, constraining their real-world application. Recently, researchers have utilized PINNs to solve inverse problems in both forward and inverse systems governed by Ordinary Differential Equations (ODEs). For example, [39], [40], and [41] concurrently estimated the states and parameters of governing linear and nonlinear ODEs representing multi-degree-of-freedom (MDOF) systems under forced vibration. PINNs have demonstrated adaptability in addressing changes in structural properties, proving advantageous for diagnosing and controlling damaged systems [42, 43].

However, these studies primarily focused on modal parameter identification, limiting their application to nonlinear systems. The conventional PINNs approach, with a single neural network as a global regularizer, is perceived to be computationally challenging due to over-parameterization, impeding its application for high-dimensional systems. This aspect will be demonstrated through numerical experimentation later in this article. The proposed method tackles these challenges by breaking down the problem into smaller algebraic segments using a parallelized PINN framework. This innovative architecture estimates both observable and hidden variables, as well as health parameters, within SHM by focusing on state and parameter estimation over time. Training optimizes the network and health parameters by applying a physics-informed loss function that leverages actual data and integrates spatial correlations in dynamic responses and boundary conditions.

The proposed parallel and interconnected PINN framework enables localized enforcement of physics-based constraints without physically partitioning the structure, improving computational efficiency and practical feasibility. Its performance is demonstrated through numerical and experimental studies involving both linear and nonlinear systems, where the estimated time histories and parameter trajectories closely match true values, confirming the approach's accuracy and responsiveness in system identification. The theoretical background on ANNs and PINNs is introduced in Section 2, followed by a detailed description of the proposed methodology in Section 3. Numerical and experimental validations are presented in Sections 4 and 5, respectively, with discussion and concluding remarks summarized in Section 6 and Section 7.

## 2. Theoretical background

Originally inspired by biological neurons, ANNs are universal function approximators, available in shallow or deep forms depending on the number of hidden layers [44, 45]. Deep Neural Networks (DNNs), with
multiple layers, enhance model expressiveness and performance by leveraging large datasets and computational power [46, 47]. Their success across domains such as computer vision, natural language processing, and medical imaging has led to widespread adoption [48]. For a given input-measurement data {$\mathbf{I}, \mathbf{y}$}, a generalized DNN with $L$ layers is defined as:

$$ \tilde{\mathbf{y}} = \mathcal{N}(\mathbf{I}; \theta) \qquad (1) $$

where, $\mathcal{N}(\mathbf{I}; \theta) = \sigma_L(\mathbf{W}_L \cdot \sigma_{L-1}(\dots \sigma_1(\mathbf{W}_1 \cdot \mathbf{I} + \mathbf{b}_1) + \dots) + \mathbf{b}_L)$. In supervised learning, the network is trained using $n$ input-measurement data pairs {$\mathbf{I}_i, \mathbf{y}_i$} by optimizing the weights $\mathbf{W}_l$ and biases $\mathbf{b}_l$, collectively denoted as $\theta$, to minimize data/measurement loss as:

$$ \hat{\theta} = \arg \min_{\theta} \left\{ \mathcal{L}(\theta) := \frac{1}{n} \sum_{i=1}^{n} |\mathbf{y}_i - \tilde{\mathbf{y}}_i|^2 \right\} \qquad (2) $$

## 2.1. Physics-Informed Neural Network

Yet, both deep and shallow neural networks face the ongoing hurdle of needing extensive training data, especially in engineering, where acquiring such measurement data through experiments or numerical simulations can be prohibitively expensive. This shortage of data can undermine performance, particularly in systems with sparse sampling. Researchers address this by utilizing accumulated information, such as differential equations, to fill data gaps. Raissi et al. [29] proposed PINNs, which integrate the physical laws represented by partial differential equations into supervised learning tasks. These networks can be used for solving PDEs directly (forward problems) or estimating parameters when solutions are available (inverse problems). Essentially, PINNs apply the broad function approximation attributes of neural networks to articulate solutions for both linear and non-linear PDEs governing system dynamics in contexts expressed as:

$$ \begin{aligned} \mathcal{F}(\mathbf{u}(\mathbf{x}, t); \lambda) &= 0, & \mathbf{x} \in \Omega, & t \in [0, T] \\ \mathcal{B}(\mathbf{u}(\mathbf{x}, t)) &= 0, & \mathbf{x} \in \partial\Omega, & t \in [0, T] \end{aligned} \qquad (3) $$

Here, $\mathcal{F}[\bullet; \lambda]$ denotes a linear or nonlinear governing differential operator (gde) defining the dynamics of a latent variable, $\mathbf{u}$, defined in the space-time continuum ($\{\mathbf{x}, t\}$). The differential setup is characterized by a set of parameters, $\lambda$. $\mathcal{B}(\bullet)$ denotes arbitrary boundary operator (e.g., Dirichlet, Neumann, Robin, etc.) that the latent variable $\mathbf{u}(\mathbf{x}, t)$ needs to adhere to in addition to the gde.

### 2.1.1. Forward PINN

Recent applications of PINN aimed to substitute expensive differential equation simulations with a forward model guided by physics. Herein, the latent variable $\mathbf{u}(\mathbf{x}, t)$ can be consistently approximated by the DNN model $\mathcal{N}(\mathbf{x}, t; \theta)$, where $\mathbf{x}$ and $t$ define the input $\mathbf{I}$ (see Equation (1)) through spatial and temporal
coordinates. The network $\mathcal{N}(\mathbf{x}, t; \theta)$ is characterized by parameters $\theta$, which can otherwise be trained using an extensive input (I) and corresponding measurement data (y) in the absence of any known physics as detailed in Equation (3). To reduce reliance on these extensive datasets, physical laws can be incorporated into the training process by adding additional constraint terms to the loss function $\mathcal{L}(\theta)$ introduced in Equation (2). These constraints, derived from various elements (such as gde, boundary conditions, etc.), ensure the training aligns with the known system physics.

$$
\begin{align}
\mathcal{L}_{\text{int}}(\theta) &= \frac{1}{N_1} \sum_{i=1}^{N_{\text{int}}} \sum_{j=1}^{N_{t1}} |\mathcal{F}(\mathcal{N}(\mathbf{x}_i, t_j; \theta); \lambda)|^2, \quad \text{with } \{\mathbf{x}_i, t_j\} \in \Omega_{\text{int}} \nonumber \\
\mathcal{L}_{\text{bc}}(\theta) &= \frac{1}{N_2} \sum_{i=1}^{N_{\text{bc}}} \sum_{j=1}^{N_{t2}} |\mathcal{B}(\mathcal{N}(\mathbf{x}_i, t_j; \theta))|^2, \quad \text{with } \{\mathbf{x}_i, t_j\} \in \Omega_{\text{bc}} \nonumber \\
\mathcal{L}_{\text{data}}(\theta) &= \frac{1}{N_3} \sum_{i=1}^{N_{\text{data}}} \sum_{j=1}^{N_{t3}} |\mathcal{H}(\mathcal{N}(\mathbf{x}_i, t_j; \theta)) - \mathbf{y}(\mathbf{x}_i, t_j)|^2, \quad \text{with } \{\mathbf{x}_i, t_j\} \in \Omega_{\text{data}} \tag{4} \\
\mathcal{L}(\theta) &= \alpha * \mathcal{L}_{\text{int}}(\theta) + \beta * \mathcal{L}_{\text{bc}}(\theta) + \gamma * \mathcal{L}_{\text{data}}(\theta) \nonumber
\end{align}
$$

Here, the set $\{\mathbf{x}_i, t_j\}$ represents all space-time points sampled within the internal domain ($\Omega_{\text{int}}$), boundary domain ($\Omega_{\text{bc}}$) and measurement data domain ($\Omega_{\text{data}}$). These points are referred to as collocation points (for the internal domain), boundary points (for the boundary domain) and measurement points (for the measurement data domain). The quantities $N_{\text{int}}$, $N_{\text{bc}}$ and $N_{\text{data}}$ denote the number of spatial locations in the internal, boundary and measurement domains, respectively, while $N_{t1}$, $N_{t2}$ and $N_{t3}$ represent the number of time instances associated with each. Consequently, the total number of collocation points, boundary points and measurement points are given by $N_1 = N_{\text{int}} \times N_{t1}$, $N_2 = N_{\text{bc}} \times N_{t2}$ and $N_3 = N_{\text{data}} \times N_{t3}$ respectively. Collocation and boundary points are used to enforce the physical laws in addition to data/measurement loss introduced in Equation (2). $\mathcal{H}(\bullet)$ is the measurement function mapping the field variable $\mathbf{u}(\mathbf{x}_i, t_j)$ to corresponding measurement data $\mathbf{y}(\mathbf{x}_i, t_j)$. For the context of the forward framework, the system parameter $\lambda$ is assumed to be known. The weighted loss function $\mathcal{L}(\theta)$ can be assumed to be a linear combination of residual loss, $\mathcal{L}_{\text{int}}$, boundary loss, $\mathcal{L}_{\text{bc}}$, and data loss, $\mathcal{L}_{\text{data}}$ as defined in Equation (4). The network optimization protocol is then enforced to estimate $\theta$, under the influence of a configurable set of hyperparameters, $\alpha$, $\beta$ and $\gamma$, acting as penalty terms, that optimally minimize $\mathcal{L}(\theta)$. It's noteworthy that the network parameters are shared by both $\mathcal{N}(\mathbf{x}, t; \theta)$ and $\mathcal{L}(\theta)$. Once the optimum $\theta$, which minimizes $\mathcal{L}$, is determined, the optimal solution $\mathbf{u}(\mathbf{x}, t)$ is also established.

The notable benefits of PINN in addressing the forward problem, as illustrated in [29] and [49], encompass its freedom from extensive dependence on the richness of simulation data, thereby ensuring computational efficiency. Additionally, PINN, trained with gdes, inherently adheres to essential physical principles such as invariance and symmetry.

### 2.1.2. Inverse PINN

The inverse method for implementing PINN in SHM and state-parameter estimation lacks extensive exploration. In system estimation's inverse domain, conventional data-driven methods focus on approximating $\mathbf{u}(\mathbf{x}, t)$ via a DNN model $\mathcal{N}(\mathbf{x}, t; \{\theta; \lambda\})$, conditioned on system parameters $\lambda$. To facilitate this,
λ is augmented with θ, forming an expanded parameter set {θ; λ}, which is inferred from existing data. Furthermore, to integrate physics-based understanding, physical constraints are embedded in the training process through loss functions:

$$
\begin{align}
\mathcal{L}_{\text{int}}(\{\theta; \lambda\}) &= \frac{1}{N_1} \sum_{i=1}^{N_{\text{int}}} \sum_{j=1}^{N_{t1}} |\mathcal{F}(\mathcal{N}(\mathbf{x}_i, t_j; \{\theta; \lambda\}); \lambda)|^2, && \text{with } \{\mathbf{x}_i, t_j\} \in \Omega_{\text{int}}, \nonumber \\
\mathcal{L}_{\text{bc}}(\{\theta; \lambda\}) &= \frac{1}{N_2} \sum_{i=1}^{N_{\text{bc}}} \sum_{j=1}^{N_{t2}} |\mathcal{B}(\mathcal{N}(\mathbf{x}_i, t_j; \{\theta; \lambda\}))|^2, && \text{with } \{\mathbf{x}_i, t_j\} \in \Omega_{\text{bc}}, \tag{5} \\
\mathcal{L}_{\text{data}}(\{\theta; \lambda\}) &= \frac{1}{N_3} \sum_{i=1}^{N_{\text{data}}} \sum_{j=1}^{N_{t3}} |\mathcal{H}(\mathcal{N}(\mathbf{x}_i, t_j; \{\theta; \lambda\})) - \mathbf{y}(\mathbf{x}_i, t_j)|^2, && \text{with } \{\mathbf{x}_i, t_j\} \in \Omega_{\text{data}} \nonumber
\end{align}
$$

wherein {xᵢ, tⱼ} ∈ Ω₉ₐᵣ denotes the space-time coordinates for which measurement (y(xᵢ, tⱼ)) is available. These samples are typically obtained from real-world sources, such as experimental data. Additionally, the overall loss function is defined as a weighted sum of the component loss functions.

$$
\mathcal{L}(\{\theta; \lambda\}) = \alpha * \mathcal{L}_{\text{int}}(\{\theta; \lambda\}) + \beta * \mathcal{L}_{\text{bc}}(\{\theta; \lambda\}) + \gamma * \mathcal{L}_{\text{data}}(\{\theta; \lambda\}) \quad (6)
$$

While a converged PINN simultaneously fits training data and satisfies physical laws via a physics-based loss, its use in inverse problems for structural dynamics, especially with coupled differential equations, remains underexplored. This study bridges that gap by proposing a novel parallel PINN framework.

3. Proposed Framework

The proposed framework focuses on the inverse estimation of a mechanical model, which is represented
in a discretized form using Finite Element Modelling (FEM). The system is characterized by mass **M**,
damping **C**, and stiffness **K**(**λ**) matrices, where only the stiffness matrix is modeled as a function of the
health parameter **λ**. In this context, the index **x** refers to the spatial positions of the nodal degrees of
freedom (DOFs), with the corresponding system responses: displacement **u**, velocity **u̇**, and acceleration
**ü**, represented as vectors defined over the discretized DOFs. The dynamic behavior of the system under
external excitation **f**(**x**, **t**) is governed in continuous time by the following equation:

$$
\begin{align}
& \mathbf{M}\ddot{\mathbf{u}}(\mathbf{x}, t) + \mathbf{C}\dot{\mathbf{u}}(\mathbf{x}, t) + \mathbf{K}(\lambda)\mathbf{u}(\mathbf{x}, t) - \mathbf{f}(\mathbf{x}, t) = 0 \tag{7} \\
& \text{or in compact form, } \quad \mathcal{F}(\mathbf{u}(\mathbf{x}, t), \lambda) = 0 \nonumber
\end{align}
$$

The inverse estimation of a mechanical model, based on discrete-time response signals, inevitably leads
to time-discrete formulations that remain spatially coupled. Such coupled MDOF formulations can pose
significant computational challenges, potentially rendering direct estimation approaches impractical. To
overcome this complexity, a parallel PINN framework is introduced, wherein the system's *gde* is algebraically
decomposed into smaller, tractable subdomains. This is achieved by isolating each row of the *gde*, effectively
creating a set of lower-dimensional systems, each governed by a reduced number of state and parameter
variables. A dedicated PINN is employed for each of these subsystems, allowing for simultaneous estimation across the entire model. Each PINN operates on an independent subset of the global health parameters and is trained concurrently with the others. Despite their parallel operation, these networks are intrinsically linked through a composite loss function that incorporates the outputs from all subsystems, ensuring consistency and coherence with the global dynamics. This modular approach not only enhances computational efficiency but also enables scalable and robust estimation of the underlying health parameters.

By parameterizing the system dynamics using a set of health indicators $\lambda$, the goal is to estimate the spatially distributed health state of the structural system. To this end, the parameter set $\lambda$ is defined as a collection of localized health or damage parameters, denoted as $\lambda_i$ for $i = 1, \dots, n_s$, where $n_s$ represents the total number of substructures resulting from the discretization of the system. Each individual parameter $\lambda_i$ corresponds to a specific discretized element of the FEM and quantifies its health condition. Practically, $\lambda_i$ reflects the degree of stiffness degradation at the associated location, where a value of 0 represents complete damage, and a value of 1 indicates an undamaged or fully healthy state. The key steps of the proposed estimation procedure are outlined as follows: Step 1: Define a set of space-time collocation points, denoted as $\Omega_{int} = \{(\mathbf{x}_i, t_j)\}$, which are distributed across spatial and temporal coordinates. When focusing on an individual substructure at a specific spatial location $x_i$, the neural network approximator for the latent variable $\mathbf{u}$ simplifies to a time-dependent form: $\mathbf{u}(t)|_{x=x_i} \approx \mathcal{N}_i(t_j; \{\theta_i, \lambda_i\})$, for all $j \in \{1, \dots, N_{t1}\}$, thus removing explicit spatial dependency. Similarly, measurement points for each isolated substructure, the relevant subset $\Omega_{i,\text{data}}$—containing measurements at space coordinate $x_i$ is used to train the corresponding network $\mathcal{N}_i(t_j; \{\theta_i, \lambda_i\})$, for all $j \in \{1, \dots, N_{t3}\}$.

Step 2: Subsequently, to train networks for the $i^{th}$ substructure independently, it is necessary to define the physics-informed loss function to quantify the physics loss as follows:

$$ \mathcal{L}_{i,int}(\{\theta; \lambda\})|_{\mathbf{x}_i} = \frac{1}{N_{t1}} \sum_{j=1}^{N_{t1}} |R_i(t_j; \{\bar{\theta}_i; \bar{\lambda}_i\})|^2 \quad \text{with} \quad \{\mathbf{x}_i; t_j\} \in \Omega_{int} \qquad (8) $$

where $R_i(t_j, \{\bar{\theta}_i; \bar{\lambda}_i\})$ represents the residual in the physical equation depicting the substructure dynamics for $i^{th}$ substructure (i.e., $\mathcal{F}_i(\cdot)$), conditioned only on a subset, $\{\bar{\theta}; \bar{\lambda}\}$, of the complete sets of network parameters $\theta = (\theta_i)$ and system parameters $\lambda = (\lambda_i)$. This can further be elaborated concerning Equation (3) as,

$$ R_i(t_j, \{\bar{\theta}_i; \bar{\lambda}_i\}) = \mathcal{F}_i(\mathbb{N}_i(t_j; \{\bar{\theta}_i; \bar{\lambda}_i\}); \bar{\lambda}_i) \qquad (9) $$

wherein $\bar{\theta}_i$ and $\bar{\lambda}_i$ are the relevant subset of $\theta$ and $\lambda$ to the $i^{th}$ residual function $R_i(\cdot)$. $\bar{\theta}_i$ and $\bar{\lambda}_i$ can be defined as:
$$
\begin{align}
\bar{\theta}_1 &= \{\theta_1, \theta_2\} \quad \text{and} \quad \bar{\lambda}_1 &= \{\lambda_1, \lambda_2\}; \quad \text{for } i=1; \nonumber \\
\bar{\theta}_{n_s} &= \{\theta_{n_s-1}, \theta_{n_s}\} \quad \text{and} \quad \bar{\lambda}_{n_s} &= \{\lambda_{n_s-1}, \lambda_{n_s}\}; \quad \text{for } i=n_s; \tag{10} \\
\bar{\theta}_i &= \{\theta_{i-1}, \theta_i, \theta_{i+1}\}; \quad \text{and} \quad \bar{\lambda}_i &= \{\lambda_{i-1}, \lambda_i, \lambda_{i+1}\} \quad \text{elsewhere}; \nonumber
\end{align}
$$

Similarly,

$$
\begin{align}
\mathbb{N}_1(t_j; \{\bar{\theta}_1; \bar{\lambda}_1\}) &= [\mathcal{N}_1(t_j; \{\theta_1; \lambda_1\}), \mathcal{N}_2(t_j; \{\theta_2; \lambda_2\})] && \text{for } i=1; \nonumber \\
\mathbb{N}_{n_s}(t_j; \{\bar{\theta}_{n_s}; \bar{\lambda}_{n_s}\}) &= [\mathcal{N}_{n_s-1}(t_j; \{\theta_{n_s-1}; \lambda_{n_s-1}\}), \mathcal{N}_{n_s}(t_j; \{\theta_{n_s}; \lambda_{n_s}\})] && \text{for } i=n_s; \nonumber \\
\mathbb{N}_i(t_j; \{\bar{\theta}_i; \bar{\lambda}_i\}) &= [\mathcal{N}_{i-1}(t_j; \{\theta_{i-1}; \lambda_{i-1}\}), \mathcal{N}_i(t_j; \{\theta_i; \lambda_i\}), \mathcal{N}_{i+1}(t_j; \{\theta_{i+1}; \lambda_{i+1}\})] && \text{elsewhere}; \tag{11}
\end{align}
$$

In the framework proposed in this study, the physics outlined by $\mathcal{F}_i(\bullet)$ essentially illustrates the dynamics of the substructure, which then functions as constraints within the residual function during parameter set optimization. Through effective optimization, the refined parameter sets guarantee adherence to the system dynamics, even at the substructure level.

Step 3: Boundary condition loss:

$$
\mathcal{L}_{i,bc}(\{\theta_i; \lambda_i\}) = |\mathcal{B}_i(\mathcal{N}_i(t_0; \{\theta_i; \lambda_i\})) - \mathbf{y}(\mathbf{x}_i, t_0)|^2 \quad \text{with} \quad \{\mathbf{x}_i, t_0\} \in \Omega_{i,bc} \quad (12)
$$

Step 4: The measurement loss is then defined by comparing network output to the measurement data $\mathbf{y}(\mathbf{x}_i, t_j)$ sampled at $\Omega_{i,\text{data}}$.

$$
\mathcal{L}_{i,\text{data}}(\{\theta_i; \lambda_i\}) = \frac{1}{N_{t3}} \sum_{j=1}^{N_{t3}} |\mathcal{H}_i(\mathcal{N}_i(t_j; \{\theta_i; \lambda_i\})) - \mathbf{y}(\mathbf{x}_i, t_j)|^2 \quad \text{with} \quad \{\mathbf{x}_i, t_j\} \in \Omega_{i,\text{data}} \quad (13)
$$

Step 5: The overall loss is defined as aggregating losses from all parallel networks as,

$$
\mathcal{L}(\{\theta, \lambda\}) = \sum_{i=1}^{n_s} (\alpha \mathcal{L}_{i,int} + (1-\alpha) \mathcal{L}_{i,bc} + (1-\alpha) \mathcal{L}_{i,\text{data}}) \quad (14)
$$

where $\alpha$ is the weight term. It has to be chosen wisely as the issue of competing terms in the loss function can arise in deep learning training, particularly when dealing with multiple physical constraints or a large number of subdomains. Each term in the loss function represents a different physical constraint or data fidelity term, and thus, they might converge at different rates, meaning one or some terms might dominate the training process. In the proposed framework, we assign a weight $\alpha$ and $(1 - \alpha)$ to each term in the loss function, as described in Equation (14), to balance their influences.

Step 6: Finally, the optimization of the neural network and structural parameters $\{\theta, \lambda\}$

$$
\{\theta^*, \lambda^*\} = \arg \min_{\{\theta, \lambda\}} \mathcal{L}(\{\theta, \lambda\}) \quad (15)
$$

Figure 1 illustrates the architecture of the proposed framework for joint state and parameter identification in structural systems. The schematic progresses from left to right. On the far left, the physical system is
modeled as MDOF structure equipped with sensors. Each DOF is associated with an independent neural network $N_i$, forming the parallel neural network (Parallel NNs) block. Sensor data is organized into three categories, as shown in the “Inputs” block: (i) measurement points $\Omega_{i,\text{data}}$ (pink), (ii) boundary condition points $\Omega_{i,\text{bc}}$ (yellow), and (iii) collocation points for enforcing physics-based constraints $\Omega_{i,\text{int}}$ (green). Each neural network $N_i$ receives time $t$ as input and outputs the corresponding displacement prediction $\tilde{u}_i(t)$. Although only displacement is directly predicted, velocity $\dot{u}_i(t)$ and acceleration $\ddot{u}_i(t)$ are computed via automatic differentiation (AD) or finite differences (FD), as shown in the “AD/FD” block. This differentiation process establishes a physical continuity chain that inherently links displacement, velocity, and acceleration. The resulting displacement, velocity, and acceleration are then used to evaluate (i) physics-informed residual loss $F_i$ (green), in addition to (ii) data loss $H_i$ (pink), based on measured acceleration, and (iii) the boundary/initial condition loss $B_i$ (yellow). Since velocities and acceleration are derived from predicted displacements, which are implicitly consistent using AD/FD, and used altogether in a physics-informed loss function $F_i$ (green), it eliminates the need to predict them separately and apply additional scaling factors. Finally, physics loss, data loss, and boundary condition loss are combined using weighting factors and aggregated into a total loss $L(\theta, \lambda)$. These weighting factors are taken as per user experience and can be automated in the future using self-adaptive tuning techniques [50]. Optimization is further carried out via a feedback loop that simultaneously updates both the neural network parameters $\theta$ and the unknown physical parameters $\lambda$.

<figure><img src="image_3.png" /><figcaption>Figure 1: Schematic diagram of the Parallel physics-informed neural networks framework for coupled dynamical systems</figcaption></figure>
The pseudo-code for the above methodology is given in Algorithm 1.

<caption>**Algorithm 1** Proposed algorithm for Parallel PINNs framework for damage detection, localization and quantification in coupled dynamical system</caption>

1: **Initialize:**
2: Divide the structural system into $n_s$ substructures.
3: Specify the inputs, including measurement points ($\Omega_{i,\text{data}} \subset \Omega_{\text{data}}$), collocation points ($\Omega_{i,\text{int}} \subset \Omega_{\text{int}}$), boundary/initial condition points ($\Omega_{i,\text{bc}} \subset \Omega_{\text{bc}}$).
4: Specify measurement data $Y_{i,\text{data}}$
5: Create neural network for each substructure ($\mathcal{N}_i(t_j; \{\theta_i, \lambda_i\})$)
6: Set hyper-parameters (no. of layers, no. of neurons, learning rate) of each neural network.
7: Initialize the neural network parameters $\{\theta_i, \lambda_i\}$
8: Set the weight of each component of the total loss in each substructure
9: Set convergence criteria.
10: **while** Convergence not reached **do**
11: &nbsp;&nbsp;&nbsp;&nbsp;**for** $i = 1$ to $n_s$ **do**
12: &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;**for** $j = 1$ to $N_{t1}$ **do**
13: &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Compute $\mathcal{N}_i(t_j; \{\theta_i, \lambda_i\})$, $\dot{\mathcal{N}}_i(t_j; \{\theta_i, \lambda_i\})$, $\ddot{\mathcal{N}}_i(t_j; \{\theta_i, \lambda_i\})$
14: &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;**end for**
15: &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Compute the collocation points loss of $i^{th}$ substructure: $\mathcal{L}_{i,\text{int}}$ ▷ Equation (8)
16: &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;**for** $j = 1$ to $N_{t2}$ **do**
17: &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Compute $\mathcal{N}_i(t_j; \{\theta_i, \lambda_i\})$, $\dot{\mathcal{N}}_i(t_j; \{\theta_i, \lambda_i\})$
18: &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;**end for**
19: &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Compute the boundary condition loss of $i^{th}$ substructure: $\mathcal{L}_{i,\text{bc}}$ ▷ Equation (12)
20: &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;**for** $j = 1$ to $N_{t3}$ **do**
21: &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Compute $\dot{\mathcal{N}}_i(t_j; \{\theta_i, \lambda_i\})$
22: &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;**end for**
23: &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Compute the data loss of $i^{th}$ substructure: $\mathcal{L}_{i,\text{data}}$ ▷ Equation (13)
24: &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;**end for**
25: Compute total loss $\mathcal{L}(\theta, \lambda)$ ▷ Equation (14)
26: Update parameters $\{\theta, \lambda\}$ by minimizing the total loss $\mathcal{L}(\theta, \lambda)$ using Adam optimizer ▷ Equation (15)
27: **end while**
28: **Deployment:**
29: The estimated structural parameters are $\lambda$;
30: The estimated full state of the system $\mathcal{N}_i(t_j; \{\theta_i, \lambda_i\})$, $\dot{\mathcal{N}}_i(t_j; \{\theta_i, \lambda_i\})$, $\ddot{\mathcal{N}}_i(t_j; \{\theta_i, \lambda_i\})$ predicted by trained neural network

# 4. Numerical validation

Before exploring the practical application of the proposed approach, it is essential to illustrate the limitations of a conventional PINN approach with a single neural network acting as a global regularizer. To demonstrate this, a 2DOF spring-mass-dashpot linear system is utilized, attempting to estimate the spring stiffnesses from its uncontaminated response to harmonic forcing. It is observed that, even with two coupled equations, the conventional inverse-PINN framework struggles to estimate latent variables and system parameters (refer to Figure 2). This inadequacy can be attributed to overparameterization, highlighting the need for a parallelized framework to address the shortcomings of conventional PINN approaches.

To demonstrate the utility of the developed Parallel PINNs framework, several classical structural systems have been selected as test cases, including one linear and two non-linear MDOF systems. It is important to note that, due to the numerical nature of the validation study, simulated responses are utilized as measurement data, which are intentionally contaminated with realistic levels of noise to mimic real-world scenarios before employing them for estimation. For practical field applications, however, real data would be employed
<figure><img src="image_4.png" /><figcaption>Figure 2: Estimation of observable variable and parameters for linear 2DOF system with conventional inverse PINN
Observable state and parameter estimation are compared against their true values with 0% snr noise</figcaption></figure>

for system estimation.

The non-linear cases involve the Duffing oscillator and Bouc-Wen model each representing different types of nonlinearity. While all of these models introduce nonlinearity in the system formulation, they differ fundamentally: the Bouc-Wen model excels at simulating asymmetric and rate-dependent hysteresis relevant for structural engineering, while the Duffing oscillator model represent stiffness nonlinearity in various other applications. Incorporating these types of nonlinearity is crucial for demonstrating the comprehensive capabilities of the proposed framework.

As illustrated in the PINN schematics in Figure 1, the proposed framework comprises several components: input data sampled from the substructured domain, parallel neural networks, physics-informed loss functions, and a feedback mechanism. The training process involves the selection of a set of uniformly chosen time instants ($\Omega_{i,int}$ and $\Omega_{i,bc}$), forming a set of time instances $\Omega_{i,phy} := \{\Omega_{i,int}; \Omega_{i,bc}\}$ where the physics of the system (pertinent to the interior and boundary domains, respectively) is enforced during learning. These collocation and boundary points serve as the training inputs for the PINN framework, where the physics and boundary loss for the PINN is determined by assessing the compatibility of the DNN output with the system physics and boundary conditions. While not mandatory for the proposed algorithm, the same sets of timestamps are used for all substructures in this study without affecting the generalization of the framework. Among these timestamps, only a randomly chosen subset ($\Omega_{i,data} \subset \Omega_{i,phy}$) is provided with corresponding measurements $\mathbf{Y}_{i,data} (= \mathbf{y}(\Omega_{i,data}))$, resulting in the measurement points and data pair $\mathcal{D} = \{\Omega_{i,data}; \mathbf{Y}_{i,data}\}$.
Before embarking on the numerical experiments, it is important to establish a standardized nomenclature for clarity in presenting plots across three case studies, each corresponding to linear, nonlinear with Duffing oscillator, and nonlinear with Bouc-Wen model, denoted as Linear, Nonlin-D, and Nonlin-BW, respectively. The nomenclature for various plots includes observation (Obs-), Hysteresis (RD-), Phase Portrait (PP-), and Parameter Estimation (Par-) histories, each specific to its corresponding case study. The noise contamination levels are indicated with the snr percentages in the plot name, and the type of excitation forces, whether harmonic or earthquake-induced, is denoted with "-H" or "-Eq" in the plots. This systematic approach ensures consistency and facilitates clear interpretation and comparison of results across different scenarios. For e.g., Nonlin-BW-H-PP-5 refers to the Phase Portrait plot of the Bouc-Wen nonlinear system under harmonic excitation with 5% snr.

<figure><img src="image_5.png" /><figcaption>Figure 3: Schematic diagram of the test structure for (a) 3DOF Linear (b) 3DOF Nonlin-D (c) 3-DOF Nonlin-BW *m*, *k*, *c*, *d* and *f* represent mass, stiffness, damping, cubic nonlinearity and forcing respectively.</figcaption></figure>

### 4.1. Numerical case study 1: Linear MDOF system

For the first case study, a linear 3DOF structural system is adopted, represented by a spring-mass-dashpot model. A schematic diagram of the same is presented in Figure 3(a). The generalized equation of motion for the system dynamics can be perceived from the generalized Equation (7), enforcing the assumed dimensionality. Because of the simplicity of the adopted system, the space can easily be discretized into three substructures without needing a complex FEM. Accordingly, the mass matrix can be idealized with $M = diag([m_1; m_2; m_3])$, and other system matrices can be defined as follows:
$$ \mathbf{K} = k_0 \begin{bmatrix} \lambda_1 + \lambda_2 & -\lambda_2 & 0 \\ -\lambda_2 & \lambda_2 + \lambda_3 & \lambda_3 \\ 0 & -\lambda_3 & \lambda_3 \end{bmatrix}; \mathbf{C} = \begin{bmatrix} c_1 + c_2 & -c_2 & 0 \\ -c_2 & c_2 + c_3 & -c_3 \\ 0 & -c_3 & c_3 \end{bmatrix}; \mathbf{f}(t) = \begin{bmatrix} f_1(t) \\ f_2(t) \\ f_3(t) \end{bmatrix}; \mathbf{u}(t) = \begin{bmatrix} u_1(t) \\ u_2(t) \\ u_3(t) \end{bmatrix} $$

Here, the elements in the mass and damping matrices are considered as $m = m_1 = m_2 = m_3 = 1$ kg, $c_1 = c_2 = c_3 = 0.32$ Ns/m. With an assumption of initial stiffness $k_0 = 30$ N/m, the system is assumed to have deteriorated in its second and third DOF with corresponding values for HIs as $\lambda_1 = 1$, $\lambda_2 = 0.833$ (deterioration of $1/6^{th}$ of original), and $\lambda_3 = 0.667$ ($1/3^{rd}$ of original). The lumped spring-mass-dashpot system is then simulated for 10sec with a sampling frequency of 50Hz under harmonic force $f_1 = f_2 = f_3 = 0.2 \sin(\pi t)$ N.

Within the parallel PINN framework, the data-driven network is composed of three distinct DNNs, each corresponding to a specific substructure and describing the dynamics of an individual mass block. The training inputs for each DNN are generated using 500 uniformly chosen timestamps for $\Omega_{i,int}$ and further selecting 100 randomly chosen timestamps for $\Omega_{i,data}$ from $\Omega_{i,int}$, respectively. The dataset $\mathcal{D}$ is further constructed by sampling $\mathbf{Y}_{i,data}$ at $\Omega_{i,data}$.

The estimation process initializes $\lambda$ with arbitrary realizations drawn from a uniform distribution within the range $[0 - 1]$ for estimating both states and parameters. A neural network architecture with 1 input, 1 output, and 6 hidden layers with 20 neurons in each layer is adopted in all the substructures. This consistency is, however, not mandatory, and the user is free to choose independent networks. A linear and non-linear (tanh) activation function is used in the output and remaining layers, respectively. To integrate physics into the inverse estimation process, the latent displacement field $\mathbf{u}(t)$, along with its temporal derivatives $\dot{\mathbf{u}}(t)$ and $\ddot{\mathbf{u}}(t)$, should satisfy the governing physical constraints as defined by Equation (7). This necessitates differentiating the neural network output $\mathbf{u}(t)$ with respect to time, which is achieved through the concurrently trained neural network model. To enforce this constraint in a tractable manner, the derivatives of the displacement field can be obtained either through automatic differentiation (AD) or via numerical approximation such as finite difference (FD) schemes, as shown in Figure 1. Although AD offers machine-precision derivatives and seamless integration into deep learning frameworks, it comes with significant computational cost, as noted in [51]. This aspect has been investigated in this study.

It can be seen from Figure 4(a) and Figure 4(b) that while Parallel PINN-AD and Parallel PINN-FD converge to the true values (indicated by the dashed lines) the FD-based model demonstrates faster convergence. Additionally, Figure 4(c) highlights the substantial computational gain achieved using the Parallel PINN with FD approach, which reduces training time by over 75% compared to the Parallel PINN with AD.

Further, the training procedure utilizes the Adam optimization scheme, incorporating momentum and
adaptive learning rate mechanisms to optimize the parameters of each neural network simultaneously. Parallel PINNs training has been performed in the Python environment using Pytorch on a computer with 12 Intel Core i7-12700K CPUs. The Adam optimizer with a learning rate of $10^{-5}$ is used to update the parameters of all neural networks. The stopping criteria for the proposed parallel PINNs are set as either reaching a maximum of $5 \times 10^6$, epochs, achieving a loss ($\mathcal{L}(\theta, \lambda)$) less than $5 \times 10^{-5}$.

Table 1 presents a quantitative evaluation in terms of the mean squared error (MSE) for displacement, velocity, and acceleration predictions, along with the percentage error (PE) in the estimated damage parameter $\lambda$. Across all DOFs, the FD-based approach consistently yields lower MSE values and more accurate damage parameter estimates, further validating its efficiency and accuracy.

<figure><img src="image_6.png" /><figcaption>Figure 4: Comparison of computational performance between Parallel PINN-AD and Parallel PINN-FD frameworks</figcaption></figure>

<table><caption>Table 1: Case study: Linear with harmonic forcing — Error analysis of states and parameters for Parallel PINN-AD and Parallel PINN-FD. (Errors in latent and observable states ($\mathbf{u}$, $\dot{\mathbf{u}}$, $\ddot{\mathbf{u}}$) are reported as Mean Squared Error (MSE), and parameter errors ($\lambda$) are reported in percent (PE).)</caption><thead><tr><th rowspan="3">DOF</th><th colspan="4">Parallel PINN-AD</th><th colspan="4">Parallel PINN-FD</th></tr><tr><th colspan="3">MSE</th><th rowspan="2">PE</th><th colspan="3">MSE</th><th rowspan="2">PE</th></tr><tr><th>u</th><th>u̇</th><th>u̇̇</th><th>u</th><th>u̇</th><th>u̇̇</th></tr></thead><tbody><tr><td>1</td><td>2.69 × 10<sup>-8</sup></td><td>5.45 × 10<sup>-7</sup></td><td>1.84 × 10<sup>-5</sup></td><td>0.21</td><td>5.84 × 10<sup>-9</sup></td><td>1.41 × 10<sup>-7</sup></td><td>1.05 × 10<sup>-5</sup></td><td>0.17</td></tr><tr><td>2</td><td>4.83 × 10<sup>-8</sup></td><td>2.95 × 10<sup>-7</sup></td><td>1.71 × 10<sup>-5</sup></td><td>0.09</td><td>1.16 × 10<sup>-8</sup></td><td>1.68 × 10<sup>-7</sup></td><td>1.46 × 10<sup>-5</sup></td><td>0.04</td></tr><tr><td>3</td><td>9.46 × 10<sup>-8</sup></td><td>6.32 × 10<sup>-7</sup></td><td>9.19 × 10<sup>-6</sup></td><td>1.36</td><td>1.22 × 10<sup>-8</sup></td><td>5.58 × 10<sup>-8</sup></td><td>5.73 × 10<sup>-6</sup></td><td>0.22</td></tr></tbody></table>

### 4.1.1. Hyperparameter Sensitivity

Although the Parallel PINN-FD framework significantly reduced computational time compared to Parallel PINN-AD, the total training duration of approximately 3.96 hours remains substantial for practical SHM applications. To further accelerate training, a hyperparameter sensitivity study was conducted. The stopping criteria were relaxed by setting the maximum number of training epochs to $5 \times 10^5$. Additionally, the learning rate was adjusted to $1 \times 10^{-4}$ to improve convergence. Among the configurations tested, the
architecture with 3 hidden layers (1–16–32–16–1) yielded the lowest training loss of $1.23 \times 10^{-5}$ with a significantly reduced training time of approximately 53 minutes. This configuration thus provides an effective balance between training accuracy and computational cost and is adopted for all subsequent studies.

<table><caption>Table 2: Number of layers, network architecture, and corresponding training loss for hyperparameter sensitivity</caption><thead><tr><td>Layers</td><td>Network architecture</td><td>Training Loss</td></tr></thead><tbody><tr><td>2</td><td>1-32-32-1</td><td>3.92 × 10<sup>-5</sup></td></tr><tr><td>3</td><td>1-16-32-16-1</td><td>1.23 × 10<sup>-5</sup></td></tr><tr><td>5</td><td>1-16-32-64-32-16-1</td><td>4.21 × 10<sup>-5</sup></td></tr><tr><td>7</td><td>1-16-32-64-128-64-32-16-1</td><td>4.16 × 10<sup>-5</sup></td></tr></tbody></table>

### 4.1.2. Noise sensitivity and forcing variability

The proposed method is further examined for its resilience against noise contamination, wherein the provided data is subjected to contamination with SWGN at signal-to-noise ratios (snr) of, 1%, 5%, and 10%. The influence of 5% noise on the quality of the estimation can be assessed based on Figure 5. The parameter estimation for both contaminated signals is additionally shown in Table 3, showcasing the framework’s ability to estimate states. Although error in the estimation of state and parameter increases with contamination, the results highlight the effectiveness of the proposed algorithm even in the presence of relatively high noise contamination of 10% snr.

<figure><img src="image_7.png" /><figcaption>Figure 5: Case study: Linear with harmonic forcing- Comparison of actual and Parallel PINN predicted (a) observable, and (b) latent states at 5%</figcaption></figure>
<table><caption>Table 3: Case study: Linear with harmonic forcing — Error analysis of states and parameters for noise levels 1%, 5%, and 10% snr. (Errors in latent and observable states (u, ū, ṻ) are reported as Mean Squared Error (MSE), and parameter errors (λ) are reported in percent (PE).)</caption><thead><tr><th rowspan="3">DOF</th><th colspan="4">1% Noise</th><th colspan="4">5% Noise</th><th colspan="4">10% Noise</th></tr><tr><th colspan="3">MSE</th><th rowspan="2">PE</th><th colspan="3">MSE</th><th rowspan="2">PE</th><th colspan="3">MSE</th><th rowspan="2">PE</th></tr><tr><th>u</th><th>ū</th><th>ṻ</th><th>u</th><th>ū</th><th>ṻ</th><th>u</th><th>ū</th><th>ṻ</th></tr><tr><th></th><th></th><th></th><th></th><th>λ</th><th></th><th></th><th></th><th>λ</th><th></th><th></th><th></th><th>λ</th></tr></thead><tbody><tr><td>1</td><td>8.82 × 10<sup>-5</sup></td><td>1.51 × 10<sup>-7</sup></td><td>1.02 × 10<sup>-5</sup></td><td>0.19</td><td>4.77 × 10<sup>-8</sup></td><td>7.72 × 10<sup>-7</sup></td><td>8.30 × 10<sup>-5</sup></td><td>0.20</td><td>1.78 × 10<sup>-5</sup></td><td>2.36 × 10<sup>-6</sup></td><td>3.00 × 10<sup>-4</sup></td><td>2.2</td></tr><tr><td>2</td><td>4.49 × 10<sup>-6</sup></td><td>3.32 × 10<sup>-7</sup></td><td>2.19 × 10<sup>-5</sup></td><td>1.21</td><td>1.89 × 10<sup>-7</sup></td><td>1.79 × 10<sup>-6</sup></td><td>1.06 × 10<sup>-4</sup></td><td>1.58</td><td>6.58 × 10<sup>-5</sup></td><td>3.68 × 10<sup>-6</sup></td><td>3.00 × 10<sup>-4</sup></td><td>0.57</td></tr><tr><td>3</td><td>5.20 × 10<sup>-8</sup></td><td>2.52 × 10<sup>-7</sup></td><td>1.13 × 10<sup>-5</sup></td><td>0.30</td><td>2.77 × 10<sup>-7</sup></td><td>1.63 × 10<sup>-6</sup></td><td>8.18 × 10<sup>-5</sup></td><td>0.71</td><td>1.21 × 10<sup>-4</sup></td><td>4.55 × 10<sup>-6</sup></td><td>2.90 × 10<sup>-3</sup></td><td>0.82</td></tr></tbody></table>

Furthermore, the system underwent simulation for non-Gaussian, non-stationary force conditions, specifically the EL-Centro earthquake, over a period of 5 seconds at a sampling frequency of 50 Hz. The results corresponding to this numerical case study, presented in Figure 6 and Table 4 demonstrate the quality of state and parameter estimation.

<figure><img src="image_8.png" /><figcaption>Figure 6: Case study: Linear with (a) harmonic and (b) earthquake excitation - Performance assessment for health parameter estimation

Parameter estimation is conducted under (a) harmonic and (b) earthquake excitation with a 5% snr noise, validating the quality of the estimation within the proposed framework.</figcaption></figure>

<table><caption>Table 4: Case study: Linear with (a) harmonic, and (b) earthquake excitation — Error analysis of states and parameters for noise level of 5%</caption><thead><tr><th rowspan="3">DOF</th><th colspan="4">Harmonic (5% snr)</th><th colspan="4">Earthquake (5% snr)</th></tr><tr><th colspan="3">MSE</th><th rowspan="2">PE</th><th colspan="3">MSE</th><th rowspan="2">PE</th></tr><tr><th>u</th><th>ū</th><th>ṻ</th><th>u</th><th>ū</th><th>ṻ</th></tr><tr><th></th><th></th><th></th><th></th><th>λ</th><th></th><th></th><th></th><th>λ</th></tr></thead><tbody><tr><td>1</td><td>4.77 × 10<sup>-8</sup></td><td>7.72 × 10<sup>-7</sup></td><td>8.30 × 10<sup>-5</sup></td><td>0.20</td><td>2.08 × 10<sup>-6</sup></td><td>3.89 × 10<sup>-5</sup></td><td>1.90 × 10<sup>-2</sup></td><td>1.67</td></tr><tr><td>2</td><td>1.89 × 10<sup>-7</sup></td><td>1.79 × 10<sup>-7</sup></td><td>1.06 × 10<sup>-5</sup></td><td>1.58</td><td>3.70 × 10<sup>-6</sup></td><td>7.81 × 10<sup>-5</sup></td><td>2.96 × 10<sup>-2</sup></td><td>3.96</td></tr><tr><td>3</td><td>2.77 × 10<sup>-7</sup></td><td>1.63 × 10<sup>-7</sup></td><td>8.18 × 10<sup>-6</sup></td><td>0.71</td><td>9.56 × 10<sup>-6</sup></td><td>7.91 × 10<sup>-5</sup></td><td>1.83 × 10<sup>-2</sup></td><td>1.62</td></tr></tbody></table>

The robustness of the algorithm is evident from the results shown in Figure 5 and Table 3. Despite increased noise levels, the errors in both observable and latent states remain within acceptable bounds. Furthermore, as demonstrated in Figure 6 and Table 4, the algorithm effectively adapts to uncertain external forces. Notably, even under an snr as high as 5%, the maximum observed error does not exceed 5%, showing
the algorithm's resilience across a range of forcing conditions.

### 4.1.3. Comparison with Kalman filtering-based SHM approaches

To benchmark the proposed Parallel PINN-FD framework, a comparison is performed against the classical Kalman filtering-based system identification approach. This serves as a meaningful reference, given the widespread use of Bayesian filtering techniques in handling uncertainty and dynamic and joint state-parameter estimation. Specifically, an Interacting Particle-Ensemble Kalman Filter (IPEnKF) is employed for this comparison.

Linear 3 DOF system is subjected to earthquake force at 50Hz and acceleration data is collected and contaminated with 5% snr SWGN to replicate the real-world situations. The IPEnKF utilizes 20 seconds of measurement data, whereas the proposed Parallel PINN-FD framework requires only 5 seconds of sampled data for training, highlighting its data efficiency. Without delving into the detailed architecture or estimation procedures, the proposed framework exhibits superior stability and smoothness in health index estimation as evident from Figures 7(a) and 7(b). Although the Parallel PINN-FD approach requires slightly more computation time than the IPEnKF, the latter requires extensive tuning of process and measurement noise parameters, posing significant challenges in practical SHM applications. In contrast, the proposed framework demonstrates enhanced robustness to measurement noise and data sparsity, making it more suitable for real-world SHM scenarios.

<figure><img src="image_9.png" /><figcaption>Figure 7: Convergence of health indices $\lambda$ for: (a) IPEnKF; and (b) Parallel PINN-FD for linear 3-DOF system under earthquake excitation, illustrating the framework's ability to capture changes in system dynamics.</figcaption></figure>

### 4.2. Numerical case study 2: Duffing MDOF system

The Duffing equation, a nonlinear second-order differential equation, characterizes oscillators exhibiting complex and often chaotic dynamics [52]. Recognized as a canonical model in chaos theory, it continues to serve in diverse applications, such as modeling ship roll dynamics and structural vibrations. In this
study, a Duffing oscillator model is employed (cf. schematic in Figure 3) to demonstrate the efficacy of the proposed estimation framework when applied to higher-order nonlinear systems. Consistent with prior investigations, this nonlinear system is analyzed under both harmonic excitation and stochastic loading, including the El-Centro earthquake. The gde for such systems retains structural similarity to their linear MDOF counterparts, with modifications to account for the system's intrinsic nonlinearity.

$$M\ddot{u}(t) + C\dot{u}(t) + K(\lambda)u(t) + Du^3(t) - f(t) = 0 \quad (16)$$

wherein all other parameters except **D** are similar to the earlier case study. **D**, for the adopted 3DOF nonlinear system, can further be elaborated as follows:

$$D = \begin{bmatrix} d_1 + d_2 & -d_2 & 0 \\ -d_2 & d_2 + d_3 & -d_3 \\ 0 & -d_3 & d_3 \end{bmatrix};$$

A similar mass matrix used for the linear case is also assumed for the nonlinear case study, while $k_0$ is assumed to be 20 N/m with $c_1 = c_2 = c_3 = 0.1$ Ns/m. For this case study, the $\lambda$ is assumed as $\lambda = (1, 0.8, 0.5)$. For **D**, cubic nonlinearity coefficients, $d_1$, $d_2$, and $d_3$, are assumed as (100, 0, 0). Accordingly, three DNNs are employed in parallel to estimate the systems and parameters from the sampled response.

300 measurement points ($\Omega_{i, data}$) for each of the three DNNs are realized uniformly within an 18-second time window $T$ sampled at a frequency of 50 Hz. For $\Omega_{i, bc}$ initial conditions, pertinent to all DOFs are sampled. $Y_{i, data}$ is realized by simulating the supporting FEM model for the time window $T$ and subsequently sampling the acceleration responses only at $\Omega_{i, data}$. 5% snr SWGN noise is further employed to contaminate the sampled response in order to replicate the real-life noise contamination. This noise contamination is undertaken for all case studies involving different forcing options at a snr level of 5% unless otherwise specified. The penalty parameter $\alpha$ is set to 0.1 while the training is performed with the Adam optimizer to estimate $\lambda$ till the criteria for convergence, i.e. epochs > 5 × 10⁵ is achieved. For initialization, $\lambda$ is randomly sampled from a uniform distribution with the range [0–1]. Explicit descriptions of the residual functions for physics and data losses, pertinent $\bar{\theta}$ and $\bar{\lambda}$, are avoided for the sake of brevity. Instead, Equations (8)-(15) defined for a linear MDOF system can be referred to.

Figure 8 offer a detailed insight into the quality of state estimation for the system subjected to harmonic forcing. The impact of noise contamination on estimation quality is perceived to be minimal in the case of harmonic forcing. The corresponding parameter estimates, illustrated in Figure 9 affirm the algorithm's proficiency in promptly and accurately estimating the **HIs** under both harmonic and earthquake forces. Notably, the algorithm demonstrates consistent efficiency even when applied to a higher-order nonlinear system (Nonlin-D) subjected to earthquake excitation. Quantitatively, the percentage error for $\lambda$ (PE) in parameter estimation remains under 10% for both harmonic and earthquake excitations. This robust performance establishes the methodology's efficacy for generalized, higher-order nonlinear systems under
407

diverse forcing conditions.

<figure><img src="image_10.png" /><figcaption>Figure 8: Case study: Nonlin-D with harmonic excitation - Performance assessment for observable and latent state estimation
*Observable and latent state estimation is compared against their true values with 5% snr noise, establishing the robustness of the proposed framework against harmonic forcing events.*</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Figure 9: Case study: Nonlin-D with (a) harmonic and (b) earthquake excitation - Performance assessment for health parameter estimation
*Parameter estimation is conducted under (a) harmonic and (b) earthquake excitation with a 5% snr noise, validating the quality of the estimation within the proposed framework.*</figcaption></figure>
### 4.3. Bouc-Wen MDOF system

The Bouc-Wen material nonlinear model surpasses the Duffing oscillator in practical applications, particularly in structural engineering. This coupled first-order ODE structure for the Bouc-Wen model is computationally more complex than the single second-order ODE definition of the Duffing oscillator model. Nevertheless, its superior ability to capture a wide range of hysteresis behaviors and complex damping effects makes it more suitable for accurately representing the nonlinear response of structures to dynamic loading. The Bouc-Wen model's flexibility in accommodating diverse material properties makes it a valuable tool for real-world engineering applications, outperforming the Duffing oscillator in practical scenarios. Nevertheless, the intricate representation of nonlinearity in the Bouc-Wen material model requires the proposed method to undergo validation against structures employing this model. This validation is crucial towards evaluating the method's effectiveness in dealing with general nonlinear systems.

To illustrate the proposed framework, a numerical case study undertaken on a 3DOF structural system modeled with Bouc-Wen nonlinearity is presented.

$$ M\ddot{u}(t) + C\dot{u}(t) + \alpha^{bw}K(\lambda)u(t) + (1 - \alpha^{bw})K(\lambda)z(t) = f(t) \quad (17) $$

with the mass, M, K, and the damping C being of the same structure as in the previous cases (cf. Equation (7)). z(t), the hysteretic displacement, with an initial condition of z(0) = 0, can further be elaborated as,

$$ \dot{z}_i^r(t) = \dot{u}_i^r(t) \left\{ A_i - \left[ \beta_i \operatorname{sign}(\dot{u}_i^r(t)z_i^r(t)) + \gamma_i \right] |z_i^r(t)|^{n_i^{bw}} \right\} \quad (18) $$

where $A_i$, $\gamma_i$, $\beta_i$, $\alpha^{bw}$, and $n_i^{bw}$ are dimensionless parameters controlling the behavior of the model. The superscripts for $z^r$ and $u^r$ denote they are relatively assessed within an MDOF framework. Pertinent to the current case study, they can be defined through inter-story drift and related non-linear hysteretic displacement as,

$$ \begin{gathered} \mathbf{z}_i^r = \mathbf{z}_i \qquad \dot{\mathbf{u}}_i^r = \dot{\mathbf{u}}_i \quad \text{for } i = 1; \\ \mathbf{z}_i^r = \mathbf{z}_i - \mathbf{z}_{i-1} \qquad \dot{\mathbf{u}}_i^r = \dot{\mathbf{u}}_i - \dot{\mathbf{u}}_{i-1} \quad \text{for } i = \text{elsewhere}; \end{gathered} \quad (19) $$

This study assumed values for $A_1 = A_2 = A_3 = 1$, $\beta_1 = \beta_2 = \beta_3 = 2$, $\gamma_1 = \gamma_2 = \gamma_3 = 2$, and $n_1^{bw} = n_2^{bw} = n_3^{bw} = 1$.

The simulated system assumes equal masses $m_1 = m_2 = m_3 = 1$ kg and damping coefficients $c_1 = c_2 = 1.5$, $c_3 = 1.2$ Ns/m, along with a stiffness coefficient of $k_0 = 50$ N/m. The model structure for mass, stiffness, and damping matrices for the system remains the same as in the previous configuration. The simulation is conducted at a sampling frequency of 100 Hz over a time series length of 1000, considering $\lambda$ as $\lambda = (1, 0.9, 1)$. Noise contamination level is maintained at 5% snr.
Moreover, there is an additional necessity to model z(t) for the Bouc-Wen model (cf. Equation (17)), and this task is undertaken in a manner similar to other latent variables. It involves the adoption of additional network models $Q_i(t_j; \bar{\eta}_i)$ for each $z_i^r(t_j)$, defined with the relevant network parameters $\bar{\eta}_i$. Consequently, the residual for such a coupled network takes the form $R_i(t_j, \{\bar{\theta}_i; \bar{\eta}_i; \bar{\lambda}_i\})$ (cf. Equation (9)), where the residual exhibits additional dependency on $\bar{\eta}_i$ due to the modeling of $z_i^r$.

The further elaboration of $R_i$ will eventually involve $z_i^r(t_j)$s from neighboring s, and accordingly, the pertinent network modules for $z_i^r$ that contribute to defining $R_i$ can be listed within $Q_i(t_j; \bar{\eta}_i)$ as $Q_i(t_j; \bar{\eta}_i) = [Q_i(t_j; \bar{\eta}_i)]$.

$$R_i(t_j, \{\bar{\theta}_i; \bar{\eta}_i; \bar{\lambda}_i\}) = \mathcal{F}_i(\mathbb{N}_i(t_j; \{\bar{\theta}_i; \bar{\lambda}_i\}), \mathcal{G}_i(\mathbb{Q}_i(t_j; \bar{\eta}_i)); \bar{\lambda}_i) \quad (20)$$

Here $\mathcal{G}_i(\cdot)$ represents the hysteritic non-linearity governed by Equation (18). Figure 10 presents the modified Parallel PINN architecture for MDOF systems exhibiting Bouc-Wen-type hysteresis. Following the same procedure as previously employed in this study for training the network, latent variables are estimated, and health parameters are initialized with initial assumptions for $\lambda$. These assumptions are generated from a uniform distribution within the range of [0, 1].

Figure 11, illustrating observable states, phase portraits of latent variables, and hysteresis forces, and Figure 12 for parameter estimation, solidly establishes the efficient performance of the proposed parallel PINN framework. This holds even for complex nonlinear models characterized by hysteretic forces, where conventional modal domain approaches are inapplicable. The estimation process is remarkably swift, effectively utilizing sparse measurement data and leveraging the intrinsic knowledge of the system's physics. The obtained estimates for both states and parameters consistently fall within acceptable bounds. Quantitatively the percentage error (PE) in parameter estimation does not exceed 10%, holding for both harmonic and earthquake excitations.

### 4.4. Performance assessment for higher dimensional systems

The proposed framework is finally applied to higher-dimensional systems to explore its applicability in such scenarios. In this experiment, 10-degree-of-freedom (DOF) systems are considered under all three modeling approaches: Linear, Nonlin-D, and Nonlin-BW. Using $\lambda = (1, 0.66, 1, 1, 0.66, 0.33, 1, 1, 0.83, 1)$, the systems are simulated, and the estimation approach is applied similarly to the previous cases. The error values for latent and parameter variables are presented in Table 5, demonstrating that even with strongly nonlinear systems and 5% snr noise contamination, the maximum deviation for parameter estimates is around 10%, while the deviations for other variables are well below this maximum error limit. This establishes the efficiency of the proposed approach in dealing with larger systems existing in reality.
<figure><img src="image_12.png" /><figcaption>Figure 10: Schematic diagram of the Parallel physics-informed neural networks framework for MDOF system with Bouc-Wen non-linearity</figcaption></figure>

<table><caption>Table 5: Error analysis of states and parameters of Linear 10DOF, Nonlin-D 10DOF, Nonlin-BW 10DOF system with 5% noise subjected to earthquake excitation.</caption><thead><tr><th rowspan="2">DOF</th><th colspan="4">Linear 10DOF</th><th colspan="4">Nonlin-D 10DOF</th><th colspan="4">Nonlin-BW 10DOF</th></tr><tr><th>u (m)</th><th>ū (m/s)</th><th>ū̇ (m/s²)</th><th>λ</th><th>u (m)</th><th>ū̇ (m/s)</th><th>ū̇ (m/s²)</th><th>λ</th><th>u (m)</th><th>ū̇ (m/s)</th><th>ū̇ (m/s²)</th><th>λ</th></tr></thead><tbody><tr><td>1</td><td>4.51 × 10<sup>-6</sup></td><td>5.73 × 10<sup>-5</sup></td><td>2.41 × 10<sup>-2</sup></td><td>1.83</td><td>3.04 × 10<sup>-3</sup></td><td>3.94 × 10<sup>-3</sup></td><td>1.52 × 10<sup>-1</sup></td><td>2.21</td><td>2.85 × 10<sup>-3</sup></td><td>3.20 × 10<sup>-3</sup></td><td>1.42 × 10<sup>-1</sup></td><td>3.12</td></tr><tr><td>2</td><td>2.62 × 10<sup>-6</sup></td><td>4.98 × 10<sup>-5</sup></td><td>1.97 × 10<sup>-2</sup></td><td>1.29</td><td>2.48 × 10<sup>-3</sup></td><td>2.40 × 10<sup>-3</sup></td><td>3.70 × 10<sup>-1</sup></td><td>1.25</td><td>5.17 × 10<sup>-4</sup></td><td>1.94 × 10<sup>-3</sup></td><td>9.08 × 10<sup>-2</sup></td><td>1.89</td></tr><tr><td>3</td><td>3.86 × 10<sup>-6</sup></td><td>6.25 × 10<sup>-5</sup></td><td>2.89 × 10<sup>-2</sup></td><td>3.12</td><td>1.67 × 10<sup>-4</sup></td><td>7.99 × 10<sup>-4</sup></td><td>1.17 × 10<sup>-1</sup></td><td>0.47</td><td>4.11 × 10<sup>-3</sup></td><td>6.18 × 10<sup>-3</sup></td><td>2.73 × 10<sup>-1</sup></td><td>4.66</td></tr><tr><td>4</td><td>6.12 × 10<sup>-6</sup></td><td>7.77 × 10<sup>-5</sup></td><td>1.91 × 10<sup>-2</sup></td><td>1.48</td><td>6.88 × 10<sup>-3</sup></td><td>5.32 × 10<sup>-3</sup></td><td>2.10 × 10<sup>-1</sup></td><td>3.85</td><td>1.23 × 10<sup>-3</sup></td><td>2.91 × 10<sup>-3</sup></td><td>1.88 × 10<sup>-1</sup></td><td>2.50</td></tr><tr><td>5</td><td>9.01 × 10<sup>-6</sup></td><td>8.22 × 10<sup>-5</sup></td><td>2.33 × 10<sup>-2</sup></td><td>2.04</td><td>4.12 × 10<sup>-4</sup></td><td>1.98 × 10<sup>-3</sup></td><td>9.77 × 10<sup>-2</sup></td><td>0.96</td><td>9.94 × 10<sup>-4</sup></td><td>4.43 × 10<sup>-3</sup></td><td>3.04 × 10<sup>-1</sup></td><td>7.35</td></tr><tr><td>6</td><td>2.95 × 10<sup>-6</sup></td><td>3.68 × 10<sup>-5</sup></td><td>1.75 × 10<sup>-2</sup></td><td>1.21</td><td>1.05 × 10<sup>-3</sup></td><td>4.75 × 10<sup>-3</sup></td><td>1.23 × 10<sup>-1</sup></td><td>6.12</td><td>6.41 × 10<sup>-4</sup></td><td>5.79 × 10<sup>-3</sup></td><td>1.10 × 10<sup>-1</sup></td><td>2.94</td></tr><tr><td>7</td><td>7.34 × 10<sup>-6</sup></td><td>6.89 × 10<sup>-5</sup></td><td>2.64 × 10<sup>-2</sup></td><td>2.76</td><td>8.91 × 10<sup>-4</sup></td><td>6.44 × 10<sup>-3</sup></td><td>2.68 × 10<sup>-1</sup></td><td>4.48</td><td>2.36 × 10<sup>-3</sup></td><td>7.81 × 10<sup>-3</sup></td><td>1.71 × 10<sup>-1</sup></td><td>5.78</td></tr><tr><td>8</td><td>1.75 × 10<sup>-6</sup></td><td>4.13 × 10<sup>-5</sup></td><td>2.15 × 10<sup>-2</sup></td><td>1.97</td><td>3.33 × 10<sup>-3</sup></td><td>1.12 × 10<sup>-2</sup></td><td>1.79 × 10<sup>-1</sup></td><td>9.91</td><td>1.01 × 10<sup>-3</sup></td><td>2.17 × 10<sup>-3</sup></td><td>8.90 × 10<sup>-2</sup></td><td>1.43</td></tr><tr><td>9</td><td>8.20 × 10<sup>-6</sup></td><td>7.33 × 10<sup>-5</sup></td><td>1.89 × 10<sup>-2</sup></td><td>1.58</td><td>2.17 × 10<sup>-3</sup></td><td>2.58 × 10<sup>-3</sup></td><td>6.25 × 10<sup>-2</sup></td><td>2.03</td><td>3.89 × 10<sup>-3</sup></td><td>6.00 × 10<sup>-3</sup></td><td>2.21 × 10<sup>-1</sup></td><td>6.91</td></tr><tr><td>10</td><td>5.47 × 10<sup>-6</sup></td><td>5.47 × 10<sup>-5</sup></td><td>2.74 × 10<sup>-2</sup></td><td>3.45</td><td>9.78 × 10<sup>-4</sup></td><td>1.09 × 10<sup>-3</sup></td><td>1.04 × 10<sup>-1</sup></td><td>0.83</td><td>8.12 × 10<sup>-4</sup></td><td>3.57 × 10<sup>-3</sup></td><td>1.35 × 10<sup>-1</sup></td><td>3.87</td></tr></tbody></table>

## 5. Experimental validation: Three-story shear frame

In this study, a laboratory experiment was conducted on a three-story aluminum frame, designed as a shear-building model (see Figure 13), under controlled ambient temperatures ranging from 20°C to 25°C. The
<figure><img src="image_13.png" /><figcaption>Figure 11: Case study: Nonlin-BW with harmonic excitation - Performance assessment for observable and latent state estimation

Observable and latent state estimation is compared against their true values with 5% snr noise establishing the robustness of the proposed framework against harmonic forcing events</figcaption></figure>

goal was to evaluate the performance of the proposed framework on a scaled structure in a controlled setting. Real-world structures inherently behave as infinite-degree-of-freedom (i-DOFs) systems, where the physics may be only partially known and measurements are often noisy posing a significant challenge for health monitoring. To emulate this complexity, the Parallel PINN framework was employed to infer the behavior of the i-DOFs system using discrete sensor data and a simplified physics-based model. The experiment aimed to assess the framework's ability to produce stable and accurate health parameter estimates, thereby demonstrating its robustness and practical relevance for real-world SHM applications.

The experimental setup replicates an idealized shear building composed of rigid aluminum floors and lightweight columns, designed to emulate real-world structural behavior. The structure is a three-story,
<figure><img src="image_14.png" /><figcaption>Figure 12: Case study: Nonlin-BW with (a) harmonic and (b) earthquake excitation - Performance assessment for health parameter estimation

Parameter estimation is conducted under (a) harmonic and (b) earthquake excitation with a 5% snr noise, validating the quality of the estimation within the proposed framework.</figcaption></figure>

single-bay moment frame, standing 1.64 meters tall, with continuous columns (25 mm × 4 mm cross-section) and aluminum floor plates (300 mm × 150 mm × 4 mm) bolted to the columns, forming a lumped-mass system. Floors are spaced at 410 mm intervals. The base is rigidly fixed to a unidirectional shake table using bolts and lock washers.

The shake table, with a platform size of 1000 mm × 1000 mm, supports up to 400 kg, with a displacement stroke of ±100 mm and input excitation up to 1 g across a frequency range of 0–15 Hz. Due to the rigid inter-floor connections, the columns experience fixed-end conditions, enabling a reduced-order 3-DOF lumped mass model. Four uni-axial ICP accelerometers are installed at each floor and the ground level. Earthquake inputs are applied using a servo-hydraulic actuator, and accelerometer data is sampled at 100 Hz via a 4-channel DAQ system using DewesoftX.

To integrate the known physics into the estimation algorithm, a 3-DOF numerical model is developed and calibrated against the real system to marginally match the modal frequencies. This involves tuning material properties (density $ρ = 2700$ kg/m³, elasticity modulus $E = 68$ GPa). The calibrated model yields mass values $m_1 = m_2 = m_3 = 1.70$ kg and stiffnesses $k_1 = k_2 = k_3 = 3.15$ kN/m, with 1% Rayleigh damping assumed.

The Parallel PINN framework was then deployed to estimate structural health indices $\lambda_i$, initialized randomly. The estimated health indices, $\tilde{\lambda}_i$, are shown in Figure 14(a), and the updated modal properties are reported in Table 6. $\tilde{\lambda}_i$ did not converge to 1, suggesting deviations due to simplifications in the numerical model and boundary assumptions. Consequently, the updated stiffness values reflecting the effective healthy state were identified as $k_1 = 2.84$ kN/m, $k_2 = 2.96$ kN/m, and $k_3 = 2.93$ kN/m.
<figure><img src="image_15.png" /><figcaption>Figure 13: **Experimental Setup for Damage Detection:** (a) Experimental setup of a 3-story aluminum frame model subjected to EL-Centro earthquake excitation using a shake table. The setup includes a DEWESoft DAQ system with a 4-channel Krypton (ICP) module, accelerometers, and the frame mounted on the shake table. Panels (b) and (c) show close-up views of the frame in the undamaged and damaged conditions, respectively, highlighting the targeted damage location.</figcaption></figure>

<table><caption>Table 6: Comparison of modal frequencies between the experimental (real undamaged) model and the calibrated numerical model. The table presents the first three modal frequencies ($\omega_1$, $\omega_2$, $\omega_3$) in Hz, alongside their corresponding relative errors. The relative error illustrates the accuracy of the numerical model calibration in replicating the real system's dynamic properties.</caption><thead><tr><td>Frequency (Hz)</td><td>Experimental</td><td>Numerical</td><td>Percentage error</td></tr></thead><tbody><tr><td>$\omega_1$</td><td>3.04</td><td>3.05</td><td>0.32</td></tr><tr><td>$\omega_2$</td><td>8.39</td><td>8.55</td><td>1.90</td></tr><tr><td>$\omega_3$</td><td>12.69</td><td>12.35</td><td>2.67</td></tr></tbody></table>
Proposed algorithm's effectiveness in detecting structural damage was validated by analyzing response data from the test structure with damage. Damage was introduced by loosening bolts on diagonal columns at the third story, reducing stiffness to simulate a compromised state. For estimation, structural parameters were maintained as same as the initial numerical model, assuming no prior information on the health state being available for the system, and measurement data from this damaged setup were processed in the Parallel PINN framework. As shown in Figure 14(b), and Table 7, this induced damage resulted in a 61.43% reduction in third-floor stiffness, while estimation errors for $k_1$ and $k_2$ remained below 5% when compared with estimates obtained for updated healthy numerical model, supporting that the estimation results are not contingent on the prior numerical assumptions.

<figure><img src="image_16.png" /><figcaption>Figure 14: Convergence of health index $\lambda$ using Parallel PINN framework for shear frame subjected to El-Centro ground motion.</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>Figure 15: Performance assessment of proposed Parallel PINN-FD for observable state estimation</figcaption></figure>

# 6. Discussion

The validation results affirm the effectiveness of the proposed parallel PINN framework, which is strategically designed to balance computational efficiency with high-precision parameter estimation. The framework demonstrates consistent performance across a variety of dynamic scenarios, with its accuracy influenced by
<table><caption>Table 7: Comparison of story stiffness values in initial and estimated (healthy and damaged) states for an aluminium frame. The table presents the numerical values of stiffness for each story level ($k_1$, $k_2$, and $k_3$) under two conditions: the estimated (healthy) state and the artificially induced damaged state. The % change in stiffness acts as an indicator of the severity of the damage at each story.</caption><thead><tr><th>Stiffness (KN/m)</th><th>Numerical (Initial)</th><th>Estimated (Healthy)</th><th>Estimated (Damaged)</th><th>% change in stiffness</th></tr></thead><tbody><tr><td>$k_1$</td><td>3.15</td><td>2.84</td><td>2.82</td><td>2</td></tr><tr><td>$k_2$</td><td>3.15</td><td>2.96</td><td>3.00</td><td>1.35</td></tr><tr><td>$k_3$</td><td>3.15</td><td>2.93</td><td>1.13</td><td>61.43</td></tr></tbody></table>

excitation characteristics, measurement noise, system dynamics, and hyperparameter settings. Notably, it exhibits stable convergence and reliable estimations under structured excitations, while maintaining adaptability to more complex input conditions showing its potential for generalized deployment in real-world applications.

To achieve this balance between fidelity and efficiency, the framework integrates finite-difference-based mappings to infer velocity and acceleration from displacement within a parallelized PINN architecture. This structure enables simultaneous training of subnetworks drawing inference from algebraically substructured equations defining the governing physics, each fine-tuned to minimize computational load while accurately estimating local states and health parameters. Global coherence is enforced through shared physics-informed constraints, ensuring that localized learning contributes meaningfully to the full-system solution. A critical factor influencing convergence and model precision is the weighting coefficient $\alpha$, which balances the data-driven and physics-based loss terms. Empirically, $\alpha = 0.1$ yielded the most favorable results across test cases. While $\alpha$ is currently selected manually, future work may incorporate self-adaptive tuning mechanisms [50] to streamline optimization and further improve training dynamics. Collectively, these architectural and algorithmic choices support a scalable, efficient, and high-fidelity solution for dynamic inverse problems. The practicality of the proposed method is verified experimentally on a three-story aluminum shear frame subjected to seismic-like base excitations. The framework demonstrates precise health assessments even under arbitrary, nonstationary dynamic inputs, with little estimation accuracy loss. These findings highlight the framework's effectiveness in real-world conditions fraught with unpredictable forces and imperfect measurements.

This study focuses on relatively smaller and controlled problems (i.e., lab-scale structures) with the primary objective of assessing the efficiency, robustness, and interpretability of the proposed physics-informed framework. By doing so, it enables developing a clear understanding of the algorithm's capabilities under various levels of uncertainty, such as noise, data sparsity, and damage variability, etc., and how these factors impact estimation accuracy. Such a baseline investigation is essential before deploying the framework in complex real-life scenarios. Authors in [53–57] highlight the pertinent practical challenges encountered in real-world SHM applications when relying solely on data-driven methods. These studies emphasize the need to address issues like environmental variability, sensor malfunctions, asynchronous data, and abnormal
signals. To address these challenges with such data-only approaches often mandates large datasets yielding
limited or no interpretability.

As an alternative, the proposed framework, capable of integrating physical laws into the learning process,
allows better generalization potential while rendering the overall network more explainable. With its perfor-
mance and reliability being firmly established through tractable problems in this study, this approach will
be extended to more complex, real-world applications in our future research. In this regard, the approaches
used for handling data time lag, sensor malfunction, and abnormal signals in [53, 54] can be investigated
during the transition to field-scale implementation.

7. Conclusion

Conventional health-assessment methods, whether based on physical models or purely data-driven ap-
proaches, often face significant limitations due to the complexity of the models and the high computational
cost, or the sparsity and noise in the data. To address these challenges, a deep-learning framework utiliz-
ing a parallel PINN-FD architecture has been proposed. This framework incorporates physical constraints
by embedding governing equations directly into the loss function, allowing for more accurate and reliable
predictions. By reformulating the global MDOF system into coupled DOF-level ODEs, each handled by
an independent network and coupled through shared system matrices, the framework efficiently estimates
both latent states and health parameters with reduced computational demands. This approach leverages
both measurement data and established physical models, enhancing prediction accuracy and reducing the
dependency on large amounts of data.

The effectiveness of the proposed framework has been demonstrated through its application to both
linear and nonlinear systems, including the Duffing and Bouc-Wen models, under harmonic and earthquake
excitations. In all cases, the framework exhibited smooth convergence, strong robustness to noisy data, and
accurate parameter estimates. Additionally, experimental validation was performed on a three-story alu-
minum shear frame, where the framework successfully identified undamaged and damaged stiffness profiles.

While the proposed framework demonstrates strong performance in identifying states and parameters
across both linear and nonlinear systems, several extensions can further enhance its scope. Current assump-
tions on known structural topology and boundary conditions can be relaxed to improve generalizability. As
system complexity increases, the parallel network design and coupled residuals raise computational costs, an
area targeted for future optimization. Additionally, automating hyperparameter selection and integrating
techniques like adaptive sampling, model order reduction, and uncertainty quantification will strengthen
scalability for real-world SHM applications.

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