Machine learning-enhanced metaheuristic optimization of lead rubber bearings for inter-story isolated buildings under seismic load

Abstract

This paper presents a machine learning-enhanced metaheuristic optimization framework to improve the seismic performance of inter-story isolated buildings equipped with lead rubber bearings (LRBs). The study focuses on three key objectives: 1) assessing the influence of isolator placement at different elevations on critical seismic responses; 2) identifying optimum LRB parameters across different isolator placements, through multi-objective optimization using particle swarm optimization (PSO) algorithm; and 3) developing an Artificial Neural Network (ANN) based surrogate model for rapid prediction of optimal LRB parameters. A 10-story shear building with LRBs installed at intermediate stories is modeled and analyzed under 175 historical earthquake records to attain these objectives. The results demonstrate that the seismic performance of inter-story isolated buildings is highly dependent on the elevation of isolator placement, with lower, mid, and upper story level inter-story isolation (ISI) configurations being most effective for displacement reduction, acceleration control, and base shear mitigation, respectively. The multi-objective optimization framework using PSO has successfully identified the story-specific optimal LRB parameters, and comparative analyses confirmed moderate performance gains over non-optimized configurations. The proposed ANN model exhibits strong predictive accuracy and generalizes well, offering a computationally efficient alternative to metaheuristic optimization runs, enabling rapid estimation of optimum LRB parameters.

Graphical abstract

Keywords

ISI / LRB / multi-objective optimization / PSO / ANN

1 Introduction

Seismic base isolation (BI) is a widely recognized passive control strategy to safeguard multi-story buildings from the devastating effects of earthquake-induced vibration. The traditional BI system involves the insertion of a laterally flexible isolation layer between the building’s superstructure and its foundation. The underlying principle of base-isolation is decoupling of the building’s fundamental frequency from the dominant frequency content of earthquake ground motions. This frequency decoupling is achieved by increasing the building’s fundamental period substantially. As a result, the dynamic response of the BI building is shifted to a region of lower spectral demand in the seismic response spectrum, thereby reducing the overall seismic forces acting on the building [1–3]. Despite its effectiveness, base-isolation can encounter some practical and economic challenges. While implementing base isolation is relatively simple in new construction, it becomes technically complex and costly for existing buildings due to the need for excavation and temporary structural supports. Moreover, the requirement for a large seismic gap around the BI buildings, intended to accommodate the expected displacement demand during earthquake events, can be a major constraint in densely populated urban settings, where providing sufficient clearance to avoid pounding against adjacent buildings is often not feasible. Additionally, base isolation is generally not suitable for medium or high-rise building structures due to their inherent flexibility and bending-dominated behavior, which reduces the effectiveness of the isolation system. The substantial weight of high-rise buildings often demands the use of a large number of isolators, further increasing costs and reducing the economic viability of the system in these cases [4–6]. In these situations, as an alternative to traditional BI, inter-story isolation (ISI) offers an effective seismic mitigation strategy for both new constructions and retrofitting existing buildings. Unlike BI, which is implemented at the foundation level, as the term implies, inter-story isolation involves placing flexible isolators between intermediate floors, typically at selected story levels, by inserting them within columns or between structural slabs. This approach transforms the portion of the structure above the isolators into a large, functional tuned mass, effectively acting as a non-conventional Tuned Mass Damper (TMD). Rather than lengthening the fundamental period like BI, inter-story isolation enhances damping by promoting energy transfer between the substructure and superstructure, using existing structural masses for both load-bearing and vibration control [5–7]. The early studies on inter-story isolated buildings are presented in Refs. [4,8–13], where the concept of installing isolators at intermediate floors was first explored. In recent years, a series of numerical and experimental investigations [14–21] have further demonstrated the effectiveness of inter-story isolation systems in enhancing the seismic performance of buildings by mitigating structural demands. A wide ranges of isolation systems have been employed in ISI buildings, including high-damping rubber bearings (HDRBs) [7,20,22–25], laminated natural rubber bearings (NRBs) [6,8,11,18], friction pendulum systems (FPS) [19,21,26], lead rubber bearings (LRBs) [6,14,16,17,27,28], etc.

Recent studies highlight that the effectiveness of the ISI systems, however, is a function of their vertical locations or positions within the structures. Ryan and Earl [14] conducted a comparative analysis evaluating the performance of ISI systems installed at the first story, mid-height, and the roof level of the building, demonstrating that the location of the isolator is critical in controlling seismic demands. The findings indicated that the mid and roof-level ISI configuration is more effective for reducing the seismic response of the superstructure. Similarly, Forcellini and Kalfas [20] evaluated the ISI system positioned at mid-height and three-quarter height of the building, concluding that the mid-height ISI configuration is most effective for accelerations and drift reductions. Complementing these findings, the experimental study by Zhang et al. [21] demonstrated that upper-story ISI configurations provide a more stable and effective vibration reduction effect. Conversely, Zheng and Xu [15] reported that lower-story isolator placement enhances damping, and it gradually reduces as the isolator is shifted toward higher stories.

Another critical aspect of designing an ISI building is the determination of optimal isolation parameters. Few studies have explored optimization techniques to identify optimal isolator parameters for ISI buildings, considering single-layer [5,16,29–31] and multi-layer [32–35] isolation systems. To achieve the optimal performance from an isolation system, the efficient tuning of its design parameters using appropriate optimization methods is essential. However, the classical optimization techniques (e.g., linear programming, gradient descent, etc.) often struggle with structural engineering optimization challenges due to nonlinearity, high dimensionality, and complex constraints. Gradient-based approaches may get trapped in local optima, especially in non-convex design spaces, and become computationally intensive as problem size grows [36–39]. In contrast, metaheuristic optimization algorithms (MHAs) excel in global searches, efficiently exploring diverse solution spaces without requiring gradient information. While MHAs generally provide near-optimal solutions, rather than exact solutions, they achieve a balance between quality and computational efficiency, making them more suitable for complex structural optimization [40–43]. Recently, several studies have successfully implemented MHAs for the optimum design of inter-story isolation systems. For instance, Zhou et al. [16] and Charmpis et al. [32,33] employed the Genetic Algorithm (GA) to determine the optimum isolation configuration in ISI buildings. Kim and Kang [31], Bernardi et al. [44] utilized Non-dominated Sorting Genetic Algorithm version II (NSGA-II) to optimize the ISI systems. Furthermore, Skandalos et al. [34] investigated three different MHAs, including the Genetic Algorithm, Particle Swarm Optimization (PSO), and Dragonfly Algorithm, to optimize single and multiple layer ISI systems in a 6-story shear frame.

Although the aforementioned literature has provided valuable insights, some notable research gaps persist. The existing studies have primarily focused on evaluating the influence of isolator placement at a few selected building heights, such as the first story, mid-height, or three-quarter height [14,20]. However, a systematic investigation considering each individual story level as a potential isolation interface remains largely unexplored in the literature. This lack of comprehensive assessment limits the understanding of how isolator location affects the seismic performance across the entire building height. Moreover, the findings reported to date are often contradictory, with some studies suggesting mid-height placement as the most effective, while others advocate upper or lower-story ISI configurations [14,15,20,23]. These inconsistencies highlight the necessity for a more detailed and systematic parametric study to develop a consistent and reliable framework to determine the optimal placement of ISI systems in buildings. Furthermore, when it comes to identifying the optimum inter-story isolator configuration, the majority of the existing works concentrated on single-objective optimization approaches, targeting a particular performance measure such as the minimization of peak floor acceleration [32,33] or the minimization of base shear [16,30]. Only a limited number of studies [31,34] have attempted multi-objective optimization to simultaneously address multiple response parameters such as isolator displacement and inter-story drift. However, even these mentioned multi-objective investigations often overlook the interactions between various other structural responses and the potential trade-offs among them, leading to solutions that may not truly represent the global optimum. Additionally, there is a lack of research regarding the systematic optimization of LRBs specifically for ISI buildings. Although few studies implemented multi-objective optimization on ISI building incorporated LRBs, their primary focus has been on optimizing supplementary devices, such as viscous dampers, rather than the LRBs themselves [6,28]. Consequently, there remains a clear need for dedicated optimization studies targeting the key design parameters of LRBs within ISI configurations. Moreover, the optimization of the ISI systems can be computationally intensive [32], highlighting the need for advanced solution strategies and metamodel-assisted predictive approaches to improve efficiency and feasibility [45]. To address this computational challenge, surrogate modeling and machine learning (ML) techniques have emerged as powerful alternatives to conventional optimization-through-simulation approaches in structural engineering. In recent years, a wide range of ML algorithms have been applied as surrogate models in structural engineering to improve computational efficiency. Linear regression-based ML approaches, such as Ordinary, Stepwise, Ridge, Lasso, and Elastic Net regressions [46,47], have been employed to predict seismic responses. In addition, the nonparametric regression model Multivariate Adaptive Regression Spline (MARS) has also been used in a few studies [48,49]. Tree-based models, including regression trees [49], bagging methods such as Bagging Trees [49], Random Forests (RF) [47,49–55], and boosting algorithms like Gradient Boosting Machine (GBM) [50–53,55] and eXtreme Gradient Boosting (XGBoost) [47, 54–57], have gained wide popularity due to their robustness and predictive accuracy in structural engineering applications. Kernel-based approaches, particularly Support Vector Machine (SVM) [47,49,54,55,58–60] and Gaussian Process Regression (GPR) [61–63], have also demonstrated strong capabilities in capturing complex nonlinear relationships even with relatively small data sets. In parallel, Artificial Neural Networks (ANNs) [47,50–53,56,59,64–66], their optimization-augmented variants such as PSO-ANN [67–69] or GA-ANN [55], Deep Neural Networks [54,70] are increasingly used for structural parameters and response prediction tasks. More recently, the emergence of Scientific Machine Learning (SciML) approaches, including Physics-Informed Neural Networks (PINNs) [71,72], neural operators [72], etc., has introduced new opportunities by embedding physical laws directly into the data-driven learning process. Neural operators such as the Variational Physics-Informed Neural Operator (VINO) [73] and the Multi-Head Neural Operator (MHNO) [74] show particular promise for efficiently solving partial differential equation-governed structural dynamics. Complementing these advances, eXplainable Artificial Intelligence (XAI) tools such as SHapley Additive exPlanations (SHAP) and Local Interpretable Model-agnostic Explanations (LIME) have been introduced in structural engineering to enhance interpretability and sensitivity analysis in surrogate modeling [52,54,57,75,76]. Further, the concept of digital twins (DTs) underscores its potential for next-generation real-time structural prediction and decision-making [77]. However, despite their proven effectiveness, the implementation of ML algorithms within the context of inter-story isolation systems has received limited attention, revealing an evident gap in the existing research landscape.

The present study aims to fill these gaps by employing a comprehensive methodology that incorporates the following aspects, which collectively highlight its novel contributions.

1) This study systematically investigates the influence of isolator placement at different story levels on the seismic performance of an inter-story isolated building, under an extensive set of earthquake excitations.

2) A multi-objective optimization framework is developed to identify the optimal values of key LRB design parameters: isolation time period (T_is), isolation damping ratio (ξ_is), and normalized yield strength (F₀) across different isolator placements, employing a metaheuristic algorithm. The optimization is performed using the PSO algorithm with the objective of minimizing critical structural responses such as roof acceleration, roof displacement, isolator displacement, and base shear simultaneously, thereby improving the effectiveness of LRB systems in ISI buildings across different story levels.

3) An ML-driven surrogate model capable of predicting optimal LRB parameters based on its placement level in the ISI building and seismic ground motion characteristics is proposed and developed using ANN. This innovation not only significantly reduces the high computational burden associated with repetitive optimization tasks for each new isolator location and earthquake record but also facilitates a practical and efficient solution for real-time or large-scale applications, making optimization-driven design strategies more feasible for engineering practice.

By incorporating these methods, this study aims to contribute to the field application of metaheuristic algorithms for the design and optimization of inter-story isolation systems. For this purpose, a 10-story shear building equipped with LRB placed at intermediate stories is modeled and analyzed under different historical earthquake ground motions using a MATLAB-based simulation environment.

2 Modeling of inter-story isolated building with lead rubber bearing

This study adopts two-dimensional multi-story shear building models schematically represented in Fig. 1, characterized by rigid floor diaphragms and mass lumped at each story level, with lateral resistance governed by story-level stiffness and damping elements, to simulate the structural responses under seismic ground excitation. The primary fixed-base building is shown in Fig. 1(a), while Figs. 1(b) and 1(c) present the corresponding BI and ISI configurations, respectively. The physical implementation of an intermediate isolation layer requires the incorporation of a dedicated lumped mass below the isolators (m_i^{\prime }), accounting for the inertial contributions of isolation devices and associated equipment required for their installation. Each lumped mass in the model (Fig. 1) is constrained to a single dynamic degree of freedom in the horizontal direction [34]. The dynamic equilibrium equation governing the response of the ISI building subjected to ground acceleration {\ddot{x}}_{\mathrm{g}} is expressed as

(1)\left[M_{\text{is}}\right]\left\{\ddot{x}\right\}\hspace{0.17em}+\hspace{0.17em}\left[C_{\text{nis}}\right]\left\{\dot{x}\right\}\hspace{0.17em}+\left[K_{\text{nis}}\right]\left\{x\right\}\hspace{0.17em}+\text{}\hspace{0.17em}\left\{\delta {\hspace{0.17em}}_{\mathrm{i}\mathrm{s}}\right\}\hspace{0.17em}{F}_{\text{is}}=-\left[M_{\text{is}}\right]\left\{I\right\}\hspace{0.17em}{\ddot{x}}_{\mathrm{g}}.

The vector \left\{x\right\}={\{x_1,x_2,..,x_{i-1},x_i^{\prime },x_i,x_{i+1},..,x_n\}}^{\text{T}} represents the floor displacements over time, with their velocities \{\dot{x}\} and accelerations \{\ddot{x}\} relative to the base of the building, respectively. Here, n represents the total number of degrees of freedom in the system. The mass matrix of the entire system [M_is] is a diagonal square matrix of order n and defined in Eq. (2). Stiffness and damping contributions, excluding the isolation system, are described by matrices [K_nis] and [C_nis], as presented in Eqs. (3) and (4), respectively.

(2)\left[M_{\text{is}}\right]=\text{diag}\hspace{0.17em}(m_1,m_2,...,m_{i-1},m_i^{\mathrm{\prime }},m_i,m_{i+1},...,m_n),

(3)\left[K_{\text{nis}}\right]=\left[\begin{array}{cc}\left[K_{\text{l}}\right]{\hspace{0.17em}}_{n_{\text{l}}\hspace{0.17em}\times n_{\text{l}}}& \\ & {\left[K_{\text{u}}\right]}_{\hspace{0.17em}{n}_{\text{u}}\hspace{0.17em}\times n_{\text{u}}}\end{array}\right],

(4)\left[C_{\text{nis}}\right]=\left[\begin{array}{cc}\left[C_{\text{l}}\right]{\hspace{0.17em}}_{n_{\text{l}}\hspace{0.17em}\times n_{\text{l}}}& \\ & {\left[C_{\text{u}}\right]}_{\hspace{0.17em}{n}_{\text{u}}\hspace{0.17em}\times n_{\text{u}}}\end{array}\right].

The matrices [K_l] and [C_l] correspond to the lower fixed-based substructure, which has n_l degrees of freedom, whereas [K_u] and [C_u] are associated with the upper isolated superstructure having n_u degrees of freedom.

(5)\left[K_{\text{l}}\right]=\left[\begin{array}{cccccc}{k}_1+k_2& -k_2& 0& \cdots & \cdots & 0\\ -k_2& k_2+k_3& -k_3& 0& \cdots & 0\\ ⋮& \ddots & \ddots & \ddots & \cdots & ⋮\\ 0& 0& 0& -k_{i-1}& k_{i-1}+k_i& -k_i\\ 0& 0& 0& 0& -k_i& k_i\end{array}\right],

(6)\left[K_{\text{u}}\right]=\left[\begin{array}{cccccc}{k}_{i+1}& -k_{i+1}& 0& \cdots & \cdots & 0\\ -k_{i+1}& k_{i+1}+k_{i+2}& -k_{i+2}& 0& \cdots & 0\\ ⋮& \ddots & \ddots & \ddots & \cdots & ⋮\\ 0& 0& 0& -k_{n-1}& k_{n-1}+k_n& -k_n\\ 0& 0& 0& 0& -k_n& k_n\end{array}\right].

The equivalent viscous damping in such structural systems exhibits non-classical damping characteristics [78,79], and therefore, distinct damping models are required for the fixed sub-structure and isolated superstructure [6,28]. The classical Rayleigh damping model, in which the damping matrix is expressed as a linear combination of the mass and stiffness terms, is adopted for the lower substructure. This approach aligns with its standard application in fixed-base structures. In contrast, a stiffness-proportional damping model is more suitable for the isolated superstructure [80,81]. The resulting damping matrices for the fixed substructure and isolated superstructure are given in Eqs. (7) and (8), respectively.

(7)\left[C_{\text{l}}\right]=a_{\text{0}}^{\text{l}}\left[M_{\text{l}}\right]+a_{\text{1}}^{\text{l}}\left[K_{\text{l}}\right],

(8)\left[C_{\text{u}}\right]=a_1^{\text{u}}\left[K_{\text{u}}\right],

where [M_l] denotes the mass matrix of the lower substructure, while a_0^{\mathrm{l}} and a_1^{\mathrm{l}} are the Rayleigh damping coefficients determined (Eq. (9)) by assigning a target damping ratio ξ_s of 2%, commonly adopted for steel buildings, at the modal frequencies ω_j and ω_k, which characterize the frequency range of interest for the lower fixed-based part of the building. For the damping matrix of the upper isolated superstructure, the proportionality coefficient a_1^{\mathrm{u}} is obtained (Eq. (10)) by linking the desired damping ratio (2%) with a representative higher-mode frequency ω_h, so that structural damping is appropriately controlled and distributed across the dominant higher modes.

(9)a_0^{\mathrm{l}}=\frac{2\xi {}_{\mathrm{s}}{\omega }_j{\omega }_k}{{\omega }_j+{\omega }_k},a_1^{\mathrm{l}}=\frac{2\xi {}_{\mathrm{s}}}{{\omega }_j+{\omega }_k},

(10)a_1^{\text{u}}=\frac{2{\xi }_{\mathrm{s}}}{{\omega }_{\text{h}}}.

The term F_is denotes the restoring force of the LRB, and {δ_is} is the corresponding influence vector. The restoring force F_is includes two contributions: a linear part and the hysteresis part. F_is can be expressed as [82]

(11)F_{\text{is}}=c_{\text{is}}{\dot{x}}_{\text{is}}+{\alpha }_{\text{r}}{k}_{\text{is}}{x}_{\text{is}}+(1-{\alpha }_{\text{r}})F_{\mathrm{y}\mathrm{s}}\hspace{0.17em}Z.

The initial elastic stiffness and viscous damping coefficient of the isolator (LRB) are represented by k_is and c_is, whereas x_{\hspace{0.17em}\text{is}} and {\dot{x}}_{\hspace{0.17em}\text{is}} denote the displacement and velocity within LRB. The rigidity ratio (α_r) is a dimensionless parameter that defines the ratio of the post-yield stiffness to the pre-yield stiffness of the isolator. The parameter F_ys denotes the yield strength of the lead plug in LRB. The hysteretic response of LRB is characterized by a non-dimensional variable Z that satisfies the following equation

(12)\dot{Z}=\left(A{\dot{x}}_{\text{is}}-\gamma |{\dot{x}}_{\text{is}}|Z|Z{|}^{\eta -1}-\beta {\dot{x}}_{\text{is}}|Z{|}^{\eta }\right)q^{-1},

where q represents the yielding displacement of the isolator. The nondimensional parameters A, β, γ, and the exponent η govern the shape of the hysteresis loop of LRB, along with the parameter α_r. The location vector {δ_is} is employed to denote the isolator influence correctly in the equation of motion and is defined by

(13)\begin{array}{rl}\left\{{\delta }_{\mathrm{i}\mathrm{s}}\right\}=& {\left\{\text{zeros}(n_{\text{l}}-1)-11\text{zeros}(n_{\text{u}}-1)\right\}}^{\text{T}}\\ =& {\left\{0\cdots 0-110\cdots 0\right\}}^{\text{T}}.\end{array}

The total restoring force of the ISI building (F_r) is governed by the sum of two contributing terms: One linear term from the elastic stiffness and viscous damping of the building (i.e., [K_nis] and [C_nis]), as well as the linear part of the F_is; the other nonlinear term due to the hysteresis part of F_is.

(14)\begin{array}{rl}{F}_{\text{r}}=& \left[K_{\text{nis}}\right]\left\{x\right\}+\left[C_{\text{nis}}\right]\left\{\dot{x}\right\}+F_{\text{is}}\\ =& \left[K_{\text{is}}\right]\left\{x\right\}+\left[C_{\text{is}}\right]\left\{\dot{x}\right\}+(1-{\alpha }_{\text{r}})F_{\mathrm{y}\mathrm{s}}Z,\end{array}

(15)\left[K_{\text{is}}\right]=\left[\begin{array}{cc}\left[\begin{array}{cccc}{k}_1+k_2& -k_2& \cdots & \cdots \\ -k_2& k_2+k_3& \cdots & \cdots \\ ⋮& \ddots & \ddots & -k_i\\ 0& 0& -k_i& k_i+{\alpha }_{\text{r}}{k}_{\text{is}}\end{array}\right]& \begin{array}{cccc}& & & \\ & & \ddots & \ddots \\ & & & \ddots \\ {\alpha }_{\text{r}}{k}_{\text{is}}& 0& & \cdots \end{array}\begin{array}{cccc}\begin{array}{cc}& \end{array}& 0& 0& \\ & ⋮& ⋮& \\ \ddots & ⋮& ⋮& \\ & 0& 0& \end{array}\\ \begin{array}{cccccccc}& 0& & 0& & \cdots & & {\alpha }_{\text{r}}{k}_{\text{is}}\\ & \cdots & & & & \\ & 0& & 0& & \cdots & & 0\end{array}& \left[\begin{array}{cccc}{k}_{\text{is}}+k_{\mathrm{i}+1}& -k_{\mathrm{i}+1}& \cdots & 0\\ -k_{\mathrm{i}+1}& \cdots & \cdots & 0\\ \cdots & & & \cdots \\ 0& 0& -k_n& k_n\end{array}\right]\end{array}\right].

The post-yield stiffness (α_rk_is) of the isolator is chosen to achieve a desired isolation time period, T_is, defined as

(16)T_{\text{is}}=2\text{π}\sqrt{\frac{M_{\text{t}}}{{\alpha }_{\text{r}}{k}_{\text{is}}}},

where M_t is the total mass of the isolated superstructure, and is calculated as the sum of all floor masses above the isolation layer. The damping ratio ξ_is of LRB is expressed as

(17){\xi }_{\text{is}}=\frac{c_{\text{is}}}{2M_{\text{t}}{\omega }_{\text{is}}},

where ω_is stands for the isolation frequency and can be calculated as ω_is = 2π/T_is.

The yield strength is normalized by the total weight of the isolated superstructure, resulting in a dimensionless yield strength, defined as

(18)F_0=\frac{F_{ys}}{M_{\text{t}}g},

where g denotes gravitational acceleration.

The physical configuration of LRB, its idealization in the ISI building, and the force–deformation behavior are shown in Fig. 2.

3 Optimization framework

3.1 Single objective model

The objective of this study is to optimally design LRBs for ISI buildings to minimize critical structural responses, including peak roof acceleration (A_r), peak roof displacement (δ_r), peak isolator displacement (δ_is), and peak base shear (V_b).

(19)\begin{array}{rl}& A_{\text{r}}=\text{max}\left\{A_{\text{r}}(t)\right\},{\delta }_{\text{r}}=\text{max}\left\{{\delta }_{\text{r}}(t)\right\},\\ & {\delta }_{\text{is}}=\text{max}\left\{{\delta }_{\text{is}}(t)\right\},V_{\text{b}}=\text{max}\left\{V_{\text{b}}(t)\right\},\end{array}

where A_r(t), δ_r(t), δ_is(t), and V_b(t) are the history of the respective structural responses.

As outlined in Section 2, modeling of LRB essentially depends on three key design parameters: 1) isolation time period (T_is); 2) isolation damping ratio (ξ_is); and 3) normalized yield strength (F₀). So, for the optimization of LRB, these parameters are considered as continuous, real-valued decision variables bounded within their specified feasible ranges

(20)T_{\text{is}}^{\text{min}}\le T_{\text{is}}\le T_{\text{is}}^{\text{ma}\text{x}};{\xi }_{\text{is}}^{\text{min}}\le {\xi }_{\text{is}}\le {\xi }_{\text{is}}^{\text{ma}\text{x}};F_{\text{0}}^{\text{min}}\le F_0\le F_{\text{0}}^{\text{ma}\text{x}}.

So, the individual response minimization objectives for a specified isolator placement height H_s (story level) can be expressed as

(21)\underset{s\left(H_{\text{s}}\right)}{\text{min}}\left\{A_{\text{r}}\left(H_{\text{s}}\right)\right\};\underset{\mathrm{s}\left(H_{\text{s}}\right)}{\text{min}}\left\{{\delta }_{\text{r}}\left(H_{\mathit{\text{s}}}\right)\right\};\underset{s\left(H_{\text{s}}\right)}{\text{min}}\left\{{\delta }_{\text{is}}\left(H_{\text{s}}\right)\right\};\underset{\mathrm{s}\left(H_{\mathit{\text{s}}}\right)}{\text{min}}\left\{V_{\text{b}}\left(H_{\text{s}}\right)\right\},

where s(H_s) is the design vector of decision variables:

(22)s\hspace{0.17em}\left(H_{\text{s}}\right)=\left\{\begin{array}{c}{T}_{\text{is}}\left(H_{\text{s}}\right)\\ {\xi }_{\text{is}}\left(H_{\text{s}}\right)\\ F_0\left(H_{\text{s}}\right)\end{array}\right\}.

3.2 Multi-objective model

Equation (21), derived from Subsection 3.1, serves as the objective functions for the respective structural responses. Since the primary aim is to minimize all these responses simultaneously, the optimization problem constitutes a multi-objective optimization challenge. Therefore, the multi-objective optimization problem can be defined as

(23)\underset{s\left(H_{\text{s}}\right)}{\text{min}}f\left(H_{\text{s}}\right)=\underset{\mathrm{s}\left(H_{\text{s}}\right)}{\text{min}}\left\{\begin{array}{c}{A}_{\text{r}}\left(H_{\text{s}}\right)\\ {\delta }_{\text{r}}\left(H_{\text{s}}\right)\\ {\delta }_{\text{is}}\left(H_{\text{s}}\right)\\ V_{\text{b}}\left(H_{\mathit{\text{s}}}\right)\end{array}\right\}.

To tackle this, the weighted linear combination (WLC) method [83] is utilized to consolidate the objectives into a single scalar function, allowing each response to contribute proportionally during optimization. The multi-objective normalized response (R_mo) using the weighted linear combination method is formalized in Eq. (24), where R₁ = A_r, R₂ = δ_r, R₃ = δ_is, and R₄ = V_b; R_i(max) and R_i(min) denote the corresponding maximum and minimum values of the corresponding peak responses, within the feasible design space. Each objective is assigned equal weights (w_i = 0.25 for i = 1,2,3,4), reflecting equal priority across all structural responses.

(24)R_{\text{m}\hspace{0.17em}\text{o}}=\sum _{i=1}^4{w}_i\hspace{0.17em}\cdot \frac{R_i-R_{i(\text{min})}}{R_{i(\text{max})}-R_{i(\text{min})}}.

The interdependence between the LRB design parameters and the R_mo is established through regression analysis. The general regression model is expressed in Eq. (25), where the response variable Y is represented as a function of the design variables, which are denoted by x_j and x_k. The regression coefficients β_j, β_jj, and β_jk correspond to linear, quadratic, and interaction terms, respectively, derived using the least squares method.

(25)

Thus, the relationship between the multi-objective response and LRB parameters is given by Eq. (26).

(26)\begin{array}{rl}{R}_{mo}& ={\beta }_0+{\beta }_1\times T_{\text{is}}+{\beta }_2\times {\xi }_{\text{is}}+{\beta }_3\times F_0+{\beta }_{11}\times (T_{\text{is}}\times {\xi }_{\text{is}})\\ & +{\beta }_{12}\times (T_{\text{is}}\times F_0)+{\beta }_{13}\times ({\xi }_{\text{is}}\times F_0)+{\beta }_{21}\times T_{\text{is}}^2\\ & +{\beta }_{22}\times {\xi }_{\text{is}}^2+{\beta }_{23}\times F_0^2,\end{array}

Therefore, the final multi-objective function for a specified isolator placement height H_s (story level) can be defined as

(27)\underset{s\hspace{0.17em}\left(H_{\text{s}}\right)}{\text{min}}\hspace{0.17em}f\left(H_{\text{s}}\right)=\hspace{0.17em}\underset{\mathrm{s}\hspace{0.17em}\left(H_{\mathit{\text{s}}}\right)}{min}\hspace{0.17em}\hspace{0.17em}\left\{R_{\text{mo}}\left(H_{\text{s}}\right)\right\},

where

(28)s\left(H_{\text{s}}\right)=\left\{\begin{array}{c}{T}_{\text{is}}\left(H_{\text{s}}\right)\\ {\xi }_{\text{is}}\left(H_{\text{s}}\right)\\ F_0\left(H_{\text{s}}\right)\end{array}\right\}.

This function is subsequently minimized using an MHA (PSO) to identify the optimal LRB parameter set. Figure 3 illustrates the formulation process of the multi-objective response function model, and the overall optimization workflow is illustrated in Fig. 4. To evaluate the performance and predictive capability of the proposed multi-objective regression model, key statistical metrics such as the coefficient of determination (R²), adjusted R² (Adj. R²), root mean square error (RMSE), standard deviation (SD), coefficient of variation (CV), etc., were determined for each model across every seismic record. Model adequacy was further validated through analysis of variance (ANOVA). Once the models demonstrated reliability and statistical robustness, the multi-objective cost functions were employed within the MHA to identify the optimal LRB design parameters.

3.3 Particle swarm optimization

In this study, the PSO algorithm is employed as a metaheuristic approach to identify the optimal parameters of the LRB for different isolator locations. PSO is among the most extensively applied MHAs in structural control due to its simplicity and flexibility. It has been widely adopted to identify the optimal parameters of TMDs [84–88] and Magnetorheological (MR) dampers [89–92]. Researchers have also used PSO to fine-tune parameters for fuzzy logic-controlled Active Tuned Mass Dampers (ATMDs) [93]and MR dampers [94]. In addition, a PSO-optimized Linear Quadratic Regulator (LQR) controller has been used to derive optimal control forces for ATMD systems [95]. Beyond these applications, PSO has been employed to determine optimal configuration and placement of viscoelastic dampers [96]. Tsipianitis and Tsompanakis [97] utilized PSO to optimize friction-based isolator parameters for a base-isolated liquid storage tank. More recently, PSO has been successfully applied to optimize MR damper parameters in base-isolated buildings [98], further extending its applicability.

PSO, among the earliest MHAs, was developed by Kennedy and Eberhart in 1995 [99,100], drawing inspiration from simplified social behaviors observed in nature, such as bird flocking, fish schooling, etc. In PSO, a population of candidate solutions, referred to as particles, simultaneously explores the solution space to find the global optimum. Each particle adjusts its velocity and position based on its individual as well as neighboring particles’ experience. The movement is guided by three key components: inertia, cognitive, and social influences. A population of N particles is initialized randomly. In a search space with D dimensions, where D corresponds to the number of parameters being optimized, the position and velocity of the ith particle at iteration t can be expressed as X_i^t=(x_{i,1}^t,x_{i,2}^t,...,x_{i,d}^t,...,x_{i,D}^t) and V_i^t=(v_{i,1}^t,v_{i,2}^t,...,v_{i,d}^t,...,v_{i,D}^t), respectively. The best position found by an individual particle during its search corresponding to the best objective value it has achieved up to iteration t, known as its personal best, is denoted as P_i^t=(p_{i,1}^t,p_{i,1}^t,...,p_{i,j}^i,...,p_{i,D}^t). Similarly, the best position discovered by any particle within the entire swarm so far at iteration t, referred to as the global best, is represented as P_g^t=(p_{g,1}^t,p_{g,2}^t,...,p_{g,j}^t,...,x_{g,D}^t). Considering a minimization problem as an example, in the original PSO algorithm, the personal best position of a particle is updated using the following rule:

(29)

Each particle updates its velocity and position using the following formulas, respectively:

(30)v_{i,j}^{t+1}=v_{i,j}^t+c_1\cdot R_1\cdot \left(p_{i,j}^t-x_{i,j}^t\right)+c_2\cdot R_2\cdot \left(p_{g,j}^t-x_{i,j}^t\right),

(31)x_{t,j}^{t+1}=x_{t,j}^t+v_{t,j}^{t+1},

where the constants c₁ and c₂ are acceleration coefficients, governing the influence of cognitive and social components, respectively. Meanwhile, R₁ and R₂ are uniformly distributed random values in the range [0,1]. Due to the limited effectiveness of the original PSO algorithm in solving optimization problems, a revised version was introduced by Shi and Eberhart in 1998 [101]. This modified PSO incorporated an inertia weight (w) into the velocity update equation, resulting in a slightly changed velocity update formula:

(32)v_{i,j}^{t+1}=w\cdot v_{i,j}^t+c_1\cdot R_1\cdot \left(p_{i,j}^t-x_{i,j}^t\right)+c_2\cdot R_2\cdot \left(p_{g,j}^t-x_{i,j}^t\right).

The iterative process defined by the aforementioned formulas continues until a termination condition is satisfied, such as reaching a preset maximum number of iterations, exceeding a specified count of iterations without any improvement in the global best solution, or attaining a predefined target fitness value.

Despite the widespread application of PSO due to its simplicity and efficiency, some limitations must be considered. Since PSO is an iterative population-based method, it requires the initial estimates of both positions and velocities of the particles. The adopted initialization strategy has a strong influence on whether particles remain within the feasible search space, thereby affecting both convergence stability and solution quality. In addition, PSO has a tendency to become trapped in local optima, especially in complex multimodal search spaces. Its performance is also highly sensitive to the tuning of its control parameters (w, c₁, and c₂); an inappropriate choice of these parameters can significantly degrade the convergence behavior. Lastly, the population size is another critical factor; a smaller population may converge prematurely, whereas larger populations enhance exploration but at the cost of computational efficiency [102–105].

4 Prediction of optimum lead rubber bearing parameters using artificial neural network

The optimization framework developed to identify optimal LRB design parameters in the ISI building requires considerable computation effort and time, primarily because the optimal LRB parameter values vary with isolator placement and different seismic records. To address the computational challenges and enhance the practical viability of the framework, an ML model needs to be implemented as a predictive tool. Among ML algorithms, ANN has been the most extensively applied in structural control, system identification, and damage detection within the field of earthquake engineering [106]. Researchers have employed ANN models to predict seismic drift in steel moment frames [56], displacement in isolation systems [47], seismic risk assessment of structures [107,108], seismic damage of shear walls [64], and reinforced concrete (RC) buildings [65,109,110], etc. Therefore, building upon the demonstrated effectiveness of ANNs in diverse earthquake engineering applications, this study employs an ANN model as a surrogate framework to predict the optimal LRB parameters for inter-story isolated buildings.

4.1 Overview of artificial neural network

ANNs are computational models inspired by the architecture and functioning of human nervous systems. They emulate the way the human brain processes information using interconnected processing units, also known as artificial ‘neurons’. It can be utilized to establish a functional mapping between input and output data [111]. Among the various alternatives of ANNs, a multi-layer feedforward neural network trained using the backpropagation (BP) algorithm is considered for this study, which involves two main processes. First, in forward propagation, data from the input layer passes through hidden layers to produce an output, and then, in BP, the estimation error is propagated backward through the network to adjust the weights, thereby minimizing the difference between predicted and target values [112,113]. Figure 5 illustrates the topological architecture of an ANN, comprising three sequential layers: an input layer, one or more hidden layers, and an output layer. Each layer comprises a set of interconnected neurons equipped with a layer-specific activation function, uniform across all neurons within that layer. Connections between the input and hidden layers are assigned weights (w), and each neuron is associated with a bias (θ or δ). The predictive capability is significantly influenced by the selection of weights and biases. The connection weights are denoted as w_ij (between input and hidden layers) and w_jk (between hidden and output layers), with input values represented as x₁,x₂,...,x_m and predicted outputs as y₁,y₂,...,y_l.

4.2 Selection of input and output parameters

In this study, a total of ten input variables were selected to train the ANN model. Among these, nine inputs represent various ground motion characteristics, chosen based on their widespread use in the established prior studies [56,65,107,109]. The tenth input variable corresponds to the isolator location within the ISI building, which significantly influences optimum LRB design configuration. The ANN model was trained to predict three target outputs, namely the optimal design parameters LRB, which were previously obtained through PSO. A comprehensive list of the input and output variables used in the model is presented in Table 1.

Before developing the ANN model, since the input variables used in this study have different units and numerical ranges, the min–max normalization is applied on the input parameters to scale them to the range of [0,1]. For each input variable X_i, the normalized value X_i^N computed as

(33)X_i^N=\frac{X_i-\text{min}\hspace{0.17em}\hspace{0.17em}\left(X_i\right)}{\text{max}\hspace{0.17em}\hspace{0.17em}\left(X_i\right)-\text{min}\hspace{0.17em}\hspace{0.17em}\left(X_i\right)},

where X_i denotes the value of the ith input parameter, while min(X_i) and max(X_i) represent the minimum and maximum values of that variable within the data set. The normalization process removes the unit disparities, ensures uniformity across parameters, and facilitates faster convergence of the surrogate model during training [75,76].

4.3 Artificial neural network architecture

In ANN modeling, the selection of a suitable network architecture is a critical step, particularly in identifying an ideal number of hidden layers and neurons in the hidden layers. The choice of the no of hidden layers and the hidden layer size plays a pivotal role in ensuring that the model not only achieves low prediction error but also maintains strong generalization capability to unseen data. For this study, a single hidden-layer architecture was adopted, based on established research [47,56], which indicates that one hidden layer is often sufficient to deliver high predictive accuracy in civil engineering applications. Additionally, careful selection of the hidden layer size, i.e., the number of hidden neurons, is equally crucial for improving prediction accuracy while avoiding overfitting. In this context, the number of hidden neurons was initially estimated using empirical formulations (Eqs. (34)–(38)) available in Refs. [114-118], providing a rational basis for the architectural design.

(34)n_{\text{h}}=X_i-1,

(35)n_{\text{h}}=\frac{\left(\sqrt{1+8X_i}-1\right)}{2},

(36)n_{\text{h}}=\frac{\left(X_i+\sqrt{Y_{\text{o}}}\right)}{L_{\text{n}}},

(37)n_{\text{h}}=\sqrt{X_i\times Y_{\text{o}}},

(38)n_{\text{h}}=\frac{4X_i^2+3}{X_i^2-8},

where n_h represents the number of neurons in the hidden layer, X_i denotes the number of input features, Y_o indicates the number of output variables, and L_n signifies the number of hidden layers used in the ANN model.

4.4 Evaluation criteria

To assess the performance of the trained ANN model, a total of five performance indicators were adopted: Mean Squared Error (MSE), Correlation Coefficient (R), RMSE, R², and Mean Absolute Error (MAE). Collectively, the metrics measure the accuracy and average prediction deviation of the ANN model. The metrics can be expressed as:

(39)MSE=\frac{1}{n}\sum _{j=1}^n{\left(t_j-y_j\right)}^2,

(40)R=\frac{{\displaystyle \sum _{j=1}^n\left(t_j-t_m\right)\left(y_j-y_m\right)}}{\sqrt{{\displaystyle \sum _{j=1}^n{\left(t_j-t_m\right)}^2{\displaystyle \sum _{j=1}^n{\left(y_j-y_m\right)}^2}}}},

(41)RMSE=\sqrt{\frac{1}{n}\sum _{j=1}^n{\left(t_j-y_j\right)}^2},

(42)R^2=1-\frac{{\displaystyle \sum _{j=1}^n{\left(t_j-y_j\right)}^2}}{{\displaystyle \sum _{j=1}^n{\left(t_j-t_m\right)}^2}},

(43)MAE=\frac{1}{n}\sum _{j=1}^n\left|t_j-y_j\right|,

where y_j and t_j are the predicted and actual output for the jth sample, respectively. Similarly, y_m and t_m are the mean values of the predicted and actual output, respectively, over the total number of n samples or data points.

The lower error values (MSE, RMSE, MAE) indicate more accurate and reliable network performance, whereas R and R² values approaching unity signify a strong correlation, demonstrating the high predictive capability of the model.

4.5 Feature important analysis

To evaluate the contribution of input features in the prediction of optimal LRB configuration, Garson’s algorithm [119] was employed. In this approach, the relative importance (RI) of each input parameter is determined based on the connection weights within the ANN architecture. During the training phase of the backpropagation neural network , the connection weights are iteratively updated to reflect the extent to which each input variable influences the corresponding output variable. By examining the distribution of these weights between the input and output layers, the relative significance of the input variables on the model output can be assessed. The Garson algorithm utilizes these connection weights within the trained network to quantify the relative contribution of each input variable (x_i) to the output variable (y_k)_. Essentially, this method performs a sensitivity-based evaluation of the network, determining how variations in each input affect the output response. The RI can be computed as:

(44)R{I}_{ik}=\frac{{\displaystyle \sum _{j=1}^n\left(|w_{ij}||w_{jk}|/{\displaystyle \sum _{i=1}^m|w_{ij}|}\right)}}{{\displaystyle \sum _{i=1}^m\sum _{j=1}^n\left(|w_{ij}||w_{jk}|/{\displaystyle \sum _{i=1}^m|w_{ij}|}\right)}},

where m represents the number of input parameters, while n denotes the number of hidden neurons. An input variable with a higher RI_ik value indicates a greater level of influence in predicting the outputs.

5 Numerical modeling and simulation setup

5.1 Structural model characteristics and configuration

5.1.1 Building characteristics

A 10-story shear building presented by Zelleke et al. [120] is adopted as the representative structure for this analysis. This configuration typically reflects a mid-rise building height, which is common in seismic-prone urban areas. It also provides a sufficiently complex system for examining the influence of isolator placement across lower, middle, and upper stories, while keeping the computational effort manageable, unlike a high-rise model. In addition, since the study involves a comparative assessment between ISI and BI systems, adopting a mid-rise structure ensures a fair assessment, as BI systems are most effective for low to mid-rise buildings.

The building is designed with uniform lateral stiffness and equal floor mass throughout all stories. Properties of the building are presented in Table 2.

5.1.2 Isolation parameters

To choose the optimum parameters for the isolation system, based on Zelleke et al. [120], the range of the different parameters of the study was selected. To investigate how the isolation time period, isolation damping ratio, and normalized yield strength affect the responses of base-isolated buildings, these parameters were varied within specific ranges:

1) 2 s to 4 s for the isolation time period (T_is);

2) 0.025 to 0.15 (i.e., 2.5% to 15%) for the isolation damping ratio (ξ_is);

3) 0.05 to 0.15 for the normalized yield strength (F₀).

The values of non-dimensional parameters of the hysteresis loop A, γ, η, and β (1, 1, 0.5, and 0.5, respectively) align with values reported in various literature sources [121–124]. The yield displacement of the isolators q is taken as 2.5 cm throughout the study [120–124].

5.2 Earthquake excitation data used

A total of 175 earthquake ground motion records, ensuring a diverse range of characteristics, were employed as external excitation input in this study. These records were obtained from the PEER Ground Motion Database. Selected earthquakes cover a moment magnitude (M_w) range of 5.5–7.6 and represent multiple fault mechanisms, including strike-slip, reverse, reverse oblique, and normal events. In addition, the data set spans a broad range of site classes (based on V_S30), from stiff soil sites (Class E) to hard rock sites (Class A), thereby ensuring representativeness of both source and site conditions [125]. To ensure a consistent basis for comparative analysis, the ground motion records were uniformly scaled to a target PGA of 0.3g [126]. The acceleration and displacement response spectra for all selected ground motions, computed with 5% critical damping, are presented in Fig. 6, while the important characteristics of the records such as event name, station, date, M_w, R_jb, PGA, site class, fault type, PEER record sequence number (RSN) are summarized in Appendix A (Table A1) in Electronic Supplementary Material.

5.3 Parametric mapping of structural response in lead rubber bearing design space

To comprehensively assess the influence of LRB design parameters on the seismic responses of the ISI building, an extensive parametric design space was developed. The parameters were discretized as follows: T_is from 2.0 to 4.0 s (increments of 0.1 s), ξ_is from 2.5% to 15% (increments of 0.5%), and F₀ from 0.05 to 0.15 (increments of 0.005), resulting in 21, 26, and 21 discrete values, respectively. By systematically combining all possible values, a total of 11466 unique parameter sets were generated. For each parameter combination, four structural responses, roof displacement, roof acceleration, base shear, and isolator displacement, were computed under individual earthquake records, forming dedicated design matrices for each seismic event. An example of the design matrix corresponding to the RSN 338 earthquake is presented later in the Results and discussion section. This comprehensive sampling enabled detailed sensitivity analysis and provided a robust foundation for regression modeling and subsequent optimization of the LRB system for ISI building.

5.4 Particle swarm optimization parameter settings and convergence criteria

In this study, the PSO algorithm is implemented with a population size of 50 and a maximum of 500 iterations. The convergence criterion was defined as either reaching the maximum number of iterations or no improvement in the best fitness value (f_best) over 25 consecutive iterations, whichever occurred first. The swarm size of 50 particles is a choice consistent with the values commonly adopted in earlier structural optimization studies [127,128]. Further, the control parameters are not tuned specifically for this problem; instead, they are chosen (w = 0.8 and c₁ = c₂ = 1.5) based on their proven efficacy in related literature [104,128].

5.5 Artificial neural network model development

5.5.1 Hidden layer architecture of artificial neural network

To identify the appropriate hidden layer size, several empirical formulas, as represented by Eqs. (34)–(38), were utilized, proposing potential hidden neuron counts of 9, 4, 12, 6, and 5, respectively. Based on these recommendations, a parametric study was conducted to assess the performance of the ANN models having varying numbers of hidden neurons ranging from 4 to 14. For each hidden layer configuration, the performance of the ANN models was evaluated using MSE and R. The configuration that exhibited an ideal balance between prediction accuracy and generalization was selected as the final network architecture.

5.5.2 Artificial neural network training setup

The ANN model was developed and evaluated using a total of 1750 data samples (corresponding to 10 optimal configurations per earthquake record). Of these, 1600 samples derived from 160 ground motion records were used for model development, which were partitioned randomly into training, validation, and test (seen) subsets. The remaining 150 samples, obtained from 15 additional earthquake records, were reserved as unseen data and employed exclusively for independent testing to assess the generalizability of the developed surrogate model.

The ANN model was trained using the Levenberg–Marquardt (LM) algorithm, a highly efficient optimization technique well-suited for nonlinear regression problems. The available data set was divided randomly into three different subsets to facilitate supervised learning. Specifically, a 70% training, 15% validation, and 15% testing split was considered, a split widely adopted in ANN-based surrogate modeling [129]. To check partition bias and confirm robustness, repeated random subsampling validation, also known as Monte Carlo cross-validation [130], was conducted, in which the data were repeatedly resampled into training, validation, and test sets across 50 independent trials. This approach ensures whether the performance of the model is sensitive to a specific data set split or not, thereby reinforcing the generalizability of the model. Furthermore, the additional data set, which was not used during model development, was tested as unseen data to assess the generalizability of the ANN surrogate.

In addition to cross-validation, the overfitting was prevented through validation monitoring and an early stopping criterion. The training process automatically terminated when the validation error did not improve for six consecutive epochs, thereby preventing the network from continuing to adapt to training noise and ensuring the model with the best generalization performance was retained. This method is a standard and effective approach to control overfitting when using the LM algorithm.

5.6 Simulation and computational environment

The seismic response of the structural models was evaluated through numerical simulations employing a full-state feedback control loop, implemented in MATLAB and Simulink, incorporating relevant toolboxes. To ensure consistency and fairness, as well as to maintain uniformity in results, all the simulations, PSO runs, and ANN models training were executed within a standardized computational setup. The PSO was programmed and run in MATLAB R2023a on a system running Microsoft Windows 11 Home, powered by an Intel® Core™ i5-12600 Central Processing Unit (3.3 GHz, 12 cores) and 8 GB of Random Access Memory. The ANN model was developed, trained, and evaluated using MATLAB’s Neural Network Toolbox under identical hardware conditions. This uniform computational setup system induced variability, ensuring the reliability and reproducibility of both optimization and predictive modeling results.

6 Results and discussion

6.1 Effect of isolator placement on seismic performance of inter-story isolated buildings

6.1.1 Reduction of structural responses

To evaluate the effect of isolator location, Figs. 7–9 present the variation in percentage reduction (control) of three critical structural responses, roof displacement (δ_r), roof acceleration (A_r), and base shear (V_b), respectively, as a function of the isolator placement within the 10-story building. The isolator is placed at different story levels, with story 0 corresponding to a conventional base-isolated system, while stories 1 through 10 signify inter-story isolated configurations, with the isolator progressively relocated higher in the building. The percentage control values are calculated with respect to the corresponding responses of a fixed-base (non-isolated) building. In each figure, subplot (a) illustrates the mean percentage control, where each data point reflects the average control percentage across all the considered earthquake excitations, while subplot (b) displays the response control achieved for some selected representative earthquake records, highlighting the variability in structural performance due to differences in seismic input.

Figure 7(a) indicates that the average percentage control of roof displacement improves progressively as the isolator is shifted upward from the base to upper stories, reaching a peak (18%–20%) at approximately the 5th to 7th story level. However, beyond the 7th story, as the isolator is placed at higher elevations (i.e., stories 8–10), the displacement control tends to plateau or reduce slightly. In case of base-isolated configuration, the average displacement control is lowest, typically around 12%–13%. The individual earthquake plots in Fig. 7(b) reveal a relatively stable and consistent trend in roof displacement control across most seismic ground motions. Despite some variability in peak control values between different earthquakes, the general trend of superior performance at mid-story (5th−7th story) levels remains evident. However, the noticeable spread in the representative individual earthquake plots in Fig. 7(b) and the higher SD across all isolator locations in Fig. 7(a) highlight that there exists considerable variability in effectiveness depending on the characteristics of the ground motion. It is also evident that under certain earthquake records, neither BI nor ISI configurations are able to adequately control the roof displacement demand. Despite these exceptions, mid-height level ISI configuration generally provides a more balanced and effective displacement control strategy, offering improved performance consistency across a broad range of seismic records.

The average percentage reduction in roof acceleration control shows a complex pattern as illustrated in Fig. 8(a). Initially, as the isolator is moved from the base to the 3rd story level, the mean control slightly decreases monotonically. The base-isolated configuration achieves about 59% control in roof acceleration, whereas placing the isolator on lower stories (e.g., story 3) leads to a local minimum. Beyond this point, from story 4, placing the isolator at higher story levels, the mean acceleration control increases consistently, reaching 63% at story 10. The roof acceleration reduction curves for individual earthquakes shown in Fig. 8(b) support this observation, with most cases exhibiting the trend of a slight drop near the 2nd or 3rd story level, but the acceleration control improved with increased isolator height remains broadly consistent. Notably, the SD decreases with increasing isolator height, implying greater robustness and predictability in acceleration control when isolators are placed near the top. Notably, the SD of roof acceleration control narrows slightly with increasing isolator placement height, implying a more consistent reduction in acceleration across different seismic events. This makes upper-story level inter-story isolation an attractive configuration for the buildings that house sensitive equipment or critical non-structural elements, such as hospitals, laboratories, or data centers, where floor acceleration is a key design concern. However, if acceleration control is a primary design criterion, then a BI building offers better performance compared to an ISI configuration with the isolator placed at lower stories (i.e., between the 1st and 3rd floors).

The base shear reduction exhibits a contrasting pattern compared to the roof acceleration control trend. Figure 9(a) indicates that BI is the most effective in terms of average base shear control, achieving a mean reduction of 79%, while as the isolator moves upward progressively, a steady decline in average base shear reduction is observed. The control percentage drops to around 50% when the isolator is installed at the top story, highlighting a significant decrease in base shear mitigation with increased isolator elevation. This pattern is physically intuitive. BI effectively decouples the entire superstructure from the ground motion, resulting in a substantial reduction in inertial forces transmitted to the base. In contrast, inter-story isolation only decouples the structure above the isolator, leaving the lower substructure fully engaged with ground motion, and allows significant force to be absorbed. Consequently, the lower substructure behaves like a fixed-base building, accumulating shear demand from both its own dynamic response and the inertial demands transmitted from the isolated superstructure. Furthermore, the variability (i.e., SD) in base shear control remains large for all the ISI configurations compared to the BI setup, especially in mid-story isolation, signifying that BI is the superior option when minimizing force demand at the base level of a building.

The trends observed across all three response controls are summarized in Table 3, which provides a more complete comparative overview of all isolator positions.

In summary, the effectiveness of an inter-story isolation system is highly dependent on its vertical placement within the building. Roof displacement control shows a modest improvement as the isolator is placed at mid-height within the building, though high variability persists across different seismic records. The maximum roof acceleration control is achieved when the isolator is placed on the upper stories. In contrast, base shear control is maximized when the isolator is positioned at the lower stories, and it consistently decreases as the isolator is moved upward. These findings collectively highlight the complex interdependence between isolator placement and structural response control and emphasize the need for a performance-based design approach, where isolator placement height should be selected based on the priority of specific response control. Thus, ISI may not be an optimal but a versatile and adaptable solution that can be tailored to meet structural or functional objectives.

6.1.2 Isolator displacement

Figure 10 presents the variation in isolator displacement (δ_is) with respect to the isolator location within the building for both the mean response across all earthquake records and representative individual earthquakes, providing insight into the expected displacement demand on the isolator at each floor level. The average isolator displacement across all the earthquakes displays a gradual increasing pattern when the isolator shifts from the base to higher stories, as illustrated in Fig. 10(a). As the isolator is moved upward, particularly beyond the mid-height of the building, the average isolator displacement begins to rise more noticeably. Additionally, the variability (SD) in the isolator deformation increases in upper-story ISI configurations. The individual earthquake responses shown in Fig. 10(b) further reinforce the observed trend. While the magnitude of displacement varies across seismic records, the overall pattern is consistent, i.e., the displacement demand of the isolator tends to increase with isolator elevation.

6.2 Optimization of lead rubber bearing design parameters for different locations

6.2.1 Adequacy of the multi-objective model

Regression-based modeling was employed based on a design space containing numerous LRB parameter combinations, as described in Subsection 5.3, to establish the multi-objective framework. The structural responses for each configuration were determined under all the considered ISI configurations and seismic records. Table 4 highlights a sample of these parameter configurations alongside their corresponding structural responses for the earthquake RSN 338, when the isolator is placed at the 1st story level. Given the extensive size of the design space, which includes distinct 11466 LRB parameter combinations, displaying each configuration may not be feasible. Therefore, only a representative subset is provided to demonstrate the structure and content of the design space. The individual structural responses are then normalized to a consistent scale, and the R_mo for each parameter combination was derived using the WLC method outlined in Subsection 3.2. Table 5 displays the normalized structural responses alongside their aggregated multi-objective response values.

The derived R_mo values facilitated the formulation of a regression-based model that represents the multi-objective function, as defined in Eq. (45).

(45)\begin{array}{rl}{f}_1& =1.820-0.776\times T_{\text{is}}-3.887\times {\xi }_{\text{is}}-0.773\times F_{\text{0}}\\ & +0.70\times T_{\text{is}}\times {\xi }_{\text{is}}-0.969\times T_{\text{is}}\times F_0-1.209\times {\xi }_{\text{is}}\times F_0\\ & +0.116\times T_{\text{is}}^{\text{2}}+3.514\times {\xi }_{\text{is}}^2+26.671\times F_0^2.\end{array}

This regression expression was subsequently integrated into the optimization framework as the objective or fitness function. The reliability, predictive accuracy, statistical significance, and goodness-of-fit of the multi-objective model were evaluated through ANOVA with a significance threshold of p ≤ 0.05 corresponding to a 95% confidence level. The summary of this analysis is presented in Table 6.

The ANOVA results show that the developed multi-objective model is statistically significant as it achieves a high F-value of 23170 with a p-value less than 0.0001, indicating that all or at least one LRB parameter has a substantial effect on the structural responses. Further, all the individual terms, i.e., linear, quadratic, and interaction, are also statistically significant with large F-values and extremely low p-values (p < 0.0001), confirming that all model terms significantly contribute to the response. Notably, among the linear terms, T_is has the largest sum of squares (8.91 × 10⁵) and F-value (1.38 × 10⁹), indicating it has the dominant effect on the structural responses. Further, the quadratic term of T_is (i.e., T_is²) is the most influential, contributing a sum of squares of 1.01 × 10⁸ and an exceptionally large F-value of 1.57 × 10¹¹, suggesting a strong nonlinear relationship between the isolation period and the structural response. In addition to the individual parameters, the interaction terms (i.e., T_is × ξ_is, T_is × F₀, and ξ_is × F₀) also exhibit higher F-values on the order of 10⁶, highlighting that the response is not solely influenced by the independent effects of the parameters but also by their combined interactions. Similarly, the remaining quadratic terms ξ_is² and F₀² contribute significantly to the model. The model demonstrates an excellent fit, with a R² of 0.9478, implying that 94.78% of the variability in response is explained by the model. The adjusted R² (0.9479) being nearly identical to R² suggests that the model is not overfitted. Furthermore, the low RMSE of 0.0254 reflects a high level of prediction accuracy. In addition, the model demonstrates good precision with a CV of 6.83% indicating low variability in the data relative to the mean of 0.3719.

Similar to Eq. (45), Eqs. (46)–(54) show the multi-objective cost function for the LRB parameter optimization when it is placed at stories 2 to 10, respectively. Table 7 presents a brief overview of the essential ANOVA statistics for the remaining isolator locations, highlighting the model adequacy across all configurations.

(46)\begin{array}{rl}{f}_2=& 1.748-0.766\times T_{\text{is}}-3.31\times {\xi }_{\text{is}}-1.139\times F_0\\ & +0.565\times T_{\text{is}}\times {\xi }_{\text{is}}-0.639\times T_{\text{is}}\times F_0-0.325\times F_0\times {\xi }_{\text{is}}\\ & +0.117\times T_{\text{is}}^2+2.786\times {\xi }_{\text{is}}^2+23.453\times F_0^2,\end{array}

(47)\begin{array}{rl}{f}_3=& 1.540-0.661\times T_{\text{is}}-3.329\times {\xi }_{\text{is}}-1.384\times F_{\text{0}}\\ & +0.668\times T_{\text{is}}\times {\xi }_{\text{is}}-1.89\times T_{\text{is}}\times F_0-6.239\times {\xi }_{\text{is}}\times F_0\\ & +0.121\times T_{\text{is}}^{\text{2}}\text{+}3.334\times {\xi }_{\text{is}}^2+47.966\times F_0^2,\end{array}

(48)\begin{array}{rl}{f}_4=& 1.637-0.66\times T_{\text{is}}-3.172\times {\xi }_{\text{is}}-3.408\times F_0\\ & +0.441\times T_{\text{is}}\times {\xi }_{\text{is}}-1.091\times T_{\text{is}}\times F_0-0.534\times F_0\times {\xi }_{\text{is}}\\ & +0.113\times T_{\text{is}}^2+3.187\times {\xi }_{\text{is}}^{\text{2}}+39.569\times F_0^2,\end{array}

(49)\begin{array}{rl}{f}_5=& 1.484-0.523\times T_{\text{is}}-3.335\times {\xi }_{\text{is}}-4.615\times F_0\\ & +0.418\times T_{\text{is}}\times {\xi }_{\text{is}}-0.859\times T_{\text{is}}\times F_0-0.338\times {\xi }_{\text{is}}\times F_0\\ & +0.086\times T_{\text{is}}^2+4.214\times {\xi }_{\text{is}}^2+48.427\times F_0^2,\end{array}

(50)\begin{array}{rl}{f}_6=& 1.592-0.538\times T_{\text{is}}-3.651\times {\xi }_{\text{is}}-4.821\times F_0\\ & +0.519\times T_{\text{is}}\times {\xi }_{\text{is}}-1.424\times T_{\text{is}}\times F_0+2.592\times {\xi }_{\text{is}}\times F_0\\ & +0.092\times T_{\text{is}}^{\text{2}}+3.125\times {\xi }_{\text{is}}^2+51.87\times F_0^2,\end{array}

(51)\begin{array}{rl}{f}_7=& 1.471-0.482\times T_{\text{is}}-3.984\times {\xi }_{\text{is}}-3.894\times F_0\\ & +0.613\times T_{\text{is}}\times {\xi }_{\text{is}}-0.487\times T_{\text{is}}\times F_0+4.991\times {\xi }_{\text{is}}\times F_0\\ & +0.067\times T_{\text{is}}^{\text{2}}+3.388\times {\xi }_{\text{is}}^{\text{2}}+32.545\times F_0^2,\end{array}

(52)\begin{array}{rl}{f}_8=& 1.556-0.557\times T_{\text{is}}-4.262\times {\xi }_{\text{is}}-2.417\times F_0\\ & +0.457\times T_{\text{is}}\times {\xi }_{\text{is}}-1.187\times T_{\text{is}}\times F_0+10.18\times {\xi }_{\text{is}}\times F_0\\ & +0.091\times T_{\text{is}}^2+4.314\times {\xi }_{\text{is}}^2+31.531\times F_0^2,\end{array}

(53)\begin{array}{rl}{f}_9=& 0.888-0.257\times T_{\text{is}}-3.046\times {\xi }_{\text{is}}+2.128\times F_0\\ & +0.371\times T_{\text{is}}\times {\xi }_{\text{is}}-1.365\times T_{\text{is}}\times F_0+1.61\times {\xi }_{\text{is}}\times F_0\\ & +0.059\times T_{\text{is}}^2+3.205\times {\xi }_{\text{is}}^2+5.541\times F_0^2,\end{array}

(54)\begin{array}{rl}{f}_{10}=& 1.159-0.379\times T_{\text{is}}-2.858\times {\xi }_{\text{is}}-0.272\times F_0\\ & +0.349\times T_{\text{is}}\times {\xi }_{\text{is}}-1.09\times T_{\text{is}}\times F_0+4.357\times {\xi }_{\text{is}}\times F_0\\ & +0.07\times T_{\text{is}}^{\text{2}}+2.974\times {\xi }_{\text{is}}^2+11.094\times F_0^2.\end{array}

6.2.2 Performance evaluation of particle swarm optimization

To evaluate the stability and reliability of the optimization results produced by PSO, as well as to examine the potential sensitivity of the optimization process to initial swarm positions, 100 independent optimization runs were performed for each story level across earthquake records. The results indicated that PSO consistently converged to the same optimal solution, with the best, worst, and average fitness values over 100 runs being identical. Moreover, the SD and CV remained extremely small (on the order of 10⁻¹⁷–10⁻¹⁶ and 10⁻¹⁴–10⁻¹³, respectively) across story levels. These findings signify that the PSO outcomes are essentially insensitive to the initial LRB parameter guesses or PSO starting positions, thereby confirming the robustness of the optimization framework.

To further validate these results, two alternative MHAs, Differential Evolution (DE) and Teaching-Learning-Based Optimization (TLBO), were also implemented under identical conditions. The outcomes of this analysis demonstrate that both DE and TLBO achieved identical optimal solutions as PSO, with negligible differences in SD and CV values observed for TLBO. The detailed comparative results are summarized in Table 8, which includes the best, worst, and average fitness values, along with the SD and CV computed across 100 independent runs. This comparative assessment indicates that although PSO exhibits some inherent limitations, its performance in the present study is equivalent to that of DE and TLBO, thereby demonstrating the reliability and robustness of the adopted optimization framework.

6.2.3 Optimum isolator parameters

The results obtained after applying PSO on the objective cost functions suggest that the optimal LRB characteristics are highly influenced by both the specific earthquake input and the location of the isolator within the building. Figure 11 presents boxplots for optimized LRB parameters across all the earthquakes and different story-level isolator placements. Each box depicts the distribution of the optimum LRB parameter values obtained for a given isolator location when the building was subjected to multiple earthquake records. The central line of each box indicates the median values, signifying the central tendency, while the box edges denote the interquartile range (IQR), displaying the central 50% of results. Further, whiskers extend to 1.5 times of IQR, and the points beyond that are plotted as outliers, representing extreme optimization outcomes. The distribution of T_is and F₀ is fluctuating noticeably within their respective ranges, indicating their considerable dependency on the seismic inputs and isolator placement configurations. In contrast, the box plot for ξ_is indicates a distinct clustering of values near the upper limit, suggesting that the optimization consistently drives the ξ_is toward its maximum permissible limit. The two alternative algorithms, DE and TLBO, produced nearly identical optimal values, with variations observed only beyond the fifth to sixth decimal places. The variation of the optimal LRB parameter values corresponding to different isolator locations is discussed in detail in Subsubsection 6.2.5.

6.2.4 Execution time and convergence behavior

The execution time of an optimization algorithm serves as a critical indicator of its computational efficiency and applicability to real-world challenges. For time-sensitive engineering applications, such as real-time structural control systems, algorithms with shorter computation times are particularly advantageous. In this study, PSO demonstrated high computational efficiency, with an average execution time of 2.5 s for all the cases, as shown in Fig. 12. Among the other two tested MHAs, DE matched PSO’s performance, achieving nearly identical computational time. In contrast, TLBO consistently required noticeably longer execution times, indicating its higher computational cost.

The convergence profiles presented in Fig. 13 depict the progression of the best objective function (f_best) values across iterations for PSO. Each curve represents the convergence efficiency of PSO toward an optimal solution for a specific configuration of isolator placement under the earthquake RSN 338. All the convergence curves exhibit a clear downward trend, indicating PSO’s ability to consistently minimize the cost function as iterations progress. During the initial phase (i.e., early iterations), a sharp decline in the objective function values, reflecting the strong global exploration capability of PSO. This phase allows the PSO to broadly search the design space, thus mitigating the risk of premature convergence. As the iteration count increases, the curves begin to flatten, signifying the transition from global exploration to local exploitation, where the algorithm progressively refines the isolator parameters. Further, the final segments of the curves show minimal variation in the objective function value, confirming that the solutions have reached convergence and additional iterations yield negligible improvements. Notably, PSO typically converged within approximately 110–120 iterations in most runs, well before reaching the maximum iteration limit set in this study. For comparison, DE exhibited a similar convergence pattern to PSO, whereas TLBO achieved convergence comparatively earlier, typically within 70–80 iterations.

Overall, PSO demonstrated a highly efficient and reliable optimization capability across all cases. It successfully identifies optimal parameter sets within a reasonable computation time and number of iterations, validating its suitability for the story-wise isolator tuning under diverse seismic records.

6.2.5 Variation of optimized isolation parameters with isolator location

The variation of the optimized LRB parameters (i.e., T_is,ξ_is, and F₀) corresponding to the isolator placement is presented in Fig. 14. The mean optimized T_is across all selected earthquake records exhibit a non-monotonic pattern with the placement of isolators, as illustrated in Fig. 14(a). For the lower story (1st to 3rd stories) level isolator placement, the average optimum T_is value remains relatively constant (3.1 s), then dips to its minimum (2.8 s) around the 4th and 5th story level, before rising again gradually toward the top stories. The mean optimum T_is for the BI building is comparatively higher than any of the ISI configurations. This observation signifies that the mid-story isolation is required to be less flexible compared to lower or upper story isolation configurations. The error distribution indicates a noteworthy variability in the optimum T_is values across earthquakes.

Figure 14(b) indicates that the mean optimum ξ_is remains relatively stable across isolator locations; a slight decrease is observed when the isolator is placed up at the 3rd story, reaching a minimum of 13.7%. Further, a gradual reduction in the optimized ξ_is again noted from the 5th to the top-story isolation. The relatively flat trend of optimized ξ_is with isolator placement suggests that the energy dissipation requirement of the structure does not vary drastically with isolator location; however, considerable variability remains across earthquakes as reflected in the error distributions, especially when the isolator is placed at the upper story level.

The average optimum F₀ progressively increases as the isolator moves to the upper stories from the lower stories. This trend indicates that as the isolator is positioned higher in the ISI building, a stiffer isolator is required to control the seismic demands effectively. For the BI building, the mean optimum F₀ is lower than all the ISI configurations.

In summary, the optimum values of T_is and F₀ are highly sensitive to isolator placement in an ISI building, while the optimum value of ξ_is shows comparatively minimal variation across different story levels. However, significant variability is observed in all the optimized isolator parameter values across different seismic records. Collectively, these trends emphasize that the optimal design parameters of LRB are dependent upon its vertical placement within the building as well as the earthquake records, and the LRB parameters should be optimized accordingly, as a single fixed set of isolator parameters is not universally optimal across isolator locations and seismic inputs

6.2.6 Comparative seismic response control––optimized vs. non-optimized lead rubber bearing

Figure 15 presents a comparative evaluation of the average reduction in various structural responses achieved utilizing optimized and non-optimized LRB configurations corresponding to different elevations in the ISI building across earthquake records. The non-optimized LRB configuration in this context refers to isolator parameters that were originally optimized specifically for the base-isolated configuration (i.e., story 0). This LRB configuration is subsequently applied to other story levels without further modification or re-optimization. The resulting structural responses are then used to calculate the corresponding percentage reductions, which are then classified as the non-optimized response in this analysis. In contrast, the optimized responses denote the values obtained by optimizing the LRB parameters for each story level.

The results indicate that optimized inter-story isolation configurations consistently achieve better performance than non-optimized ones, particularly when isolators are placed in mid and upper stories. However, the difference between optimized and non-optimized response control is comparatively small for lower-story isolator placements (i.e., story 1 to 3). The most significant improvements from story-specific optimization are observed in roof displacement and base shear control at mid-story levels. Roof acceleration control also shows modest improvement due to optimization at the lower story levels. However, at higher stories, the optimized values become slightly lower than the non-optimized case, though the difference remains negligible. Figures 16–18 illustrate the comparison of the time histories between the fixed-based, non-optimized, and optimized ISI buildings across isolator locations under the earthquake RSN 171 for roof displacement, roof acceleration, and base shear, respectively.

Figure 19 compares the displacement experienced by the LRB when optimized and non-optimized parameters are used across different story levels. In both optimized and non-optimized cases, a general upward trend in isolator displacement is observed as the isolator is placed at higher story levels. However, the optimized isolator configurations consistently demonstrate reduced displacement demands, particularly when the isolator is placed at upper-story levels. This clearly signifies that by tuning the isolation parameters for the specific story levels, the isolation system is better equipped to manage relative deformations between the isolated upper-structure and fixed substructure of the building. The difference between the average optimized and non-optimized isolator displacement becomes significantly pronounced beyond mid-height stories, confirming that location-specific optimization noticeably enhances the performance of upper-story level ISI configuration in terms of reducing deformation demand on the isolator. Conversely, minor differences in isolator displacement demand are observed at the lower story level ISI configuration, indicating that optimized parameters for the BI configuration are reasonably effective near the base or lower third of the building, where the dynamic behavior of the ISI system remains similar to that of a BI system.

Overall, these findings emphasize the importance of story-specific optimization of LRB parameters in ISI buildings. It is critical for minimizing structural responses like displacement or base shear, as well as ensuring the mechanical feasibility and safety of the isolator itself by limiting excessive deformation.

6.3 Artificial neural network-based prediction of optimized lead rubber bearing parameters

6.3.1 Selection of the hidden neuron number

The appropriate selection of the hidden layer size, i.e., number of neurons in the hidden layer, plays a critical role in the performance and generalization ability of an ANN. A comprehensive parametric analysis was conducted by testing different neuron configurations ranging from 4 to 14 to identify the most effective architecture for the hidden layer. For each configuration, the network was trained and evaluated based on MSE and R values across training, validation, and test data sets.

Table 9 summarizes the performance of the ANN models for each neuron configuration. The results indicate that an increase in the number of neurons generally enhances training accuracy, as reflected by lower training MSE and higher training R values. However, superior training performance does not always guarantee better generalization to unseen data. Therefore, validation and testing metrics were critically analyzed to identify the optimal hidden layer configuration for the proposed ANN model.

The models with fewer hidden neurons (4–6) demonstrated relatively higher validation and test MSE, along with lower R values, suggesting underfitting. This indicates insufficient model capacity to capture the underlying data patterns effectively. As the neuron count increased from 7 to 11, a noticeable improvement was observed in both MSE and R values, which suggests enhanced learning capability and improved generalization. However, among the tested network configurations, the model comprising 11 neurons in the hidden layer demonstrated the most effective generalization capability. Specifically, this model achieved the lowest test MSE of 0.1146 and the highest test R-value of 0.9712, indicating strong predictive accuracy on unseen data. Additionally, it produced robust validation performance, with an MSE of 0.1198 and an R value of 0.9700, closely matching the test results. This close alignment of test and validation results implies good generalization and minimal overfitting. Conversely, the models with hidden layer configurations beyond 11 neurons (i.e., 12 to 14 neurons) exhibited signs of overfitting. The larger networks achieved lower training MSE and higher training R, which may initially seem advantageous. However, their validation and test MSEs increased, and R values slightly declined, indicating that the models memorized the training data at the expense of generalization to unseen inputs.

Therefore, the ANN architecture with 11 hidden neurons was selected in this study as the optimal configuration, as it provides high predictive accuracy, without significant evidence of underfitting or overfitting.

6.3.2 Performance evaluation of the artificial neural network model

A comprehensive analysis was carried out to assess the generalization and predictive performance of the developed ANN surrogate model. To address potential partition bias and verify the reliability of the used data, Monte Carlo cross-validation was conducted by repeatedly dividing the data set into training, validation, and testing subsets across 50 independent runs. The resulting distributions of RMSE, MAE, and R² across these repetitions are presented in the form of scatter plots (Fig. 20) and box plots (Fig. 21). The scatter plots indicate that the performance metrics consistently cluster around stable mean values across repeated trials. The boxplots further confirm that the metrics exhibit narrow IQRs with only a few outliers, demonstrating consistency and robustness of the ANN predictions under multiple random partitions. Collectively, these results demonstrate that the model maintains reliable performance independent of specific data splits. The detailed mean ± SD values and their corresponding 95% confidence intervals (CIs) for each data set are summarized in Table 10.

Beyond this, a detailed diagnostic evaluation was performed using the MATLAB Neural Network Toolbox, which utilizes statistical indicators and graphical diagnostics generated via the Neural Network Fitting Tool. The evaluation included key performance indicators such as MSE values across training, validation, and testing data sets, along with R-values. Additionally, graphical diagnostics, including error histogram, MSE convergence curve, regression plots, and training state indicators, were analyzed. These measures collectively offer a detailed assessment of learning dynamics, convergence behavior, prediction accuracy, and generalization ability of the developed ANN model.

The performance curve shown in Fig. 22 illustrates the MSE for the training, validation, and test data sets over epochs. The model achieved its lowest validation error at epoch 10, with an MSE of 0.1198, indicating optimal early stopping to prevent overfitting. Post-epoch 10, MSE slightly fluctuated but remained low, representing training stability and no overfitting. The MSE curves for the training, validation, and test sets demonstrate a consistent downward trend before stabilizing, indicating effective learning and convergence behavior of the network. The close alignment of these curves further implies robust generalization across all data sets, with no evidence of significant overfitting or underfitting.

The error histogram in Fig. 23 demonstrates the distribution of prediction errors (Target output−Predicted output) across the training, validation, and testing (blue, green, and red) phases. It shows that the majority of prediction errors are concentrated near zero, forming a sharp peak centered around the zero-error line. This tight clustering near zero error implies that the trained model has learned the underlying mapping between input and output variables effectively. Additionally, this symmetric distribution of error reflects the model’s high prediction accuracy and low bias of overestimation or underestimation across training, validation, and testing phases.

The R-values as presented in the regression plots (Fig. 24) were notably high, 0.9711 (training), 0.9700 (validation), 0.9712 (testing), and 0.9709 (overall), indicating a strong linear correlation among the predicted and actual target values. These results confirm that the network is highly effective in estimating the optimal LRB parameters from a given data set accurately. Additionally, the regression lines for all data subsets exhibit slopes close to unity with minimal offsets, further reinforcing the reliability and predictive accuracy of the developed model.

The training state parameters, as presented in Fig. 25, offer a deeper insight into the optimization process and convergence behavior of the model during training. The progressively decreasing gradient over epochs indicates that the model is approaching a local minimum, signifying effective optimization. Additionally, the small gradient of 0.011308 at epoch 16 suggests that further weight updates have minimal impact, confirming the network has effectively converged. The adaptive learning rate (Mu), which is associated with the LM algorithm, stabilized at 0.0001 by epoch 16 after an initial sharp drop from 10⁵, indicating a transition to a more efficient optimization phase with minimal regularization. The validation check count reached its threshold of 6 at epoch 16, indicating that the validation error failed to improve for six consecutive epochs, triggering early stopping to prevent overfitting and preserve the generalization capability of the model.

The predictive capability of the developed ANN surrogate model was validated using an independent data set comprising unseen earthquake records that were not utilized during model training, validation, or testing. The regression plot shown in Fig. 26(a) illustrates a high correlation (R = 0.9663) between the predicted and target optimal LRB configurations for the unseen earthquake scenarios. This high R value indicates that the developed surrogate model accurately reproduces the nonlinear mapping between seismic input features and optimal LRB parameters, even for earthquake data not encountered during model development. In addition, a comparative summary of performance metrics across seen test data, independent unseen data, and Monte Carlo cross validation (MCCV) trials is presented in Fig. 26(b). The close agreement between the seen and unseen data sets further confirms that the model’s predictive accuracy remains unchanged when applied to unseen data. The absence of any significant deviation or bias between seen and unseen data confirms that the model generalizes effectively to new seismic scenarios.

In summary, the trained ANN model is highly effective in predicting the optimal LRB parameters based on input variables. The model exhibited low error, high correlation of predicted and target values across both known and unseen data, tightly clustered prediction errors, and smooth convergence behavior across all the data sets, suggesting the robustness of the network in identifying the complex relationships between input and target output variables, with strong generalization capacity and minimal risk of overfitting.

6.3.3 Performance of the artificial neural network model surrogate model across isolator placements

To further evaluate the robustness of the developed ANN model, its predictive accuracy was examined separately across each individual story level. The model was tested using the test data sets corresponding to the respective story levels, and the regression plots for each story level are shown in Fig. 27. In addition, the story-specific performance metrics (R², RMSE, MAE) are summarized in Fig. 28. The results demonstrate that the model exhibits high accuracy and exceptional consistency across all ten stories. The story-level metrics cluster tightly around the overall model performance, with R² values ranging narrowly between 0.9388 and 0.9487. Similarly, the error values show minimal variation across story levels. The RMSE values range from 0.3316 to 0.3414, and the MAE values range from 0.167 to 0.188. These deviations of the performance indicators are marginal, and more importantly, do not exhibit any systematic trend with story elevation, indicating that the developed model generalizes well, regardless of whether the isolator is placed at lower, mid, or upper stories. This uniform performance suggests that the ANN surrogate model successfully captures the relationships between input features and optimal LRB parameters without being biased toward specific isolator elevations.

6.3.4 Evaluation of the relative importance of input features

To enhance the interpretability of the ANN surrogate model, Garson’s algorithm was employed to quantify the relative contribution of each input feature to the model predictions. The computed Garson’s RI values, presented in Fig. 29, provide valuable insight into the internal decision process of the trained network.

The results indicate that effective duration (D_eff), PGA, and isolator location (L_iso) are the most influential parameters, contributing 12.9%, 12.4%, and 11.9%, respectively, to the ANN predictions. This implies that both the ground motion intensity and placement dynamics of the isolator significantly govern the optimal LRB configuration. Other ground motion characteristics, such as PGD and PGV, also exhibit a notable influence, highlighting the sensitivity of the model to ground motion characteristics.

Other input features, including the spectral acceleration (S_a,1), moment magnitude (M_w), and source-to-site distance (R_jb), exhibit relatively moderate importance. The dominant frequency ratio (f_D) and the arias intensity (I_A) of the earthquake records contribute modestly but remain relevant, as they capture the frequency content and energy potential of the ground motion, which influence the structural responses of the ISI building.

Overall, these findings align with engineering intuition: the surrogate model appropriately emphasizes isolator placement and excitation-intensity-related input features as dominant predictors, while assigning lesser weight to broader ground motion characteristics. This weight-based interpretability thus confirms that the internal reasoning or the decision process of the surrogate model is physically consistent and not purely data-driven, thereby enhancing its trustworthiness and credibility.

6.3.5 Computational efficiency: Particle swarm optimization framework vs. artificial neural network prediction

The proposed multi-objective optimization framework based on PSO requires extensive computational effort. To obtain a particular optimal configuration of LRB, an exhaustive evaluation of 11665 parameter combinations (as detailed in Subsection 5.3) is required. For each parameter combination, the structural response to be optimized is computed through time-history analysis, with an average runtime of 1 s per evaluation. In addition, the PSO search needs an average of 2.5 s to complete the optimization cycle, and an average computational cost of 1 s is considered for the WLC calculation to evaluate the objective function. This results in a computational demand of

{\mathit{T}}_{\mathrm{o}\mathrm{p}\mathrm{t}}=(11665\times 1\mathrm{s})\mathit{+}1\mathrm{s}\mathit{+}2.5\mathrm{s}\mathit{=}11665.5\mathrm{s}\approx 3.2\mathrm{h}\mathrm{r}\mathrm{s}.

In sharp contrast, the trained ANN surrogate model is capable of predicting the optimal LRB configuration in approximately 1 to 1.5 s, completely bypassing the need for repetitive structural response evaluations. This corresponds to a computational cost saving of nearly four orders of magnitude relative to the full PSO-based framework. Such a drastic reduction highlights the substantial computational advantage of the ANN surrogate model for the rapid and efficient design of LRB in ISI buildings, thereby enhancing its suitability for real-world applications.

{\mathrm{T}}_{\mathrm{A}\mathrm{N}\mathrm{N}}=1\mathrm{t}\mathrm{o}1.5\mathrm{s}.

6.4 Practical implications and scalability of the proposed framework

The computational efficiency of the proposed ML-assisted optimization framework opens up avenues for its use in real-time or near-real-time design assistance. Since the trained ANN surrogate model accurately predicts optimal LRB configurations within seconds, practicing engineers may employ it as a rapid design assistance tool during preliminary design stages by exploring numerous design scenarios, such as varying isolator placement, seismic hazard characteristics, or even structural configurations. A step ahead, given its near-instantaneous prediction capability, the trained surrogate model could be embedded into a user-friendly software tool or mobile application to assist engineers in the optimal design and retrofitting of ISI buildings. Through a simple graphical interface, engineers would be able to specify the desired isolator placement story, site-specific ground motion data, and other relevant details, and the underlying model proposed in this study would then return optimal LRB parameters in real-time. Such an implementation would enable practitioners to rapidly evaluate design alternatives, explore different isolator configurations, and obtain near-real-time feedback, without the need to perform repeated time-history analyses or complex optimization runs.

However, for the framework to be translated into real-world practice, some important extensions are necessary to capture the complexities of actual structures and site conditions. Although the proposed framework is implemented in a two-dimensional ten-story shear building with regular geometry, the underlying methodology is inherently scalable and adaptable. Extending it to buildings with more stories or irregular geometries requires, first, analysis and optimization considering a larger number of stories and irregular features such as setbacks, soft stories, or torsional irregularities, and then train ANN model on the new data set with additional input features such as building height or normalized story height, fundamental period, and metrics for plan or vertical irregularity.

Furthermore, for full-scale practical application, the framework must eventually address three-dimensional (3D) structural behavior and nonlinear soil–structure interaction (SSI). Extending the framework to 3D buildings must account for torsional modes, bidirectional ground motion inputs, bidirectional isolator coupling, etc., necessitating an expanded input parameter space for the ML model that includes torsional eccentricity and the directionality of the ground motion. Consideration of nonlinear SSI is essential to capture more realistic boundary conditions. However, SSI introduces additional complexity, requiring a more sophisticated numerical model that includes a nonlinear soil domain or simplified soil springs and dashpots. This, in turn, further increases the input features related to soil characteristics and foundation type. As the input feature space expands to accommodate additional structural and soil parameters, more sophisticated ANN architectures or advanced surrogate models may be required to effectively capture the increased complexity. Nevertheless, the fundamental concept of the proposed framework, which employs metaheuristic optimization to generate optimal LRB configurations for diverse structural scenarios and subsequently uses this data set to train an ML model, remains a robust strategy to deliver rapid seismic design assistance.

7 Conclusions

This study comprehensively analyzed the seismic performance of ISI buildings by evaluating the influence of isolator placement across story levels, identifying optimal LRB parameters for ISI buildings through PSO-based multi-objective optimization, and developing an ANN-driven surrogate model for rapid prediction of optimal LRB parameters. The key outcomes derived from this extensive analysis are summarized as follows.

1) The seismic performance of ISI buildings is strongly influenced by the vertical placement or location of the isolator within the building. The peak roof displacement control was observed with isolators installed at mid-story levels, achieving reductions of approximately 18%–20%. In contrast, the reduction in roof acceleration improved monotonically from mid-stories to upper stories, achieving up to 63% control at higher story level isolator placements. Base shear reduction was maximized when the isolator was placed at the lower stories and declined steadily as the isolator was moved higher. These trends underscore the importance of strategic isolator placement in ISI buildings based on the targeted performance objective or specific structural response to be controlled. The ISI system is therefore not a one-size-fits-all solution; rather, it is a flexible design strategy that can be tailored to meet specific structural and functional goals, where designers must carefully consider the operational requirements, occupancy type, and seismic performance criteria of the building to determine the suitable isolator location.

2) The displacement demand of the isolator increases progressively as it is placed at higher stories, particularly beyond mid-stories, signifying the necessity for careful consideration of isolator deformation capacity, especially in upper-story ISI configurations, to ensure structural integrity under seismic loading.

3) The significant variability in structural responses across different earthquake records highlights the importance of considering a wide range of seismic inputs in the design of a robust and reliable inter-story isolation system.

4) The developed multi-objective optimization model to identify optimal LRB design parameters exhibited excellent robustness and reliability, supported by its statistical adequacy. Furthermore, the PSO algorithm demonstrates computational efficiency across all optimization cases and consistently converges to identical optimal solutions in multiple independent runs, with negligible variability, thereby validating the stability of the proposed framework. In addition, a comparative assessment with two alternative MHAs (DE and TLBO) further reinforced the reliability of PSO.

5) The comparative evaluation between optimized and non-optimized LRB configurations across various isolator elevations clearly demonstrates that the story-specific optimum design of LRB leads to notable improvements of seismic performance in terms of roof displacement control, base shear control, and isolator deformation, particularly in mid-story and upper-story ISI configurations. In contrast, for the lower stories, improvements from story-specific optimization are insignificant, indicating that LRBs optimized for the BI system remain reasonably effective for lower-story level ISI configurations.

6) The optimum values of LRB parameters, particularly T_is and F₀, exhibit considerable sensitivity to the vertical location of the isolator in ISI buildings. The results indicated that the mid-story level isolator placements required shorter T_is compared to both lower and upper story ISI configurations, suggesting a less flexible isolator for the mid-story ISI configuration. In contrast, the optimized F₀ values gradually increase as the location of isolator placement moves upward, indicating that stiffer isolators are needed at higher elevations. However, the optimum ξ_is remains relatively stable across different story-level isolator placements, implying that energy dissipation demand does not vary significantly with the location of the isolator. Further, the significant variability observed in all the optimum LRB parameters across seismic inputs highlights the necessity of ground-motion-sensitive optimal design of LRB to ensure consistent isolator behavior under diverse seismic scenarios.

7) The optimal design of LRB in ISI buildings cannot therefore rely on a single and uniform set of design parameters. Instead, story-specific and earthquake-specific optimization is essential to achieve effective and reliable performance across different isolator locations and earthquake records. However, performing case-specific individual optimizations for each configuration can be highly time-consuming and computationally demanding for real-time applications.

8) The proposed ANN surrogate model demonstrated strong capability in predicting the optimal LRB parameters from the input features. The low MSE and high R across data sets, minimal error distribution, and stable convergence behavior collectively confirm the robustness, predictive accuracy, and generalization ability of the trained ANN model.

9) The trained network exhibited consistently high predictive accuracy across all isolator placements, with minimal variation in performance metric values. The absence of any systematic trend with story elevation further confirms that the model generalizes effectively, regardless of whether the isolator is positioned at lower, mid, or upper stories.

10) The feature importance analysis indicated that effective duration, PGA, and isolator location are the most influential input parameters, confirming that both the seismic excitation intensity and the elevation of isolator placement govern the optimal LRB configuration. This, in turn, signifies that the internal decision-making of the surrogate model aligns with physical understanding, thereby enhancing its credibility and suitability for engineering design applications.

11) The proposed ANN surrogate achieves a significant reduction in computational demand, predicting optimal LRB configurations within 1–1.5 s compared to 3.2 h required by the full optimization framework. This represents a cost saving of nearly four orders of magnitude, providing a highly efficient alternative to the time-consuming optimization process and enabling near-real-time estimation of the optimal LRB parameters in practical design scenarios for ISI buildings.

12) The proposed machine-learning-enhanced optimization framework establishes a versatile paradigm that can be extended well beyond the specific structural configuration presented in this study, offering a pathway for its application to a broad spectrum of complex civil structures.

8 Limitations and future scope

While this study successfully establishes an ML-enhanced optimization framework for inter-story isolated buildings, its scope is defined by several limitations that open avenues for future research and further development in this field.

The proposed framework considers a two-dimensional ten-story shear building with regular geometry to simplify the analysis and reduce computational effort. Although this configuration adequately captures the influence of story-specific isolator placement, its scalability to taller, irregular, or 3D structures remains an important direction of future studies. Incorporating structural irregularities, bi-directional seismic excitations, and SSI will enhance the surrogate model’s generalizability and applicability to realistic scenarios.

In terms of optimization, while the present study primarily focused on PSO and two alternative MHAs, which provided reliable convergence, exploring more advanced algorithms such as the Covariance Matrix Adaptation Evolution Strategy (CMA-ES), Asymmetric Genetic Algorithm (AGA) [131], or other advanced hybrid MHAs may further enhance optimization robustness and efficiency.

Regarding surrogate modeling, although the proposed ANN model exhibits excellent predictive accuracy, future research should include a comparative assessment with other existing surrogates and reduced order models such as RF, MARS, GBM, XGBoost, SVM, GPR, Response Surface Method, etc. Additionally, emerging SciML approaches, including PINNs and neural operators such as VINO and MHNO, present a promising future direction for integrating physical laws directly within data-driven frameworks.

Another limitation is that the present study employed a deterministic structural model, neglecting the parameter uncertainties, such as material properties, which may influence the robustness of the optimization results. In future studies, incorporating uncertainty quantification and sensitivity analysis, for instance, through Monte Carlo estimation based on the Sobol’s indices [75,76], may help evaluate the influence of parameter variability on system performance.

Finally, although this study has employed Garson’s algorithm, which provides a preliminary level of interpretability for ANN decisions, in the future, the inclusion of advanced XAI tools, such as SHAP and LIME, can further enhance the explainability and interpretability of the surrogate model decision.

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