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Embedding Model-form Uncertainty in Probabilistic Calibration of Digital Twins for Bridges

Daniel Andrés Arcones, Martin Weiser, Phaedon-Stelios Koutsourelakis, Jörg F. Unger

arXiv:2609.10171v1cs.CE

TL;DR

Digital-twin bridge predictions need calibrated physics-based models that account for discrepancies between models and monitored structures. This paper embeds MFU as stochastic parameter uncertainty in a simplified Nibelungenbrücke thermal model and uses staged calibration and uncertainty analysis. The reported variance decomposition distinguishes uncertainty sources and shows that embedded uncertainty accounts for the discrepancy at the top sensor, while other sensors retain larger aleatoric contributions.

  • Problem

    Bridge digital twins need physics-based models for full-field prediction, but model calibration must account for discrepancies caused by model inadequacy and simplifying assumptions.

    Source: §1 · p. 1

  • Method

    The study calibrates a simplified Nibelungenbrücke thermal model using stochastic parameter embedding for MFU, a staged calibration strategy, variance decomposition, and discrepancy metrics.

    Source: §1 · p. 2, §5.1 · p. 13, §8 · p. 23

  • Results

    The embedded model has variance comparable to the baseline, with most predictive variance epistemic; the top sensor’s discrepancy is accounted for by embedding, while other sensors retain larger aleatoric contributions.

    Source: §6.1.2 · pp. 18–19

  • Takeaways & Limitations

    Separating measurement variability from model inadequacy allows physically interpretable uncertainty to be propagated from calibration to unmeasured quantities.

    Source: §8 · pp. 23, 25

    The embedded representation is limited to the domain explored by available observations and does not guarantee that all uncertainty sources affecting future observations are captured.

    Source: §2.2 · p. 6

Abstract

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Digital twins of bridges rely on physics-based models to infer full-field structural responses from sparse monitoring data. The reliability of these predictions depends on the calibration of model parameters while accounting for discrepancies between the model and the physical system. Such discrepancies, commonly referred to as model-form uncertainty (MFU), arise from simplifying assumptions and incomplete representations of physical processes, and can substantially affect predictive reliability. Explicitly quantifying MFU during calibration is therefore essential for trustworthy digital-twin predictions. This work presents a calibration framework that explicitly represents MFU through stochastic parameter embedding. The framework is demonstrated using a simplified two-dimensional cross-sectional thermal model of the Nibelungenbr\"ucke calibrated against temperature measurements from its monitoring system. By explicitly accounting for MFU, the proposed methodology enables computationally efficient models to be reliably employed within digital-twin environments while maintaining predictive credibility. The framework introduces a three-step calibration strategy that progressively addresses sources of discrepancy while maintaining a clear distinction between identified uncertainties. Variance decomposition is used to characterize the structure of predictive uncertainty and its propagation to quantities of interest. Remaining discrepancies between predictive distributions and observations are quantified using Kolmogorov-Smirnov metrics to assess predictive consistency across seasonal conditions. The results demonstrate that explicitly accounting for MFU improves the interpretability and reliability of predictive uncertainties and validate the methodology for calibration of physics-based models supporting digital twins in structural health monitoring.

1 Introduction

The introduction frames a need for physics-based bridge models in digital twins, while noting that high-fidelity models are costly and simplified models introduce discrepancies that must be quantified. The study proposes probabilistic calibration with embedded model-form uncertainty (MFU), demonstrated on the Nibelungenbrücke thermal model.

1 Introduction

  • Physics-based models remain important for extrapolation, interpretability, and scenario analysis, but high-fidelity models are costly and computationally demanding for continuously updated digital twins.Surrogates reduce computational cost but are limited by their training domain and may be less interpretable.Source: §1 · p. 1
  • Model simplifications create discrepancies that can grow as model complexity is reduced, so their sources—including MFU—need explicit quantification and attribution.Source: §1 · p. 1
  • Thermal models can reconstruct the full temperature field and estimate thermal deformations at arbitrary locations, whereas local sensor pairing neglects spatial temperature variability.The study uses probabilistic model updating based on maximum likelihood estimation to quantify uncertainty after calibration.Source: §1 · p. 2
  • The study proposes a probabilistic calibration framework that explicitly embeds MFU in a simplified transient thermal model of the Nibelungenbrücke and propagates uncertainty to quantities of interest.The framework is validated using monitoring data from the real bridge and is presented as extendable to other system classes.Source: §1 · p. 2

2 Thermal model and representation of Model-Form Uncertainty

The section defines a computationally efficient two-dimensional thermal model of the bridge cross-section and describes how its parameters and sensor predictions are represented. It contrasts additive discrepancy with embedded MFU, then details the bridge monitoring data and the mismatch between physical and virtual sensors.

2.1 Thermal model definition

  • The model is a two-dimensional cross-sectional transient heat-conduction solution operator that maps parameters and environmental boundary conditions to a temperature field.Finite elements approximate the cross-section, and virtual sensor observations are extracted from the resulting temperature fields.Source: §2.1 · pp. 2, 4
  • The boundary model includes wind-driven exterior convection, interior natural convection, and shortwave radiation, while correction factors c_c and c_r modify boundary heat-transfer terms to represent aggregated unresolved effects.These effects include simplified geometry, shadowing, changing solar incidence, cloud cover, and traffic-induced thermal loads.Source: §2.1 · pp. 2–3, §2.2 · p. 6
  • The model prediction for sensor k is obtained by evaluating its temperature field at the corresponding virtual sensor location, which may differ from the physical sensor location.The virtual geometry is schematic, and the real-to-virtual sensor-position discrepancy is particularly apparent near the top surface because the asphalt layer is omitted.Source: §2.1 · p. 4, Figure 2 · p. 4, §2.3 · p. 7
  • The dimensional simplification is motivated by slender geometry, slowly varying cross-sections, sparse monitoring, and dominant cross-sectional thermal gradients, reducing repeated-simulation cost.The omitted effects of modelling assumptions are addressed through explicit MFU treatment.Source: §2.1 · pp. 4–5

2.2 Model-form uncertainty formulation

  • The additive formulation writes observations as model predictions plus a systematic discrepancy term and measurement noise; the discrepancy can represent unresolved environmental, structural, loading, or geometric effects.The account identifies environmental processes that are only partly represented as the dominant MFU source and models their aggregated effects with boundary correction factors.Source: §2.2 · pp. 5–6
  • The embedded MFU approach incorporates discrepancy into model parameters, allowing uncertainty to propagate through governing equations; an additive discrepancy term sits outside the model.The additive formulation is described as less suitable for forward prediction under new conditions because it cannot naturally propagate through the equations.Source: §2.2 · pp. 5–6

2.3 Data and preprocessing

  • The calibration and validation observations come from bridge monitoring sensors; raw data are cleaned and downsampled from 10 Hz to 30-minute intervals for the bridge’s slow thermal dynamics.The sensors’ manufacturer-stated measurement uncertainty is below 0.1 °C; wind speed records are hourly and interpolated to the 30-minute timestep.Source: §2.3 · p. 6
  • The June 2024 calibration set contains 2,209 entries with nine sensor measurements per entry, while the multi-year seasonal-validation set contains 44,641 entries.The calibration period runs from May 15 to June 30, 2024; the validation period spans June 15, 2023 to December 31, 2025.Source: §2.3 · p. 7

3 Initial condition analysis

The analysis tests how an arbitrary initial thermal field affects transient predictions and identifies a burn-in period after which its influence is negligible. Under June 2024 thermal loads, 800 discarded steps—roughly two weeks—are selected for that period.

3 Initial condition analysis

  • The actual initial temperature field cannot be prescribed from the operational observations, and inferring it as an additional parameter is outside the study’s scope.The authors note that adding it would increase the inverse problem’s complexity and ill-posedness.Source: §3 · p. 7

3.1 Methodology

  • The simulations start from a homogeneous field with arbitrary T0 and no prior thermal history, varied over 270–300 K to test when initial-condition effects become negligible.The burn-in steps are discarded while time-varying boundary conditions develop a heterogeneous temperature field.Source: §3.1 · p. 7
  • Rmax measures the largest across-time spread between ensemble predictions at each sensor and provides a conservative estimate of error due to the initial-condition choice.The sensitivity analysis uses June 2024 thermal loads at 30-minute resolution and default parameters without MFU.Source: §3.1 · p. 8, §3.2 · p. 8

3.2 Results

  • After 800 burn-in steps, Rmax is below 10^-6 at every sensor, and the estimated maximum squared initial-condition error is below machine precision.The 800 steps correspond to roughly two weeks of simulated data and are adopted as the burn-in period.Source: §3.2 · p. 8, Figure 4 · p. 9

4 Variance-based sensitivity analysis pre-calibration

Variance-based sensitivity analysis (VBSA) ranks parameters by their influence on model outputs to inform calibration and uncertainty attribution. In the June 2024 analysis, convection dominates several sensor predictions, while first- and total-order indices are nearly identical, indicating limited interactions.

4 Variance-based sensitivity analysis pre-calibration

  • VBSA ranks parameter influence to guide calibration, helping identify negligible parameters to fix and influential ones to prioritize or update in stages.The analysis also helps distinguish output variability attributable to meaningful parameter variation from potential numerical or structural artifacts.Source: §4 · p. 9

4.1 Methodology

  • Sobol’ variance decomposition separates individual-parameter contributions from interactions, and a total-order index includes a parameter’s main effect and all interactions involving it.Source: §4.1 · pp. 9–10
  • Saltelli sampling uses paired input matrices to estimate first- and total-order indices efficiently, with a convergence study needed to check estimator accuracy.The June 2024 analysis used 128 Saltelli samples, reported as sufficient to preserve the parameter-importance ranking.Source: §4.1 · p. 10, §4.2 · p. 10
  • Sensitivity indices depend on the assumed input distributions and ranges: overly broad ranges can inflate apparent influence, while narrow ranges can suppress meaningful variability.The reported analysis uses uniform, non-informative distributions within the calibration domains, so its rankings are conditional on those choices.Source: §4.1 · p. 10, §4.2 · p. 10

4.2 Results

  • The convection closure coefficient c_c dominates predictions at TBottom, TNorth, and TSouth and contributes significantly to TTop, making it a candidate for stochastic embedding.The authors associate its influence with convective heat exchange as the system’s primary heat input.Source: §4.2 · pp. 10–11
  • Thermal diffusivity α has limited, sensor-specific influence and is not considered suitable for stochastic embedding; its apparent sensitivity may be inflated by a broad sampling range.Including sensor-position parameters substantially reduces α’s relative importance compared with c_c.Source: §4.2 · p. 11
  • Total-order Sobol’ indices are nearly identical to first-order indices, indicating limited parameter interactions; sensor positions dominate sensitivity but are omitted from Figure 5.The figure shows time-resolved first-order indices for the main model parameters across sensors, without sensor-position parameters.Source: §4.2 · p. 10, Figure 5 · p. 11

5 Parameter calibration

The calibration separates geometric, physical-parameter, and residual uncertainty through sequential steps, embedding model-form uncertainty (MFU) in stochastic parameters. Results show that this embedding changes inferred corrections and noise attribution, while predictive uncertainty remains conditional on the adopted model class.

5.1 Methodology

  • A subset of parameters is treated as random variables, so structural discrepancy contributes to stochastic model-response variability and predictive variance.A Polynomial Chaos Expansion propagates the embedded parameters; the likelihood also assumes additive normally distributed noise.Source: §5.1 · pp. 12–13
  • The three steps infer sensor geometry and diffusivity, calibrate physical parameters with embedded MFU, then estimate sensor-specific residual noise.The sequence is intended to reduce parameter interactions; the final residual is tied to the training sensors and observations and is not straightforwardly propagated to unseen conditions.Source: §5.1 · pp. 13–14

5.2 Results

  • In the first calibration step, diffusivity reaches its lower bound and the southern sensor depth reaches its lower bound, reflecting unresolved model discrepancies.The authors also note limited identifiability between diffusivity and sensor depth, and caution that the inferred diffusivity is not representative for future predictions where its influence may be larger.Source: §5.2 · p. 14
  • The second step embeds c_c as a lognormal variable and uses June 2024 measurements; its variance serves as a proxy for MFU, not proof of parameter miscalibration.Source: §5.2 · p. 15
  • 7% below nominal, the inferred mean c_c correction accompanies an approximately 13% reduction in heat input through c_r.The authors suggest embedded variance absorbs part of the discrepancy otherwise corrected by c_c; the c_r correction may also reflect omitted pavement and asphalt layers.Source: §5.2 · p. 15
  • For TTop, stochastic embedding yields noise estimates comparable to other sensors, while the classical approach assigns the discrepancy largely to noise.Embedding has limited effect on bottom, north, and south noise estimates; the south sensor still requires substantially more noise relative to its predictive variance.Source: §5.2 · pp. 15–16

5.3 Discussion on model prediction and parameter interpretation

  • Under the adopted interpretation, predictions combine epistemic uncertainty from parameter inference with aleatoric residual noise, but remain conditional on the model structure.The embedded parameters represent only the component of MFU captured by the chosen stochastic embedding.Source: §5.3 · pp. 16–17
  • Calibrated parameters are specific to the adopted model formulation and may compensate for its deficiencies, limiting direct transfer to alternative formulations.Calibration targets parameter distributions that provide reliable predictions within that framework, rather than model-independent parameter values.Source: §5.3 · p. 17

6 Variance decomposition, uncertainty attribution and model validation

The section separates predictive uncertainty into parameter-induced epistemic uncertainty and sensor-noise aleatoric uncertainty, then evaluates predictive-distribution agreement across training and seasonal conditions. Together, variance decomposition and KS deviations support model assessment that considers both uncertainty structure and distributional fit.

6.1 Variance decomposition

  • The hierarchical formulation separates parameter-induced epistemic uncertainty from aleatoric sensor noise, enabling analytic variance decomposition without repeated observations.This is useful in the monitoring setting, where only one observation is available per sensor at each time step.Source: §6.1 · p. 17, §6 · p. 17

6.1.2 Results and discussion

  • The embedded model has variance comparable to the unembedded model, but most embedded-model variance is epistemic; aleatoric noise is nearly absent at TTop and dominates uncertainty at the southern sensor.The top sensor is also where embedding most effectively represents observed variability, while remaining discrepancies are suggested for other sensors.Source: §6.1.2 · pp. 18–19, Table 5 · p. 18, Figure 8 · p. 19

6.2.1 Formulation

  • The KS deviation quantifies how much uncertainty is required for predictive distributions to remain consistent with observations; values near zero indicate plausible reproduction, while values near one indicate evidence against adequacy.The one-sided version penalizes underdispersion but intentionally disregards overestimation of predictive uncertainty.Source: §6.2 · p. 19, §6.2.1 · p. 20
  • Overlapping observation windows yield deviations over time, and rolling quantiles over consecutive deviations summarize their central tendency and spread while reducing sensitivity to window-specific outliers.The observation-window size and the number of deviation values aggregated for rolling statistics are distinct quantities.Source: §6.2.1 · p. 20

6.2.2 Results

  • After noise calibration, embedded models have training-month deviations comparable to or below the reference while retaining smaller total variance; outside training, winter deviations remain higher at TTop.The classical nominal-noise model has the largest deviations across the annual period, and the embedded model’s higher winter deviation at TTop may indicate training-set overfitting or missing cold-weather dynamics.Source: §6.2.2 · p. 21, Table 6 · p. 21, Figure 9 · p. 22

6.3 Discussion on model comparison

  • Model comparison uses complementary diagnostics: variance decomposition characterizes uncertainty sources, while KS deviations assess agreement between predictive distributions and observations.The framework frames the baseline-versus-humidity-augmented comparison as an example of MFU-aware analysis guiding model improvement.Source: §6.3 · pp. 21–23

7 Quantity of Interest propagation

The section uses thermal curvature as a quantity of interest (QoI) for connecting the cross-sectional temperature field to bridge deformation. It compares a sensor-based estimate with curvature reconstructed from the calibrated model, whose uncertainty can be propagated to the QoI.

7 Quantity of Interest propagation

  • Thermal curvature is the selected QoI because the vertical temperature gradient can be related to deformation under small-deflection beam kinematics.For constant curvature over bridge length L, displacement is proportional to κ_T L^2 and rotation to κ_T L.Source: §7 · p. 23
  • The model-based curvature integrates the predicted temperature field over the cross-section and propagates embedded-parameter uncertainty using PCEs.Thermal expansion and the second moment of area are treated as deterministic in this analysis to isolate uncertainty from the thermal model.Source: §7 · p. 23
  • The relative curvature error exceeds 100% around 60 h, despite both approaches capturing the same overall temporal evolution.The full-field reconstruction generally yields lower curvature and provides confidence intervals reflecting calibration uncertainty.Source: §7 · p. 23

8 Conclusions

The conclusions show that stochastic MFU embedding distinguishes model-response uncertainty from measurement variability and carries interpretable uncertainty into unmeasured predictions. Predictive reliability is conditional on environmental conditions, calibration data, and the simplified model’s physical scope.

8 Conclusions

  • Explicit MFU embedding attributes some discrepancy to uncertain model parameters, unlike observational noise alone, which does not change the model response or distinguish discrepancy sources.The resulting epistemic component represents uncertainty in the model response itself.Source: §8 · pp. 23–24
  • Embedded-parameter uncertainty propagates through the physical model to unmeasured or derived quantities, providing an interpretable representation for downstream predictions.The benefit is not necessarily lower discrepancy metrics on calibration data, but propagation of uncertainty beyond observed quantities.Source: §8 · p. 24
  • Predictive performance remains stable under conditions comparable to calibration data, while larger winter discrepancies indicate neglected physical processes under colder regimes.MFU quantification can identify where simplified models remain reliable and where additional physical modelling may be needed.Source: §8 · p. 24
  • Identifiability depends on data information and embedding structure; limited calibration periods and simplified two-dimensional geometry constrain seasonal accuracy and physical representation.Embedding few parameters can aggregate multiple discrepancy sources, reducing attribution interpretability.Source: §8 · p. 24
  • Future extensions include multiple seasons and conditions, time-dependent parameters, three-dimensional thermomechanics, and continuous updates, each bringing additional inference or discrepancy challenges.Operational updates must distinguish physical changes, model inadequacy, and sensor degradation.Source: §8 · p. 24
  • The framework separates measurement-related aleatoric uncertainty from epistemic uncertainty arising from model inadequacy and supports propagation of physically interpretable uncertainty to unmeasured quantities.It combines variance decomposition, distributional discrepancy metrics, and uncertainty propagation in the Nibelungenbrücke thermal-model demonstration.Source: §8 · pp. 23, 25

A Time-resolved Sobol’ indices with sensors

The June 2024 analysis presents time-resolved first Sobol’ indices with sensor positions and initial temperature included among the parameters. Values above one are identified as sampling artefacts, and total indices match the presented first indices.

A Time-resolved Sobol’ indices with sensors

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