PSI - Issue 78

Michele Mattiacci et al. / Procedia Structural Integrity 78 (2026) 1159–1166

1162

2.3. Selection of Competing Model Classes for Structural Identification

The second proposed approach is based on the theory of model class selection and aims to construct a digital twin comprising a population of competing structural models Herna´ndez-Gonza´lez and Garc´ıa-Mac´ıas (2024). Each model class represents a specific damage scenario relevant to the typology of the monitored masonry structure. Within a continuous SHM framework, the digital twin enables structural identification by iteratively acquiring in-field data and calibrating the digital models to reflect the observed behavior. To ensure computational e ffi ciency, surrogate models are used in place of full FE simulations. These surrogates are trained and validated to replicate the input–output behavior of their FE counterparts, significantly reducing the cost associated with repeated evaluations during the calibration process. Once updated using experimental data, each model class is assessed using a statistical model selection criterion to determine the most likely structural scenario and estimate the corresponding damage severity. The pivotal aspect of this framework is the model updating Za´rate and Caicedo (2008) phase where it stands for the process of calibrating a set of model parameters x = [ x 1 , x 2 , . . . , x m ] T within a physically admissible design space D = { x ∈ R m | a i ≤ x i ≤ b i } , where [ a i , b i ] denotes the allowable range for each parameter x i . The objective is to minimize the discrepancy between experimentally observed data values and those predicted by the numerical model. This inverse problem is formulated as a constrained optimization problem as x ∗ = argmin x ∈D J ( x ), where J ( x ) is an objective function quantifying the mismatch between observed and simulated monitoring data. Due to uncertainties arising from modeling assumptions, material properties, and boundary conditions, J ( x ) is generally non-convex, and global optimization techniques are recommended for its solution. After model updating, a model selection criterion is used to determine the most plausible model class among the candidates. Given the diversity of potential damage scenarios in real structures, a one-size-fits-all model is impractical. Instead, models representing specific failure mechanisms are grouped into classes denoted with MC i . The Bayesian Information Criterion (BIC) Neath and Cavanaugh (2012) is adopted in this work due to its balance between accuracy and complexity. For a model class MC i , the BIC is defined as BIC i = 2ln( L i ) + c ln( l ), where L i = p ( y | x , MC i ) is the maximized likelihood of the observed data y given the calibrated parameters x , c = m is the number of parameters estimated, namely it describes the model complexity, and l is the number of sensors. The BIC penalizes models that are either over-complex or poorly predictive, therefore the model class with the lowest BIC value is selected as the optimal representation of the identified damage mechanism. The proposed methodology leverages time-dependent MLP regressors integrated within the framework of nonlinear cointegration theory for novelty detection in strain time series, as schematically depicted in Figure 1. This approach is tailored to the SHM of masonry structures instrumented with a network of strain sensors. Due to the influence of environmental variability, strain measurements often display non-stationary behavior, potentially concealing structural changes induced by damage-related load redistribution. To overcome these challenges, the strategy first involves assessing the stationarity and the integration order of each strain time series using the Augmented Dickey-Fuller (ADF) test. Time series exhibiting the same integration order are grouped accordingly. Within each group, nonlinear regressions are carried out using time-dependent MLP networks to extract damage-sensitive features that are robust against environmental fluctuations. A dedicated MLP model is trained for each sensor, accounting for the specific characteristics of the target as well as input time series. The architecture of each MLP is optimized through a grid search procedure with k -fold cross-validation, ensuring generalizability. Formally, let n denote the number of nonstationary time series of strain data sharing the same integration order I ( d ), collected from the sensor network. For each sensor i , the nonlinear regression is defined as ϵ i ( t ) = g i ϵ − i ( t ) , for i = 1 , 2 , . . . , n , where ϵ i ( t ) represents the predicted strain for the i -th sensor, and ϵ − i ( t ) = { ϵ j ( t ) ∈ R | j i } denotes the set of explanatory variables excluding the i -th component. The function g i ( · ) is modeled as a time-dependent MLP regressor trained using the remaining n − 1 time series. A proper selection of the training period is critical for accurate modeling. This period must include su ffi cient environmental variability while ensuring that the structure 3. Neural Network-Driven Monitoring via Nonlinear Cointegration 2.4. Selection Criterion