Utility-Driven Adaptive Model Selection for Digital Twinning

Utility-Driven Adaptive Model Selection for Digital Twinning Konstantinos Vlachasa,∗, Antonios Kamariotisa, Eleni Chatzia aDepartment of Civil, Environmental, and Geomatic Engineering, ETH Zürich, Zürich, Switzerland Abstract Digital twins have become central to modern engineering by providing interconnected virtual representations that mirror the behavior of physical assets. Their effectiveness, however, depends critically on the fidelity and efficiency of the underlying computational models used for decision support. This work introduces an adaptive Bayesian framework for quantifying and optimizing the fidelity, efficiency, and overall utility of such virtual representations when approximating system responses in decision-oriented contexts. By integrating reduced-order models (ROMs) within a Bayesian inference and decision-theoretic setting, the framework identifies, at any point in time, the lowest-cost model capable of delivering the required predictive accuracy across all quantities of interest, while rigorously accounting for the uncertainty associated with each model resolution. The approach maximizes an expected-utility function that balances two competing attributes: (i) precision, reflecting the effect of overor under-estimating target quantities on decision quality, and (ii) computational efficiency, ensuring feasibility for real-time inference. Leveraging continuously assimilated monitoring data, the framework performs this model selection recursively, enabling the virtual twin to evolve in tandem with the physical system. The resulting methodology supports automated decisions on whether to retain the current model, switch to an alternative resolution, or trigger retraining—thus establishing a pathway toward adaptive, trustworthy digital twins for engineering decision support. Keywords: Digital Twinning, Reduced Order Models (ROMs), Bayesian Model Updating, Bayesian Inference, uncertainty quantification 1. Introduction Infrastructure assets are inherently complex, featuring large-scale models, nonlinear dynamics, and considerable2 uncertainty. As next-generation systems come to the forefront, there is a clear shift toward digital and hybrid twin representations that provide virtual counterparts of physical assets [1, 2]. The process of creating these representa-4 tions—commonly referred to as virtualization or twinning—aims to replicate and interact with the behavior of real systems [3, 4]. To ensure such virtual assets operate effectively, they must rely on computational models that are both6 accurate and efficient [5]. Without this balance, real-time applications central to Operations and Management (O&M), Structural Health Monitoring (SHM), and decision support become infeasible [6, 7].8 In the existing literature, numerous frameworks aim to develop low-dimensional models or numerical surrogates that enable rapid computation while retaining the essential physics of high-fidelity systems [8, 9, 10]. Broadly, these10 approaches fall into two categories: data-driven methods and physics-aware methodologies. Data-driven models are particularly effective for systems with complex or chaotic dynamics [11, 12]. However, they often rely on ad hoc12 training procedures that capture only limited input–output relationships and tend to degrade in performance under environmental or parametric variability [13]. Moreover, such methods typically struggle to extrapolate across time,14 input conditions, or spatial discretizations of the target system [14]. As a result, many data-driven surrogates remain problem-specific, mesh-dependent, and unable to generalize beyond the variable used for training [15, 16]. These16 limitations restrict their practicality for developing actionable digital models suitable for integration into higher-level O&M, SHM, or decision-making frameworks.18 ∗Corresponding author Email address: [email protected] (Konstantinos Vlachas ) This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=5682281 Preprint not peer reviewed In contrast, physics-based and physics-aware approaches preserve the internal structure of the dynamical system and embed physical knowledge through explicit constraints or inductive biases within the virtual model [17, 18].20 This improves interpretability, stability, and accuracy in extrapolation and generalization tasks, thereby enhancing the applicability and reliability of the resulting digital representations for higher-level engineering workflows [19]. The ef-22 fectiveness of such strategies has been demonstrated in several studies, including [20] and its extension in [21], where reduced-order models (ROMs) serve as forward simulators in inverse problem settings for fault detection and input24 estimation in real time. Similarly, [22] employs stochastic ROMs for damage detection in a mock-up aircraft wing, while [23] showcases the integration of ROMs with the Unscented Kalman Filter for system identification and uncer-26 tainty quantification under environmental and parametric variability. Beyond model-based inference, related advances using physics-informed surrogates have also emerged [24, 25, 26, 27, 28]. In this work, we focus on model-based28 methodologies, which provide a structured, physics-grounded framework to support engineering decisions across the operation and maintenance (O&M) life-cycle of infrastructure assets [29, 30]. The core of such frameworks lies in30 numerical models that condense complex systems into computationally tractable forms while retaining their essential dynamic behavior, variability, and uncertainty.32 However, the usefulness of virtual models ultimately depends on balancing computational cost against model fidelity, particularly when their outputs support high-stakes engineering decisions. Developing digital representations34 that maintain this balance is inherently challenging, as achieving consistent efficiency without compromising accuracy becomes increasingly difficult for evolving and data-rich systems [31]. In this work, we address this challenge through36 an adaptive, data-informed framework that leverages continuous monitoring information to dynamically select the most suitable model using Bayesian decision theory. By evaluating a hierarchy of models with different levels of38 refinement and fidelity, the proposed approach allows the virtual representation to adapt and evolve in tandem with the physical system.40 Within this context, a number of hybrid methodologies have been proposed to develop representations that adapt to changes in the system’s dynamics. For instance, several studies [32, 33, 34, 35] employ physics-based indicators42 to trigger model retraining or refinement steps, often supported by data-driven enrichment of the projection basis for problems such as fracture dynamics. Other approaches, such as [36], introduce basis enrichment through de-44 composition, splitting selected reduced-basis vectors into disjoint components and employing vector-space sieving or refinement tree techniques to adapt the reduced-order space. Extensions of these ideas address online model updates,46 either through basis compression [37] or additive low-rank updates of reduced spaces [38]. Finally, active learning strategies [39, 40] and probabilistic schemes based on Gaussian Processes and Bayesian networks [41, 42, 43, 44, 45]48 have been proposed to assimilate sensor data and build adaptive, data-driven surrogates capable of refining their predictions as new information becomes available.50 However, existing approaches only partially address the challenge of creating dynamically evolving digital assets, as model retraining or refinement is typically triggered by response-based error indicators. Such schemes neglect the52 overall utility of the virtual representation and cannot explicitly assess the Value of Information (VoI)—that is, the benefit gained from knowledge that improves decision-making [46, 47]. In contrast, this work introduces a Bayesian54 decision-theoretic framework that quantifies the expected fidelity and utility of available virtual models when used for generalization or extrapolation. The proposed approach tackles a key challenge in the virtualization process: the56 recursive selection of the lowest-cost model capable of achieving the required predictive accuracy for a given engineering context [4]. In doing so, it establishes a pathway toward fit-for-purpose digital representations that maximize58 decision value and reliability throughout the asset’s life-cycle [48]. In this context, selecting the optimal model resolution for predicting the response of engineered systems from60 a pool of candidate models represents a persistent and critical challenge [49, 50, 51]. These candidate models—or virtual representations—can range from simplified analytical formulations to high-fidelity finite element models, each62 offering a different trade-offbetween accuracy, computational cost, and uncertainty [23]. Effective model selection enables reliable and efficient decision-making for system design and operation by exploiting available monitoring data64 to continuously reduce uncertainty [52]. Through Bayesian model updating (BMU) and the recursive assimilation of new information [53, 54], one can estimate the posterior uncertainty associated with each candidate model or66 resolution, thereby supporting data-informed and transparent decisions. Consequently, the model selection problem can be naturally formulated as a task of decision-making under uncertainty [55]. The ultimate goal is to identify the68 fit-for-purpose model that offers the best balance between accuracy in recovering quantities of interest (QoIs) and computational efficiency. To achieve this, the proposed framework approaches the model selection task through the70 2 This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=5682281 Preprint not peer reviewed lens of Bayesian decision theory, enabling rational and adaptive choices among competing virtual representations [56]. The proposed framework begins by deriving hybrid model representations of varying fidelity levels. Leverag-72 ing the extrapolation and generalization capabilities of Reduced-Order Models (ROMs) [57], these are formulated alongside a Full-Order Model (FOM) following [58, 59], with a generic parametrization that accounts for environ-74 mental variability, operational changes, and evolving phenomena such as damage or deterioration. The ROMs are then embedded in a sequential Bayesian inference framework that quantifies the posterior uncertainty of each model76 resolution and guides the selection of the most suitable representation [60]. By recursively assimilating monitoring data, the framework adaptively tracks the evolving state of the physical asset and selects the model that maximizes78 expected utility, balancing prediction accuracy and computational efficiency for real-time applications. As illustrated in Figure 1, the method autonomously decides whether to retain the current model, switch to another, or trigger80 retraining—thus enabling adaptive, uncertainty-aware digital twins tailored to the decision context. Continuous monitoring Bayesian Inference (or any other equivalent framework) Mopt|d=argmax {M1,M2,...,Mn} Ep|d,Mi[U] Pretrained parametric resolutions Mi(p) Hyper ROM ROM FOM Utility function U(fidelity,efficiency,p|Mi) Optimal model at any given time ˆ p→Model Updating Response QoIs Expected Utility Intermediate estimates Highest efficiency Lowest fidelity Lowest efficiency Highest fidelity Figure 1: Graphical abstract of the proposed adaptive digital-twin framework. Reduced-order and full-order models of varying fidelity are integrated within a Bayesian decision-theoretic scheme that recursively assimilates monitoring data, updates parameters, and selects the model that maximizes expected utility, ensuring an adaptive and efficient representation of the physical asset. This paper is organized as follows: In section 2, the reduced-order modeling methodology employed to derive the82 pretrained model resolutions in Figure 1 is presented in short. Section 3 outlines a Bayesian decision-theoretic approach to model selection. In section 4, we describe the numerical case studies used for validation, and the numerical84 results that highlight the efficiency and effectiveness of the proposed framework. Lastly, section 5 concludes the paper by summarizing our contributions, offering insights, and suggesting directions for future research.86 2. Parametric Reduced Order Models The first component of the proposed framework involves developing parametric reduced-order models (ROMs)88 through physics-based reduction, yielding low-dimensional surrogates of the full-order model (FOM). These ROMs, illustrated in Figure 1, balance computational speed with predictive accuracy under varying conditions, making them90 well-suited for downstream tasks in Structural Health Monitoring (SHM) and Prognostics and Health Management (PHM) [61]. The following subsections outline the problem formulation via the governing nonlinear equations of92 motion and describe the employed ROM methodology. 3 This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=5682281 Preprint not peer reviewed 2.1. Problem Statement94 We assume availability of a Full Order Model (FOM), which corresponds to a Finite Element (FE) model that is suitable for nonlinear structural dynamics simulations. The corresponding dynamical system is dependent on the96 input vector p=[p1, ..., pk]T∈Ω⊂Rk, which captures all systemand excitation-relevant parameters. Thus, the response is described by the following set of governing equations:98 M¨ u(t)+g(u(t),˙ u(t),p)=F(t,p),(1) where u(t)∈Rndenotes the response in terms of displacements, M∈Rn×nthe mass matrix, and F(t,p)∈Rnthe induced excitation. For simplicity, the parametric dependence of the mass matrix is omitted. Nonlinear behavior100 is captured through the restoring force term g!(u(t),˙ u(t),p)∈Rn, which may represent effects such as plasticity, hysteresis, or interface nonlinearities, depending on the system response and parameter vector p. The finite element102 (FE) model serving as the full-order model (FOM) is a discretized form of Equation (1). Its full-order dimension n defines the size of the coordinate space—and thus the total number of degrees of freedom—governing the model’s104 computational cost, as numerical operations scale with n. 2.2. Projection-based model order reduction106 The reduction approach employed herein begins with a Galerkin projection scheme, with more general extensions such as the Petrov–Galerkin method discussed in [62, 63]. As outlined in section 1, a physics-based reduc-108 tion is adopted to enhance interpretability and applicability within higher-level Structural Health Monitoring (SHM) frameworks. This approach assumes that the system’s dynamics—i.e., the solution of Equation (1)—reside in a low-110 dimensional subspace of rank r≪n, where nis the full-order dimension. Accordingly, the system response can be expressed as:112 u(p)≈V(p)q(2) where V∈RN×rrepresents the projection basis that expresses the aforementioned subspace of the Reduced Order Model (ROM) and q∈Rris the low-order coordinate vector. Via substitution of uinto Equation (1) and after114 multiplying the governing set with uT, thus performing a Galerkin projection, the following can be derived: ˜ M¨ q(t)+˜ g(¨ q,˙ q,p)=˜ F(p,t)(3) where ˜ M=VTMV,˜ g=VTgand ˜ F=VTF. Several techniques exist for constructing the projection basis V[8]. In116 this work, the Proper Orthogonal Decomposition (POD) is employed, using a set of FOM simulations over a training parameter space to assemble the solution snapshots as:118 ˆ S=hˆ Up1ˆ Up2. . . ˆ UpNs i (4) where ˆ Upi∈RN×Ntcontains the displacement time history for a given parametric realization pi, henceforth termed as a snapshot, and, as a result, ˆ S∈RN×(Nt×Ns)is termed the snapshot matrix. The variable Ntrepresents the number120 of simulation (time) steps and Nsis the total number of snapshots. Via Singular Value Decomposition (SVD) of ˆ Sthe ROM projection basis can be assembled as follows122 ˆ S=ΛΣZT(5) and after truncating Λ: V=hΛ1Λ2. . . Λri(6) where Λiis the icolumn of matrix Λ, loosely defined as a Proper Orthogonal Decomposition (POD) mode. Since the124 low-order dimension of Equation (2) is defined as r, the above truncation is applied to obtain the first rorthonormal components of V. To define r, a suitable error measure is employed based on the singular value decay [64, 65].126 4 This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=5682281 Preprint not peer reviewed 2.3. VpROM: A conditional Variational Autoencoder (cVAE)-boosted ROM The dynamic behavior of the system, governed by the equations in Equation (1), is highly dependent on the128 parameter vector p. As a result, attempting a projection-based reduction with a single basis, as described in Equation (2), might necessitate a large number of modes, resulting in a prohibitively large dimension rand a ROM that130 proves inefficient or impractical. To handle parametric variability and enable response inference across different operating conditions, a common strategy is to construct a set of local projection bases Vi, each derived from FOM132 snapshots ˆ U(pi) corresponding to a specific parameter realization pi. Subsequent interpolation [66] or clustering [65] methods can then approximate system responses at unseen parameter values [67]. Building upon prior work by the134 authors [68, 58, 59], the proposed framework employs a conditional Variational Autoencoder (cVAE) as a nonlinear generative model to learn a smooth, generalizable mapping between parameters and basis representations. The re-136 sulting VpROM not only accelerates response prediction but also quantifies confidence in its estimates, enhancing its value for SHM applications. Further details on the framework’s components can be found in [69] and related138 references. To address parametric variability at the ROM level, the two-stage process introduced in [59] is used. First, a140 collection of local bases is obtained as described and the following system is solved in the least-squares sense: Vipi=Vglobal ∗Xipi(7) where Vi∈Rn×ris an instance of the collection of local bases, Vglobal ∈Rnטrcaptures the dynamics across the entire142 domain and Xi∈R˜r×ris a coefficient matrix. ˜rsignifies the total number of truncated odes retained on Vglobal and can be computed similarly to r. Second, interpolation is performed on the coefficient matrices Xi. These comprise a144 reduced size (˜r≪r), thus removing any dependency on the FOM dimension noffering additional efficiency. After interpolating the coefficient matrices Xi, the corresponding local ROM basis Vcan be obtained for any validation146 parametric sample. In [59] the local bases Vare projected first to the tangent space of the proper Grassmannian manifold, where the interpolation operations performed will output a local basis that retains key properties like or-148 thogonality [70, 71]. Alternatively, a re-orthogonalization step can be performed after interpolating the coefficient matrices and recovering the projection basis.150 This work builds upon the framework introduced in [58], which employs a conditional Variational Autoencoder (cVAE) as a generative model capable of recursively inferring the local basis from features extracted from monitoring152 data. The cVAE acts as a nonlinear generator and approximator of the coefficient matrices X, and consequently of the local projection subspaces Vand associated ROMs. Parametric variability is incorporated directly at the ROM level154 by conditioning the cVAE on known system parameters or on features derived from measured responses. In previous formulations [69, 58], the VpROM was designed in a generalized manner, assuming no prior knowledge of the true156 parameter vector pduring training or deployment; instead, the cVAE infers the local basis from observed response features. When the parameters pare known, they can directly serve as conditioning inputs. Finally, the probabilistic158 nature of the cVAE allows the VpROM to quantify prediction uncertainty: the latent space defines a learned probability distribution whose sampling enables propagation of uncertainty through the ROM, yielding confidence bounds and160 error estimates on the predicted responses. The overall framework is illustrated in Figure 2. The conditioning features Ware concatenated with the input X162 and the latent variables Z. Formally, the encoder learns the conditional distribution qθ(Z|X,W), while the decoder reconstructs the data via pϕ(X|Z,W). For simplicity, we set W=p, since parameter inference is handled separately164 within the Bayesian inference framework shown in Figure 1. Accordingly, the system variability in Equation (1) is represented by p, and the mapping from pto the local bases Vin Equation (3) is learned through the relationship166 between pand the reduced coefficient matrices Xin Equation (7). Once trained, the conditional VAE serves as a generative model capable of sampling the reduced basis coefficients168 Xidefined in Equation (7). Given an inferred parameter realization ˆ pfrom Bayesian inference, samples drawn from the latent distribution are decoded to generate the corresponding Xvalues. In this work, a diagonal Gaussian prior is170 adopted for the latent space, as depicted in Figure 3. 2.4. Uncertainty Bounds via the Unscented Transform172 Once the decoder is trained, predictions are obtained by sampling the inferred variational distribution of the latent space and decoding these samples. A realization of the random variable ϵis first drawn and transformed into Z174 5 This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=5682281 Preprint not peer reviewed X W +Encoder qθ(Z|X,W) µθ σθ ε ⊙ +Z W N(0,1) +Decoder pϕ(ˆ X|Z,W)ˆ X Figure 2: The architecture of the employed cVAE. The conditioning features Ware injected via concatenation with the input vector Xand the latent space Z. µθ σθ ε ⊙ +Z ˆ p Obtained via Bayesian Inference + N(0,1) Decoder pϕ(ˆ X|Z,p)ˆ Xˆ V ˆ V=Vglobal ∗ˆ X Figure 3: Architecture of the cVAE in basis generation mode: The prior distribution ϵis sampled, and the latent vectors are taken after concatenating with the (inferred) vector ˆ p. Dimensionality serves only illustrative purposes. using the deterministic mean µθand standard deviation σθinferred by the encoder. The latent vector Zis then concatenated with the inferred parameters ˆ pand passed through the decoder to generate samples from pϕ(X|Z,p)—each176 representing a possible realization of the reduced coefficients X. By repeating this process, the mean and variance of the estimated coefficients and projection bases can be evaluated. Subsequent parallel ROM simulations using these178 sampled bases, as defined in Equation (3), enable uncertainty propagation from the latent space to the output response. The resulting ensemble statistics yield numerical error bounds that quantify prediction uncertainty, as demonstrated180 in [68, 58]. The sampling procedure described above can be computationally expensive, as hundreds of samples may be re-182 quired to capture the system’s response uncertainty [58]. To improve efficiency, we adopt the Unscented Transform (UT) [72, 73], which exploits the Gaussian nature of the latent distributions to generate a deterministic set of sigma184 points instead of numerous random samples. These sigma points are propagated through the decoder, and the resulting outputs are combined using predefined weights that preserve the mean and covariance of the original Gaussian186 distribution, yielding accurate and efficient uncertainty estimates. 2.5. Hyper-reduction188 Hyper-reduction provides a second-level approximation that alleviates the computational cost of updating and reconstructing the nonlinear term ˜ gin Equation (3) [74]. Here, we employ the Energy Conserving Mesh Sampling and190 Weighting (ECSW) method [75], a well-established physics-based technique. Comprehensive discussions of ECSW 6 This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=5682281 Preprint not peer reviewed and related approaches can be found in [76, 77, 78]. In brief, ECSW selects a subset of finite elements from the192 FE discretization—used here as the FOM—by solving a nonlinear optimization problem that ensures the weighted projections of the nonlinear terms accurately approximate the total internal work. Once the optimal subset and corre-194 sponding weights are determined, a Hyper-ROM is obtained, as depicted in Figure 1, offering a substantial reduction in computational cost. Among the available model resolutions, the Hyper-ROM achieves the highest efficiency, the196 FOM the highest fidelity, and standard ROMs occupy the intermediate range. 3. A Bayesian decision-theoretic approach to model selection198 The framework outlined in the previous section can be employed to generate different model resolutions to assist with decision-making and SHM tasks. Specifically, we have highlighted three resolutions in Figure 1, each one200 corresponding to a trade-offbetween computational efficiency and fidelity: a) The full-order FE model of the system, termed FOM, which offers the highest possible precision, b) the ROM that strike a balance between precision and202 efficiency, and c) the Hyper ROM, which corresponds to the developed ROM equipped with hyper-reduction, and potentially offers the highest efficiency. Furthermore, the proposed framework allows for the development of virtual204 models conditioned on system dependencies, thus capturing parametric variability. As physical infrastructure assets evolve over time, a recursive challenge emerges: determining which parametric206 instance and model resolution provide the best balance between computational cost and predictive accuracy. This model selection task, conditioned on incoming response data, can be naturally formulated as a decision-making under208 uncertainty (DMUU) problem, where the optimal choice is the model that maximizes the expected utility [79, 80]. In decision theory, utility serves as a formal measure to evaluate the relative desirability of alternative actions. It210 is defined through an objective function that maps the attributes of a decision to a numerical value representing its overall benefit. In the present context, two primary attributes govern the utility of a model:212 1. Model fidelity, in terms of recovering target Quantities of Interest (QoIs), and 2. Model cost, in terms of computational complexity and runtime.214 These attributes are central to the problem, as virtual models used in decision-making and SHM applications inherently balance precision against computational cost. They are also competing by nature: higher precision typi-216 cally entails higher cost. Bayesian decision theory [56, 79] offers a rigorous framework for solving decision-making under uncertainty (DMUU) problems, where information becomes available recursively and progressively reduces218 uncertainty in decision-making. Let θdenote the random vector of uncertain model parameters. For simplicity, we assume these coincide with220 the modeling parameters of the FOM and ROMs in Eqs. 1, 3, and Figure 3, i.e., θ=ˆ p. This assumption is not restrictive—the framework remains valid provided θ⊆p. Hence, the model parametrization introduced in section 2222 is kept intentionally general, as the parameter vector directly governs model selection. In turn, a prior probabilistic model πpr(ˆ p) needs to be assigned. The incoming recorded signals, collectively denoted as monitoring data d, can224 be leveraged to perform inference and Bayesian model updating (BMU), which outputs the posterior distribution of ˆ pgiven d, denoted by πpos(ˆ p|d). The optimal model to be selected Mopt conditional on the monitoring data dcan be226 then obtained as: Mopt|d=argmax {M1,M2,...,Mn} Eˆ p|d,Mi[Ui],(8) where Ui=U(fidelity|Mi,efficiency|Mi,ˆ p|Mi) where Uis the considered utility function that relates the uncertain parameter vector ˆ pand models M, thus mapping228 possible outcomes to their utility as illustrated in Figure 4. The expectation Ein Equation (8) is evaluated with respect to the posterior distribution πpos(ˆ p|y,Mi). The possible decisions are summarized in the set {M1,M2,...,Mk}in the230 case of kcompeting models, where Mirepresents the decision to employ the i-th model for inferring the parameter and recovering the dynamic behavior of the engineered system of interest.232 As discussed, two competing attributes define the utility function Uin this work: model fidelity and computational efficiency. Illustrative examples of the resulting utility branches are shown in Figure 4. In Figure 4a the considered234 utility curve with respect to the ability of the model to recover the quantity of interest is depicted, also accounting for 7 This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=5682281 Preprint not peer reviewed −20 −10 0 10 20 Measured QoI - Prediction (%) 0.0 0.2 0.4 0.6 0.8 1.0 Utility U Underestimation Overestimation (a) Utility Ufas a function of the model fidelity. 0.1 1 10 Evaluation time (secs) 0.0 0.2 0.4 0.6 0.8 1.0 Utility Ue (b) Utility Ueas a function the model runtime. Figure 4: Example utility functions for the considered competing attributes, namely i) model fidelity, which accounts for over/under-estimating target quantities, and (ii) computational runtime to ensure real-time inference when possible. overand under-estimation effects. Underestimation of quantities of interest is here penalized more as it might lead236 to unexpected failures and catastrophic events, which are considered more critical in asset management compared to potential additional costs caused by overestimation. Since the Bayesian inference framework is also employed for pa-238 rameter estimation, the utility with respect to the model’s precision is evaluated using readings from redundant sensor measurements that are not employed on the inference of ˆ p. In Figure 4b the implemented relationship between the240 evaluation time of the model and its utility is reported. The overall utility can be computed as a weighted combination of the competing attributes, allowing the decision-maker to weigh the considered attributes according to the decision242 at hand. The corresponding mathematical formulation reads: Ui(fidelity|Mi,efficiency|Mi,ˆ p|Mi)=γ∗Ui(fidelity|Mi,ˆ p|Mi)+(1−γ)∗Ui(efficiency|Mi,ˆ p|Mi) (9) Ui(fidelity|Mi,ˆ p|Mi)=Uf(ˆ p|Mi)=             1−w1∗yf−ˆyf yf2 ∗(w1∗ϵmax)−1,yf−ˆyf>0 1−w2∗yf−ˆyf yf2 ∗(w2∗ϵmax)−1,yf−ˆyf≤0 (10) Ui(efficiency|Mi,ˆ p|Mi)=Ue(ˆ p|Mi)=1−log2tMi log2(tmax)(11) where yfdenotes the observed values for the QoI and ˆythe model-approximated ones using the inferred ˆ pand the se-244 lected model resolution Mi. The variable tMidenotes the evaluation time of model resolution Mi,tmax is the maximum acceptable evaluation time that corresponds to zero utility and ϵmax the maximum prediction error that is penalized246 with zero utility. The weights w1,w2are chosen properly to account for overand under-estimation effects (w1>w2) and to normalize the computed discrepancy so that the maximum possible utility is equal to 1, as illustrated in Fig-248 ure 4, where w1=3,w2=1,tmax =10s, ϵmax =20%. The factor γweighs the importance of fidelity in the task at hand, also reflecting a balancing act with the corresponding efficiency.250 For the Bayesian model updating (BMU) task, the improved Transitional Markov Chain Monte Carlo (iTMCMC) method is employed [49, 81], adapted from the open-source implementation available at https://github.com/252 ERA-Software/Overview. An initial ensemble of ns=1000 samples is drawn from the prior distribution πpr(ˆ p). The updating step is then performed conditionally on the observed data yto obtain the posterior distribution of the254 uncertain parameters, πpos(ˆ p|y). The discrepancy between the measured acceleration signals yand the model-predicted 8 This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=5682281 Preprint not peer reviewed responses ˆ yis modeled probabilistically as:256 nRMS E(ˆ p,Mi)=v u u tPnf j=1yj−ˆ yj(ˆ p,Mi)2 Pnf j=1y2 j ∈Rnf∼ N0,Σ=diag(c2||y||2)(12) where N(0,Σ) denotes the multivariate normal (MVN) distribution with a zero-mean vector and covariance matrix Σ. A diagonal covariance matrix is assumed, with the variance of each component assumed proportional to the L2-norm258 of y. The factor ccan be regarded as a coefficient of variation, and its chosen value reflects the total prediction error. It is here assumed that c=0.10. The likelihood function can be then written as:260 L(p;y)∼ Nη;0,Σ,(13) where N(·;0,Σ) denotes the value of the MVN density function at a specified location. Thus, the expected fidelity utility Ufto employ model Mican be approximated using nsposterior samples from πpos(ˆ p|y) as:262 Eˆ p|y,Mi[Uf(ˆ p|Mi)] ≈1 ns ns X j=1 UfMi,ˆ pj,(14) The equivalent is implemented for Ue. 4. Numerical case studies264 The proposed framework is validated in two structural dynamics case studies, involving hysteretic, geometric, and material nonlinearities. In both cases, the approach performs recursive model selection to identify the optimal266 resolution for structural assets whose state, parameters, and health condition evolve over time. The considered model resolutions are summarized in Table 1.268 Table 1: Reference table for the considered model resolutions. Reference name Description FOM The full-order FE model VpROM ROM according to the formulation in subsection 2.3. H–VpROM VpROM additionally equipped with hyper-reduction Perturbed FOM The full-order FE model with a finer mesh and perturbed material properties. Used for generating response signals with a 8% noise level for testing. Given the limited availability of experimental or field monitoring data, simulated measurements are used for validation. These are generated from an independent numerical finite element (FE) model, referred to as the perturbed270 FOM, as summarized in Table 1. To clearly distinguish the monitoring data used for testing from the FOM employed in constructing the ROMs, the following measures are adopted:272 •The perturbed FOM employs a finer finite element mesh than the in-house FOM used to train the ROMs. •The Young’s modulus Eis randomly perturbed, with element-wise values drawn from a normal distribution274 with mean Eref and standard deviation 5,GPa. •Measurement noise corresponding to 8% of the root-mean-square (RMS) value of each signal is added.276 The perturbed FOM is employed to generate the testing data in order to avoid the inverse crime [82, 83], which arises when the same or nearly identical model is used both to generate and to infer system responses in an inverse278 problem [84]. The performance of the proposed framework in parameter and response inference (see Figure 1) is assessed by280 reproducing the time-history response of selected quantities of interest or their spatial distribution at specific simulation snapshots. Parameter inference is based on vibration measurements from accelerometers assumed to be installed282 9 This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=5682281 Preprint not peer reviewed 4.2. Simplified fuselage panel The second numerical example concerns a simplified fuselage panel, a representative load-carrying component in418 both maritime and aerospace structures, and therefore of broad practical relevance. This example is loosely inspired by [93], where dynamic substructuring was applied to analyze cracking in thin-walled aeronautic structures. Here,420 we adopt and extend the implementation proposed in [20] for model assembly, modifying it to account for large deformations and the resulting geometric nonlinearities.422 Figure 11: Simplified fuselage panel geometry, loading, and sensor locations. The blue sensors are used for inference while the red redundant sensor is employed for performance and utility evaluation. The fuselage panel geometry, boundary conditions, loading, and sensor layout are shown in Figure 11. The panel is cylindrical with a radius of 2.5 m and a uniform thickness of 3 mm. An aluminum material is assumed, with a variable424 Young’s modulus Epintroduced to amplify large deformations and thus capture geometric nonlinearities. Unlike previous studies employing free–free or fully fixed conditions, the present setup applies fixed boundary conditions426 only along the left edge, while the right edge is connected through springs of variable stiffness kbto represent bolted joint loosening. The stiffness parameter ranges from 0 (free end) to 1 (fully fixed). The fuselage is discretized428 using 4,393 MITC4 elements, resulting in 47,334 degrees of freedom. A dynamic pressure load is applied uniformly, following a white-noise-like signal with parametric magnitude pin the frequency range [1, 250] Hz over a duration430 of 2 s. Time integration is performed using the implicit Newmark scheme with a step size of ∆t=2×10−4s. Table 5: System properties and range of parametric traits. Parameter: Ep(×25 GPa)kb(×1e16)p(kN/m2) Poisson’s ratio Density (kg/m3) Range: [1.0,2.0] [0.0,1.0] [0.1,1.0] ν=0.3ρ=2700 The system properties and the parametric dependencies of the model are summarized in Table 5. The parameter432 vector to be inferred is p=[Ep,kb,p]. The material parameter Epcan simulate potential damage on the fuselage and influence the geometrically nonlinear behavior directly. Similarly, parameter kbcontrols the rigidity of the boundary434 and can model fixed-free or fixed-fixed conditions and bolt loosening or equivalent damage events in a simplified manner. Lastly, the amplitude of the excitation pinfluences the deformation amplitude, and thus, the nonlinear436 behavior. For training the ROMs listed in Table 1, r=32 basis modes are employed along with 500 training samples of p438 generated via Latin Hypercube Sampling (LHS). Testing is performed using 200 parametric samples. The monitoring setup, summarized in Table 6, includes eight acceleration sensors placed at distinct nodes (blue dots in Figure 11),440 each recording acceleration along the excitation direction. These measurements are used for the parameter inference task, as described in Figure 1. A redundant sensor (sensor 9, shown in red in Figure 11) provides independent vibration442 data to trigger adaptive model selection and assess model fidelity. The utility functions follow the same formulation 16 This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=5682281 Preprint not peer reviewed as in Equations 9–11 and Figure 4, except that in the efficiency utility, the maximum penalized evaluation time (zero444 utility) is set to 100 instead of 10 seconds. Table 6: Details of the monitoring setup for the fuselage panel. Scenario Panel exhibits geometric nonlinearity and parametric material and boundary traits while experiencing 2s excitation windows of varying amplitude. Objective Select model to be used for SHM-related downstream tasks Monitoring Data Nodal accelerations Eight nodes monitored (blue dots in Fig. 11) QoIs for Utility computation Fidelity: Max. acceleration at redundant sensor 9 (Normalized RMSE) Efficiency: Avg. model evaluation time (s) Utility functions Equations 9-11, tmax =100s,w1=3,w2=1, ϵmax =25% Monitoring window 0.5 seconds rolling windows As summarized in Table 6, the testing scenarios consist of repeated 2-second windows with evolving parametric446 traits. The excitation amplitude pvaries freely within each sequence, while the material modulus Epand boundary stiffness kbdecrease progressively to emulate structural deterioration. For instance, Epmay decrease from 1.65 to448 1.42, 1.12, and so forth. Performance validation includes 50 such combinations, with parameter realizations drawn via LHS. A representative example of the Bayesian parameter inference results for the excitation amplitude and boundary450 stiffness is shown in Figure 12; similar trends are observed for the material parameter Ep. The reported results correspond to the best-performing 2-second window among the rolling sequences. Despite minor discrepancies, the452 proposed framework successfully reconstructs the evolving parametric configuration from the sensing data, enabling adaptive model updates in real time.454 0.2 0.4 0.6 0.8 1.0 0 2 4 6 8 10 12 14 Density True Value Prior H-VpROM VpROM FOM (a) Inference for excitation amplitude coefficient kb. 0.2 0.4 0.6 0.8 1.0 0 2 4 6 8 10 12 Density True Value Prior H-VpROM VpROM FOM (b) Inference for boundary stiffness coefficient p. Figure 12: Bayesian parameter inference for obtaining the posterior distribution of the uncertain model parameters. A representative average performance example is shown. As shown in Figure 12, the FOM exhibits the highest precision in capturing the underlying parameters, as expected. This trend is confirmed in Figure 13a, where the FOM achieves the greatest fidelity utility Uf. However, this456 accuracy comes at a substantial computational cost, as reflected by the very low efficiency utility Uein Figure 13b. Conversely, the H-VpROM provides hyper-accelerated evaluations, yielding high Uevalues while maintaining accept-458 able accuracy. The VpROM demonstrates a similar trend, attaining a different but still favorable accuracy–efficiency trade-off. These results highlight the importance of the utility-based formulation, which quantitatively balances com-460 peting attributes and enables data-informed model selection and validation. ollowing Equation (9), the competing utility attributes can be balanced through an appropriate choice of the462 weighting coefficient γ. This parameter can be defined by the decision maker, allowing task-specific prioritization 17 This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=5682281 Preprint not peer reviewed 23456 0.4 0.6 0.8 1 Time (s) Utility Uf H-VpROM VpROM FOM (a) Utility Ufwrt model’s precision. 23456 0 0.5 1 Time (s) Utility Ue H-VpROM VpROM FOM (b) Utility Uewrt model’s efficiency. Figure 13: Evaluations of the fidelity and efficiency utility functions for the considered model resolutions for a representative example testing scenario. Red dotted lines indicate parameter changes. between fidelity and efficiency. Two illustrative examples of the resulting total utility for different γvalues are pre-464 sented in Figure 14. As shown in Figure 13, the H-VpROM consistently achieves the highest utility under the equal weighting scheme, owing to its superior computational performance, and is thus identified as the optimal model for466 the given task (see Figure 14a). When precision is prioritized (γ=0.75), however, the VpROM becomes more favorable, as shown in Figure 14b. These results emphasize the value of utility-based model selection, which enables468 context-aware adaptation by quantifying trade-offs between competing objectives. 23456 0.4 0.6 0.8 1 Time (s) Utility U H-VpROM VpROM FOM (a) Equally important utility attributes (γ=0.50). 23456 0.6 0.7 0.8 0.9 Time (s) Utility U H-VpROM VpROM FOM (b) Precision prioritized over efficiency (γ=0.75). Figure 14: Total utility evaluations for different weighted combinations of the considered attributes in Figure 13.Red dotted lines indicate parameter changes. Following the model updating and selection process, and based on the posterior distributions in Figure 12, the470 H-VpROM is employed for response prediction. Owing to its physics-aware formulation, the model can recover quantities of interest across all physical fields. Samples drawn from the posterior distributions are propagated through472 the H-VpROM to quantify confidence bounds on the predicted responses. By further combining these samples with draws from the model’s probabilistic latent space via the Unscented Transform (see subsection 2.4), the resulting474 bounds also account for uncertainty in the reduced representation itself. The corresponding H-VpROM predictions are shown in Figures 15–16 for the system’s displacement and acceleration responses. The reported average and maximum476 error estimates are computed across all testing configurations, using the highest-utility H-VpROM identified in the 18 This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=5682281 Preprint not peer reviewed initial rolling windows.478 0 0.5 1 1.5 2 −1 0 1 2 Time (s) Displacement (mm) Pert. FOM H-VpROM Conf. bounds (a) Average approximation. 0 0.5 1 1.5 2 −1 0 1 2 Time (s) Displacement (mm) Pert. FOM H-VpROM Conf. bounds (b) Maximum error approximation. Figure 15: Predicted displacement response at the redundant node with associated confidence bounds. Reported values include the average and maximum approximation errors across all testing samples. As shown in Figure 15a, the framework effectively balances computational efficiency and accuracy, delivering highly consistent response estimates on average. Moreover, the probabilistic formulation and embedded Bayesian480 components enable the framework to provide confidence bounds that enclose the true response, even under the maximum error scenario depicted in Figure 15b. Similar trends are observed for the acceleration response in Figure 16. As482 discussed in the previous case study, the decision-maker can define a precision utility threshold Ufto automatically trigger model retraining in real time, avoiding delays associated with full FOM evaluations. This approach mitigates484 local minima in Figure 13 and improves high-error cases such as that in Figure 15b. Alternatively, by adjusting the weighting scheme between competing attributes, the framework can prioritize efficiency and select the VpROM486 instead, as illustrated in Figure 14b. 0.5 1 1.5 2 −50 0 50 Time (s) Acceleration (m/s2) Pert. FOM H-VpROM Conf. bounds (a) Average approximation. 0 0.5 1 1.5 2 −100 −50 0 50 100 Time (s) Acceleration (m/s2) Pert. FOM H-VpROM Conf. bounds (b) Maximum error approximation Figure 16: Framework’s acceleration response approximation at the redundant node and corresponding error bounds. The average and maximum error performance across testing samples is reported. 19 This preprint research paper has not been peer reviewed. Electronic copy available at: https://ssrn.com/abstract=5682281 Preprint not peer reviewed 5. Discussion488 The developed framework addresses a central challenge in real-time Operations and Management (O&M), Structural Health Monitoring (SHM), and decision support—the adaptive selection of the lowest-cost model capable of490 delivering predictions of the required fidelity under evolving conditions [4]. At its core, the proposed framework integrates a generative reduced-order modeling (ROM) scheme, conditioned on parametric inputs, with a Bayesian in-492 ference layer that recursively estimates the underlying system parameters from sensing data. This combination enables continuous model adaptation, quantifies prediction confidence, and supports decision-making through a utility-based494 formulation that balances two competing attributes: (i) model fidelity and (ii) computational efficiency. By maximizing the expected utility, the framework identifies the fit-for-purpose model at each time step, determining whether to496 accept, switch, or retrain a candidate representation. Two numerical case studies—a hysteretic shear frame with uneven damage and a geometrically nonlinear fuselage498 panel—demonstrate the framework’s ability to select optimal model resolutions and adapt to changing system conditions. The approach reliably identifies when reduced-order models lose validity and retraining is required, particularly500 during extreme events beyond the training domain. Moreover, the probabilistic formulation provides uncertainty quantification, generating response confidence bounds that capture the true behavior even under low-precision conditions.502 Several limitations outline avenues for future research. First, the current iTMCMC-based Bayesian updating remains computationally demanding for high-dimensional problems; future work will explore more efficient inference504 strategies for real-time deployment. Second, the framework assumes that all relevant physics are represented within the available models. A fully adaptive system should also handle unforeseen or unmodeled phenomena, where even506 high-fidelity models may yield low utility—a challenge pointing toward self-evolving digital representations. Third, while this study uses perturbed simulation data to avoid inverse crime, experimental validation will be essential to508 assess robustness in real-world conditions. Finally, the parametric formulations employed may not fully capture effects such as excitation frequency content or mode interactions; these aspects will be explored alongside the extension510 toward self-evolving model structures. Data Availability512 The models and data supporting this study’s findings are available from the author KV upon request. Declaration of Competing Interest514 The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.516 Acknowledgements The author KV, EC gratefully acknowledges the funding from the European Commission under the Horizon Eu-518 rope funding guarantee, for the projects ‘INBLANC - INdustrialisation of Building Lifecycle data Accumulation, Numeracy and Capitalisation’ (grant agreement No: 101147225) and ‘TURING - Trustworthy Unified Robust Intel-520 ligent Generative Systems’ (grant agreement No: 101215032). References522 [1] F. Chinesta, E. Cueto, Empowering engineering with data, machine learning and artificial intelligence: a short introductive review, Advanced Modeling and Simulation in Engineering Sciences 9 (1) (2022) 21. doi:10.524 1186/s40323-022-00234-8. 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