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MoRF-AST: Calibrated Probabilistic Virtual Sensing for Structural Monitoring under Changing Operating Conditions

Wingho Feng, Quanwang Li, Ming Zhong, Jingyu Yang, Chen Wang

arXiv:2608.24531v1cs.CEcs.LG

TL;DR

Sparse sensing makes full-field structural reconstruction underdetermined, while changing operating conditions can miscalibrate posterior intervals. MoRF-AST combines residual flow matching with sensor-history-based spread transport, reducing cross-domain coverage error across eight shifted traffic domains.

  • Problem

    Probabilistic virtual sensing lacks effective adaptation to changing operating conditions, despite sparse measurements and differing on-site response scales threatening interval calibration.

  • Method

    MoRF-AST learns posterior-whitened residuals around an analytic Gaussian reference and adjusts posterior spread from operating-condition sensor histories while preserving its mean.

  • Results

    Across eight shifted domains, AST reduces MoRF’s mean ACE from 0.0535 to 0.0236, while MoRF achieves 7.20% NRMSE with 8,000 training events.

  • Takeaways & Limitations

    Observation-only deployment calibration supports probabilistic full-field reconstruction and safer operation under changing conditions within the evaluated settings.

  • Takeaways & Limitations

    MoRF-AST assumes the source representation and observation relation remain valid and may fail under new modes, structural changes, anisotropic shifts, or sensor faults.

Abstract

from arXiv · show

Probabilistic full-field reconstruction provides uncertainty-aware response evidence for structural reliability assessment, yet inference from sparse and noisy measurements remains underdetermined. Most existing methods overlook shifts between offline training and operational distributions. Under such shifts, posterior intervals may become miscalibrated, causing the reported uncertainty to lose its probabilistic meaning. This study proposes Modal Residual Flow Matching with Context-Conditioned Affine Spread Transport (MoRF-AST) for calibrated structural virtual sensing under changing operating conditions. MoRF constructs an analytic Gaussian reference posterior in normalized modal coordinates and trains a conditional flow only on posterior-whitened residuals. At deployment, AST estimates response scale from historical measurements at installed sensors and uses gated, mean-preserving Bures-Wasserstein transport to adjust posterior spread. On a bridge-deck benchmark, MoRF achieves a posterior-mean normalized root-mean-square error (NRMSE) of 7.20%, compared with 16.1% and 17.9% for two direct conditional flows. Across eight shifted traffic domains, AST reduces MoRF's cross-domain average coverage error from 0.0535 to 0.0236, a 55.9% reduction, while preserving posterior-mean accuracy. The same transport does not improve the tested alternatives in aggregate, showing that calibration gains require its direction to match the base posterior's dispersion bias. MoRF-AST provides a data-efficient framework for probabilistic full-field reconstruction whose uncertainty remains interpretable under scale-dominated operational distribution shifts. More broadly, this work highlights the need to calibrate uncertainty under changing operational distributions, thereby supporting trustworthy probabilistic modeling and reliability-informed decision-making in civil and infrastructure engineering.

1 Introduction

Virtual sensing addresses the gap between sparse instrumentation and the distributed structural responses needed for assessment and maintenance, but evolving operating conditions can shift response distributions and undermine uncertainty calibration. MoRF-AST addresses this setting by learning around an analytic posterior and adapting posterior spread from deployment-time sensor histories while preserving the posterior mean.

  • Motivation: Virtual sensing reconstructs uninstrumented structural responses when sensor count and spatial coverage are limited by access, infrastructure, maintenance, and reliability constraints.These reconstructions support condition assessment, anomaly localization, and maintenance planning.
  • Deployment challenge: Changing loads, environmental conditions, and operating states can shift deployment response distributions away from the source domain, requiring uncertainty that reflects how current sparse measurements constrain the full field.Sparse and noisy measurements leave multiple compatible high-dimensional response fields, motivating a conditional-posterior formulation whose mean gives a point reconstruction and whose spread represents remaining uncertainty.
  • Deployment challenge: Offline regime balancing embeds sampling weights into the learned source distribution, so differing onsite regime frequencies and response scales can make posterior intervals too wide or too narrow without adaptation.The introduction identifies this calibration problem as affecting probabilistic deep learning models that provide uncertainty intervals.
  • Proposed method: MoRF-AST integrates source-domain posterior learning with deployment-time uncertainty adjustment by combining an analytic Gaussian reference with residual learning and historical-sensor-based spread adaptation.AST adjusts posterior spread while preserving the posterior mean.
  • Contributions: MoRF learns residual distributions around an analytic Gaussian posterior, while AST uses effective-scale estimation, truncation-aware gating, and mean-preserving Bures transport without target full fields or generator retraining.The method is designed to improve data efficiency and provide shared posterior coordinates under a fixed sensing setup.

2 Related Works

Structural virtual sensing has progressed from model-driven reconstruction to data-driven and generative posterior modeling, while operational adaptation has mainly emphasized prediction or representation robustness. The remaining gap is probabilistic adaptation of posterior spread and coverage using limited target-domain information, addressed here by AST.

  • Structural virtual sensing: Structural virtual sensing reconstructs unmeasured response fields from limited measured channels using modal, dynamic, reduced-order, regularized, and mechanics-based methods.These approaches can also infer structural parameters, unknown loads, and unmeasured responses.
  • Structural virtual sensing: Data-driven methods learn nonlinear measurement-to-response maps, while GANs, VAEs, and conditional diffusion models generate stochastic response ensembles beyond deterministic estimates.MLPs, DNNs, and CNNs support regression, sensor selection, and exploitation of temporal or cross-channel dependence.
  • Generative posterior modeling: Generative posterior models include VAEs, conditional GANs, normalizing flows, diffusion, and flow matching, which learn conditional distributions and draw posterior samples for new observations.Inference-time guidance can instead combine unconditional priors with measurements across inverse problems when likelihood or physics information is available.
  • Operational adaptation: Operational adaptation uses supervised target labels or unlabeled sensor records, but existing objectives center mainly on point prediction or representation robustness rather than full-field posterior calibration.Probabilistic adaptation additionally requires addressing posterior spread and coverage under distribution shift.
  • Operational adaptation: AST uses installed-sensor histories as its only additional target-domain information to adapt scale-dominated posterior spread while preserving the base-posterior mean and source-trained generator.Its approach differs from supervised full-field retraining and representation alignment, drawing on Bures-Wasserstein geometry for covariance transport.

3 Methodology · 3.1 Engineering Problem and Method Overview

The methodology formulates probabilistic virtual sensing as conditional full-field reconstruction from sparse, noisy measurements in a normalized POD modal space. MoRF-AST combines a Gaussian posterior and residual flow with context-based posterior-spread calibration under operating-condition shifts.

  • 3.1 Engineering Problem and Method Overview: Probabilistic virtual sensing reconstructs the conditional full-field response distribution p(X | y) from limited noisy sensor measurements.The response field X has P entries, while y contains p scalar sensor readings.
  • 3.1 Engineering Problem and Method Overview: Source-fixed channel scaling maps the physical response X to a dimensionless field z, while the inverse operator restores physical units.The scaling operator divides each field entry by its channel-specific source-fixed scale.
  • 3.1 Engineering Problem and Method Overview: Installed sensors observe selected components of z through a binary selection matrix, with measurement error modeled by positive-definite covariance R.The measurement model uses H ∈ R^p×P and error covariance R ∈ R^p×p.
  • 3.1 Engineering Problem and Method Overview: Because sensor data are sparse, reconstruction occurs in a retained d-dimensional POD subspace using an encoder that produces normalized modal coefficients.The encoder E centers and projects z into c = E(z), where d is the number of retained modes.
  • 3.1 Engineering Problem and Method Overview: The affine decoder maps modal coefficients back to physical response fields, including inverse scaling, and posterior samples are decoded to represent p(X | y).Samples c^(n) are transformed as X^(n) = D(c^(n)) within the retained POD subspace.
  • 3.1 Engineering Problem and Method Overview: Operating-condition shifts can miscalibrate posterior spread despite accurate reconstructed fields, so deployment calibration uses historical measurements from the same sensors as context C.The context C contains historical sensor measurements and avoids requiring full-field reference data.
  • 3.1 Engineering Problem and Method Overview: MoRF-AST combines Modal Residual Flow Matching with Context-Conditioned Affine Spread Transport to construct and calibrate probabilistic virtual-sensing posteriors.Source full-field data establish channel scaling, the POD representation, a Gaussian reference, and the decoder, while MoRF learns conditional residuals beyond that reference.

3.2 Gaussian Reference Model

The Gaussian reference model normalizes structural responses into retained modal coordinates and formulates sparse sensing as a linear observation model. It supplies a fixed-covariance analytic posterior and trains MoRF on posterior-whitened departures from that reference.

  • Gaussian Reference Model: POD normalization produces retained orthonormal modes whose source coefficients have zero empirical mean and identity covariance, while decoding restores the physical field units.The normalized encoder uses the retained singular vectors and modal variances; the affine decoder reconstructs the dimensionless field before applying S−1.
  • Gaussian Reference Model: The installed sensors induce the linear observation model y = b + Ac + ϵ, where A records each normalized mode’s sensor response and b is the restricted source mean.This representation separates sensor geometry and modal observability from the latent coefficient state.
  • Gaussian Reference Model: The reference posterior combines POD-derived second-order structure with a standard-normal Gaussian assumption to define an analytic conditional law with mean µG(y) and covariance ΣG.The covariance has a lower Cholesky factor LG, completing the posterior parameterization in retained modal coordinates.
  • Gaussian Reference Model: For fixed sensors and noise, ΣG and LG are measurement-independent and computed before deployment, while only the posterior mean µG(y) changes with each measurement.This provides MoRF with a common posterior coordinate system across source and operating events without measurement-specific covariance inversion.
  • Gaussian Reference Model: The whitened residual r measures modal departure from the analytic estimate after posterior-scale normalization, and under the reference satisfies r | y ~ N(0, Id).MoRF therefore learns observed source departures from the Gaussian reference rather than modeling the full posterior directly.

3.3 Modal Residual Flow Matching

MoRF learns a conditional flow that transports Gaussian draws into measurement-conditioned residuals, using standardized analytic posterior information and noise-aligned training. The learned residuals are mapped back to modal coordinates to form the base posterior for full-field decoding, with AST modifying coefficient spread at deployment.

  • Modal Residual Flow Matching: MoRF flow matching transports independent Gaussian starts to residual samples conditioned on the current sensor measurement.The residual endpoint follows the conditional data distribution, while straight-path velocity equals endpoint displacement.
  • Modal Residual Flow Matching: The neural velocity minimizes a flow-matching loss to approximate the condition-dependent path velocity.At sampling, an ordinary differential equation is integrated from t = 0 to t = 1 to generate residual samples.
  • Modal Residual Flow Matching: The conditioning vector uses a source-standardized analytic posterior mean, with training statistics computed once from noise-augmented source observations.For fixed A and R, the Gaussian conditional law depends on measurements through µG(y), which h(y) preserves in standardized form.
  • Modal Residual Flow Matching: Fresh measurement noise during training determines both the analytic center and flow condition, aligning optimization with noisy deployment measurements.The residual endpoint captures structure beyond the Gaussian reference, including residual bias, modal dependence, and non-Gaussian structure.
  • Modal Residual Flow Matching: MoRF transports standard Gaussian points through the frozen residual flow, maps terminal residuals back to modal coordinates, and decodes full-field samples.The identity branch decodes X^(n) = D(c^(n)), while active AST modifies coefficient spread before decoding.

3.4 Context-Conditioned Affine Spread Transport

AST uses historical measurements from installed sensors to detect operating-scale changes and selectively transport the MoRF posterior spread. Its gated, mean-preserving Bures-Wasserstein correction preserves learned covariance structure while adjusting uncertainty without current full-field data.

  • Scale estimation and gating: AST estimates a shared operating-response scale from centered historical sensor measurements and uses an identity gate to retain, contract, or expand posterior spread.It requires no full-field data from the current operating condition and leaves the posterior mean unchanged.
  • Scale estimation and gating: The gate estimate includes POD truncation covariance, whereas the transport estimate excludes it because general sensor-space residuals lack a unique retained-coefficient representation.The identity band [τ −1, τ] retains the MoRF base posterior; values outside it activate transport.
  • Bures spread transport: When active, AST uses the symmetric Bures map, which is the W2-optimal transport between same-mean Gaussian covariances, to define the scaled-reference posterior.The correction direction depends on conditioning rather than only the scalar scale factor.
  • Bures spread transport: AST acts most strongly along weakly observed directions and is attenuated where measurements already constrain the response.The variance-change sign is determined by bαv −1, while its magnitude decreases with sensor information ρi.
  • Mean preservation and covariance structure: The affine correction preserves the empirical coefficient mean and posterior mean field while changing the reference spread without discarding MoRF’s reference-normalized covariance structure.The complete procedure is specified in Algorithm 1.

4 Numerical Validation

The bridge-deck validation evaluates probabilistic reconstruction from sparse observations across controlled traffic-regime shifts, focusing uncertainty analysis on the demanding moment fields. MoRF-AST improves MoRF’s shifted-domain coverage while preserving mean accuracy, because transport direction aligns with MoRF’s dispersion bias but not those of the alternatives.

  • Benchmark and evaluation: The benchmark reconstructs seven response fields from sparse multichannel bridge-deck observations, with quantitative uncertainty analysis focused on Mx, My, and Mxy.Moment fields provide a demanding full-field uncertainty test and support stress recovery using section properties.
  • Benchmark and evaluation: All methods share the source scaling, POD basis, observation operator, measurement covariance, decoder, evaluation events, and score implementation in the primary 40-location domain comparison.Only the traffic regime changes in the primary domain comparison; reduced sensor designs use nested prefixes of the same master ordering.
  • Calibration results: The identity gate retains the base posterior in E0, V1, and V2, while Bures transport activates in N0, N1, and V3 to V6.E0 is reported separately as an independent source-envelope control.
  • Calibration results: 55.9%: MoRF-AST reduces mean ACE from 0.0535 to 0.0236 across eight shifted domains while mean NRMSE remains 7.72%.nCRPS changes only from 2.66% to 2.64%; all six activated domains improve, while identity-routed domains retain base values.
  • Directional compatibility: The same Bures map increases shifted-domain aggregate ACE from 0.135 to 0.154 for Deep Ensemble and from 0.0841 to 0.102 for GMR.Thus estimated operating scale alone does not determine calibration improvement; the base posterior’s spread bias also matters.
  • Directional compatibility: MoRF-AST improves calibration because N0 and N1 require contraction whereas V3 to V6 require expansion, matching MoRF’s signed coverage bias in each regime.The same expansion worsens alternatives because Deep Ensemble and GMR already overcover in the expansion domains; MoRF’s paired-bootstrap probability of lowering ACE is at least 0.957 in every activated domain.

5 Ablation, Sensitivity, and Operational Studies

The studies show that MoRF’s analytic reference and residual formulation drive data efficiency, while mean-preserving Bures transport, identity gating, and relevant operational history sustain calibration under distribution shifts. Sensitivity analyses further show robustness to sensor coverage and noise, although calibration depends on transport design and operating-state relevance.

  • Flow-training sensitivity: At 250 flow-training events, MoRF obtains 9.35% posterior-mean NRMSE versus 9.34% for the Gaussian reference, while direct flows remain near 90.5%.At 8,000 events, MoRF reaches 7.20%, compared with 16.1% for direct-y and 17.9% for the parameter-matched direct-µ model.
  • Integration-step sensitivity: Four Heun integration steps remove most integration-induced calibration error and approach the 100-step result with one twenty-fifth as many steps.Further refinement to 100 steps changes the result only slightly.
  • Posterior modeling and guidance: In V5, active AST restores calibration while conditional MoRF retains its reconstruction advantage over likelihood-guided unconditional sampling.On source validation, direct conditional MoRF gives the best mean reconstruction and calibration among the displayed flow constructions.
  • Centering and covariance transport: The symmetric Bures map attains a nine-domain mean ACE of 0.0225, compared with 0.0270 for the Cholesky map.Mean preservation is necessary because replacing MoRF’s center increases mean NRMSE from 7.65% to 10.1% and mean ACE from 0.0495 to 0.171.
  • Identity gating: The truncation-aware scale estimate activates less often, while carrying its correction into transported covariance raises aggregate ACE in most domains.The final design uses bαg for gating and bαv for retained-space spread transport, protecting against unnecessary adaptation and unresolved POD energy.
  • Operational sensitivity: The eight-domain mean ACE stabilizes by 400 historical events at 0.0239, close to 0.0236 with the full 1,500-event context.AST improves calibration across sensor counts and noise ratios, while HSMM weighting outperforms pooled history by matching the current operating state.

6 Discussion

AST estimates an effective scalar response scale to adjust posterior spread under operating-condition shifts, avoiding unsupported high-dimensional covariance estimation. Its calibration is conditional: it cannot correct posterior-mean errors and may fail when source validity, dispersion-bias alignment, or response structure breaks down.

  • Method: AST estimates an effective scale α rather than a general high-dimensional covariance, reducing unsupported degrees of freedom when p < d and history is limited.Distinct modal covariances can produce the same sensor covariance, while the Gaussian conditional covariance and Bures map combine α with sensor geometry.
  • Limitations: AST adjusts posterior spread but cannot correct an erroneous posterior center, so its expansion or contraction must match the base model’s dispersion bias.Its effect also diminishes as observations become more informative, with the Bures map approaching the identity as the posterior concentrates.
  • Applicability: MoRF-AST is best suited when the source representation, observation relation, source-domain coverage, and effective response-scale approximation remain valid.The effective scale may also reflect operating-mean shifts and noise misspecification.
  • Limitations: New response modes, structural changes, pronounced anisotropic shifts, or sensor faults may exceed both AST and the base model.AST targets overall response-scale shifts inferred from sensor-response second moments rather than specific traffic parameters, with possible relevance to wind, wave, and rotating-machinery loads.

7 Conclusion

MoRF-AST enables probabilistic full-field reconstruction from sparse, noisy measurements under changing operating conditions. MoRF improves posterior-mean accuracy, while AST calibrates posterior spread when its transport direction matches the base posterior’s bias.

  • MoRF reconstruction: MoRF combines an analytic Gaussian reference with flow matching in posterior-whitened residual coordinates to focus learning on residual structure.With 8,000 training events, MoRF achieves an NRMSE of 7.20%, less than half the corresponding NRMSEs of the two direct conditional flows.
  • AST calibration: Across eight shifted domains, AST reduces MoRF’s mean ACE from 0.0535 to 0.0236 through sensor-only operating-history scale estimation and mean-preserving Bures transport.Its truncation-aware identity decision correctly avoids unnecessary adaptation.
  • Calibration conditions: Transport improves calibration only when its direction matches the base posterior’s spread bias.MoRF’s bias aligns with the required action in both contraction and expansion regimes, unlike the other baselines.
  • Calibration conditions: MoRF provides AST with a calibrated base posterior whose spread bias is correctable under domain shifts.Further ablation and sensitivity studies support the MoRF-AST design choices across the evaluated deployment settings.
  • Deployment significance: Observation-only calibration during deployment supports long-term probabilistic full-field reconstruction and safer operation under changing conditions.The study offers design insights for diffusion- and flow-based generative methods in engineering probabilistic virtual sensing.

CRediT authorship contribution statement

The authors contributed across data curation, analysis, methodology, validation, supervision, project administration, resources, writing, visualization, and conceptualization.

  • CRediT authorship contribution statement: Wingho Feng led data curation, formal analysis, methodology, validation, writing, and visualization.Feng contributed to both the original draft and review and editing.
  • CRediT authorship contribution statement: Quanwang Li contributed supervision, project administration, and resources.
  • CRediT authorship contribution statement: Ming Zhong and Jingyu Yang contributed project administration and resources, while Chen Wang contributed conceptualization, methodology, and writing–review & editing.
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