Physics informed Long Short-Term Memory neural network for dual state-parameter estimation of linear dynamical systems robust to input forces

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**

Structural Health Monitoring (SHM) of structures is significantly challenged by the presence of unknown input forces, e.g, external unmeasured and non-stationary excitations applied to the system such as wind or loads, which hinder accurate damage assessment and system identification. Traditional model-based approaches face inherent limitations related to model inversion, observability, and identifiability, all of which are impaired by the need for input force knowledge. Data-driven methods offer advantages in terms of scalability and automation but still depend on known inputs and often lack physical interpretability. To address these issues, an input-robust Physics-Informed Long Short-Term Memory (rPi-LSTM) framework is introduced, integrating input-robust physical modeling of system dynamics with the temporal learning capabilities of LSTM networks. The framework employs an output injection strategy to reject unknown input forces, enabling estimation of system states and spatial health parameters without requiring input force information. By preserving temporal dependencies with the LSTM network, an aspect that is often neglected in conventional physics-informed networks, the method ensures stable and accurate system estimation while complying with the system physics. Validation through numerical simulations and real laboratory-scale experiments confirms its robustness to input forces, noise, data sparsity, and varying damage scenarios, demonstrating strong potential for real-world SHM applications under uncertain conditions.

**Keywords:** Physics informed deep learning model, Frame structure, Long-short term memory, Dual state-parameter identification, Output injection.

**1. Introduction**

Effective structural health monitoring (SHM) is essential to ensure the safety and operational integrity of structures. A critical challenge in SHM is the precise assessment of structural conditions in the presence of unknown input forces, e.g, external unmeasured and non-stationary excitations applied to the system such as wind or loads. Conventional model-based techniques, such as updating the finite element (FE)

<footnote>*Corresponding author; E-mail address: subhamoy@iitmandi.ac.in</footnote>
model, depend on detailed numerical models that predict structural performance based on known input forces. These methods utilize measured vibration data to refine FE models by minimizing the discrepancies between the predictions of the model and actual measurements [1, 2]. To bypass the requirement for unknown input forces, most approaches to model updating focus on comparing modal domain responses. In the context of time-domain approaches, filtering algorithms are typically employed, with a recursive Bayesian estimation framework that compares time-domain responses. Significant progress has been made with advanced filtering techniques, such as non-linear variants of Kalman filters and particle filters, improving computational efficiency and parameter estimation [3–6]. However, these algorithms inherently require that the input forces be known explicitly or statistically, which is often impractical for real-world applications.

Typically, these filtering-based approaches idealize unknown input forces as Stationary White Gaussian Noise (SWGN). However, this assumption is often invalid for exogenous forces capable of damaging civil infrastructure, such as seismic, wind, and wave forces, which are mostly unknown and do not conform to this statistical model. Recent advancements, as demonstrated in [7], have introduced an input rejection technique via output injection for systems with both linear and non-linear sensor measurements. This method involves injecting known measurement output into the process equation to transform the process model into a state evolution model that is robust to input forces. Nonetheless, the effectiveness of such approaches heavily depends on the accuracy of the system model used for state evolution. Moreover, the noise models representing the modeling inaccuracy, forcing uncertainty, and measurement noise are required to be tuned or estimated, which is a cumbersome process acknowledged by the pertinent research community [8–10]. Moreover, if the system complexity is not adequately represented with sufficient discretization or if the system is not sufficiently observed through measurements, the approach may become severely ill-posed. Furthermore, the extensive computational demands and the need for dense instrumentation pose significant barriers to the practical application of these algorithms in real-world scenarios.

In contrast to methods grounded in physics-based models, data-driven approaches rely entirely on statistical models. Traditional data-driven techniques include identification tools formulated in the time domain, such as the eigensystem realization algorithm [11], and those in the frequency domain, such as frequency domain decomposition [12]. Further, the convergence of advancements in sensing technologies and the evolution of Machine Learning (ML) methods [13–16] has led to the emergence of ML-based approaches with wide-ranging applications across various fields [17–21]. Data-driven methods are particularly beneficial for addressing scientific problems involving processes that are not fully understood. However, these methods employ non-mechanistic (black box) models, which often require substantial data sets and exhibit limited capacity to generate physically consistent results [22].

Eventually, both data-driven and model-based approaches have their respective advantages and limitations, rendering them insufficient for addressing complex problems under real-life constraints. To overcome these limitations, researchers have introduced Physics-Informed Neural Networks (PINNs) as a viable solu-
tion [23, 24]. PINNs integrate the physics of the system, encapsulated within the physics-based model, with the deviations of data-based model predictions from real measurements into a loss function. This synthesis combines the strengths of artificial neural networks (ANNs) with physical laws, enabling more precise modeling in scenarios with limited data and simplified system representations.

PINNs have shown remarkable adaptability, exemplified by innovations such as conservative PINNs (cPINNs) and variational PINNs (VPNNs), tailored for specific problem domains [25, 26]. These advancements efficiently showcase PINNs' versatility across diverse fields, including fluid mechanics and solid mechanics [27, 28]. Their proficiency in solving differential equations, which are commonly used to describe physical system behaviors, further enhances their utility [29].

In recent years, researchers have explored the application of PINNs in estimating state parameters within structural dynamics. For example, studies [30–32] have concurrently estimated the states and parameters of the governing linear and nonlinear ordinary differential equations (ODEs) representing multidegree-of-freedom (MDOF) systems under forced vibration. This demonstrates the potential of PINNs to provide robust solutions in complex, real-world scenarios. However, these studies focused on the estimation of modal parameters, which may restrict the application of the approaches to linear systems only. Further, modal properties (typically mode shapes and frequencies), being global parameters, are observed to be less sensitive towards localized damages.

Despite the significant potential of PINNs, recent research has identified inherent failure modes, particularly when applied to ODEs/PDEs characterized by high-frequency or multiscale features [33–35]. These challenges have prompted extensive research to address these limitations through various approaches, including the development of alternative model architectures, novel learning schemes, and advanced data interpolation techniques [36–38].

Although most efforts to enhance the generalization ability and address failure modes in PINNs have concentrated on conventional PINNs, which primarily rely on multi-layer perceptron (MLP) architectures, these approaches often overlook significant temporal dependencies present in the responses sampled from real-world physical systems. Finite element methods (FEM), for example, inherently incorporate temporal dependencies by sequentially propagating the global solution, relying on the principle that the state at time $t + \Delta t$ depends on the state at time $t$. In contrast, PINNs, as a point-to-point framework, do not explicitly model temporal dependencies within ODEs/PDEs. Neglecting these temporal dependencies poses challenges for globally propagating initial condition constraints within PINNs. As a result, PINNs often exhibit failure modes where the approximations remain accurate near the initial conditions but degrade into overly smooth or naive approximations as time progresses.

Some examples of the use of PINN for solving state estimation and structural identification problems of discrete structural systems taking into account temporal dependence governed by ODEs can be found in the literature. For example, [39] introduced physics-guided CNNs and physics-informed Long Short-
Term Memory (LSTM) networks [40], showcasing the utility of PINNs in predicting building responses to earthquake excitation. [32] 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 (*dof*), constraining their real-world application.

To include the temporal dependencies within the estimation framework, a natural approach is to employ LSTM-based models [41], which are known for their ability to capture long-term dependencies in sequential data through gates. LSTM-based models have demonstrated substantial success across various domains. Recent work by [42] employed a physics-informed LSTM model to simulate the structural response of both single and MDOF systems subjected to seismic force. Building upon this framework, [43] further expanded the application of the physics-informed LSTM model for predicting seismic responses and estimating the parameters of a frame structure equipped with base isolation fluid interter system. Nevertheless, both applications depend on a known input condition that cannot always be assured. To the best of the authors' knowledge, the application of physics-informed LSTM model for simultaneous state and parameter estimation in real-world structures, while ensuring robustness to input forces, has not yet been investigated in the existing literature. In general, the mathematical formulation in this study requires a discrete structural system and temporal dependencies between states, for which vanilla-PINN cannot be used; therefore, Pi-LSTM can provide an efficient solution for state-parameter identification problems under consideration.

Consequently, this study proposes a new Pi-LSTM-based estimation framework that integrates both model-based and data-driven approaches for the estimation of latent states of the system. Further, to enforce the robustness in the system estimation against unknown input forces, an output injection approach is adopted to develop a robust version of Pi-LSTM (named rPi-LSTM). In this framework, the governing differential equation and its subsequent discretized state-space model of system dynamics are transformed into an input-robust model, eliminating the need for known input forces. To achieve this, the output measurements are injected into the process model via a suitably chosen gain matrix, ensuring that the process equation remains orthogonal to the input force. In addition, a linear transformation is applied to the measurement model to make it independent of input forcing. The detailed methodology (Section 3) and subsequent validation through numerical simulations (Section 4) and real-world experiments (Section 5) are presented in the following sections. Before that, the LSTM approach that is integrated into the proposed physics-informed estimation framework is detailed in the following (Section 2).

## 2. Standard Long short-term memory model

In sequence-to-sequence modeling, deep LSTM networks are utilized to map an input sequence to a corresponding output sequence over temporal space. The architecture consists of an LSTM layer and a fully connected network (FCN) layer in addition to the input and output layers, facilitating the temporal pairing of the input and output sequences (as illustrated in Figure 1(a)).
An LSTM cell, which functions as a neural node within the network, contains a unique set of weights and biases shared in the temporal dimension. Each LSTM cell is composed of four primary components: an internal cell, an input gate, a forget gate, and an output gate. The internal cell retains the cell state from the previous time step via a self-recurrent connection. The input gate controls the flow of input activation into the internal cell state, the forget gate adjusts the internal cell state by enabling the cell to selectively forget or reset its memory, and the output gate regulates the flow of output activation from the LSTM cell. Through the coordinated actions of these gates, the LSTM cell effectively captures long-term dependencies in sequential data.

In a discrete-time framework, at an arbitrary time step $t_k$ ($k = 1, \dots, l$), with input-output pairs $\{\mathbb{I}_k; \mathcal{O}_k\}$, each LSTM cell encompasses several elements: a forget gate $f_k$, an input gate $i_k$, an output gate $o_k$, a cell state $c_k$, and a hidden output $h_k$. These elements also interact with prior values $c_{k-1}$ and $h_{k-1}$ from the $(k-1)^{th}$ step within the cell. Their interactions are governed by the equations, illustrated in Figure 1(b):

$$ 
\begin{aligned}
f_k &= \sigma (W_{\mathbb{I}f} \mathbb{I}_k + W_{hf} h_{k-1} + b_f), \\
i_k &= \sigma (W_{\mathbb{I}i} \mathbb{I}_k + W_{hi} h_{k-1} + b_i), \\
\tilde{c}_k &= \tanh (W_{\mathbb{I}c} \mathbb{I}_k + W_{hc} h_{k-1} + b_c), \\
o_k &= \sigma (W_{\mathbb{I}o} \mathbb{I}_k + W_{ho} h_{k-1} + b_o), \\
c_k &= f_k \odot c_{k-1} + i_k \odot \tilde{c}_k, \\
h_k &= o_k \odot \tanh (c_k)
\end{aligned}
\quad (1) $$

In this context, $W_{\alpha\beta}$ where $\alpha = \{\mathbb{I}, h\}$ and $\beta = \{f, i, c, o\}$, denote the input-specific weight matrices for each gate type, such as input gate or forget gate. Concurrently, $b_{\beta}$ refers to the associated bias vectors. As an illustration, $W_{\mathbb{I}f}$ and $W_{hf}$ are the weight matrices for input vectors $\mathbb{I}_k$ and $h_{k-1}$ within the forget gate. The variable $\tilde{c}_k$ signifies an intermediate candidate vector generated by a tanh layer; meanwhile, $\sigma$ is the logistic sigmoid, tanh represents the hyperbolic tangent, and $\odot$ indicates the Hadamard product, the element-wise multiplication. Furthermore, LSTM cell output $h_k$ is given as input to fully connected neural network (FCNN) to predict output $\mathcal{O}_k$.

The intricate connections within each LSTM cell empower the LSTM neural network to model complex sequences effectively, while the fully connected layers aid in mapping the temporal features to the desired output space.

## 2.1. Relevance of the proposed data-based approach

Typical for SHM problems, this study also addresses the challenge of estimating the mechanical system parameters that represent system health from a set of measured acceleration responses. Various alternative approaches, including optimization-based, output-only subspace identification methods, stochastic subspace identification method (SSI), or Bayesian filtering-based approaches, have been utilized to tackle similar
<figure><img src="image_1.png" /><figcaption>(a) LSTM neural network</figcaption></figure>

<figure><img src="image_2.png" /><figcaption>(b) Single LSTM cell structure</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>(c) Fully connected neural network architecture
(FCNN)</figcaption></figure>

<caption>Figure 1: Schematic of unfolded LSTM neural network and its corresponding LSTM cell and FCNN</caption>

160 problems. Although various approaches have been developed and applied, the problems that continue to arise are the accuracy of the identification, the required computational time, and the requirement of the input force information [44–47]. Operational modal analysis (OMA) based on subspace identification is a widely used technique for identifying modal parameters that benefit from not requiring input force information due to its reliance on output-only data. However, OMA’s accuracy is often contingent upon accumulating extensive measurement data, leading to significant computational costs [44, 45, 48]. The large matrix decompositions and the necessity for user-defined parameters in OMA pose challenges to the efficient and accurate identification of modal parameters [49, 50].
In this regard, the relative advantages of using data-driven DL networks with LSTM architecture compared to traditional subspace- or filtering-based identification techniques to estimate the system parameters are apparent. LSTM networks have been perceived to offer different benefits, especially in managing long-term temporal correlations for temporally ordered sequences of information sampled from the system with data-based prediction models. LSTM outperforms subspace methods in capturing non-linearities and long-term dependencies due to their inherent design, which in turn enhances robustness against noise and suits complex environments. Conversely, subspace techniques need to integrate the noise through modeling and, as a consequence, often face accuracy degradation under high or variable noise levels, making LSTMs a more reliable option for maintaining prediction accuracy in such conditions.

Additionally, LSTM networks benefit from modern hardware accelerations that facilitate the efficient handling of large datasets and complex computations. Their straightforward retraining process enhances adaptability and allows for continuous improvement as new data becomes available. In contrast, subspace methods demand significant domain expertise, extensive computation, and detailed mathematical modeling for updates, which limits their flexibility. The capability of LSTMs to seamlessly integrate into automated monitoring systems and provide real-time insights further emphasizes their superiority in SHM applications, particularly for accurate real-time parameter estimation for systems with limited understanding.

Furthermore, this proposed approach enables the development of a physics-informed data-based estimation framework that not only leverages the intricate information embedded in the data but also incorporates physical knowledge about the system through a conceptual physics-based model. Ultimately, this can offer a highly practical solution for real-world SHM of complex structures, for which often little to no prior knowledge is available.

### 3. Proposed methodology

The proposed methodology focuses on state-parameter identification for structural systems that predominantly operate within the linear elastic region. Under this assumption, the dynamics of the system can be effectively represented using the linear equations of motion, which govern the relationship among inertial effects, input forces, and boundary conditions. These governing equations can be discretized spatially into a finite number of degrees of freedom (DOFs) using a Finite Element (FE) approach. The resulting reduced order model can be represented as:

$$M\ddot{x}(t) + C\dot{x}(t) + Kx(t) = u(t) \qquad (2)$$

Here $x(t) \in \mathbb{R}^{n \times 1}$ and $\dot{x}(t) \in \mathbb{R}^{n \times 1}$ denote the displacement and velocity response vectors, respectively. The input force vector is given by $u(t) \in \mathbb{R}^{n \times 1}$, representing the unknown and unmeasured external excitation applied to all DOFs. The mass matrix $M \in \mathbb{R}^{n \times n}$, damping matrix $C \in \mathbb{R}^{n \times n}$, and stiffness matrix $K \in \mathbb{R}^{n \times n}$ characterize the dynamic properties of the system.
Further, for a general linear system with time-invariant (LTI) dynamics, the continuous-time dynamics in state-space form can be expressed as:

$$ \dot{z}(t) = \mathbf{F}_c z(t) + \mathbf{E}_c u(t) \quad (3) $$

$\mathbf{z}(t) = \begin{bmatrix} \mathbf{x}(t) \\ \dot{\mathbf{x}}(t) \end{bmatrix}_{2n \times 1}$ represents the state vector. $\mathbf{F}_c$ is the state matrix, and $\mathbf{E}_c$ is the input matrix defined in the continuous-time domain. These matrices can be elaborated as follows:

$$ \mathbf{F}_c = \begin{bmatrix} \mathbf{0}_{n \times n} & \mathbf{I}_{n \times n} \\ -\mathbf{M}^{-1}\mathbf{K} & -\mathbf{M}^{-1}\mathbf{C} \end{bmatrix}_{2n \times 2n} ; \quad \mathbf{E}_c = \begin{bmatrix} \mathbf{0}_{n \times n} \\ \mathbf{M}^{-1} \end{bmatrix}_{2n \times n} \quad (4) $$

The system states are observed through a measurement mapping, defined as the measurement model:

$$ y(t) = \Gamma \{ \mathbf{H}_c z(t) + \mathbf{L}_c u(t) \} \quad (5) $$

where $\mathbf{H}_c = [-\mathbf{M}^{-1}\mathbf{K} \quad -\mathbf{M}^{-1}\mathbf{C}]_{n \times 2n}$ is the output matrix, $\mathbf{L}_c = [\mathbf{M}^{-1}]_{n \times n}$ denotes the direct transmission matrix, and $\Gamma_{p \times n}$ is Boolean selection matrix that selects $p$ measurements from $n$ based on sensor availability.

Then, numerical methods are used to incrementally advance the simulation in discrete time [51]. In the context of estimating such a real-world LTI system using discrete-time responses sampled from the system, the continuous-time representation of Eq. (3) and (5) must be converted to discrete-time form as follows:

$$ \mathbf{z}_k = \mathbf{F} \mathbf{z}_{k-1} + \mathbf{E} \mathbf{u}_k \quad (6) $$

$$ \mathbf{y}_k = \mathbf{H} \mathbf{z}_k + \mathbf{L} \mathbf{u}_k \quad (7) $$

In these equations, the variables represent their corresponding continuous-time counterparts, now expressed in discrete-time form. The influence of the selection matrix is presumed to be integrated into the discretized equivalents.

With this state-space formulation of the system dynamics, the system can be estimated through inverse estimation using real responses sampled from the system. In this regard, the state matrix is typically parameterized to assess structural health with a set of location-based health indices, $\lambda$, which are subsequently estimated from the available measurements. With $\mathbf{F}$ parameterized as $\mathbf{F}(\lambda)$, the unobserved states are propagated in time to be observed through $\mathbf{y}_k$. However, this approach requires explicit knowledge of the input force $\mathbf{u}_k$, which is rarely available in practice.

A similar issue, framed in the context of inverse stochastic estimation of the state of the system [52] under uncertainty in the process and the measurement model, has employed an injection technique to make the process model robust against input forces. Literature [7] documents the use of this approach for parameter
estimation within a nonlinear filtering environment. This study seeks to apply a similar method, described below, to enhance robustness in the process model against input forcing.

Given the measurement equation (see Eq. (7)), the following holds true for an arbitrary bounded matrix $G$:

$$0 = G(y_k - Hz_k - Lu_k) \qquad (8)$$

The state equation in Eq. (6) can therefore be rewritten as:

$$z_k = Fz_{k-1} + Eu_k + G(y_k - Hz_k - Lu_k) \qquad (9)$$

Setting $\mathcal{L}$ as $\mathcal{L} = I - GH$, Eq. (9) can be written in compact form as,

$$z_k = \mathcal{F}z_{k-1} + \mathcal{E}u_k + \mathcal{G}y_k \qquad (10)$$

where $\mathcal{F} = \mathcal{L}F$ and $\mathcal{E} = \mathcal{L}E - GL$. By choosing $G = E(HE + L)^{-1}$, $\mathcal{E}_k$ becomes a null matrix in Eq. (10), eliminating the dependency of the state evolution on the input force, resulting in:

$$z_k = \mathcal{F}z_{k-1} + \mathcal{G}y_k \qquad (11)$$

In many situations, $G$ may vary in time due to changes concerning $k$. In this work, it is assumed that such variations are effectively captured through $\lambda$, as discussed in the following sections. Due to its role in injecting the output in the modified process equation, $G$ is termed the output injection matrix for further reference. Readers may refer the appendix for the complete derivation.

### 3.1. Physics informed loss function for rPi-LSTM

The proposed framework first parameterized the system with certain location-based health parameters $\{\lambda_i\}$, $i = 1, \dots, n$, where $n$ denotes the number of monitored locations. Each $\lambda_i$ is defined to signify the localized stiffness deterioration level, with 0 indicating complete damage and 1 denoting a healthy state. This eventually renders the state and output injection matrix to be functions of $\lambda$. In the subsequence, the proposed rPi-LSTM approach inversely estimates the assumed health parameters, $\lambda$, drawing inference from the sampled response $y_k$. For elucidation, Eq. (11) is here reformulated to reflect the dependence of the system matrices on the health indices $\lambda$:

$$z_k = \mathcal{F}(\lambda)z_{k-1} + \mathcal{G}(\lambda)y_k \qquad (12)$$

The system architecture employs high fidelity acceleration data $y_k$ as the primary input $I_k$ (Eq. (1)) for the network, facilitating the training for displacement prediction $\tilde{x}_k$ as $\mathcal{O}_k$ (Eq. (1)) at each interval. It integrates an LSTM layer to produce intermittent hidden state $h_k$ and a fully connected layer to convert
these hidden states $\mathbb{I}_k$ into displacement outputs $\mathbb{O}_k$. Thus, the pair $\{\mathbf{y}_k; \tilde{\mathbf{x}}_k\}$ operates as the input-output pair $\{\mathbb{I}_k; \mathbf{O}_k\}$ in the LSTM section of the rPi-LSTM network. Mathematically, prediction of displacement is formulated as follows:

$$ \tilde{\mathbf{x}}_k = \mathcal{N}(\mathbf{y}_k, \{\tilde{\theta}, \tilde{\lambda}\}) \qquad (13) $$

In this equation, $\mathcal{N}(\cdot)$ represents the neural network and $\tilde{\mathbf{x}}_k$ represents the predicted displacement in the time step $k$, while $\mathbf{y}_k$ denotes the high-fidelity acceleration data as input at the same time step. The parameters $\tilde{\theta}$ and $\tilde{\lambda}$ are current estimates of the network parameters and the health parameters, respectively. The goal is to optimize these parameters to enhance the accuracy of the network's predictions. By refining $\tilde{\theta}$ and $\tilde{\lambda}$, both the physics-based loss (related to adherence to physical laws) and the data-based loss (related to discrepancies between predictions and actual data) can be minimized. Although the dependence of the network prediction $\tilde{\mathbf{x}}_k$ on these parameters is implied, it is not explicitly detailed here for simplicity.

The data-based prediction is further validated against physics laws and real response data using a collective loss function. To compare the prediction with data or physics, the predicted displacement, $\tilde{\mathbf{x}}_k$, is numerically differentiated ($N_D$) through a Finite Difference Filter (FDF) to yield corresponding velocity ($\dot{\tilde{\mathbf{x}}}_k$) and acceleration ($\ddot{\tilde{\mathbf{x}}}_k$) at the same time step. Furthermore, the velocity ($\dot{\tilde{\mathbf{x}}}_{k-1}$) and acceleration ($\ddot{\tilde{\mathbf{x}}}_{k-1}$) from the previous time step are used to ensure that the prediction adheres to the governing differential equation. Accordingly a residual $r_k^p$ is defined as:

$$ r_k^p = \left\| \tilde{\mathbf{z}}_k - (\mathcal{F}(\tilde{\lambda})\tilde{\mathbf{z}}_{k-1} + \mathbf{G}(\tilde{\lambda})\mathbf{y}_k) \right\|_2 \qquad (14) $$

Here, $\tilde{\mathbf{z}}_k$ and $\tilde{\mathbf{z}}_{k-1}$ are formed by combining displacement and velocity estimates at the $k^{th}$ and $(k-1)^{th}$ time steps, such as

$$ \tilde{\mathbf{z}}_i = \begin{bmatrix} \tilde{\mathbf{x}}_i \\ \dot{\tilde{\mathbf{x}}}_i \end{bmatrix} \qquad (15) $$

$\mathcal{F}(\tilde{\lambda})$ and $\mathbf{G}(\tilde{\lambda})$ are calculated using current estimates of $\lambda$ (i.e. $\tilde{\lambda}$), and $\mathbf{y}_k$ represents the real response (high-fidelity data) injected into the process equation to ensure robustness against input force. Eq. (14) computes the Euclidean norm (L2 norm) of the difference between the network prediction for the state ($\tilde{\mathbf{z}}_i$) and the prediction from the physics-based model (i.e., $\mathcal{F}(\tilde{\lambda})\tilde{\mathbf{z}}_{k-1} + \mathbf{G}(\tilde{\lambda})\mathbf{y}_k$).

This loss is assessed at all the collocation points defined in time. For simplicity, it is assumed that the system is sampled uniformly in time. Therefore, all instances where responses are available ($N_h$) are considered as collocation points ($N_c$), implying $N_h = N_c$. The physics loss is accordingly estimated at all
collocation points, and the residuals from all collocation points are integrated as follows:

$$ \mathcal{L}_1(\{\tilde{\theta}, \tilde{\lambda}\}) = \frac{1}{N_h} \sum_{k=1}^{N_h} r_k^p \qquad (16) $$

Here, $\mathcal{L}_1(\tilde{\theta}, \tilde{\lambda})$ is defined as the mean of the collocation losses (or physics losses), dependent on the current estimates of the network parameters $\tilde{\theta}$ and the health parameters $\tilde{\lambda}$.

To assess the data loss, the pretrained network prediction is compared against the true response. The network prediction for the displacement $\tilde{x}_k$, needs to be mapped to the corresponding measurement. The FDF step discussed previously provides the necessary mapping of states to the corresponding measurement predictions $\tilde{y}_k$ in terms of $\tilde{\ddot{x}}_k$, which can then be compared with the true measurement $y_k$ sampled from the system. Consequently, the measurement data loss, $\mathcal{L}_2(\{\tilde{\theta}, \tilde{\lambda}\})$, is calculated as follows:

$$ r_k^d = \|\mathbf{y}_k - \tilde{\mathbf{y}}_k\|_2 \qquad (17) $$

Again, the L2 norm is used to quantify discrepancies at different *dofs* of the predicted measurement compared to the true measurement in a single scalar value. The total measurement loss, termed data loss, is further calculated as:

$$ \mathcal{L}_2(\{\tilde{\theta}, \tilde{\lambda}\}) = \frac{1}{N_h} \sum_{k=1}^{N_h} r_k^d \qquad (18) $$

The total loss function is defined as:

$$ L(\{\tilde{\theta}, \tilde{\lambda}\}) = \alpha \mathcal{L}_1(\{\tilde{\theta}, \tilde{\lambda}\}) + \beta \mathcal{L}_2(\{\tilde{\theta}, \tilde{\lambda}\}) \qquad (19) $$

where $\alpha, \beta$ are hyperparameters. For this study $\alpha=\beta=1$ is taken, however, self adaptive techniques also can be used in future to speed up convergence [34, 53]. Finally, the optimization problem to determine the neural network and structural parameters $\{\tilde{\theta}, \tilde{\lambda}\}$ is formulated as follows:

$$ \{\tilde{\theta}^*, \tilde{\lambda}^*\} = \arg \min_{\{\tilde{\theta}, \tilde{\lambda}\}} L(\{\tilde{\theta}, \tilde{\lambda}\}) \qquad (20) $$

This hybrid framework ensures that network predictions adhere to both physical laws and real response data, optimizing the parameter set to improve prediction accuracy and reduce overall loss. The approach presented schematically in Figure 2, leverages the strengths of LSTM networks in sequence modeling and the rigor of physics-informed learning. By combining data-driven techniques with physical constraints, the network can potentially produce more reliable and physically plausible predictions. The FDF plays a crucial role in approximating derivatives, ensuring the network captures the system's essential dynamics. The physics-informed loss functions guide the network toward solutions that fit the data while respecting the underlying physical principles. This architecture represents a significant advancement in modeling
<figure><img src="image_4.png" /><figcaption>Figure 2: Schematic diagram of proposed rPi-LSTM framework for state-parameter estimation</figcaption></figure>

complex dynamical systems, addressing the limitations of purely data-driven methods and enhancing the interpretability and robustness of the predictions. The pseudo-code for the above methodology is given in Algorithm 1.

<caption>**Algorithm 1** Proposed rPi-LSTM algorithm for damage detection, localization, and quantification</caption>

1: Specify high-fidelity input data, input collocation points, and high-fidelity measurement data ($\mathbf{y}_k$)
2: Set hyperparameters (no. of layers, no. of neurons, learning rate, maximum iteration (maxitr)) of neural network.
3: Initialize the network parameters: $\{\tilde{\theta}, \tilde{\lambda}\}$
4: **for** $j = 1$ to maxitr **do**
5:     **for** $k = 1$ to $N_h$ **do**
6:         Compute predictions for collocation data: $\tilde{\mathbf{x}}_k := \mathcal{N}(\mathbf{y}_k; \{\tilde{\theta}, \tilde{\lambda}\})$
7:         Compute numerical differentiation: $\dot{\tilde{\mathbf{x}}}_k = N_D(\tilde{\mathbf{x}}_k)$, $\ddot{\tilde{\mathbf{x}}}_k = N_D(\tilde{\mathbf{x}}_k)$
8:     **end for**
9:     Compute total process model loss: $\mathcal{L}_1(\tilde{\theta}, \tilde{\lambda})$ ▷ Eq. (16)
10:    Compute total measurement loss: $\mathcal{L}_2(\tilde{\theta}, \tilde{\lambda})$ ▷ Eq. (18)
11:    Compute total loss $L(\tilde{\theta}, \tilde{\lambda})$ ▷ Eq. (19)
12:    Update parameters $\{\tilde{\theta}, \tilde{\lambda}\}$ by minimizing the total loss $L(\tilde{\theta}, \tilde{\lambda})$ using Adam optimizer ▷ Eq. (20)
13: **end for**
14: **Deployment :**
15: The estimated structural parameters: $\tilde{\lambda}$
16: The estimated full state of the system: $\tilde{\mathbf{x}}, \dot{\tilde{\mathbf{x}}}, \ddot{\tilde{\mathbf{x}}}$
## 4. Numerical validation: Linear 10-dof system

The proposed approach is validated using numerical experiments to investigate its ability to handle noisy signals of different severity levels of noise contamination and damages of different intensity levels. For the numerical case study, a linear 10-dof structural system is adopted, represented by a spring-mass-dashpot model. A schematic diagram of the same is presented in Figure 3. The generalized equation of motion for the dynamics of the system can be perceived from the generalized equation in Eq. (3) that imposes the assumed dimension. Accordingly, the mass matrix can be idealized with $M = diag([m_1; m_2; ... m_{10}])$, and other system matrices can be defined as follows:

$$ K = k_0 \begin{bmatrix} \lambda_1 + \lambda_2 & -\lambda_2 & 0 & \cdots & 0 \\ -\lambda_2 & \lambda_2 + \lambda_3 & -\lambda_3 & \cdots & 0 \\ 0 & -\lambda_3 & \lambda_3 + \lambda_4 & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & \lambda_{10} \end{bmatrix}_{10 \times 10}, \quad F_c = \begin{bmatrix} \mathbf{0}_{10 \times 10} & \mathbf{I}_{10 \times 10} \\ \mathbf{M}^{-1}\mathbf{K} & \mathbf{M}^{-1}\mathbf{C} \end{bmatrix}_{20 \times 20}, \quad E_c = \begin{bmatrix} \mathbf{0}_{10 \times 10} \\ \mathbf{M}^{-1} \end{bmatrix}_{20 \times 10} $$

Furthermore, utilizing Eq. (11) and initializing the system dynamics in discrete form robust to input with $\tilde{\lambda}$ can be written as:

$$ z_k = \mathcal{F}(\tilde{\lambda})z_{k-1} + \mathcal{G}(\tilde{\lambda})y_k \quad (21) $$

<figure><img src="image_5.png" /><figcaption>Figure 3: Schematic diagram of building model for numerical experiment</figcaption></figure>

Here, the elements in the diagonal mass matrix are considered as $m_1, m_2, ..., m_{10} = 1$ kg each. The initial undamaged stiffness for all springs is assumed to be $k_{0,i} = 20$ N/m for all $i$. Damping coefficients are
determined using Rayleigh damping criteria with a damping ratio $\zeta = 2\%$. The system is considered to have
deteriorated in its second, third, fifth, seventh, and ninth dof. These damages are simulated by defining
a set of health indices (HIs), represented as $\lambda$, which indicate the level of deterioration of spring stiffness
within the range $[0, 1]$. Here, 1 denotes a completely healthy spring, while 0 represents complete damage.
The health indices modify the initial stiffness according to $k_i = \lambda_i k_{0,i}$, where $k_i$ is the stiffness of the $i^{th}$
spring, $\lambda_i$ is the health index, and $k_{0,i}$ is the initial undamaged stiffness.

With this configuration, the true values for the health indices at all locations are represented by the
vector $\lambda = \{1, \frac{1}{2}, \frac{1}{3}, 1, \frac{1}{4}, 1, \frac{1}{5}, 1, \frac{1}{6}, 1\}$, indicating induced damage at various points. The lumped spring-
mass-dashpot system is subsequently simulated under two different excitation scenarios: (i) earthquake
excitation using El Centro earthquake data from May 18, 1940 (North-South Component) [54], and (ii)
ambient vibration modeled as stochastic white Gaussian noise forcing (SWGN) applied to all nodes of the
system. Training data, consisting of acceleration time histories over 5 seconds also sampled at a sampling
frequency of 100 and 1000 Hz, is generated for parameter identification for the above two cases.

To replicate real-life response sampling, SWGN of varying severity levels is added to the simulated
response signal to introduce contamination. The signal-to-noise ratio (snr) measure is employed for this
purpose, where the standard deviations of the SWGN noise are determined based on the standard deviation of
the uncontaminated simulated signal and the desired snr level for noise contamination. The contaminated
response is then processed using the proposed rPi-LSTM framework to estimate the stiffness values at
different story levels. By introducing different damage levels at various structural locations within a single
numerical experiment, the proposed algorithm's capability to detect multiple damage scenarios and assess
varying levels of damage is effectively demonstrated.

The adopted model architecture comprised of LSTM cell with 10 input neurons, and a fully connected
network after the LSTM cell with 256 neurons in 2 hidden layers and 10 neurons in the output layer. The
weights for process model loss and measurement loss are taken as unity, meaning they have been equally
valued. Since time is not used as input to the network, the automatic differentiation tool (AD) cannot be
used directly for numerical differentiation, therefore, a finite difference scheme is adopted to get derivatives of
the output of the neural network. Authors in [40, 42] utilized FD in place of AD when time is not explicitly
provided as input to network. Furthermore, AD is computationally more expensive [55], motivating the
application of the finite difference scheme in this study. The stopping criteria for the proposed learning are
set as either reaching a maximum of $10^6$ epochs, achieving a loss ($L(\tilde{\theta}, \tilde{\lambda})$) less than 0.005, or the difference
between the losses of two successive epochs becoming less than $10^{-4}$. The training procedure utilizes the
Adam optimization scheme, incorporating momentum and individual learning rate mechanisms to optimize
the parameters of the network. Pi-LSTM-based training has been performed in the Python environment
using Pytorch on a computer with 12 Intel Core i7-12700K CPUs.
<figure><img src="image_6.png" /><figcaption>Figure 4: Schematic diagram of Pi-LSTM framework for state-parameter estimation</figcaption></figure>

## 4.1. Comparison between robust and non-robust Pi-LSTM

Proposed framework elaborated in previous section is compared with non-robust Pi-LSTM (Pi-LSTM). The Pi-LSTM framework serves as a baseline for comparison with the proposed robust Pi-LSTM (rPI-LSTM). The Pi-LSTM directly incorporates input forces into the process model loss function, assuming their availability during training as shown in Figure 4. However, this reliance on explicit input force data makes the framework sensitive to unknown inputs, leading to diminished accuracy in real-world scenarios. In contrast, the rPI-LSTM framework introduces the output injection matrix $G(\lambda)$, allowing the process model to utilize output measurements ($\mathbf{y}_k$) directly to mitigate the influence of unknown input forces. These distinctions emphasize the key contributions of the rPI-LSTM framework over its non-robust counterpart. Initial success with the proposed algorithm in a base numerical case study (not demonstrated here) has proved the efficacy of the rPI-LSTM framework. The augmented robustness is highlighted by its capability to accurately estimate system health even when the input forces are unknown—an achievement that surpasses previous data-based non-robust algorithms while circumventing typical issues for any model-based robust SHM approaches [7, 56]. This advantage becomes particularly clear when comparing two methodologies: one that uses the proposed robust algorithm (rPi-LSTM), which can accommodate unknown input forces, and another that does not accommodate unknown input forces (Pi-LSTM).
Table 1 shows the comparison between true and estimated health parameters and for brevity Figure 5 shows the convergence of Pi-LSTM for two health parameters without prior knowledge of the input force $u_k$ and without incorporating the term $Gy_k$ into the process model. The results illustrate that, without the proposed output injection method, health parameter estimates exhibit significant inaccuracies. This problem is exacerbated since the earthquake excitation as forcing involves severe ill impacts on the non-robust estimation approach, as evidenced by the results presented. Earthquake excitation introduces complex and unpredictable dynamics, which may have led to the divergence and unreliable estimates with the non-robust Pi-LSTM approach.

<table><caption>Table 1: Comparison of true (λ) and estimated (λ̃) health parameters with Pi-LSTM (without the inclusion of robustness ($Gy_k$)) and rPi-LSTM under unknown earthquake excitation. The absolute relative error (%) is shown in brackets.</caption><thead><tr><th>True (λ)</th><th>Pi-LSTM (λ̃)</th><th>rPi-LSTM (λ̃)</th></tr></thead><tbody><tr><td>1.000</td><td>-0.033 (103.3%)</td><td>0.994 (0.58%)</td></tr><tr><td>0.500</td><td>0.0046 (99.08%)</td><td>0.499 (0.08%)</td></tr><tr><td>0.333</td><td>0.0052 (98.44%)</td><td>0.329 (0.11%)</td></tr><tr><td>1.000</td><td>0.0879 (91.21%)</td><td>0.97 (2.83%)</td></tr><tr><td>0.250</td><td>0.0083 (96.68%)</td><td>0.249 (0.09%)</td></tr><tr><td>1.000</td><td>-0.0655 (106.55%)</td><td>0.992 (0.72%)</td></tr><tr><td>0.200</td><td>0.0002 (99.9%)</td><td>0.199 (0.10%)</td></tr><tr><td>1.000</td><td>0.0494 (95.06%)</td><td>0.978 (2.16%)</td></tr><tr><td>0.166</td><td>-0.0042 (102.53%)</td><td>0.158 (0.65%)</td></tr><tr><td>1.000</td><td>0.0513 (94.87%)</td><td>0.996 (0.39%)</td></tr></tbody></table>

<figure><img src="image_7.png" /><figcaption>Figure 5: Estimation of health indices $\tilde{\lambda}$ using Pi-LSTM (without the inclusion of robustness ($Gy_k$)) under unknown earthquake excitation. (Firm lines correspond to parameter estimations while the dotted lines of the same colour correspond to their respective true values.)</figcaption></figure>

Evidently, the input robustness of the proposed approach stems from its unique design to handle unknown input forces, a capability not found in any existing ML-based algorithms. The existing methods inherently require prior information about the input forces for their problem formulation and operation, making a direct comparison infeasible. The proposed framework, by contrast, eliminates the need for such prior knowledge, addressing a critical gap in the current state-of-the-art. The capability to estimate parameters, therefore, automatically establishes its relative efficiency over other PINN-based inverse estimation approaches.

## 4.2. Comparison with other input-robust SHM approaches

Attempts to accommodate unknown forcing within an input-robust filtering framework have been explored in literature [56], and the proposed approach is compared against such existing input-robust SHM methodologies. This comparison enables a meaningful evaluation, as Bayesian filtering techniques are widely
recognized in the literature for their capability to handle uncertainty and dynamic state estimation. The competing method is adopted from [57], where the same structural system is assessed using the proposed rPi-LSTM-based framework.

The study evaluates a comparative estimation approach for a 6-dof structural system with the following characteristics: masses $m_1, m_2, ..., m_6 = 100$ kg, initial undamaged stiffness values $k_0 = \{13, 13, 11, 11, 9, 9\}$ kN/m, and damping coefficients $c_1 = c_2 = 0.75$ Ns/m, $c_3 = c_4 = 0.5$ Ns/m, and $c_5 = c_6 = 0.25$ Ns/m. A 40% damage level is introduced at the top story, leading to a health index vector $\lambda = \{1, 1, 1, 1, 1, 0.6\}$. The adopted details are assumed to be the same as those presented in [57].

An Interacting Particle-Ensemble Kalman Filter (IPEnKF)-based estimation approach is employed for comparison. While the IPEnKF utilizes 30 seconds of measurement data from six sensors, the proposed rPi-LSTM framework requires only 5 seconds of data sampled from six sensors for training. Without delving into the intricate details of network architecture or estimation processes for brevity, the proposed rPi-LSTM approach demonstrates superior performance. Further, it can be observed that, while [57] relies on adopting a model for the unknown input force, the proposed approach operates without such assumptions, making it more flexible and widely applicable. Its relative smoothness and stability in health parameter assessment are clearly evident when comparing Figure 6(a) and Figure 6(b), that establishes its efficiency and effectiveness. In addition, IPEnKF required 48 minutes to complete state-parameter estimation, the rPi-LSTM framework completed the estimation in 28 minutes.

Further, inverse stochastic estimation methods are heavily dependent on the quality and fidelity of the underlying model, as well as on addressing challenges related to observability and scalability. Moreover, the inherent need to fine-tune process and measurement noise parameters in Bayesian filter-based SHM frameworks poses significant challenges for practical, real-world applications. In contrast, the proposed DL-based approach demonstrates greater robustness in handling measurement noise and accommodating model inaccuracies [43, 58, 59]. This capability positions it as a more practical and effective solution for real-world SHM scenarios, where uncertainties are often poorly characterized or not fully understood.

### 4.3. Adaptability against forcing variability

Eventually, the efficacy of the proposed network is validated against both types of forcing: the earthquake ground motion which is typically a band-limited colored noise, and the generalized ambient forcing that can be represented as SWGN. Figure 7 demonstrates that the proposed rPi-LSTM successfully detected damage under both excitation scenarios employed in this study. However, it should be noted that the epochs required for convergence under SWGN excitation are substantially larger than those needed for earthquake excitation. This difference can be attributed to the data-based prediction effort required for response states corresponding to ambient vibration forcing. Due to the highly random nature of SWGN force, the response is more force-dominated than inertia-dominated. In contrast, the base excitation in the
<figure><img src="image_8.png" /><figcaption>Figure 6: Convergence of health indices $\tilde{\lambda}$ for: (a) IPEnKF; and (b) rPi-LSTM, illustrating the framework's ability to capture changes in system dynamics. (Firm lines correspond to parameter estimations while the dotted lines of the same colour correspond to their respective true values.)</figcaption></figure>

earthquake scenario results in a more inertia-dominated response, which is more easily replicated by the FDF-based state prediction network utilized in this algorithm. Consequently, for random vibration in the second numerical case study, a considerably higher number of epochs was required to stabilize the prediction network. Nevertheless, this should not pose practical limitations in real-life applications, as more time can typically be allocated for system estimation under normal operating conditions under ambient vibration compared to adverse loading conditions such as earthquake excitation.

Within this numerical investigation, an information content sensitivity test was also conducted, wherein training data comprising acceleration time histories for 5, 10, and 15 seconds were recorded at a 100 Hz sampling rate under earthquake excitation. The learning rate was set at $10^{-4}$. As observed from Table 2, 5 seconds of data is sufficient for estimation purposes. Estimated states ($\tilde{x}_k$, $\dot{\tilde{x}}_k$, $\ddot{\tilde{x}}_k$) and parameter ($\tilde{\lambda}$) are compared against true states ($x_k$, $\dot{x}_k$, $\ddot{x}_k$) and parameters ($\lambda$) using the mean squared error (MSE) and relative absolute percentage error (PE) metric. Consequently, data from these 5-second recordings are used for further sensitivity analysis. This establishes the detection promptness of the proposed algorithm at a maximum of 5 seconds, which is practical and beneficial for real-life applications.

### 4.4. Sensitivity to initialization

The algorithm's sensitivity to the initial assumptions regarding the health condition of the structure is also investigated. Ideally, any SHM algorithm should converge to the respective true value of the health
<figure><img src="image_9.png" /><figcaption>Figure 7: Convergence of rPi-LSTM for health parameter estimation under (a) Earthquake excitation (El-Centro), (b) Stochastic White Gaussian Noise (SWGN) ( Firm lines correspond to parameter estimations while the dotted lines of the same colour correspond to their respective true values. )</figcaption></figure>

432 state, λ, regardless of the initially assumed value. To verify this, two sets of estimations are performed with different initial conditions for ˜ λ. It can be seen from Figure 8 that the random initialization of ˜ λ does not affect the quality of the estimation. The estimates consistently converged to their respective true values, regardless of their starting points. This robustness in convergence demonstrates the algorithm's reliability in accurately assessing the structural health state, even when starting with different initial assumptions.

<table><caption>Table 2: Comparison of performance of rPi-LSTM in state-parameter estimation for 10 – dof linear system with different window lengths of response time-history</caption><thead><tr><th rowspan="3">dof</th><th colspan="6">T=5sec</th><th colspan="6">T=10sec</th><th colspan="6">T=15sec</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><th colspan="3">MSE</th><th rowspan="2">PE</th></tr><tr><th>y</th><th>ŷ</th><th>ŷ̃</th><th>λ</th><th>y</th><th>ŷ</th><th>ŷ̃</th><th>λ</th><th>y</th><th>ŷ</th><th>ŷ̃</th><th>λ</th></tr></thead><tbody><tr><td>1</td><td>3.54E-07</td><td>3.18E-05</td><td>8.82E-05</td><td>0.58</td><td>1.02E-06</td><td>7.43E-05</td><td>3.38E-05</td><td>1.61</td><td>8.31E-07</td><td>5.82E-05</td><td>3.62E-05</td><td>1.09</td><td>4.87E-07</td><td>3.11E-05</td><td>7.87E-05</td><td>0.08</td></tr><tr><td>2</td><td>4.87E-07</td><td>3.11E-05</td><td>7.87E-05</td><td>0.08</td><td>9.94E-07</td><td>7.57E-05</td><td>3.38E-05</td><td>0.13</td><td>7.52E-07</td><td>5.97E-05</td><td>4.13E-05</td><td>0.01</td><td>5.99E-07</td><td>3.20E-05</td><td>8.13E-05</td><td>0.11</td></tr><tr><td>3</td><td>5.99E-07</td><td>3.20E-05</td><td>8.13E-05</td><td>0.11</td><td>1.29E-06</td><td>7.32E-05</td><td>3.58E-05</td><td>0.31</td><td>8.36E-07</td><td>5.89E-05</td><td>3.41E-05</td><td>0.30</td><td>5.39E-07</td><td>3.18E-05</td><td>7.82E-05</td><td>2.83</td></tr><tr><td>4</td><td>5.39E-07</td><td>3.18E-05</td><td>7.82E-05</td><td>2.83</td><td>1.04E-06</td><td>7.38E-05</td><td>4.16E-05</td><td>1.89</td><td>8.40E-07</td><td>5.87E-05</td><td>4.32E-05</td><td>1.30</td><td>4.90E-07</td><td>3.14E-05</td><td>8.87E-05</td><td>0.09</td></tr><tr><td>5</td><td>4.90E-07</td><td>3.14E-05</td><td>8.87E-05</td><td>0.09</td><td>1.23E-06</td><td>7.86E-05</td><td>3.20E-05</td><td>0.06</td><td>8.84E-07</td><td>6.00E-05</td><td>3.16E-05</td><td>0.03</td><td>6.19E-07</td><td>3.17E-05</td><td>8.09E-05</td><td>0.72</td></tr><tr><td>6</td><td>6.19E-07</td><td>3.17E-05</td><td>8.09E-05</td><td>0.72</td><td>1.50E-06</td><td>7.92E-05</td><td>3.53E-05</td><td>0.74</td><td>9.17E-07</td><td>5.77E-05</td><td>3.09E-05</td><td>0.38</td><td>5.04E-07</td><td>3.12E-05</td><td>7.80E-05</td><td>0.10</td></tr><tr><td>7</td><td>5.04E-07</td><td>3.12E-05</td><td>7.80E-05</td><td>0.10</td><td>1.29E-06</td><td>7.52E-05</td><td>3.30E-05</td><td>0.01</td><td>9.22E-07</td><td>6.00E-05</td><td>4.06E-05</td><td>0.01</td><td>5.67E-07</td><td>3.17E-05</td><td>7.13E-05</td><td>2.16</td></tr><tr><td>8</td><td>5.67E-07</td><td>3.17E-05</td><td>7.13E-05</td><td>2.16</td><td>1.24E-06</td><td>7.51E-05</td><td>3.62E-05</td><td>0.75</td><td>9.30E-07</td><td>5.73E-05</td><td>4.82E-05</td><td>0.22</td><td>6.49E-07</td><td>3.25E-05</td><td>8.48E-05</td><td>0.65</td></tr><tr><td>9</td><td>6.49E-07</td><td>3.25E-05</td><td>8.48E-05</td><td>0.65</td><td>1.22E-06</td><td>7.37E-05</td><td>3.64E-05</td><td>0.67</td><td>9.56E-07</td><td>5.85E-05</td><td>4.63E-05</td><td>0.68</td><td>6.88E-07</td><td>3.21E-05</td><td>9.17E-05</td><td>0.39</td></tr><tr><td>10</td><td>6.88E-07</td><td>3.21E-05</td><td>9.17E-05</td><td>0.39</td><td>1.41E-06</td><td>7.44E-05</td><td>3.67E-05</td><td>0.28</td><td>9.55E-07</td><td>5.87E-05</td><td>4.95E-05</td><td>0.31</td><td></td><td></td><td></td><td></td></tr></tbody></table>
<figure><img src="image_10.png" /><figcaption>Figure 8: Convergence of rPi-LSTM subjected to initialization of $\tilde{\lambda}$ as (a) 1 or (b) random. The prediction performance of the network for parameter estimation is observed not to be affected by initialization schemes. (Firm lines correspond to parameter estimations while the dotted lines of the same colour correspond to their respective true values.)</figcaption></figure>

4.5. Sensitivity against noise severity

To assess the robustness of the proposed rPi-LSTM framework under varying levels of noise contamination, a series of experiments were performed using synthetic data with snr levels of 1%, 2%, and 5%. For a consistent comparison across all noise sensitivity scenarios, network architecture and learning rate are kept the same as briefed in earlier studies, however $\tilde{\lambda}$ was initialized with 1, and the stopping criterion was set to a maximum of $10^5$ training epochs. The outcomes of these tests are summarized in Table 3 and illustrated in Figures 9 and 10. As shown in Figure 9, the state predictions remain stable and within acceptable error bounds across all noise levels. Furthermore, the percentage error in the estimated parameters for the damaged elements, visualized in Figure 10, consistently remains below 2%, showing the accuracy and reliability of the proposed approach under real conditions.

In some scenarios, a slight increase in parameter estimation errors is observed under higher levels of noise contamination, a behavior that aligns with the well-established understanding that noise can influence estimation accuracy. When sensor noise levels are known, it is common practice to incorporate probabilistic measures that reflect the reliability of estimation outcomes across varying noise intensities and damage severities. For instance, lower confidence is typically associated with minor damage estimations in high-noise environments, whereas estimations of more significant damage under low-noise conditions are generally regarded with higher confidence. This insight points toward a broader and compelling research direction centered on uncertainty quantification, which extends beyond the scope of the present study but holds strong
<figure><img src="image_11.png" /><figcaption>Figure 9: Convergence of rPi-LSTM for health parameter estimation subjected to snr contamination of (a) 0% (b) 1% (c) 2% (d) 5%. The prediction performance of the trained network for damaged parameters is observed to be affected minimally by noise contamination. (Dotted lines correspond to parameter estimations while the solid lines of the same colour correspond to their respective true values.)</figcaption></figure>

456 **4.6. Computational performance**

In most cases, training PINNs for complex problems is notoriously challenging, often requiring days of computation on high-end GPUs. This is primarily due to the multi-objective loss functions, which inherently
<figure><img src="image_12.png" /><figcaption>Figure 10: Comparison of actual and rPi-LSTM-predicted states of the first story at different snr levels (a) 0% (b) 1% (c) 2% (d) 5%. The prediction performance of the trained network is validated against actual values under different noise severities, which can be perceived to be affected minimally by increasing contamination levels.</figcaption></figure>

459 compete during the minimization process [33, 60]. Given these computational demands, it is crucial to report the computational cost associated with state-parameter identification when using the proposed algorithm.
<table><caption>Table 3: Comparison of computed parameters and states of 10-DOF linear system identified using rPi-LSTM with ground truth under various snr levels</caption><thead><tr><th colspan="2">DOF</th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th><th>6</th><th>7</th><th>8</th><th>9</th><th>10</th></tr></thead><tbody><tr><td rowspan="3">0% Noise</td><td rowspan="2">MSE</td><td>y</td><td>1.25E-06</td><td>1.18E-06</td><td>1.31E-06</td><td>1.33E-06</td><td>1.80E-06</td><td>1.69E-06</td><td>1.72E-06</td><td>1.72E-06</td><td>1.72E-06</td><td>1.71E-06</td></tr><tr><td>y&#775;</td><td>1.06E-04</td><td>1.10E-04</td><td>1.08E-04</td><td>1.10E-04</td><td>1.12E-04</td><td>1.09E-04</td><td>1.12E-04</td><td>1.08E-04</td><td>1.12E-04</td><td>1.12E-04</td></tr><tr><td>y&#775;</td><td>4.22E-05</td><td>3.25E-05</td><td>3.40E-05</td><td>3.50E-05</td><td>3.29E-05</td><td>3.36E-05</td><td>3.28E-05</td><td>3.28E-05</td><td>3.38E-05</td><td>3.25E-05</td></tr><tr><td rowspan="3">1% Noise</td><td rowspan="2">MSE</td><td>λ</td><td>0.62</td><td>0.09</td><td>0.61</td><td>2.13</td><td>0.42</td><td>0.40</td><td>0.20</td><td>0.01</td><td>0.89</td><td><b>5.18</b></td></tr><tr><td>y</td><td>1.28E-06</td><td>1.14E-06</td><td>1.17E-06</td><td>1.24E-06</td><td>1.52E-06</td><td>1.27E-06</td><td>1.40E-06</td><td>1.30E-06</td><td>1.55E-06</td><td>1.62E-06</td></tr><tr><td>y&#775;</td><td>1.09E-04</td><td>1.11E-04</td><td>1.11E-04</td><td>1.11E-04</td><td>1.14E-04</td><td>1.13E-04</td><td>1.13E-04</td><td>1.13E-04</td><td>1.13E-04</td><td>1.13E-04</td><td></td></tr><tr><td rowspan="3">2% Noise</td><td rowspan="2">MSE</td><td>y&#775;</td><td>4.61E-05</td><td>4.54E-05</td><td>4.68E-05</td><td>4.19E-05</td><td>3.94E-05</td><td>3.91E-05</td><td>4.37E-05</td><td>2.97E-05</td><td>3.28E-05</td><td>3.18E-05</td></tr><tr><td>λ</td><td>1.35</td><td>0.01</td><td>1.36</td><td>5.47</td><td>1.72</td><td><b>8.47</b></td><td>1.77</td><td>8.39</td><td>5.97</td><td>0.93</td></tr><tr><td rowspan="3">5% Noise</td><td rowspan="2">MSE</td><td>y</td><td>1.36E-06</td><td>1.33E-06</td><td>1.51E-06</td><td>1.04E-06</td><td>2.50E-06</td><td>2.55E-06</td><td>2.59E-06</td><td>1.99E-06</td><td>2.63E-06</td><td>1.74E-06</td></tr><tr><td>y&#775;</td><td>1.11E-04</td><td>1.12E-04</td><td>1.11E-04</td><td>1.12E-04</td><td>1.14E-04</td><td>1.13E-04</td><td>1.17E-04</td><td>1.14E-04</td><td>1.13E-04</td><td>1.13E-04</td></tr><tr><td>y&#775;</td><td>5.51E-05</td><td>4.57E-05</td><td>4.42E-05</td><td>4.37E-05</td><td>3.82E-05</td><td>3.98E-05</td><td>4.24E-05</td><td>3.44E-05</td><td>3.30E-05</td><td>3.14E-05</td><td></td></tr><tr><td rowspan="3">5% Noise</td><td rowspan="2">MSE</td><td>λ</td><td>1.52</td><td>0.09</td><td>0.26</td><td>3.28</td><td>0.83</td><td>4.74</td><td>1.70</td><td>1.78</td><td>0.35</td><td><b>9.37</b></td></tr><tr><td>y</td><td>1.56E-06</td><td>1.42E-06</td><td>1.53E-06</td><td>1.68E-06</td><td>2.28E-06</td><td>2.97E-06</td><td>2.51E-06</td><td>2.06E-06</td><td>2.84E-06</td><td>2.02E-06</td></tr><tr><td>y&#775;</td><td>1.12E-04</td><td>1.16E-04</td><td>1.15E-04</td><td>1.17E-04</td><td>1.19E-04</td><td>1.19E-04</td><td>1.19E-04</td><td>1.18E-04</td><td>1.19E-04</td><td>1.18E-04</td><td></td></tr><tr><td rowspan="2">5% Noise</td><td rowspan="2">MSE</td><td>y&#775;</td><td>1.28E-04</td><td>1.25E-04</td><td>1.32E-04</td><td>6.70E-05</td><td>6.77E-05</td><td>6.74E-05</td><td>6.78E-05</td><td>6.72E-05</td><td>4.85E-05</td><td>4.66E-05</td></tr><tr><td>λ</td><td>1.49</td><td>0.28</td><td>0.27</td><td>5.50</td><td>0.00</td><td>6.90</td><td>0.66</td><td><b>14.80</b></td><td>1.93</td><td>8.19</td></tr></tbody></table>

While previous efforts to enhance PINNs' computational efficiency have predominantly focused on optimizing the multi-objective loss functions [34, 61], this study takes an alternative approach by employing a point-to-point LSTM network. Key hyperparameters, including the number of neurons, network layers, learning rate, and stopping criteria, significantly impact both the accuracy and computational time during training.

Through extensive experimentation with these hyperparameters, the learning rate and stopping criteria have been identified as the primary factors influencing both accuracy and computation time. The network architecture was fixed with a single LSTM cell containing 10 input neurons, followed by a fully connected network (FCN) with a 1-256-256-10 configuration. To explore the impact of these factors, an ablation study was conducted with varying learning rates and a tolerance criterion of $L(\tilde{\theta}, \tilde{\lambda}) < 0.1$.

As shown in Table 4, reducing the learning rate improves accuracy but at the cost of increased computation time. Additionally, adjusting the stopping criteria from $L(\tilde{\theta}, \tilde{\lambda}) < 0.005$ to $L(\tilde{\theta}, \tilde{\lambda}) < 0.1$ significantly reduces computation time. This trade-off between accuracy and computational efficiency must be carefully considered based on the specific objectives of the study.

<table><caption>Table 4: Computation time and accuracy comparison in identified damaged parameters with change in learning rate of 10-dof linear system subjected to El-Centro earthquake excitation with 1% snr</caption><thead><tr><th>Learning rate</th><th>Accuracy (%)</th><th>Epochs</th><th>Compute time (min)</th></tr></thead><tbody><tr><td>0.001</td><td>97.13</td><td>12321</td><td>18.42</td></tr><tr><td>0.005</td><td>98.38</td><td>19456</td><td>27.35</td></tr><tr><td>0.0001</td><td>99.94</td><td>26366</td><td>41.12</td></tr></tbody></table>

### 4.7. Impact of data sparsity

To evaluate the performance of the proposed rPi-LSTM framework under sensor sparsity, a comprehensive analysis was performed, with results illustrated in Figures 11 and Table 5. The sensor configuration was guided by prior findings in the literature, particularly the work of [59], which demonstrated that PINNs
<table><caption>Table 5: Damage detection results for different damaged stories and uninstrumented degrees of freedom (DOFs)</caption><thead><tr><th rowspan="2">Uninstrumented<br>DOF</th><th>10th story damaged</th><th rowspan="2">False positives</th><th>6th story damaged</th><th rowspan="2">False positives</th></tr><tr><th>Accuracy</th><th>Accuracy</th></tr></thead><tbody><tr><td>3,4,7,8</td><td>98.72</td><td>no</td><td>99.21</td><td>no</td></tr><tr><td>2,3,4,7,8</td><td>90.72</td><td>no</td><td>94.94</td><td>no</td></tr><tr><td>2,3,4,5,7,8,9</td><td>84.23</td><td>yes</td><td>85.13</td><td>yes</td></tr></tbody></table>

can accurately reconstruct the full state and identify parameters of a 20-DOF Duffing system using only two sensors. Building on this insight, the present study assumes selection of sensor locations neither fully strategic nor entirely random. Insights from real-world SHM applications suggest that sensors placed at locations with stronger vibration responses tend to be less affected by sensor noise. Accordingly, one sensor is consistently placed at the top of the structure, where dynamic responses are typically most prominent. Additionally, based on our experience with the proposed algorithm, learning from sensors located near the base of the structure, though more challenging, provides valuable information. Hence, the bottom node closer to the ground is also consistently instrumented in most cases. Under this configuration, the rPI-LSTM framework effectively estimated health parameters and detected structural damage with a minimum of 40% instrumentation density, corresponding to four active sensor nodes. The estimated health indices ($\tilde{\lambda}_i$) converged with reasonable accuracy, achieving satisfactory detection performance. However, as the instrumentation density decreased to 30%, the algorithm's performance declined, characterized by slower convergence and less accurate parameter recovery.

The study highlights that while the rPI-LSTM framework can handle moderate sensor sparsity, there is a clear trade-off between sensor density and the accuracy or efficiency of the damage estimation process. For the system under consideration, a minimum of six measurement channels strategically placed at {1, 2, 5, 6, 9, 10} story of the structure is recommended to ensure robust and reliable health parameter estimation. Lower sensor densities result in longer convergence times and reduced accuracy, emphasizing the importance of adequate instrumentation for optimal performance.

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

A laboratory experiment was further conducted on a three-story aluminum frame, representing a shear building model (cf. Figure 12), with ambient temperatures maintained within 20-25°C. This study aimed to evaluate the effectiveness of the proposed rPI-LSTM on a real-life structure. Real-life structures, considered as infinite dof (i-dof) systems, accurately represent the real-world conditions along with pertinent challenges wherein the system physics is never well understood and the force-time history or the noise statistics contaminating the measurement are not known a priori. Estimating the health of such a system by drawing inferences from sparse and discrete measurements sampled from such i-dof systems provides a reality check
<figure><img src="image_13.png" /><figcaption>Figure 11: Convergence of rPi-LSTM for health parameter estimation of 10-dof linear system with 60% damage on top story subjected to earthquake excitation with 1% snr under (a) 60% (b) 40% (c) 30% instrumentation density. (Firm lines correspond to parameter estimations while the dotted lines of the same colour correspond to their respective true values.)</figcaption></figure>

for the capability of any proposed algorithm. In this study, the *i-dof* system was estimated using the proposed approach, with the necessary physics supplied through a simplified conceptual physics-based support model. The objective was to determine if the proposed framework could reliably converge to a stable and practical estimate of the system's health.

The structure replicates the behavior of an idealized shear building with rigid floors and columns of
<figure><img src="image_14.png" /><figcaption>Figure 12: (a) Experimental setup of a 3-story damaged aluminum frame model subjected to El-Centro earthquake excitation in (b) Undamaged, (c) Damaged condition</figcaption></figure>

insignificant mass, with input excitation and response measurements closely resembling real-world conditions. The experimental setup consists of a three-story, single-bay moment frame structure. The frame is 1.64 m tall, with continuous columns measuring 25 mm × 4 mm in cross-section. At each floor level, aluminum plates measuring 300 mm × 150 mm × 4 mm are bolted to the columns to idealize a lumped mass system. The bottom columns are securely attached to a shake table using bolts and lock washers to ensure a rigid connection. The floor levels are uniformly spaced, resulting in a nominal story height of 0.41 meters (410 mm). The experimental setup is placed on a unidirectional shake table capable of generating ground motion. The mounting platform of the shake table has a dimension of 1000mm × 1000mm and is capable of exciting structures of loads up to 400kg with a base excitation of maximum acceleration of 1g and a maximum stroke of 100mm. The frequency of the input waveform can range from 0 – 15Hz.

Given the rigid connections at each floor level, the columns can be assumed to have a fixed boundary condition at each level, facilitating the computation of the equivalent stiffness and allowing the use of a 3-dof reduced-order lumped mass model. The mass at each floor level is significantly larger than the mass of the columns, allowing the assumption of lumped masses at each floor. Four uni-axial ICP (Integrated Circuit Piezoelectric) accelerometers are placed at floor levels including the ground floor, on the structure. The shaking table is subsequently used to impart earthquake excitation to the frame in a controlled environment
via a servohydraulic actuator actuated by a control panel. The accelerometer data are sampled using a 4-channel data acquisition (DAQ) system at a constant frequency of 1000Hz and DewesoftX software is used to view and record the experimental data. Detailed specifications of the instruments can be found in [62].

To integrate the known physics of the real frame structure into the estimation algorithm, it is essential to develop a numerical model that accurately replicates the experimental structure's behavior. This involves developing a 3-dof lumped mass model of the system, parameterized with story stiffness values to provide a rough representation of the real structure without requiring prior calibration of the stiffness parameters. While it has been previously established that the proposed algorithm is insensitive to initialization, a preliminary calibration was conducted to obtain an approximate assessment of the probable story stiffness. This allows for meaningful comparison with the rPi-LSTM estimates.

Consequently, the undamaged frame, mounted on a shake table, is subjected to earthquake excitation (El-Centro earthquake) using a servo-hydraulic actuator. The time-domain data is converted to the frequency domain using the Frequency Domain Decomposition (FDD) approach, allowing for the extraction of experimental natural frequencies.

Through calibration, the aluminum frame properties were crudely tuned to align the experimental modal properties with the numerical ones. This initial tuning resulted in a mass density of $ρ = 2700$ kg/m³ and a modulus of elasticity of $E = 65$ GPa. Based on these assumptions, the theoretical values for mass and stiffness in the numerical model were calculated as $m_1 = m_2 = m_3 = 1.70$ kg and $k_1 = k_2 = k_3 = 3.01$ kN/m, respectively. The experimental modal damping was employed in the Rayleigh damping model with a 2% damping ratio.

Table 6 presents a comparison between the modal parameters from the real experiment and the initial numerical model. This comparison highlights the need for further calibration to address geometric and boundary irregularities, as well as non-homogeneous material properties that the initial tuning did not fully account for. To enhance the model, the rPi-LSTM is further applied, initialized with random story stiffness values.

With the initial calibrated stiffness values denoted as $k_{0,i}$, $λ_i$s are obtained as initial values for the parameters corresponding to the randomly adopted stiffness values. Subsequently, the rPi-LSTM framework was used to estimate the health indices of the test structure. The adopted model architecture comprises an LSTM cell with 3 input neurons, followed by a fully connected network with two hidden layers containing 64 neurons each, and an output layer with 3 neurons. Acceleration data acquired from DAQ is preprocessed and 5 second data is segmented to be further used for estimation purpose in rPi-LSTM framework. Consequently, acceleration data is fed into the rPi-LSTM model to generate predicted displacement data. This predicted displacement is further differentiated to obtain the corresponding predicted velocity and acceleration, which are collectively utilized in the data-based and physics-based loss functions to minimize the total loss. The training process is governed by stopping criteria defined as reaching a maximum of $10^6$ epochs, achieving
<table><caption>Table 6: Comparison between modal properties of the real undamaged and the calibrated and estimated numerical models</caption><thead><tr><th>Frequency (Hz)</th><th>Experimental</th><th>Numerical (Initial)</th><th>Relative error (%)</th><th>Numerical (Estimated)</th><th>Relative error (%)</th></tr></thead><tbody><tr><td>f<sub>1</sub></td><td>3.04</td><td>2.88</td><td>5.18</td><td>2.98</td><td>1.65</td></tr><tr><td>f<sub>2</sub></td><td>8.39</td><td>8.07</td><td>3.74</td><td>8.72</td><td>3.99</td></tr><tr><td>f<sub>3</sub></td><td>12.69</td><td>11.67</td><td>8.03</td><td>12.14</td><td>4.29</td></tr></tbody></table>

a loss value ($L(\tilde{\theta}, \tilde{\lambda})$) below 0.005, or observing a difference between the losses of two successive epochs smaller than $10^{-4}$. The model parameters are optimized using the Adam optimization algorithm, which incorporates momentum and adaptive learning rate mechanisms to ensure efficient convergence.

The estimation process yielded convergent health indices that reflect the healthy state of the structure. The updated modal properties, compared to experimental and initial values, are shown in Table 6. The estimated health indices ($\tilde{\lambda}$) are presented in Figure 13(a). The algorithm demonstrates its capability to converge effectively even with a random initial guess. Contrary to initial assumptions, $\tilde{\lambda}$ values did not converge to 1 or similar values. This minor deviation in estimated $\tilde{\lambda}$ can largely be attributed to variations in boundary conditions, which affect the story stiffness and are not uniform across all story levels. The estimated health indices ($\tilde{\lambda}$) were further used to assess the story stiffnesses, calculated as $\hat{k}_i = \hat{\lambda}_i k_{0,i}$, where $\hat{k}$ denotes the estimates. These updated story stiffnesses have been adopted as the undamaged benchmark for the experimental aluminum frame.

<figure><img src="image_15.png" /><figcaption>Figure 13: Convergence of the rPI-LSTM framework under experimental conditions: (a) undamaged state, demonstrating stability and accuracy in predictions; and (b) damaged state, illustrating the framework's ability to capture changes in system dynamics.</figcaption></figure>

To further verify the efficacy of the proposed algorithm for damage detection, responses sampled from
<table><caption>Table 7: Comparison between experimentally and numerically measured modal parameters for the damaged condition</caption><thead><tr><td>Frequency (Hz)</td><td>Experimental</td><td>Numerical</td><td>Relative error (%)</td></tr></thead><tbody><tr><td>f<sub>1</sub></td><td>2.88</td><td>2.85</td><td>0.81</td></tr><tr><td>f<sub>2</sub></td><td>7.46</td><td>7.19</td><td>3.60</td></tr><tr><td>f<sub>3</sub></td><td>12.18</td><td>10.99</td><td>9.71</td></tr></tbody></table>

the damaged state of the test structure are used with the proposed estimation approach to check if the proposed approach can detect significant variations in the health indices to denote damage. Accordingly, the bolts on one set of diagonal columns at the third story are loosened to artificially reduce the story stiffness, thereby creating a damaged condition. Using the benchmarked stiffness values as the initial stiffness (i.e., $k_{0,i} = \hat{k}_i$) for the estimation, the proposed rPi-LSTM network is employed to estimate the current status of the story stiffnesses. In Figure 13(b) it can be observed that with loosened bolts, the third floor experienced a substantial 67.38% reduction in floor stiffness corresponding to the actual conditions. The updated frequencies are further compared against their experimental counterparts and presented in Table 7. Thus, this experiment ultimately validates the efficacy of the proposed approach in assessing the structural health of structural systems and demonstrates its practicality for real-life implementation.

## 6. Discussion

The results presented in this study demonstrate the effectiveness of the proposed rPi-LSTM framework in identifying structural parameters and inferring dynamic states under challenging conditions such as unknown input forces, limited sensor data, and varying noise levels. By embedding physical constraints directly into the network's training via the loss function, the model shows robust performance across multiple configurations. This physics-informed approach enhances generalizability and interpretability, key traits for real-world SHM applications.

One key insight from our numerical analyses is that accurate identification and prediction are possible even with sparse instrumentation, provided the sensor configuration captures critical dynamic information. Semi-strategic sensor placement approach, informed by domain knowledge and partially randomized to ensure spatial diversity, proved effective for both synthetic and real data. However, it can be observed that optimizing sensor placement using observability metrics or data-driven adaptive strategies remains a scope for future research.

The current framework has demonstrated promising results, particularly in the context of linear system dynamics, a common operational scenario in civil infrastructure. While its initial applications have focused on this domain, the approach is inherently versatile and holds strong potential for extension to nonlinear systems such as those found in aerospace and wind turbine vibration analysis. Future directions include exploring the impact of colored noise and broadening the framework to address high-dimensional stochastic systems, where parameters are better represented through probabilistic rather than deterministic measures.
To this end, reformulating the algorithm within a probabilistic domain forms an exciting avenue of ongoing
and future research.

7. Conclusion

Traditional health assessment methods, whether model-based or model-agnostic, often encounter chal-
lenges like model complexity and significant computational costs. Furthermore, the identification of struc-
tural health in SHM is further complicated by unknown input forces acting on structures. This study
confronts these issues by introducing a novel physics-informed data-driven framework, an ODE-constrained
neural network designed for state and parameter identification for linear structural systems, utilizing high-
fidelity data alongside partially known physical models.

The research presents a robust Pi-LSTM (rPi-LSTM) approach for SHM, specifically tailored to han-
dle acceleration measurements and remain resilient to unknown and arbitrary input forces. By integrating
system physics into the LSTM framework, the model is constrained to produce realistic and practical pre-
dictions, avoiding the pitfalls of purely data-driven methods. Input robustness is achieved through an
innovative output injection technique embedded within the rPi-LSTM architecture. To enhance compu-
tational efficiency during the training phase, numerical differentiation is employed in place of automatic
differentiation, coupled with an early stopping criterion. The effectiveness and robustness of the proposed
method are validated through rigorous testing, including numerical simulations and experimental data. The
rPi-LSTM demonstrates consistently high performance across a range of conditions, such as variations in
input force types, random initialization strategies, noise intensities, and data sparsity scenarios. While the
method operates effectively without prior knowledge of input forces, the characteristics of forces and system
influence computational efficiency, particularly affecting the smoothness of the data and the convergence of
the estimation process, a common consideration in data-driven estimation.

To address the identified limitations, future research could focus on developing a gradient-free variant
of the algorithm to further improve computational speed, stability and reduce costs. Additionally, the lack
of standardized procedures for optimizing network architecture, learning rates, stopping criteria, and the
weighting of data versus physics loss terms remains a critical concern. Furthermore, while this study focused
on linear time-invariant systems, extending the framework to encompass linear and nonlinear time-variant
systems could yield valuable insights into the broader applicability and robustness of the proposed SHM
algorithm.

8. Funding

This study was partially funded by Science & Engineering Research Board (SERB), New Delhi, India,
through grant file No. CRG/2023/005267
## 9. Declaration of generative AI and AI-assisted technologies in the writing process

During the preparation of this work, the authors used ChatGPT to correct grammatical mistakes. After using ChatGPT, authors reviewed and edited the content as needed and takes full responsibility for the content of the published article.

## 10. Appendix

The continuous-time representations of the system shown in Eqs. (3), (5) are converted to their discrete-time counterparts as follows:

$$z_k = Fz_{k-1} + Eu_k, \qquad (22)$$

$$y_k = Hz_k + Lu_k, \qquad (23)$$

where all variables now represent their discrete-time equivalents. The system's states $z_k$ evolve over time and are observed through $y_k$. However, this approach requires explicit knowledge of the input force $u_k$, which is rarely available in practice.

To address this, we adopt an injection technique inspired by [52] and [7], enhancing robustness against input forces. Starting from the measurement equation (see Eq. (23)), we introduce an arbitrary bounded matrix $G$:

$$0 = G(y_k - Hz_k - Lu_k). \qquad (24)$$

Substituting this into the state equation (see Eq. (22)), we rewrite:

$$z_k = Fz_{k-1} + Eu_k + G(y_k - Hz_k - Lu_k). \qquad (25)$$

Expanding and rearranging terms:

$$z_k = Fz_{k-1} + Eu_k + Gy_k - GHz_k - GLu_k. \qquad (26)$$

Substituting $z_k$ from Eq. (22) into (26), we get:

$$z_k = Fz_{k-1} - GHFz_{k-1} + Eu_k + Gy_k - GHEu_k - GLu_k. \qquad (27)$$

Rearranging further:

$$z_k = (I - GH)Fz_{k-1} + (I - GH)Eu_k + Gy_k - GLu_k. \qquad (28)$$

Introducing $\mathcal{L} = I - GH$, the state equation simplifies to:

$$z_k = \mathcal{L}Fz_{k-1} + \{\mathcal{L}E - GL\}u_k + Gy_k. \qquad (29)$$

To eliminate the dependency on the input force $u_k$, we set $G = E(HE + L)^{-1}$, making $\mathcal{L}E - GL = 0$. This results in:

$$z_k = \mathcal{L}Fz_{k-1} + Gy_k, \qquad (30)$$
where $\mathcal{L} = \mathbf{I} - \mathbf{G}\mathbf{H}$ and $\mathcal{F} = \mathcal{L}\mathbf{F}$. This final form ensures robustness against input forces while maintaining consistency with the governing equations.

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