probabilistic uncertainty estimation
in structural health monitoring. These
methods integrate probabilistic models
with multi-sensor data to quantify uncer-
tainty more robustly than deterministic
approaches. For instance, Xue et al. [96]
demonstrated how Monte Carlo simula-
tions derive probability density functions
for multi-sensor fusion, enabling precise
error prediction and measurement
uncertainty minimization. Similarly,
Monte Carlo variance propagation has
been applied to model lidar-based
system uncertainties, improving point
cloud accuracy in dynamic environ-
ments [97].
Computational efficiency improve-
ments are critical for real-time applica-
tions. Adaptive Monte Carlo localization
optimizes particle filtering to reduce
computational overhead while main-
taining accuracy. These advancements
enable practical implementations in
robotics, where robustness to sensor
noise and environmental variability
is essential [98]. To reduce noise and
enhance signal clarity, variational auto-
encoders (VAEs) apply Monte Carlo–
based denoising, filtering out sensor
inconsistencies and environmental
disturbances in ultrasonic guided wave
inspections [99].
PHYSICS-INFORMED AI FOR HYBRID
SENSOR DATA INTERPRETATION
Physics-informed machine learning
(PIML) integrates acoustic, electromag-
netic, and thermal wave propagation
models into AI-based discontinuity rec-
ognition systems, significantly improving
uncertainty-aware discontinuity sizing
and shape reconstruction. Unlike purely
data-driven AI models, PIML combines
the physical laws governing NDE wave
interactions with deep learning method-
ologies to enforce physical consistency.
It is especially effective in high-stakes
applications such as turbine diagnostics,
smart factory inspections, and nuclear
component evaluations.
PIML combines the physical laws
governing NDE wave interactions with
deep learning methodologies, ensuring
that inspection results are not only data-
driven but also physics-constrained.
This approach is widely used in smart
factory inspection systems, real-time
turbine engine diagnostics, and nuclear
reactor component evaluations, where
high-confidence discontinuity detection
is essential. Şahin et al. [100] demon-
strated how physics-informed neural
networks outperform purely data-driven
methods in civil structure surrogate
modeling by fusing sensor data with
elasticity equations. Bayesian probabi-
listic neural networks (BPNNs) further
advance uncertainty quantification
by incorporating real-time posterior
updates, supporting adaptive predic-
tive maintenance strategies [101]. These
advancements improve decision confi-
dence in automated robotic NDE plat-
forms and digital twin–enabled quality
control systems, ensuring more reliable
and adaptive inspection processes.
The Role of Intelligent Maintenance
in Integrating UA&UQ
Unlike traditional manual inspections,
NDE 4.0 and intelligent maintenance
leverage automation and data-driven
decision-making, improving disconti-
nuity detection accuracy and reducing
operational downtime. The transition
to NDE 4.0 integrates cyber-physical
systems, AI-driven decision-making,
real-time analytics, and digital twins,
creating intelligent maintenance ecosys-
tems. UA&UQ are central to this evolu-
tion, ensuring that automated inspec-
tions are uncertainty-aware, adaptive,
and risk-informed.
PREDICTIVE AND CONDITION-BASED
MAINTENANCE WITH UA&UQ
Predictive and condition-based mainte-
nance (CBM) integrates UA to provide
probabilistic insights into equipment
degradation, enabling more informed
maintenance scheduling. These models
High-resolution and multi-
spectral sensors for enhanced
uncertainty quantification
Improved flaw sensitivity, depth-resolved
imaging, material-adaptive precision
The workflow for TFM imaging of composites
including four steps: analyzing the material’s
shape, measuring sound speed, tracking sound
paths, and creating images using PAUT [88].
The probe setup connects to the measurement
system, with two AMR sensors, generating
signals from magnetic flux leakage and
eddy current testing [95].
Multi-sensor data fusion and
Bayesian inference
Cross-modal integration,
confidence-weighted decision-making,
adaptive defect localization
Uncertainty-consistent variational model
ensembling (UCVME) is a semi-supervised deep
regression approach that improves pseudo label
quality by prioritizing samples with lower
uncertainty through higher weighting [98].
AI-enhanced signal processing
Real-time probabilistic reasoning,
physics-constrained interpretation,
uncertainty-aware automation
Figure 6. Advances in UA&UQ on industry adoption: (a) high-resolution and multi-spectral sensors (b) multi-sensor data fusion and Bayesian
interference (c) AI-enhanced signal processing.
NDT TUTORIAL
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UA&UQ
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M AT E R I A L S E V A L U AT I O N • A U G U S T 2 0 2 5

optimize inspection timing, reducing
unnecessary interventions while improv-
ing asset longevity. They are extensively
applied in aircraft fleet maintenance,
offshore wind turbine monitoring, and
nuclear plant condition tracking, sup-
porting data-driven decision-making for
critical infrastructure. For instance, the
benefits of UA in CBM are highlighted by
Bott et al. [102], where Bayesian frame-
works and cloud-ready pipelines are
applied for remaining useful life (RUL)
prediction in bearings and ball screws,
improving accuracy while quantifying
predictive uncertainty. Yardimci and
Cavus [103] introduced the Rashomon
perspective to address model multiplicity
in survival analysis, revealing how cen-
soring levels impact uncertainty in RUL
predictions. By embedding UA&UQ into
CBM, industries achieve smarter, cost-ef-
fective, and data-driven maintenance,
improving long-term structural integrity
and operational efficiency.
ROBOTICS AND AUTONOMOUS NDE
INSPECTION SYSTEMS
Robotics and autonomous systems are
transforming NDE by integrating UQ
into advanced perception and planning
algorithms, enabling them to navigate
complex and uncertain environments.
UAV-based structural health monitor-
ing is increasingly used in large-scale
inspections, where drones equipped
with thermographic, ultrasonic, and lidar
sensors provide real-time discontinuity
probability distributions [104]. Pairet et
al. [105] demonstrated how probabilistic
safety and online mapping allow auton-
omous underwater vehicles to perform
inspections in unpredictable conditions.
Innovations like the EELS robot illus-
trate how autonomous systems adapt to
unknown terrains, such as icy moons, by
quantifying risks and planning motions
under uncertainty [106]. Additionally, a
semantic-aware framework is introduced
for inspection in unknown environ-
ments, emphasizing active uncertainty
reduction. These advancements demon-
strate how robotics and autonomous
systems, empowered by UA, are revolu-
tionizing NDE inspections.
Outlook: Digital Twins,
AI-Enhanced UA&UQ, and
Automation
The next phase of UA&UQ in NDE will
be defined by advanced AI modeling,
digital twin integration, and fully auton-
omous robotic inspection systems.
These advancements will enhance real-
time discontinuity detection, predictive
maintenance, and adaptive inspection
planning—improving safety, efficiency,
and cost-effectiveness across industries
such as aviation, nuclear energy, and
offshore drilling.
DIGITAL TWIN–BASED PREDICTIVE
MAINTENANCE AND SELF-ADAPTIVE
INSPECTION
Digital twins will integrate multi-source
sensor data with AI-driven probabilistic
models, providing real-time updates on
structural health and optimizing inspec-
tion schedules based on failure risk
predictions. These self-adaptive mainte-
nance systems will reduce unnecessary
inspections, extend asset lifespan, and
improve failure risk forecasting [107].
Industries such as aerospace, energy,
and offshore operations will benefit
from dynamic, risk-informed inspection
strategies, minimizing downtime while
ensuring greater structural reliability.
AI-ENABLED EDGE COMPUTING FOR
INSTANTANEOUS UNCERTAINTY
ESTIMATION IN AUTONOMOUS NDE
AI-enabled edge computing, combined
with uncertainty-aware models, is revolu-
tionizing autonomous NDE by enabling
real-time discontinuity detection, risk
assessment, and adaptive inspection
without dependence on cloud infra-
structure [108]. Through local processing,
these systems minimize latency, reduce
bandwidth demands, and enable imme-
diate decision-making—essential for
predictive maintenance in safety-critical
environments. Self-learning AI and
quantum-enhanced sensors further
optimize inspections in complex, noisy,
or remote settings such as aerospace
composites, buried pipelines, and harsh
industrial facilities. IoT-connected smart
sensors enhance system adaptability by
forming probabilistic networks that adjust
inspection parameters based on real-time
risk modeling. Together, these edge-AI
and IoT innovations deliver scalable, resil-
ient, and uncertainty-aware NDE systems,
improving operational efficiency, safety,
and lifecycle reliability [109].
FULLY AUTONOMOUS INSPECTION
WITH SELF-OPTIMIZING UA&UQ
MODELS
The next generation of autonomous
NDE systems will feature self-optimizing
UA&UQ models, allowing robots to
perform real-time discontinuity assess-
ments with minimal human inter-
vention [110]. Bayesian-driven sensor
networks will enable proactive inspec-
tion scheduling, improving discontinu-
ity detection accuracy while reducing
maintenance costs. These self-learning,
real-time adaptive inspection workflows
will minimize unexpected failures and
enhance operational efficiency, setting
new standards for automated NDE and
structural health monitoring.
Conclusions
Uncertainty analysis and quantification
(UA&UQ) are essential for enhancing the
reliability and decision-making of NDE
applications across aerospace, nuclear
energy, transportation, and infrastruc-
ture sectors. This tutorial reviewed key
UA&UQ methodologies, addressing
factors such as measurement noise,
material variability, environmental
influences, and model uncertainties.
By examining probabilistic, statisti-
cal, simulation-based, and AI-driven
approaches, UQ contributes to improved
discontinuity detection, greater con-
fidence in inspection outcomes, and
more effective predictive maintenance
planning.
Advances in sensor technology
enable real-time uncertainty assess-
ment and failure probability estimation,
paving the way for adaptive inspections
and proactive maintenance. NDE 4.0,
with real-time sensor networks, digital
twins, and AI automation, is revolu-
tionizing discontinuity detection, while
UQ in advanced manufacturing miti-
gates process-induced uncertainties to
enhance discontinuity classification,
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