[1]Linda-Sophie Schneider

# Data-driven Modeling in Metrology - A Short Introduction, Current
Developments and Future Perspectives

Patrick Krauss  University Hospital Erlangen, Neuroscience Lab, Erlangen,
Germany, e-mail<EMAIL_ADDRESS>Nadine Schiering  Zentrum für
Messen und Kalibrieren & ANALYTIK GmbH, Bitterfeld-Wolfen, Germany, e-mail:
<EMAIL_ADDRESS>Christopher Syben  Development Center X-ray
Technology, Fraunhofer Institute for Integrated Circuits, Fürth, Germany,
e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>Richard Schielein  Development Center
X-ray Technology, Fraunhofer Institute for Integrated Circuits, Fürth,
Germany, e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>Andreas Maier  Friedrich-Alexander-
University Erlangen-Nuremberg, Chair of Pattern Recognition, Erlangen,
Germany, e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Mathematical models are vital to the field of metrology, playing a key role in
the derivation of measurement results and the calculation of uncertainties
from measurement data, informed by an understanding of the measurement
process. These models generally represent the correlation between the quantity
being measured and all other pertinent quantities. Such relationships are used
to construct measurement systems that can interpret measurement data to
generate conclusions and predictions about the measurement system itself.
Classic models are typically analytical, built on fundamental physical
principles. However, the rise of digital technology, expansive sensor
networks, and high-performance computing hardware have led to a growing shift
towards data-driven methodologies. This trend is especially prominent when
dealing with large, intricate networked sensor systems in situations where
there is limited expert understanding of the frequently changing real-world
contexts. Here, we demonstrate the variety of opportunities that data-driven
modeling presents, and how they have been already implemented in various real-
world applications.

##  1 Introduction

Due to increasing digitalization, processes in industry are changing at an
accelerating pace [1, 2]. In the future, entirely digitized production
processes are envisaged, in which complex networks of sensors, which can be
adapted to specific tasks, are used for monitoring, control, and prediction
[3, 4, 5]. In addition, there will be internet-based connectivity of the
sensors, similar to the Industrial Internet of Things (IIoT) [6]. Due to the
increasing availability of cost-effective IIoT-capable measurement devices,
the sensor networks are much larger and more complex than in traditional
measurement applications [7]. Since this also significantly increases the
amount of data collected, machine learning methods are increasingly used [8].
In addition, the role of modeling in the „Factory of Future“ is becoming
increasingly important through the use of digital twins, where a model of a
physical object is updated based on an evolving data set [9, 10, 11, 12]. In
light of these advancements in measurement data processing and analysis, it is
crucial to re-evaluate conventional metrological modeling methodologies.
Moreover, constructing accurate models of these processes is a necessity for
efficiently managing production and supply chain logistics, such as ensuring
quality control , as it enables precise decision-making based on reliable
data, minimizing errors and optimizing operational performance [13, 14].
Mathematical modeling is generally about translating all the relevant
components of a system or process into the language of mathematics. An
important part of this is the derivation of mathematical equations that
specify the relationships between the components [15]. In the context of
measurements, a distinction is made between mathematical terms that refer to
known or measured quantities and those that are unknown and must be estimated
from the measured data. The former includes quantities representing the
response and calibration of individual sensors, as well as applied corrections
to account for environmental effects. Unknown quantities comprise the stimuli
to the sensors and representations of system properties derived from these
stimuli [16, 17, 18, 19].
A model can be assembled by utilizing the governing physical laws, often
incorporating modular modeling approaches like port-Hamiltonian systems to
ensure a comprehensive and structured representation of the underlying
processes [20]. Expert knowledge and the underlying theory of the process can
guide the definition of dependencies among the considered quantities and
identify those relevant for modeling [21, 22, 23]. Conversely, one can employ
a data-driven model that discerns the relationships among variables based
solely on data, with no underlying theoretical backing [24, 25]. Numerous
models blend both these physical and data-driven elements, often referred to
as hybrid models [26, 27, 28].
Mathematical modeling is a crucial instrument that aids metrology and
applications dependent on measurements [29]. It underpins the creation of
sensors and measurement systems, allows for deductions and forecasts about
relevant quantities, and enhances comprehension of actual processes or
systems. A model furnishes a simplified yet abstract representation.
Furthermore, models are useful to gain an understanding of the functioning and
structure of the system under consideration. Such an understanding can be used
to generalize the functioning of the system to other scenarios [30, 31, 32].
Thus, models can form the basis for the simulation of a system. In systems
theory, for example, models are the basis for the description, analysis, and
synthesis of dynamic system behavior [33, 34, 35]. Crucially, models provide
the basis for the evaluation of uncertainty. Faulwasser et al. [36] provide a
comprehensive review aimed at enhancing systems theory by integrating
stochastic elements into the behavioral approach, drawing upon classical
concepts from Willems and Wiener [37, 38]. Their review aims to establish a
robust framework for data-driven control and analysis of stochastic systems,
bridging theoretical underpinnings with practical applications. This is
necessary to understand the quality of estimates of quantities, to use these
estimates for decision making and to ensure the traceability of measurement
results [39].
The current trend towards the digital transformation of the manufacturing
industry is driven by the use of large-scale networks, so-called „smart “
sensors and artificial intelligence to automatically make decisions about and
control production processes [4, 5]. This change brings several challenges for
modeling. In particular, the lack of physical models is compensated by the
availability of large amounts of sensor data, leading to a dependence on data-
driven models [24, 25]. However, it is still necessary that the decisions of a
data-driven approach are explainable and understandable [40], and that the
evaluation of uncertainty is possible in order to ensure trust in the
measurement results [41]. Further challenges arise from sensor redundancies
[42, 43, 44], synchronisation problems [45, 46, 47] or sensors of different
quality [48, 49, 50], among others [51].
Here, we provide an overview of the application of traditional analytical
parametric modeling to dynamic and distributed measurement systems.
Subsequently, the application of data-driven modeling to such systems is
motivated, and the different methods of data-driven modeling are discussed.
Finally, intelligent, adaptive systems are discussed and the concept of
digital twins is described in detail.

##  2 Modeling in Metrology

###  2.1 Aims of Modeling

A model is understood to be a representation of reality reduced to its
essential components - usually the representation of a process or system. A
model can be used to determine a measurement result and to explain, simulate
or evaluate a process or system. In addition, a model can also be used to
design a new sensor and, with its help, systematically and comprehensibly
design and build a system or process flow [22, 25, 24].

To understand and quantitatively assess complex systems and processes,
requires to simplify and abstract them significantly [52, 53, 54, 55, 56].
This leads to the inherent limitations and imperfections of such models,
constraining their ability to depict real-world behaviors accurately. However,
it's critical to ensure these models consider all vital elements and factors
impacting the intended results. In general, a model should meet three major
criteria [57]: First, a model should be falsifiable, i.e. there should exist
experimental paradigms to assess a candidate model. This is in line with
Popper’s ideas [58]. Second, a model should make quantitative predictions, as
opposed to purely qualitative, predictions [59]. Third, a model should be as
simple as possible, i.e., contain the smallest possible number of parameters
and assumptions. Hence, if two models explain a given system or process
equally well, the simpler one is considered to be the better one, a principle
referred to as Ockham’s razor [60].

Generally, a model consists of a set of mathematical equations involving at
least two quantities of interest. Furthermore, a model can generally be
represented graphically, in tabular form, as a flow chart or schedule, as
explanatory text, or in other ways [57].

###  2.2 Modeling of Complex Systems

Modeling of complex systems requires a process of system reduction,
abstraction, and decomposition that, in particular, involves the following
steps [61, 62, 63]:

First, _delimitation_ involves defining the scope of applicability for the
model. Models usually apply to specific situations or systems, and the
conditions under which they apply must be clearly stated.

Second, _abstraction_ which means that the model is developed to apply not
only to a very specific, individual or even unique system but instead to a
class of systems that behave in a similar way. Typically, this process
involves parameter adjustments to ensure the model can cover a range of
similar systems.

Third, in the _reduction_ step, the model is simplified to make it more
understandable. This comprises eliminating details or variables that have
little influence on the system's behavior. Usually, this process leads to
deviations from the real-world behavior of the system, which must be evaluated
and accounted for.

Fourth, in the step of _decomposition_ , the complex systems are broken down
into simpler subsystems, which can be more easily modeled. The challenge here
is to define the boundaries of these subsystems so that they form meaningful
functional units that are easier to understand and model [59].

Subsequently, in a process called _aggregation_ , the sub-systems (or sub-
models) created in the decomposition step need to be combined and re-assembled
again in order to form a complete model of the system under consideration.
This process must ensure that the sub-models fit together via adequate
interfaces to form a valid overall-model. Any deviations or errors occurring
at this stage may require revising the model.

Finally, in the _verification_ step, the resulting model is empirically tested
against real-world data, with the aim of identifying and evaluating any
discrepancies between model and reality. This may involve adjusting the model
to match the desired capabilities, taking into account system dynamics, time
variability, and response to external disturbances.

The described modeling approach is particularly suitable for complex systems
where understanding the interactions between various sub-systems is crucial.
It also highlights the iterative nature of modeling, where the model is
continually tested, revised, and refined to improve its accuracy and
reliability [57].
In metrology, modeling complex systems involves simplifying complicated
phenomena in order to better understand, predict, and control their behavior.
To illustrate the steps in this process, we examine a study on mathematical
and physical modeling of complex biological systems [64]. The model focuses on
the simulation of biological fluid flow in ophthalmology, with specific
conditions outlined for its applicability. Abstraction transforms complex
fluid dynamics into mathematical equations, facilitating the study of flow
patterns in similar systems. Reduction simplifies structures and interactions
into key variables for clearer understanding. Decomposition divides the system
into manageable subsystems, while aggregation integrates these models to
capture overall system behavior. Verification involves testing the integrated
model against experimental data to improve predictive accuracy.

###  2.3 Measurement Methods and Model Structure

####  2.3.1 Forward Models and Inverse Models

The traditional (cognitive-systematic) method of understanding a system or
process follows a path from cause to effect, or, in terms of time, from the
beginning to the conclusion of a process [65]. This method is frequently used
to create explanations, justifications, educational content, and more. It
mirrors the prevalent modeling approach in both science and engineering.
Consequently, it is the standard mode of depicting measurement systems and the
progression of signals, starting from the cause (the metric with its mostly
undetermined value) to its outcome, which is the shown or resultant signal
value of the measuring system [34].

In measurement technology, the initial step often involves a sequence of
measurements. The individual components, derived from breaking down the
complete measurement, are known as transmission elements. The quantity to be
measured is the input, the first in the series of transmission elements are
sensors or transducers, and the sequence concludes with the display or signal
output. Interfering or disruptive factors might be viewed as secondary input
variables. This cause-and-effect model of a measurement is useful for
simulating the core and critical behavior of a developing sensor or
measurement system [66, 67].

In the realm of metrology, forward modeling, also known as cause-effect
modeling, typically relies on foundational physical principles and associated
material measurements to produce or replicate reference values for specific
quantities. While the primary objective in metrology is to ascertain the most
precise value of a quantity and gauge its uncertainty, this often involves
tracing back from the displayed output to the original measured value. A
reverse model facilitates the extraction of details about the initial
measurement quantity from the presented value, insights into potential major
influencing factors, and the modeled procedure. Thus, measuring assessment
entails addressing what is termed an inverse problem [68, 69]. The resulting
inverse model is also called measurement equation [70], evaluation model or
measurement model [71]. It is obtained by inverting the cause-effect model or
the forward model, respectively, and is the starting point for applying the
uncertainty assessment methods described in the Guide to the Expression of
Uncertainty in Measurement (GUM) [71] and its supporting documents [72]. Data-
driven models, in particular adaptive and learning models, can usually be
taught directly in the "inverse direction". Whereas inversion of forward
models is generally possible, it often may result in numerically unstable
equation systems. To reduce this problem, regularisation techniques allow for
more robust inversion procedures [73].

####  2.3.2 White-, Grey- and Black-Box Models

Beyond categorizing models based on their directionality into forward and
inverse, another differentiation can be drawn considering the transparency-
opacity spectrum, leading to white-box, grey-box, and black-box models [74,
75, 76, 77, 78].

The mathematical terms used to represent the relevant components or quantities
of a system or process generally include model parameters that serve to
describe the dependencies between these terms. The nature and significance of
such parameters depend on the nature of the model. For example, in a model
derived from physics, the model parameters usually refer to specific physical
properties, such as temperature coefficient, mass or time. Such models are
called white-box models [78]. Sometimes, however, the dependencies cannot be
derived from physical arguments alone. For example, linear interpolation
through a point cloud may be derived from correlation arguments rather than
purely physical considerations. However, the parameters obtained in this way
can still carry a physical interpretation. Models of this type are thus a
mixture of a direct relationship to a known quantity (physical or otherwise)
and a more abstract relationship. Such models are called grey-box models [79,
80, 81, 82, 83]. If there is no direct relationship derived from physical or
other arguments, the model is usually called a black-box model [81, 78]. In
Figure 1 an overview of these three types of models is provided.

Modeling in Metrology Input: – Data with unknown meaning Model: – Known
physical relations – Prior knowledge Analytical Modeling (White-Box Modeling)
Input: – Data with limited known meaning Model: – Arbitrary functional
relations – mathematical – statistical – learned – neuronal Data-driven
Modeling (Black-Box Modeling) Input: – Data with unknown or only partially
known meaning Model: – Mixture of physical and functional relationship –
mathematical – statistical – learned – neuronal Mixed Modeling (Grey-box
Modeling) System characterisation – Measurement uncertainties –
Reproducibility and comparability – Bayesian description of rules –
Explainability of decision making Fig. 1: White, grey and black box models
with their basic properties [84].

The general principles described above apply equally to white-, grey- and
black-box models. Depending on the choice of method, generalization and
transferability can be challenging, as generally, learning only guarantees
stability if the training data is drawn from the same distribution as the test
data. In particular, parameter-rich black-box models typically suffer from
such problems, as described in section 4. To compensate for such data
maldistribution, the discipline of transfer learning or adaptation has been
developed, which can contribute to the generalization of black-box models [85,
86, 87, 88, 89, 90, 91]. However, domain transfer is more difficult than
classical modeling methods [92] due to the limited interpretability of the
learned parameters [85].

##  3 Analytical Modeling: White-Box Models

###  3.1 Overview

Analytical modeling is essentially the study of the structure of a system and
is therefore also known as structural modeling [61]. The aim is to represent
the (physical) properties of the system, including all relevant influencing
factors and interactions, as accurately as possible in a mathematical model
[62]. The result is a model with fully known properties, often referred to as
a white box model. Typically, the quantities used for such modeling are
tangible physical quantities such as pressure, temperature, flow rate,
irradiance, or angle [63].  Moreover, physics-based approaches like port-
Hamiltonian systems offer a unified framework for modeling complex systems
across various physical domains in analytical modeling. These approaches
emphasize energy as a universal language, enabling robust analysis and control
across diverse domains such as mechanical, electrical, and hydraulic systems
[20].

The construction of an analytical model requires an adequate physical
understanding of the system and all its influencing elements. However,
unknowable or unobservable influences can be represented as random variables,
serving as a form of model reduction [93, 94, 95, 96, 97, 98]. This modeling
approach inherently involves aspects of decomposition and aggregation through
the chosen selection and merging of the subsystems used [99].

This approach is useful when the components of a system, together with their
properties, can be explicitly defined for the synthesis of a new system. The
basic behavior of the system is then directly determined by the selected
system components, their respective behavior, and the way they are combined
(e.g., by a block diagram). Finite element models, where the interaction
between elements is represented by specific physical relationships, also fall
into the category of analytical models [100].

While the fundamental behavior (e.g. dynamics) of the system is determined by
specifying the system structure, an analytical model can be tuned to physical
reality by modifying model parameters (e.g., gains, time constants). The
identification of these parameters can be achieved through specific
experiments (such as the application of impulse or step functions or
harmonics) or during system operation by comparing expected and actual system
responses. If this step of parameter adjustment fails to produce the desired
quality of system modeling, structural modifications can be applied to the
model to account for the causes of deviations that have not been previously
considered [101, 102, 103].

Prior knowledge of the system structure is essential for the development of
analytical models. The level of detail in the known system structure
determines the accuracy achievable by the model. In addition, incorporating
prior knowledge of the model parameters can be beneficial; typically,
empirical optimization of these parameters (in a data-driven manner) improves
the quality of the model fitting [101, 102, 103].

Projection-based model reduction methods complement the principles of white
box modeling by simplifying detailed, physics-based models into more
computationally efficient forms while maintaining their interpretability and
accuracy [104]. These reduction methods start with high-fidelity models
grounded in the system's physical laws and use projection techniques to create
lower-dimensional models that still reflect the essential dynamics and
parameters of the original systems. This process preserves the transparency
and interpretability inherent in white box models, allowing for meaningful
insights into the reduced model's behavior in relation to known physical
principles. Moreover, just like in white box modeling, these reduced models
undergo rigorous verification against full-scale models or empirical data to
ensure their fidelity, making projection-based model reduction a natural
extension of white box modeling that balances detail with computational
efficiency.

###  3.2 Uncertainty Evaluation

In white box modeling, uncertainty evaluation typically involves deterministic
methods where uncertainties in input parameters are propagated through the
model to assess their impact on the output. This process often utilizes
sensitivity analysis, which quantitatively determines how changes in input
variables influence the model's predictions. A prominent method in this
category is the application of Taylor series expansions to approximate the
effect of input uncertainties on the output, allowing for a clear
understanding of how each input contributes to the overall uncertainty [105].
This approach provides a direct link between the model inputs and outputs,
facilitating the identification and minimization of key sources of
uncertainty. An example of this method in metrology is the development of a
simulation framework designed to evaluate measurement uncertainty in detailed
real-life measurements, supporting structured modeling scenarios and
introducing various techniques to optimize these scenarios, as discussed by
Wolf [106].

###  3.3 Advantages and Limitations

The advantages of analytical modeling are primarily in its interpretability,
as the quantities and functional blocks involved typically allow a tangible
physical interpretation [94]. This often allows an easy plausibility check of
the fundamental model properties. By representing specific physical system
properties, analytical models often facilitate an accurate system description
over a wide range of values of the modeled quantities [101, 107]. Of course,
the validity range defined in the delimitation step of model generation must
be taken into account [108].

A disadvantage of analytical models is the requirement for a comprehensive
physical understanding of the system in terms of its structure, function and
any subsystems present, which typically requires specialized knowledge of the
system [109]. If the structure of the system isn't known in advance,
considerable effort may be required to analyze the system and parameterize the
model [110]. This is essential in order to identify relevant influencing
factors and cause-effect relationships or to carry out appropriate experiments
[101, 111].

##  4 Data-driven Modeling: Black-Box Models

###  4.1 Overview

Data-driven modeling is based on the analysis of the data that characterize
the system under study. A model is then defined based on the relationships
between the state variables (input, internal, and output variables) of the
system [24]. Data-driven modeling is essentially implemented with machine
learning or pattern recognition methods. Behind both terms is the problem of
making automated decisions, for example, distinguishing apples from pears. In
the traditional literature [112], this process is outlined using the pattern
recognition system (cf. fig. 2). Since the internal structure of the model is
not known, such a model is also called a black box model.

Fig. 2: Diagram illustrating the conventional pattern recognition system used
for automated decision making. The sensor data is pre-processed and "hand-
crafted" features are extracted during both the training and testing phases.
In the training phase, a classifier is developed, which is then used in the
testing phase to automatically determine the classes (Figure reprinted under
CC BY 4.0 [112]).

Within a training phase, the so-called _training data set_ is processed and
meaningful _features_ are extracted. While pre-processing remains in the
original data space and involves operations such as noise reduction and image
rectification, feature extraction is faced with the task of determining an
algorithm that would be able to extract, for example, a distinctive and
complete feature representation. This task is usually difficult to generalize.
Therefore, it is necessary to redesign such features for each new application
essentially. In the deep learning literature, this approach is often referred
to as ``hand-crafted features''. Based on the feature vector $\bm{x}\in^{n}$,
the _Classifier_ must predict the correct _Class_ $y$. This is typically
estimated by a function $\hat{y}=\hat{f}(\bm{x})$, which leads directly to the
classification result $\hat{y}$. The parameter vector $\bm{\theta}$ of the
classifier is determined during the training phase and later evaluated at an
independent _test data set_. A detailed description of this process is given
in [113].

Data-driven modeling eliminates the need to analyze and map already known
structures, physical interactions, and similar properties of the system.
Instead, the system is described only by the interaction with its environment
at the system's inputs and outputs [20, 36]. Since no knowledge of the
internal structure of the system is required to create the model, this method
can also be applied in cases where the system structure and the interaction of
the components in the system are not or insufficiently known, e.g. in existing
systems whose documentation is incomplete. Data-driven models can also be
advantageous when the interactions in the system are difficult to describe or
parameterize, e.g., in the case of strongly non-linearly coupled state
variables. An analysis of the system and a complex parameter identification
thus become superfluous [84].

Building upon the advantages of data-driven modeling, particularly in
scenarios with strongly non-linearly coupled state variables, the Extended
Dynamic Mode Decomposition (EDMD) method offers a robust framework for
effectively capturing and predicting system behavior [114]. This approach
leverages the Koopman operator to linearize complex dynamics in an infinite-
dimensional space, subsequently approximated by EDMD in a finite-dimensional
subspace, enabling linear analysis and control methods to be applied to
nonlinear systems. The essence of EDMD in data-driven modeling lies in its
ability to use observable functions, such as sensor data, to construct a data-
driven surrogate model for predicting system dynamics. Error analysis, such as
finite-data error bounds, plays a crucial role in assessing the accuracy and
reliability of Koopman-based predictions and controls, providing a
probabilistic measure of the approximation and prediction errors [115]. The
theoretical foundation of this method is rooted in the Koopman framework,
which adeptly manages strong non-linearities by transforming them into a
linear, albeit infinite-dimensional, model, thus simplifying analysis and
expanding the scope for advanced control strategies, including kernel-based
methods [116].

In data-driven models, an implicit reduction takes place insofar as only those
variables that are observable as input or output variables in the interaction
with the environment can be considered in the system modeling. The internal
structure and the internal variables of the system, on the other hand, are not
included in the modeling.

Data-driven models, like analytical models, generally contain internal state
variables. However, unlike analytical models, these variables are usually not
physical quantities. Rather, the model variables are synthetic quantities,
which are therefore not always unambiguously interpretable.

As the most important prior knowledge for the creation of a data-driven model,
the determining input and output variables must first be known. In addition,
these variables must be observed in all relevant states, whereby the dynamics
of the variables must also be represented. The selection of data for typical
states used in model building always leads to a limitation of the model
created. The learning of the system behavior on the basis of such typical
observations implicitly leads to the storage of this behavior as a "good
state" of the system, whereby, for example, deviations of the observations
from the created model become recognizable.

###  4.2 Uncertainty Evaluation

Black box models, due to their empirical nature, rely heavily on data-driven
methods for uncertainty evaluation. Monte Carlo simulations are widely used in
this context, providing a means to generate a distribution of possible
outcomes based on repeated random sampling. This approach is particularly
valuable for complex systems where analytical methods may not inspire
confidence or may be difficult to apply [117]. Monte Carlo Simulation allows
for the assessment of the uncertainty of measurement systems by simulating a
range of possible outcomes using random variables. The method is beneficial
for evaluating the uncertainty of various measurement systems, including those
that cannot be readily addressed by conventional analytical means, due to its
ability to take account of partially correlated measurement input
uncertainties and its adaptability to complex, nonlinear measurement
equations. The key advantage of this approach is its ability to handle non-
linear relationships and model interactions without a detailed understanding
of the underlying processes. Brando et al. [118] explored the addition of
uncertainty scores to black-box predictions, a crucial development for real-
world applications where predictive models are often used as black boxes. This
work demonstrates the feasibility of quantifying uncertainty in black box
models without accessing their internal workings.

###  4.3 Advantages and Limitations

An advantage of data-driven modeling is that no explicit physical
understanding of the system is required, so modeling can also be performed for
complex systems or systems whose internal structure is unknown. By limiting
the learning process to observed system states, data-driven models are well
suited for the detection of anomalies, i.e. significant deviations of the
current system state from a desired or normal state [119].

However, data-driven modeling also has limitations and drawbacks. When
observing the system, typical inputs and outputs of the system must be
available in all relevant forms. If these observations are not available, the
model may be incomplete, i.e. the simplification is too strong, or the
delineation is not correct. To ensure that the relevant variables are observed
and that typical observations and system states are selected in the data
selection, a minimum level of system knowledge is required so that expert
knowledge cannot be completely dispensed with. A drawback in the plausibility
check of models is the aforementioned difficulty in interpreting the model
variables [120]. In addition, as mentioned above, data-driven models tend to
have comparatively low generalisability, since the restriction of the learning
process to an observed operational domain corresponds to a narrowing down to
exactly that domain.

However, low generalisability may be desirable: In anomaly detection, the
restriction to a certain (error-free) operating range is intended, so that in
this case a targeted restriction of the learning process to an observation
dataset that is rated as good takes place. Furthermore, data-driven models are
usually only slightly abstract. Since the internal structure of the system is
not known or used in the modeling, the formation of classes of similar systems
is difficult and can only be achieved phenomenologically by analyzing the
interactions of the system with its environment.

##  5 Hybrid modeling: Grey-Box Models

###  5.1 Overview

Analytical and data-driven modeling are not necessarily in competition with
each other, but can rather be combined to exploit the advantages of both
approaches. An important reason for this mixing is that the choice of
approaches often depends on the professional background of the actors. For
example, while engineers are more interested in understanding a system and
therefore tend to use analytical models, computer scientists are used to
dealing with data and therefore often prefer data-driven models [84].

Although in analytical modeling, the physical structure of the system is
defined and the model parameters are initially determined by analyzing the
system, the quality of the parameter fit can usually be improved by later
data-driven optimization. This approach of "data-driven optimized analytical
modeling" can be used advantageously, for example, if the internal state
variables of the system are known but are coupled by unknown or difficult-to-
identify (e.g., strongly non-linear) interactions. In a pure white-box model
approach, the unknown interactions would have to be included using
probabilistic arguments or as model uncertainties. A combined approach of
analytical and data-driven modeling can reduce the overall uncertainty.

If a system modeling is to be essentially data-driven instead, it can still be
advantageous to first structurally decompose the overall system into a few,
easily identifiable subsystems. Depending on the type of subsystems, this
approach of "analytically structured data-driven modeling" can have the
advantage that the emerging subsystems can be described more easily in several
models that are initially considered separately.

In a recent paper, it was found that the inclusion of known subsystems is
generally favorable with regard to the maximum error bounds [Maier 2019b]. The
paper shows that the inclusion of prior knowledge in the form of
differentiable modules always reduces the maximum error that can be generated
by the learning problem. The authors demonstrate the applicability of their
theory in grey-box models that incorporate differentiable modules in the
trainable CT reconstruction, in the measurement of retinal vessels using
hybrids between deep learning and classical image processing, and in the
application of image re-binning. The trend towards integrating differentiable
known modules into deep networks can thus also be traced back to a solid
theoretical basis and shows that classical theory and novel data-driven
methods are not in contradiction. Instead, solutions that combine the best of
both worlds are preferable. Deep networks can even be reverse-engineered to
identify relevant processing modules within the deep network, as shown in
[121].

Today, we see a steady rise of deep learning models. Obviously, this paper is
not able to summarize the totality of deep learning methods. Even focusing
only on the domain of medical image analysis would be far beyond the scope of
a single paper. However, quite successful attempts to do so can be found in
the literature. Ker et al. [122] explore the diverse applications of deep
learning in medical imaging, demonstrating its ability to improve both
diagnostic accuracy and operational efficiency. Litjens et al. [123] provide
an in-depth examination of the transformative impact of deep learning on
medical imaging, highlighting improvements in accuracy and processing speed.
Maier et al. [124] explore the combination of traditional and deep learning
approaches, suggesting innovative ways to advance the field. Despite the
comprehensive insights provided by these studies, they do not fully capture
the full spectrum of deep learning research in medical imaging. In particular,
areas such as image synthesis, multimodal image fusion, and patient-specific
modeling and prediction remain less explored in these reviews. This limitation
underscores our focused approach, in which we discuss the concepts of grey-box
modeling at a high level and illustrate them with selected examples.

The following sections briefly summarise important developments in the field
of grey-box models and their relation to hybrid models. It will first look at
the field of Deep Learning from an overarching perspective and show how
different approaches succeed in introducing an inductive bias.

A competing approach, as already mentioned by Sutton [125], is meta-learning.
Therefore, a brief review of the literature will be provided, summarising
relevant methods that are current research at the time of publication of this
paper. Finally, we will look at methods that implicitly or explicitly
introduce prior knowledge into deep networks and summarise their strengths and
weaknesses.

####  5.1.1 Deep learning

Deep learning is generally regarded as an artificial neural network with many
layers [126]. However, interpretations of "many" vary widely in the literature
from "more than three" to thousands of layers. The network that made a major
breakthrough in image classification was Alex-Net [127]. Alex-Net was then
able to approximately halve the error in the image net challenge. In
particular, the introduction of specialized operations in networks such as
convolutional layers seems to be a key factor for the success of Deep
Learning. It seems that the invariances introduced by such layers are
beneficial both in terms of parameter reduction and in terms of incorporating
_inductive bias_ in the network.

While convolutional layers are probably the most popular layer used in Deep
Learning, many other invariances can also be encoded in network layers.
Bronsteinet al. summarised several important invariances in a recent
publication [128]. As they demonstrate, lattices, groups, graphs, geodesics
and meters provide suitable invariances that lead to certain types of layers,
ranging from convolutions to graph layers to recurrent layers. Each of these
layers is capable of describing a particular data invariance that can be
exploited in a particular mathematical operation. However, there are many
other types of layers that use such invariances, such as tensor networks,
which also allow the encoding of invariances corresponding to higher-order
tensor products. Such techniques are also already used in medical imaging,
e.g. in image segmentation [129]. A question that currently cannot be answered
adequately in the literature is which invariances should be used in which
specific order and configuration. A key method to address the problem seems to
be the „graduate student descent“method.

####  5.1.2 Meta-learning

Meta-learning attempts to learn the _inductive bias_ directly from the data
itself. The general approach aims to learn similarities across different
tasks. Although a full summary of meta-learning methods is beyond the scope of
this paper, only some highlights of the past years are described in this
section.

A model-independent approach to meta-learning is presented in [130]. Here, the
authors propose to separate the meta-task parameters from the task-specific
ones. This is achieved by modeling two sets of parameters: one for the generic
task and one for the specific task. The developed training strategy produces
stable parameters for the meta-task and only a few gradient iterations with
few samples allow the adaptation to a specific application. Several other
approaches can be found in the literature, ranging from learning invariances
[131] to learning prototypical networks [132] to neural architecture search
[133]. However, the differences between the methods are small and vary from
dataset to dataset.

Unfortunately, a recent study suggests that simply learning the task-dependent
nonlinear distance in the feature space [134] may prove to be as effective as
the above meta-learning approaches without modeling the meta-task at all. The
paper shows that common architectures used in Deep Learning are sufficient to
accomplish this task.

It is clear that meta-learning has a high potential for future applications in
modeling inductive bias. However, none of the methods found in the literature
can demonstrate this for practical applications while outperforming current
deep learning methods.

####  5.1.3 Pre-knowledge as regularisation

Pre-knowledge as regularisation is an approach that attempts to incorporate
prior knowledge into deep learning models. The most common approach is to use
the loss function to embed prior knowledge using a regularisation term [26].
As such, the loss is extended by an additive part that penalizes the model if
it deviates from a priori knowledge given as an analytical model during
training. A major drawback of this approach is that the deviation from the
physical model is only penalized during training, and there is no mechanism to
actually guarantee the plausibility of the model during the testing period.
Another very common approach to making deep learning models plausible is to
embed them in a constrained learning framework. This is often done through
reinforcement learning or imitation learning. This approach has proven to be
very effective in games, as the rules of the game can be used and
sophisticated search algorithms such as Monte Carlo tree search [135] can be
embedded in the machine learning algorithms. This approach is also very
popular for medical imaging applications, as shown in [136]. Unfortunately,
the reinforcement learning setup is computationally expensive during training,
and many events of the present learning task have to be repeated to train a
good network.

####  5.1.4 Known operator learning

Known operator learning, a term introduced in earlier work, is an approach to
embed analytical models directly into deep networks already known from
classical theory. It is based on the assumption that the function to be
learned can be decomposed into modules, some of which are known and some of
which have to be learned from data. The term's specific use in our context
aims to highlight this unique integration, distinguishing it from the broader,
more general use of "operator learning" in a functional-analytic sense.

Deep networks are often observed to organize themselves into such modular
configurations, supporting the viability of this approach [137]. Incorporating
known operations into the network architecture not only reduces the maximum
error bound [138] but also decreases the number of parameters that need to be
trained. This reduction in trainable parameters can lead to a smaller required
training dataset. Moreover, this method has been shown to enhance the
generalization capabilities of the network, a benefit that has been
empirically verified [139].

Fig. 3: Today, several methods are commonly used to introduce an _inductive
bias_ into machine learning models. They range from "graduate student descent"
in Deep Learning to Meta Learning and regularisation methods. The image is
reprinted under CC BY 4.0 [124].

####  5.1.5 Physics-Informed Learning

Physics-informed learning, particularly through models like Physics-Informed
Neural Networks (PINNs), incorporates physical laws, often in the form of
differential equations, directly into the learning process [140]. This
approach ensures that the learned models adhere to known physical principles,
which is particularly useful when data is scarce or noisy. For instance,
Physics-Informed Neural Operators (PINOs) utilize both data and physics
constraints to learn the solution operator of parametric Partial Differential
Equations (PDEs), offering a hybrid solution that overcomes the limitations of
purely data-driven methods and purely physics-based approaches. PINOs can
incorporate data and PDE constraints at different resolutions, allowing for
the combination of coarse-resolution data with higher-resolution PDE
constraints without loss in accuracy, a property known as discretization-
invariance. This enables efficient learning even in scenarios where no data is
available, by optimizing PDE constraints across multiple instances rather than
a single instance, as seen in other approaches like PINNs [141].  A recent
paper [142] showed that learning the entire hydrogen dissociation curve is
possible and experimentally validated the generalization properties predicted
by the well-known operator learning theory. The approach is also suitable for
extracting symbolic equations using graph networks, as shown in [143].

###  5.2 Uncertainty Evaluation

In grey box modeling, uncertainty evaluation often involves statistical
methods to estimate the uncertainty associated with empirically derived model
parameters. This approach effectively bridges the gap between theoretical
models and empirical data, making it particularly useful in systems lacking
complete theoretical understanding. Bayesian inference is a common technique
used in this context, allowing for the incorporation of prior knowledge about
the parameters into the uncertainty evaluation process [144]. This approach is
based on Bayes' theorem, which updates the probability for a hypothesis as
more evidence or information becomes available. It provides a principled way
to combine new data with prior beliefs, and is particularly useful in complex
systems where the true state may not be directly observable. For example,
Bayesian methods have been applied to the evaluation of measurement
uncertainty, where they can fully integrate prior and current sample
information, determine the prior distribution based on historical data, and
deduce the posterior distribution by integrating the prior distribution with
current sample data using the Bayesian model. This enables the optimization
estimation of uncertainty, reflecting the latest information on the accuracy
of the measurement system. Boumans discussed a more objective Type B
evaluation achieved through model-based uncertainty evaluations involving
grey-box modeling and validation, highlighting the application of grey-box
approaches in metrology [145].

###  5.3 Advantages and Limitations

Hybrid modeling offers a balanced approach by integrating both data-driven
elements and physical knowledge of the system. This dual approach enables the
modeling of complex systems where some aspects of the internal structure may
be understood, while others might be too intricate or unknown. By
incorporating physical laws or system dynamics into the models, grey-box
approaches can enhance interpretability and reliability, particularly in
extrapolating beyond the range of observed data.

Unlike purely data-driven models that rely entirely on observed states and can
struggle with generalizability and interpretability, grey-box models leverage
known system equations or behaviors to structure the model. This can
significantly improve the model's ability to generalize from limited data by
grounding the predictions in established physical principles. For instance, in
a mechanical system, equations of motion could guide the model structure,
ensuring that predictions adhere to fundamental laws like conservation of
energy or momentum.

However, grey-box modeling also requires careful consideration. The model's
effectiveness hinges on the accuracy and relevance of the incorporated
physical knowledge. Incorrect assumptions about the system's dynamics can lead
to models that are no more reliable than their purely data-driven
counterparts. Moreover, the process of integrating physical knowledge into the
model requires a deep understanding of the system, which might not always be
available or might necessitate significant expertise.

In metrology, the application of Grey-Box Models enhances the precision and
reliability of measurements in complex systems where direct quantification can
be challenging. By integrating physical principles with empirical data, these
models are instrumental in calibrating measurement instruments, dynamically
compensating for environmental influences, and quantifying measurement
uncertainties more accurately. For instance, in scenarios where environmental
conditions such as temperature and humidity affect measurement accuracy, Grey-
Box Models can adjust measurements in real-time, ensuring consistent accuracy.
Additionally, they prove invaluable in non-invasive measurement techniques,
allowing for the estimation of inaccessible quantities through indirect
measurements and known physical relationships. This hybrid approach not only
improves the direct application of metrological instruments but also aids in
proactive maintenance by predicting performance degradation due to wear and
tear, thereby maintaining the integrity of precision measurements. While
offering significant advantages in enhancing measurement precision and
reliability, they also present limitations, particularly in their dependency
on the accuracy of the embedded physical models. If the incorporated physical
principles do not fully capture the system's dynamics or if they are based on
incorrect assumptions, the model's predictions can be misleading, potentially
compromising measurement accuracy. This limitation underscores the importance
of comprehensive system understanding and the need for meticulous validation
of the physical models used. Grey-Box models, therefore, enhance metrology by
improving measurement capabilities and reliability, but their effectiveness
hinges on the accuracy of the integrated physical models to ensure measurement
integrity.

##  6 Intelligent Adaptive Systems

Intelligent adaptive systems refer to advanced technological frameworks that
autonomously adjust and optimize their operations in response to varying
environmental conditions and user requirements. These systems combine
artificial intelligence, machine learning, sensors, actuators, and control
logic to create dynamic solutions that can learn from and adapt to new
situations without human intervention. In the context of metrology,
intelligent adaptive systems play a transformative role.

Intelligent adaptive systems in metrology involve the integration of sensors,
actuators, and control logic into measurement systems to enhance their
adaptability, precision, and responsiveness. These systems can dynamically
adjust their behavior or operation in response to changes in the environment
or measurement conditions, improving the accuracy and reliability of
metrological processes. For instance, in aerospace metrology, intelligent
structures equipped with sensors and actuators can modify their mechanical
states or characteristics—like position, velocity, stiffness, or damping—to
maintain measurement accuracy under different operational conditions [146].

This section explores the transformative impact of intelligent adaptive
systems and digital twins in metrology, highlighting their integration of AI,
machine learning, and real-time sensor data to improve measurement accuracy
and adaptability. It explores the seamless fusion of physical and digital
entities through digital twins, enabling a continuous feedback loop that
refines metrological practices. It also outlines advances in intelligent
learning systems, such as the artificial neural twin and reservoir computing,
and their role in improving predictive accuracy and system adaptability. Using
examples such as soft sensors in bioprocess control and the Frangi filter in
advanced imaging, this section illustrates the multiple benefits and
challenges of these innovative technologies in metrology.

###  6.1 Intelligent Adaptive and Autonomous Systems

In the specialized domain of Intelligent Adaptive and Autonomous Systems, the
adaptive capabilities of intelligent systems are extended to include autonomy
in decision making and task execution. These systems are self-sufficient,
using advanced AI to react to changes and independently handle complex
situations. After learning, these systems can be either static, optimized for
a single, predefined task, or flexible, capable of generalizing their learned
knowledge to a variety of related challenges. Incorporating principles of safe
and continual learning further ensures that these adaptations occur without
compromising system integrity or performance over time, thereby broadening
their applicability and effectiveness in dynamic settings [147]. To illustrate
this, we take a closer look at two examples here.

The first example discusses the use of soft sensors in industrial processes.
Soft sensors, also known as virtual sensors, are a type of machine learning
that aims to infer unmeasurable process variables from available sensor data.
These algorithms are designed to learn meaningful representations from
operational data, often without requiring explicit human-crafted features or
extensive domain knowledge. We use the example of applying soft sensors to
bioprocess control, specifically by applying a soft sensor for sequential
filtering of metabolic heat signals [148]. The approach uses temperature
sensor signals from a bioreactor's cooling system to estimate a microbial
culture's metabolic heat. This estimation allows for the derivation of
specific growth rates and active biomass concentrations, which can enhance
bioprocess monitoring and control capabilities. The robustness of the soft
sensor is significantly enhanced by implementing sequential digital signal
filtering. This improvement makes it suitable for industrial applications
where cultures generate low metabolic heat in environments with high noise
levels.

Fig. 4: Illustration of the bioprocess control using soft sensors based on
sequential filtering of metabolic heat signals [148].

This method relies on a data-driven structure, which is beneficial for the
specific task of bioprocess control. However, the representations learned are
highly specialized, making applying them directly to unrelated tasks almost
impossible without substantial retraining or adaptation. This underscores the
importance of customized approaches to the application of soft sensors in
specific domains.

The situation is different when using a trained Frangi filter [149]. The
adaptation of the Frangi filter into a trained neural network model has
notably advanced the automatic segmentation of the retinal vascular tree, a
task complicated by the intricate nature of retinal vessels and often
compromised image quality. Originating from Frangi et al.'s [150] vascular
enhancement methodology, which relies on the eigenvalue analysis of the
Hessian matrix at various Gaussian scales, the trained Frangi filter
incorporates the strengths of neural networks to refine segmentation accuracy.
This hybrid approach not only preserves the inherent vessel-identifying
capabilities of the original Frangi filter but also capitalizes on the
adaptive learning potential of neural networks, resulting in significant
improvements in identifying vessels of varying sizes in retinal images.

(a)

(b)

Fig. 5: Results of the vessel segmentation before and after training. The
colors red, green, and yellow represent the manual annotations, the
segmentation results, and the overlaps between them, respectively [149].

Expanding its utility beyond ophthalmic imaging, the Frangi filter has also
been adapted for segmenting fat and fascia in ultrasound images, as
demonstrated in Rybakov et al.'s study [151]. This application underscores the
filter's versatility, effectively identifying fascia—which appears similar to
white vessels on ultrasound images—and fat as a thick, dark, tubular
structure, thereby streamlining the segmentation process for biological
research and clinical diagnostics.

Furthermore, the versatility of the Frangi filter extends beyond medical
imaging. Its application has proven effective in the automation of book page
extraction from high-resolution 3D scans [152]. Here the biggest challenge
lies in accurately distinguishing and separating the individual pages within
the tightly compressed and highly detailed 3D volume.

In the domain of metrology, the Frangi filter finds a novel application in the
enhancement of vessel-like structures within optoacoustic imaging, a technique
that merges optical and acoustic imaging to provide detailed insights into the
properties of vascular structures. A pivotal study by Longo et al. [153]
rigorously assessed the Hessian-based Frangi vesselness filter's efficacy in
optoacoustic images, validating its capability to accentuate vessel-like forms
in phantom models and further examining its performance in vivo with the aid
of gold nanorods. This research illuminated the significant impact of factors
such as contrast, filter scales, and angular tomographic coverage on image
quality, while also addressing the potential for artifact generation that
could lead to misinterpretations. The findings offer valuable recommendations
for the judicious application of the Frangi filter and similar vesselness
filters in optoacoustic imaging, ensuring the accuracy of measurements and
interpretations crucial in the metrological evaluation of vascular structures
[153].

The Frangi filter is a compelling example of how intelligent adaptive and
autonomous systems can evolve. It demonstrates the potential for these systems
to perform predefined tasks with increased efficiency and apply their
capabilities in novel, dynamic environments, expanding the horizons of what
autonomous systems can achieve in the realm of intelligent adaptation.

###  6.2 Role and Potential of Digital Twins

In the field of metrology, precise measurements and analysis are of vital
importance. The concept of digital twins represents an important innovation as
it enables the accurate reproduction of an object's condition, behavior, and
changes in real-time. Digital twins create a digital counterpart to a physical
entity. This continuous synchronization throughout an object's lifecycle
enables deep integration of physical measurements with digital simulations,
improving metrology applications with high accuracy and efficiency.

The foundation of digital twins lies in the idea that a physical object and
its digital representation can be interconnected, allowing for real-time data
transmission and analysis. This concept is not tied to a specific technology
but is a versatile framework that can be realized through various advanced
technological solutions [154]. A central problem that the digital twin model
should solve is the contradiction between the simplified virtual model and the
complex behavior of the physical object. The origin of the Digital Twin is
attributed to Michael Grieves and his work with John Vickers of NASA, with
Grieves introducing the concept in 2003 in a lecture on product lifecycle
management [155]. The original description defines a Digital Twin as a virtual
representation of a physical product that contains information about the
product. In an early paper [155] Grieves expanded this definition and
described the Digital Twin as consisting of a physical product made up of
three components, a virtual representation of that product and the
bidirectional data links that transfer data from the physical to the virtual
representation and information and processes from the virtual representation
to the physical product. Grieves described this flow as a cycle between the
physical and virtual states (mirroring or twinning).

Virtual SpaceReal Space Information/ Processes Data Fig. 6: Reflection or
twinning between the physical and virtual spaces [156].

Since the inception of the Digital Twin in 2003, the concept has gained
interest and has been listed by Gartner as a key strategic technology trend
for 2019 [156]. This growth is largely driven by advances in related
technologies and initiatives such as Internet-of-Things, Big Data, multi-
physical simulation and Industry 4.0. In addition, there are real-time sensors
and sensor networks, data management, data processing and the drive towards a
data-driven and digital future of manufacturing.

Data is the foundation of digital twins. Sensors, measuring devices, RFID tags
and readers, cameras, scanners, etc. are selected and integrated to collect
data for the digital twin. The data is then to be transmitted in real-time or
near real-time. However, such data is usually very large and diverse, making
it difficult and costly to transfer to the digital twin on, for example, a
cloud server. Therefore, edge computing is an ideal method for pre-processing
the collected data to reduce the network load and eliminate the risk of data
loss [156].

The model is the core of the digital twin. Digital twin models consist of
semantic data models and physical models. Semantic data models are trained on
known inputs and outputs using AI methods. Physical models require a
comprehensive understanding of physical properties and their mutual
interaction. Therefore, multi-physical modeling is essential for the realistic
modeling of the digital twin. Simulation is an important aspect of the digital
twin in this regard. Digital twin simulation allows the virtual model to
interact with the physical entity in real-time. The structure of digital twins
described below is based on the results of the literature review by Jones et
al. [156].

####  6.2.1 Construction of a Digital Twin

In general, the digital twin consists of a physical entity, a virtual
counterpart and the data connections in between. In total, digital twins can
be divided into 12 components. First, a distinction can be made between two
environments, the physical environment and the virtual environment. The
physical environment refers to the "real" space in which the physical entity
is located. Aspects of these environments are measured and transferred to the
virtual twin to ensure an accurate virtual environment upon which simulations,
optimizations, and/or decisions are made. Physical processes refer to the
activities performed by the physical entity in the physical environment. The
virtual environment exists in the digital domain and is a mirror of the
physical environment, where twinning is achieved through physical measurement
technology (i.e. sensors), transferring important measurements from the
physical to the virtual environment. Virtual processes refer to the activities
performed using the virtual entity within the virtual environment. The term
parameters in this context refers to the types of data, information and
processes that are passed between the physical and virtual twins.

The physical-virtual links are the means by which the state of the physical
entity is transferred to and realized in the virtual environment - i.e.,
updating the virtual parameters so that they reflect the values of the
physical parameters. These include Internet-of-Things sensors, web services,
5G, and other customer requirements [156]. The connection itself consists of a
measurement phase, in which the state of the physical entity is recorded, and
a realization phase, in which the delta between the physical and the digital
entity is determined, and the virtual entity is updated accordingly. For
example, the temperature change of a physical engine is measured with an
Internet-of-Things thermometer (measurement phase), the temperature
measurement is transmitted to the virtual environment via a web service, and a
virtual process determines the temperature difference between the physical
engine and the virtual engine and updates the virtual engine so that both
measurements are equal (realization phase). This continuous connection between
the physical and virtual engine is a distinguishing feature between the
Digital Twin and more traditional simulation and modeling approaches, where
analysis is often performed "off-line". The physical-virtual link allows
monitoring of state changes that occur both in response to conditions in the
physical environment and state changes that occur in response to interventions
by the digital twin itself. Thus, if a change in engine speed were to be made
as a result of temperature measurements, the physical-digital link would also
measure the effects of this intervention. Grieves describes the virtual-
physical connection as the flow of information and processes from the virtual
to the physical, i.e. the digital twin contains the functionality to realize a
change in the physical state physically.

The link from virtual to physical environment mirrors the link from physical
to virtual environment in that it also contains measurement as well as
realization phases. Virtual processes and measurement methods determine and
measure an optimal set of parameter values within a physical entity or
environment, and realization methods determine the delta between these new
values and the state and update the state of the physical entity accordingly.
For example, in response to an increased engine temperature that exceeds a
certain threshold, the effect of engine speed can be modeled. This allows a
speed to be calculated that sufficiently reduces the temperature, thus
adjusting the physical engine speed.

Virtual SpaceReal Space Information/ Processes DataVirtual UnitVirtual UnitPhysical UnitPhysical UnitMeasurementRealizationRealizationMeasurementVirtual ProcessVirtual ProcessPhysical ProcessPhysical ProcessVirtual EnvironmentPhysical EnvironmentPhysical-virtual connection/TwinningVirtual-physical connection/TwinningTwinning RateTwinning Parameter 1: | Value
---|---
Parameter 2: | Value
… | …
Parameter n: | Value
Fig. 7: The physical-virtual and virtual-physical twinning process [156].

Twinning describes the synchronization of the virtual and physical states,
e.g., measuring the state of a physical entity and translating this state into
the virtual environment so that the virtual and physical states are "the
same," i.e., all virtual parameters have the same value as the physical
parameters. A change that occurs in either the physical or virtual entity is
measured before it is realized to the corresponding virtual/physical twin. If
both states are the same, the units are " twinned ". The twinning rate is then
the frequency at which twinning occurs.

In Fig. 7 it is shown how physical/virtual processes act on the corresponding
physical/virtual entity, these processes causing a change in the state of this
entity via its parameters. This change of state is captured by measurement
methods, transmitted via physical-virtual and virtual-physical connections,
and realized in the other (virtual/physical) environment by synchronizing all
parameters. Both the virtual and physical environments contain the means to
measure and realize changes of state. The process (change → measurement →,
realization) is the twinning process and runs in both directions, from virtual
to physical and from physical to virtual.

###  6.3 New Directions for Intelligent Learning Systems

In the rapidly changing field of Intelligent Autonomous Systems, particularly
in metrology, new developments are constantly emerging, reshaping the way
these systems learn and adapt. This section aims to highlight two significant
advancements in this field. The first is the concept of the Artificial Neural
Twin, an innovative extension of digital twins that integrates neural network
capabilities to enhance the simulation and prediction of physical systems. The
second is reservoir computing, a recently popularized approach within the
recurrent neural network framework that is characterized by its efficiency in
processing complex time-series data.

####  6.3.1 Reservoir Computing

Reservoir computing represents a significant leap forward in machine learning,
combining ideas from recurrent neural networks and echo state networks. At its
core, it introduces an efficient way to handle data that changes over time by
focusing on a "reservoir" - a complex, high-dimensional structure. Unlike
traditional neural networks, which require intensive training of all layers,
reservoir computing simplifies the process by training only the output layer,
making it much more efficient. This simplicity is particularly beneficial in
several fields, such as optoelectronics, as highlighted by Paquot et al. in
2011 [157].

In the field of metrology, reservoir computing has led to remarkable
improvements, especially in dealing with complex, fluctuating signals where
standard techniques struggle. Przyczyna et al. [158] demonstrated its
promising role in improving the sensitivity and speed of chemical sensors and
measurement systems.

As an example of its application in metrology, research by Xia et al. [159]
explores quantum reservoir engineering to better estimate the interaction
strength in atom-cavity systems. Quantum reservoir computing (QRC) combines
the principles of quantum and reservoir computing, using the dynamics of
quantum systems to process time-sensitive data. This quantum version takes
advantage of quantum properties such as superposition and entanglement,
potentially offering superior performance over classical approaches.

Fig. 8:  Framework of the quantum reservoir consisting of a sequence of qubits
initialized at the same state [159] .

Significant advances in QRC include Nakajima et al.'s strategy of using
spatial multiplexing [160]. This approach uses multiple quantum systems
simultaneously, increasing computational power without the need for a large,
singular quantum system. This method fits well with current experimental
setups, such as those using nuclear magnetic resonance, allowing for more
qubits while keeping experiments manageable. In addition, Chen, Nurdin, and
Yamamoto in 2020 introduced a quantum reservoir computing model that exploits
complex quantum dynamics [161]. Their work suggests that even small, imperfect
quantum systems, like those in today's quantum computers, can effectively
tackle complex, time-based tasks. This discovery opens up the possibility of
using emerging quantum computers in a wide range of applications, from neural
modeling to language processing.

By incorporating reservoir computing into metrology, we can overcome the
limitations of traditional signal analysis and enable real-time tracking and
evaluation of dynamic chemical processes. This advancement not only improves
the accuracy and speed of measurement tools, but also paves the way for new
diagnostic and sensing technologies that push the boundaries of what is
possible in measurement science.

####  6.3.2 Artificial Neural Twin

In the field of data-driven metrology, the concept of Artificial Neural Twins
(ANT) is a key innovation that bridges the gap between real-time data
acquisition and predictive analytics. By mirroring physical systems in a
digital framework, ANT enables unprecedented levels of precision and
adaptability in measurement technologies. To seamlessly integrate an entire
measurement hardware setup into a neural network that can be optimized toward
a universal quality criterion, it is essential to ensure that each process
step and each sensor is compatible with a state-of-the-art AI interface. This
interface is designed with two main goals in mind: first, to facilitate full
error backpropagation, allowing deviations from the overall quality benchmark
to be tracked and corrected through all stages of the measurement process. The
second is to dynamically adjust the settings of each component in the system
based on these errors, thereby minimizing deviations from the desired quality
standard.

To achieve these goals, the AI interface must meet several critical criteria.
It should be able to compute the partial derivative of each component's output
with respect to its input signal, as well as the partial derivative of the
component's output with respect to its adjustable parameters. In addition, the
interface must preserve the current signal for updating downstream components
and compute a local quality metric based on the prevailing material flow. This
structured approach allows the precise quantification of each component's
contribution to the overall error and enables targeted adjustments to reduce
these discrepancies.

For the practical implementation of the first two properties, the theoretical
foundation laid out in [138] is utilized. This approach establishes a robust
basis for the AI interface, ensuring that each element within the system
contributes effectively to achieving the global quality measure.

Building on this foundation, the ANT concept emerges as a key innovation for
system adaptation, aiming for superior performance by leveraging advances in
AI and digital partnership. The ANT paradigm seeks to transcend conventional
methodologies by applying deep learning innovations, particularly through the
use of specialized neural network layers, such as convolutional layers, that
introduce valuable invariances. These invariances are critical for
streamlining the model by reducing the number of parameters and incorporating
inherent biases, thereby increasing the efficiency of the network. The main
challenge in this area is to identify the most effective arrangement and
configuration of these invariances in order to develop powerful deep learning
architectures. This task is complex and requires considerable effort. To
address this, the concept of known operator learning proposes a blend of
analytical models from traditional theories and deep learning approaches,
resulting in hybrid or "gray box" models. These models combine data-driven
insights with established invariances, effectively bounding errors and
reducing the need for large training datasets. This synergy significantly
increases the ability of models to generalize, especially in areas where
domain expertise is critical, such as signal reconstruction and image
processing.

Fig. 9: Artificial Neural Twin illustrated in the context of waste management
from the K3I-Cycling Project funded by the German Federal Ministry of
Education and Research: Each processing step fulfills the requirements for a
decentralized AI interface.

The ANT concept is pivotal in the current digital transformation, advocating
for the creation of digital twins that accurately reflect physical entities to
facilitate their evaluation, optimization, and prediction. The rapid evolution
of technologies such as the Internet of Things, Big Data, multi-physics
simulation, Industry 4.0, alongside real-time sensors and sensor networks, has
catalyzed the growth of digital twins. These virtual models represent a
convergence of vast amounts of information from the physical world,
necessitating advanced solutions for their processing and synchronization.
Machine learning, particularly deep learning, has shown promise in enhancing
these digital representations, whether it's through refining data extraction
from physical environments or modeling complex systems as black box models
when faced with high complexity or scant information. An example of an ANT
implementation is given in Figure 9 for the scenario of a complex waste
management system.

The widespread integration of AI within digital twins is still nascent, with
its application mainly confined to specific modules or as a tool within
limited domains. The vision of a seamless, end-to-end integrated digital twin
system is yet to be realized. ANT advocates for a strategic application of
black, grey, and white box models to achieve this comprehensive integration.
However, one of the barriers to the adoption of ANT is the lack of
standardization in digital twinning methodologies. Overcoming this obstacle is
essential for the advancement and establishment of ANT as a benchmark for the
future of digital and physical world synthesis.

The recent innovation in artificial intelligence, particularly in deep
learning, has led to AI methods becoming standard solutions in various
domains. However, these methods are often implemented as isolated solutions,
addressing single, specific problems, especially in fields like medicine and
industrial production. This isolated approach means that AI methods typically
do not integrate information from preceding or subsequent steps in a process,
such as separating signal reconstruction from AI-based evaluations like
classification or segmentation. This limits the potential of AI, as only parts
of a process are enhanced, not the entire system. Most AI methods operate as a
black box, requiring extensive training data, making solutions heavily data-
dependent. To address this, the ANT aims to digitize entire process chains,
including all components and material flows. Unlike traditional digital twins,
ANT is a fully differentiable digital representation capable of optimizing
both individual components and the entire system towards a global quality
measure while considering local conditions. A key innovation is the
development of a standardized interface for system components, ensuring
modularity and compatibility. This interface requires components to provide
their own derivations relative to the input, allowing for domain-specific
implementation by component manufacturers without revealing proprietary
information. This approach ensures the protection of sensitive information and
is essential for industry acceptance, overcoming obstacles like the lack of
standardization and privacy concerns [162], which have previously hindered
digitalization and AI innovation in industrial settings.

##  7 Conclusion and Outlook

In metrology, the mathematical model that relates the measured values to the
calculated value of the measurand plays an essential role. With the help of a
model, a relationship can be established between the observed values and the
underlying mechanisms. This relationship can be established for simple
measurements of a univariate quantity as well as for complex measurements. The
theory of mathematical modelling in measurement is well established and a wide
range of methods can also be found in the literature.
In the future, there will be more and more networked, distributed measurement
systems, large amounts of volatile data and data-driven modelling approaches.
The combination of these characteristics poses challenges to established
measurement modeling approaches. In addition, many sensors are typically
combined with other data sources to form a complex network of sensors.
In this paper, we have briefly reviewed existing concepts from modeling and
discussed their potential use for data analysis scenarios in the future.
White-box, black-box and grey-box models are already known in many fields. The
use of white- and grey-box models is quite well known in metrology, with e.g.
the recent addition to the GUM focusing on these model types. The extension
and further development of the underlying principles for handling data-driven
black-box models will build on this knowledge. Therefore, this paper sets out
the basic principles for white- and grey-box models and shows how they differ
from black-box models. Furthermore, this paper offers insight into the
versatile applications of data-driven modeling. In particular, adaptive and
intelligent autonomous systems, which include the concept of digital twins,
are described in detail and their potential use is highlighted using various
examples. Due to the complex nature of real sensor networks and the volatility
of the measured data, these approaches will combine several of the modeling
principles outlined in this paper and will lead to AI systems implemented in
distributed hardware as outlined in the ANT concept.

##  8 Acknowledgements

A main part of this work was financed by the „SmartCT - Artificial
Intelligence Methods for an Autonomous Robotic CT System“ project (project nr.
DIK-2004-0009).
Additionally, part of this work was funded by German Ministry of Education and
Research (BMBF) under grant number 033KI201.
Also, part of this work has been developed within the Joint Research project
17IND12 Met4FoF of the European Metrology Programme for Innovation and
Research (EMPIR). The EMPIR is jointly funded by the EMPIR participating
countries within EURAMET and the European Union.

Our special thanks go to our author colleagues with whom we jointly prepared
the manuscript „Modelling of Networked Measuring Systems - From White-Box
Models to Data Based Approaches“ [84] within the EMPIR project 17IND12
Met4FoF, namely Andonovic, I., University of Strathclyde, UK; Dorst, T., ZeMA
- Zentrum für Mechatronik und Automatisierungstechnik gGmbH, Saarbrücken;
Eichstädt, S., Physikalisch-Technische Bundesanstalt, Braunschweig and Berlin;
Füssl, R., Technische Universität Ilmenau; Gourlay, G., University of
Strathclyde, UK; Harris, P. M., National Physical Laboratory, Teddington, UK;
Heizmann, M., Karlsruhe Institute of Technology; Luo Y., National Physical
Laboratory, Teddington, UK; Schütze, A., Saarland University,
Saarbrücken;Sommer, K.-D., Ilmenau University of Technology; Tachtatzis, C.,
University of Strathclyde, UK.

## References

  * [1] Michael Rüßmann, Markus Lorenz, Philipp Gerbert, Manuela Waldner, Jan Justus, Pascal Engel, and Michael Harnisch.  Industry 4.0: The future of productivity and growth in manufacturing industries.  Boston consulting group, 9(1):54–89, 2015.
  * [2] Mohd Aiman Kamarul Bahrin, Mohd Fauzi Othman, Nor Hayati Nor Azli, and Muhamad Farihin Talib.  Industry 4.0: A review on industrial automation and robotic.  Jurnal teknologi, 78(6-13), 2016.
  * [3] Dionisis Kandris, Christos Nakas, Dimitrios Vomvas, and Grigorios Koulouras.  Applications of wireless sensor networks: an up-to-date survey.  Applied System Innovation, 3(1):14, 2020.
  * [4] Hugo Landaluce, Laura Arjona, Asier Perallos, Francisco Falcone, Ignacio Angulo, and Florian Muralter.  A review of iot sensing applications and challenges using rfid and wireless sensor networks.  Sensors, 20(9):2495, 2020.
  * [5] Mamoona Majid, Shaista Habib, Abdul Rehman Javed, Muhammad Rizwan, Gautam Srivastava, Thippa Reddy Gadekallu, and Jerry Chun-Wei Lin.  Applications of wireless sensor networks and internet of things frameworks in the industry revolution 4.0: A systematic literature review.  Sensors, 22(6):2087, 2022.
  * [6] Jeretta Horn Nord, Alex Koohang, and Joanna Paliszkiewicz.  The internet of things: Review and theoretical framework.  Expert Systems with Applications, 133:97–108, 2019.
  * [7] Lalit Chettri and Rabindranath Bera.  A comprehensive survey on internet of things (iot) toward 5g wireless systems.  IEEE Internet of Things Journal, 7(1):16–32, 2019.
  * [8] Puneet Kumar Aggarwal, Parita Jain, Jaya Mehta, Riya Garg, Kshirja Makar, and Poorvi Chaudhary.  Machine learning, data mining, and big data analytics for 5g-enabled iot.  Blockchain for 5G-Enabled IoT: The new wave for Industrial Automation, pages 351–375, 2021.
  * [9] Stefan Boschert and Roland Rosen.  Digital twin—the simulation aspect.  Mechatronic futures: Challenges and solutions for mechatronic systems and their designers, pages 59–74, 2016.
  * [10] Fei Tao, Jiangfeng Cheng, Qinglin Qi, Meng Zhang, He Zhang, and Fangyuan Sui.  Digital twin-driven product design, manufacturing and service with big data.  The International Journal of Advanced Manufacturing Technology, 94:3563–3576, 2018.
  * [11] Guodong Shao, Sanjay Jain, Christoph Laroque, Loo Hay Lee, Peter Lendermann, and Oliver Rose.  Digital twin for smart manufacturing: the simulation aspect.  In 2019 Winter Simulation Conference (WSC), pages 2085–2098. IEEE, 2019.
  * [12] Mengnan Liu, Shuiliang Fang, Huiyue Dong, and Cunzhi Xu.  Review of digital twin about concepts, technologies, and industrial applications.  Journal of Manufacturing Systems, 58:346–361, 2021.
  * [13] Hasan Tercan and Tobias Meisen.  Machine learning and deep learning based predictive quality in manufacturing: a systematic review.  Journal of Intelligent Manufacturing, 33(7):1879–1905, 2022.
  * [14] Arif Sanjid Mahammad.  Using ai in dimensional metrology.  In Handbook of Metrology and Applications, pages 1025–1042. Springer, 2023.
  * [15] Grégoire Allaire.  Numerical analysis and optimization: an introduction to mathematical modelling and numerical simulation.  OUP Oxford, 2007.
  * [16] Pedram Ghamisi, Behnood Rasti, Naoto Yokoya, Qunming Wang, Bernhard Hofle, Lorenzo Bruzzone, Francesca Bovolo, Mingmin Chi, Katharina Anders, Richard Gloaguen, et al.  Multisource and multitemporal data fusion in remote sensing: A comprehensive review of the state of the art.  IEEE Geoscience and Remote Sensing Magazine, 7(1):6–39, 2019.
  * [17] Jamil Fayyad, Mohammad A Jaradat, Dominique Gruyer, and Homayoun Najjaran.  Deep learning sensor fusion for autonomous vehicle perception and localization: A review.  Sensors, 20(15):4220, 2020.
  * [18] De Jong Yeong, Gustavo Velasco-Hernandez, John Barry, and Joseph Walsh.  Sensor and sensor fusion technology in autonomous vehicles: A review.  Sensors, 21(6):2140, 2021.
  * [19] Jiaxin Li, Danfeng Hong, Lianru Gao, Jing Yao, Ke Zheng, Bing Zhang, and Jocelyn Chanussot.  Deep learning in multimodal remote sensing data fusion: A comprehensive review.  International Journal of Applied Earth Observation and Geoinformation, 112:102926, 2022.
  * [20] Arjan Van Der Schaft, Dimitri Jeltsema, et al.  Port-hamiltonian systems theory: An introductory overview.  Foundations and Trends® in Systems and Control, 1(2-3):173–378, 2014.
  * [21] Balvant Rajani and Yehuda Kleiner.  Comprehensive review of structural deterioration of water mains: physically based models.  Urban water, 3(3):151–164, 2001.
  * [22] Yujiang Xiang, Jasbir S Arora, and Karim Abdel-Malek.  Physics-based modeling and simulation of human walking: a review of optimization-based and other approaches.  Structural and Multidisciplinary Optimization, 42:1–23, 2010.
  * [23] Adrian Cubillo, Suresh Perinpanayagam, and Manuel Esperon-Miguez.  A review of physics-based models in prognostics: Application to gears and bearings of rotating machinery.  Advances in Mechanical Engineering, 8(8):1687814016664660, 2016\.
  * [24] Dimitri Solomatine, Linda M See, and RJ Abrahart.  Data-driven modelling: concepts, approaches and experiences.  Practical hydroinformatics: Computational intelligence and technological developments in water applications, pages 17–30, 2008.
  * [25] Simon Arridge, Peter Maass, Ozan Öktem, and Carola-Bibiane Schönlieb.  Solving inverse problems using data-driven models.  Acta Numerica, 28:1–174, 2019.
  * [26] Jared Willard, Xiaowei Jia, Shaoming Xu, Michael Steinbach, and Vipin Kumar.  Integrating physics-based modeling with machine learning: A survey.  arXiv preprint arXiv:2003.04919, 1(1):1–34, 2020.
  * [27] Jorge Pulpeiro González, King Ankobea-Ansah, Qian Peng, and Carrie M Hall.  On the integration of physics-based and data-driven models for the prediction of gas exchange processes on a modern diesel engine.  Proceedings of the Institution of Mechanical Engineers, Part D: Journal of Automobile Engineering, 236(5):857–871, 2022.
  * [28] Jie Chen, Min Zhao, Xiuheng Wang, Cédric Richard, and Susanto Rahardja.  Integration of physics-based and data-driven models for hyperspectral image unmixing: A summary of current methods.  IEEE Signal Processing Magazine, 40(2):61–74, 2023.
  * [29] Dominique Placko et al.  Metrology in industry: The key for quality.  John Wiley & Sons, 2013.
  * [30] Markku Rummukainen.  State-of-the-art with regional climate models.  Wiley Interdisciplinary Reviews: Climate Change, 1(1):82–96, 2010\.
  * [31] Matt J Keeling and Ken TD Eames.  Networks and epidemic models.  Journal of the royal society interface, 2(4):295–307, 2005.
  * [32] Matthew Ellis, Helen Durand, and Panagiotis D Christofides.  A tutorial review of economic model predictive control methods.  Journal of Process Control, 24(8):1156–1178, 2014.
  * [33] W Ross Ashby.  An introduction to cybernetics.  1957\.
  * [34] Ludwig von Bertalanffy.  General system theory: Foundations, development, applications.  G. Braziller, 1968.
  * [35] Norbert Wiener.  Cybernetics or Control and Communication in the Animal and the Machine.  MIT press, 2019.
  * [36] Timm Faulwasser, Ruchuan Ou, Guanru Pan, Philipp Schmitz, and Karl Worthmann.  Behavioral theory for stochastic systems? a data-driven journey from willems to wiener and back again.  Annual Reviews in Control, 2023.
  * [37] Jan C Willems.  The behavioral approach to open and interconnected systems.  IEEE control systems magazine, 27(6):46–99, 2007.
  * [38] Norbert Wiener.  The homogeneous chaos.  American Journal of Mathematics, 60(4):897–936, 1938.
  * [39] David Laner, Helmut Rechberger, and Thomas Astrup.  Systematic evaluation of uncertainty in material flow analysis.  Journal of Industrial Ecology, 18(6):859–870, 2014.
  * [40] Alejandro Barredo Arrieta, Natalia Díaz-Rodríguez, Javier Del Ser, Adrien Bennetot, Siham Tabik, Alberto Barbado, Salvador García, Sergio Gil-López, Daniel Molina, Richard Benjamins, et al.  Explainable artificial intelligence (xai): Concepts, taxonomies, opportunities and challenges toward responsible ai.  Information fusion, 58:82–115, 2020.
  * [41] Michael Kläs.  Towards identifying and managing sources of uncertainty in ai and machine learning models-an overview.  arXiv preprint arXiv:1811.11669, 2018.
  * [42] Yong Gao, Kui Wu, and Fulu Li.  Analysis on the redundancy of wireless sensor networks.  In Proceedings of the 2nd ACM international conference on Wireless sensor networks and applications, pages 108–114, 2003.
  * [43] Daniel-Ioan Curiac, Constantin Volosencu, Dan Pescaru, Lucian Jurca, and Alexa Doboli.  Redundancy and its applications in wireless sensor networks: A survey.  WSEAS Transactions on Computers, 8(4):705–714, 2009.
  * [44] Tu N Nguyen and Sherali Zeadally.  Mobile crowd-sensing applications: Data redundancies, challenges, and solutions.  ACM Transactions on Internet Technology (TOIT), 22(2):1–15, 2021\.
  * [45] Saurabh Ganeriwal, Ram Kumar, and Mani B Srivastava.  Timing-sync protocol for sensor networks.  In Proceedings of the 1st international conference on Embedded networked sensor systems, pages 138–149, 2003.
  * [46] Fikret Sivrikaya and Bülent Yener.  Time synchronization in sensor networks: a survey.  IEEE network, 18(4):45–50, 2004.
  * [47] Michael Kevin Maggs, Steven G O'keefe, and David Victor Thiel.  Consensus clock synchronization for wireless sensor networks.  IEEE sensors Journal, 12(6):2269–2277, 2012.
  * [48] Amitabha Ghosh and Sajal K Das.  Coverage and connectivity issues in wireless sensor networks: A survey.  Pervasive and Mobile Computing, 4(3):303–334, 2008.
  * [49] GaoJun Fan and ShiYao Jin.  Coverage problem in wireless sensor network: A survey.  Journal of networks, 5(9):1033, 2010.
  * [50] Seapahn Meguerdichian, Farinaz Koushanfar, Miodrag Potkonjak, and Mani B Srivastava.  Coverage problems in wireless ad-hoc sensor networks.  In Proceedings IEEE INFOCOM 2001. Conference on computer communications. Twentieth annual joint conference of the IEEE computer and communications society (Cat. No. 01CH37213), volume 3, pages 1380–1387. IEEE, 2001.
  * [51] Stergios I Roumeliotis and George A Bekey.  Distributed multirobot localization.  IEEE transactions on robotics and automation, 18(5):781–795, 2002\.
  * [52] Nino Boccara and Nine Boccara.  Modeling complex systems, volume 1.  Springer, 2010.
  * [53] Julio M Ottino.  Engineering complex systems.  Nature, 427(6973):399–399, 2004.
  * [54] Mark EJ Newman.  Complex systems: A survey.  arXiv preprint arXiv:1112.1440, 2011.
  * [55] Yang-Yu Liu and Albert-László Barabási.  Control principles of complex systems.  Reviews of Modern Physics, 88(3):035006, 2016.
  * [56] Stefan Thurner, Rudolf Hanel, and Peter Klimek.  Introduction to the theory of complex systems.  Oxford University Press, 2018.
  * [57] Achim Schilling, William Sedley, Richard Gerum, Claus Metzner, Konstantin Tziridis, Andreas Maier, Holger Schulze, Fan-Gang Zeng, Karl J Friston, and Patrick Krauss.  Predictive coding and stochastic resonance as fundamental principles of auditory phantom perception.  Brain, page awad255, 07 2023.
  * [58] Karl R Popper.  Science as falsification.  Conjectures and refutations, 1(1963):33–39, 1963.
  * [59] Yuri Lazebnik.  Can a biologist fix a radio?—or, what i learned while studying apoptosis.  Biochemistry (Moscow), 69:1403–1406, 2004.
  * [60] Nicole Lazar.  Ockham's razor.  Wiley Interdisciplinary Reviews: Computational Statistics, 2(2):243–246, 2010.
  * [61] F Kozin and HG Natke.  System identification techniques.  Structural Safety, 3(3-4):269–316, 1986.
  * [62] Lennart Ljung.  Perspectives on system identification.  Annual Reviews in Control, 34(1):1–12, 2010.
  * [63] Karel J Keesman and Karel J Keesman.  System identification: an introduction, volume 2.  Springer, 2011.
  * [64] R. Sacco, G. Guidoboni, and A. Mauri.  Mathematical and physical modeling principles of complex biological systems.  Ocular Fluid Dynamics, 2019.
  * [65] KD Sommer, A Weckenmann, and BRL Siebert.  A systematic approach to the modelling of measurements for uncertainty evaluation.  In Journal of Physics: Conference Series, volume 13, page 052. IOP Publishing, 2005.
  * [66] John P Bentley.  Principles of measurement systems.  Pearson education, 2005.
  * [67] Ernest O Doebelin and Dhanesh N Manik.  Measurement systems: application and design.  2007\.
  * [68] James V Beck and Keith A Woodbury.  Inverse problems and parameter estimation: integration of measurements and analysis.  Measurement Science and Technology, 9(6):839, 1998.
  * [69] Charles W Groetsch and CW Groetsch.  Inverse problems in the mathematical sciences, volume 52.  Springer, 1993.
  * [70] BIPM, IEC, IFCC, ILAC, ISO, IUPAC, IUPAP, and OIML.  International vocabulary of metrology - basic and general concepts and associated terms (vim), 2012.
  * [71] BIPM, IEC, IFCC, ILAC, ISO, IUPAC, IUPAP, and OIML.  Guide to the expression of uncertainty in measurement, 2008.
  * [72] BIPM, IEC, IFCC, ILAC, ISO, IUPAC, IUPAP, and OIML.  Evaluation of measurement data — supplement 1 to the “guide to the expression of uncertainty in measurement” — propagation of distributions using a monte carlo method, 2008.
  * [73] Curtis R. Vogel.  Computational methods for inverse problems.  Society for Industrial and Applied Mathematics, 2007.
  * [74] Mohd Ehmer Khan and Farmeena Khan.  A comparative study of white box, black box and grey box testing techniques.  International Journal of Advanced Computer Science and Applications, 3(6), 2012.
  * [75] Zhuo Yang, Douglas Eddy, Sundar Krishnamurty, Ian Grosse, Peter Denno, Yan Lu, and Paul Witherell.  Investigating grey-box modeling for predictive analytics in smart manufacturing.  In International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, volume 58134, page V02BT03A024. American Society of Mechanical Engineers, 2017.
  * [76] Emmanuel Pintelas, Ioannis E Livieris, and Panagiotis Pintelas.  A grey-box ensemble model exploiting black-box accuracy and white-box intrinsic interpretability.  Algorithms, 13(1):17, 2020.
  * [77] Ophélie Lo-Thong, Philippe Charton, Xavier F Cadet, Brigitte Grondin-Perez, Emma Saavedra, Cédric Damour, and Frédéric Cadet.  Identification of flux checkpoints in a metabolic pathway through white-box, grey-box and black-box modeling approaches.  Scientific reports, 10(1):13446, 2020.
  * [78] Octavio Loyola-Gonzalez.  Black-box vs. white-box: Understanding their advantages and weaknesses from a practical point of view.  IEEE access, 7:154096–154113, 2019.
  * [79] Tim J Rogers, Geoffrey R Holmes, EJ Cross, and Keith Worden.  On a grey box modelling framework for nonlinear system identification.  In Special Topics in Structural Dynamics, Volume 6: Proceedings of the 35th IMAC, A Conference and Exposition on Structural Dynamics 2017, pages 167–178. Springer, 2017.
  * [80] Leifur Þ Leifsson, Hildur Sævarsdóttir, Sven Þ Sigurðsson, and Ari Vésteinsson.  Grey-box modeling of an ocean vessel for operational optimization.  Simulation Modelling Practice and Theory, 16(8):923–932, 2008.
  * [81] K Worden, CX Wong, Ulrich Parlitz, A Hornstein, David Engster, Tegoeh Tjahjowidodo, Farid Al-Bender, DD Rizos, and SD Fassois.  Identification of pre-sliding and sliding friction dynamics: Grey box and black-box models.  Mechanical systems and signal Processing, 21(1):514–534, 2007.
  * [82] Daniel J Pitchforth, Timothy J Rogers, Ulf T Tygesen, and Elizabeth J Cross.  Grey-box models for wave loading prediction.  Mechanical Systems and Signal Processing, 159:107741, 2021.
  * [83] RHA Hellsen, Georgo Z Angelis, MJG Van de Molengraft, AG De Jager, and JJ Kok.  Grey-box modeling of friction: An experimental case-study.  European Journal of Control, 6(3):258–267, 2000.
  * [84] Klaus-Dieter Sommer, Peter Harris, Sascha Eichstädt, Roland Füssl, Tanja Dorst, Andreas Schütze, Michael Heizmann, Nadine Schiering, Andreas Maier, Yuhui Luo, Christos Tachtatzis, Ivan Andonovic, and Gordon Gourlay.  Modelling of networked measuring systems – from white-box models to data based approaches, 2023.
  * [85] Sinno Jialin Pan and Qiang Yang.  A survey on transfer learning.  IEEE Transactions on knowledge and data engineering, 22(10):1345–1359, 2009.
  * [86] Lisa Torrey and Jude Shavlik.  Transfer learning.  In Handbook of research on machine learning applications and trends: algorithms, methods, and techniques, pages 242–264. IGI global, 2010\.
  * [87] Karl Weiss, Taghi M Khoshgoftaar, and DingDing Wang.  A survey of transfer learning.  Journal of Big data, 3(1):1–40, 2016.
  * [88] Oscar Day and Taghi M Khoshgoftaar.  A survey on heterogeneous transfer learning.  Journal of Big Data, 4:1–42, 2017.
  * [89] Chuanqi Tan, Fuchun Sun, Tao Kong, Wenchang Zhang, Chao Yang, and Chunfang Liu.  A survey on deep transfer learning.  CoRR, abs/1808.01974, 2018.
  * [90] Fuzhen Zhuang, Zhiyuan Qi, Keyu Duan, Dongbo Xi, Yongchun Zhu, Hengshu Zhu, Hui Xiong, and Qing He.  A comprehensive survey on transfer learning.  Proceedings of the IEEE, 109(1):43–76, 2020.
  * [91] Mohammadreza Iman, Hamid Reza Arabnia, and Khaled Rasheed.  A review of deep transfer learning and recent advancements.  Technologies, 11(2):40, 2023.
  * [92] Yaroslav Ganin and Victor Lempitsky.  Unsupervised domain adaptation by backpropagation.  In International conference on machine learning, pages 1180–1189. PMLR, 2015.
  * [93] RH Gardner and RVa O’Neill.  Parameter uncertainty and model predictions: a review of monte carlo results.  Uncertainty and forecasting of water quality, pages 245–257, 1983\.
  * [94] George Klir and Bo Yuan.  Fuzzy sets and fuzzy logic, volume 4.  Prentice hall New Jersey, 1995.
  * [95] Josefa Mula, Raul Poler, Jose P García-Sabater, and Francisco Cruz Lario.  Models for production planning under uncertainty: A review.  International journal of production economics, 103(1):271–285, 2006\.
  * [96] Morteza Aien, Ali Hajebrahimi, and Mahmud Fotuhi-Firuzabad.  A comprehensive review on uncertainty modeling techniques in power system studies.  Renewable and Sustainable Energy Reviews, 57:1077–1089, 2016.
  * [97] Chao Jiang, Jing Zheng, and Xu Han.  Probability-interval hybrid uncertainty analysis for structures with both aleatory and epistemic uncertainties: a review.  Structural and Multidisciplinary Optimization, 57(6):2485–2502, 2018.
  * [98] Erdem Acar, Gamze Bayrak, Yongsu Jung, Ikjin Lee, Palaniappan Ramu, and Suja Shree Ravichandran.  Modeling, analysis, and optimization under uncertainties: a review.  Structural and Multidisciplinary Optimization, 64(5):2909–2945, 2021.
  * [99] Karl Johan Åström and Peter Eykhoff.  System identification—a survey.  Automatica, 7(2):123–162, 1971.
  * [100] Thomas JR Hughes.  The finite element method: linear static and dynamic finite element analysis.  Courier Corporation, 2012.
  * [101] Karl Johan Astrom and Bjorn Wittenmark.  Adaptive control 2nd edition.  Addison-Wesley Pub Co., 1994, 1994.
  * [102] George EP Box, Gwilym M Jenkins, Gregory C Reinsel, and Greta M Ljung.  Time series analysis: forecasting and control.  John Wiley & Sons, 2015.
  * [103] Norman S Nise.  Control systems engineering.  John Wiley & Sons, 2020.
  * [104] Peter Benner, Serkan Gugercin, and Karen Willcox.  A survey of projection-based model reduction methods for parametric dynamical systems.  SIAM review, 57(4):483–531, 2015.
  * [105] Chih-Ming Wang and H. Iyer.  Propagation of uncertainties in measurements using generalized inference.  Metrologia, 42:145 – 153, 2005.
  * [106] Marco Wolf.  A modeling language for measurement uncertainty evaluation.  2009\.
  * [107] Jerry Banks, II Carson, Nelson BL JS, and DM Nicol.  Discrete-event system simulation, international version, 2010.
  * [108] Andrea Saltelli, Marco Ratto, Terry Andres, Francesca Campolongo, Jessica Cariboni, Debora Gatelli, Michaela Saisana, and Stefano Tarantola.  Global sensitivity analysis: the primer.  John Wiley & Sons, 2008.
  * [109] Alex Simpkins.  System identification: Theory for the user, (ljung, l.; 1999)[on the shelf].  IEEE Robotics & Automation Magazine, 19(2):95–96, 2012.
  * [110] George EP Box and Norman R Draper.  Empirical model-building and response surfaces. John Wiley & Sons, 1987.
  * [111] Gene F Franklin, J David Powell, Abbas Emami-Naeini, and J David Powell.  Feedback control of dynamic systems, volume 4.  Prentice hall Upper Saddle River, 2002.
  * [112] Heinrich Niemann.  Pattern analysis and understanding, volume 4.  Springer Science & Business Media, 2013.
  * [113] Andreas Maier, Christopher Syben, Tobias Lasser, and Christian Riess.  A gentle introduction to deep learning in medical image processing.  Zeitschrift für Medizinische Physik, 29(2):86–101, 2019.
  * [114] Steven L. Brunton and J. Nathan Kutz.  Data-Driven Science and Engineering: Machine Learning, Dynamical Systems, and Control.  Cambridge University Press, 2019.
  * [115] Feliks Nüske, Sebastian Peitz, Friedrich Philipp, Manuel Schaller, and Karl Worthmann.  Finite-data error bounds for koopman-based prediction and control.  Journal of Nonlinear Science, 33(1):14, 2023.
  * [116] Guoru Ding, Qi hui Wu, Yu dong Yao, Jinlong Wang, and Yingying Chen.  Kernel-based learning for statistical signal processing in cognitive radio networks: Theoretical foundations, example applications, and future directions.  IEEE Signal Processing Magazine, 30:126–136, 2013.
  * [117] Martin Basil and A. Jamieson.  Uncertainty of complex systems by monte carlo simulation.  Measurement and Control, 32:16 – 20, 1999.
  * [118] Axel Brando, Damià Torres, José A. Rodríguez-Serrano, and Jordi Vitrià.  Building uncertainty models on top of black-box predictive apis.  IEEE Access, 8:121344–121356, 2020.
  * [119] Octavio Loyola-González.  Black-box vs. white-box: Understanding their advantages and weaknesses from a practical point of view.  IEEE Access, 7:154096–154113, 2019.
  * [120] Andreas Holzinger, Markus Plass, Katharina Holzinger, Gloria Cerasela Crisan, Camelia-Mihaela Pintea, and Vasile Palade.  A glass-box interactive machine learning approach for solving np-hard problems with the human-in-the-loop.  CoRR, abs/1708.01104, 2017.
  * [121] Weilin Fu, Katharina Breininger, Roman Schaffert, Nishant Ravikumar, and Andreas K. Maier.  A divide-and-conquer approach towards understanding deep networks.  CoRR, abs/1907.06194, 2019.
  * [122] Justin Ker, Lipo Wang, Jai Rao, and Tchoyoson Lim.  Deep learning applications in medical image analysis.  IEEE Access, 6:9375–9389, 2018.
  * [123] Geert Litjens, Thijs Kooi, Babak Ehteshami Bejnordi, Arnaud Arindra Adiyoso Setio, Francesco Ciompi, Mohsen Ghafoorian, Jeroen AWM van der Laak, Bram Van Ginneken, and Clara I Sánchez.  A survey on deep learning in medical image analysis.  Medical image analysis, 42:60–88, 2017.
  * [124] Andreas Maier, Harald Köstler, Marco Heisig, Patrick Krauss, and Seung Hee Yang.  Known operator learning and hybrid machine learning in medical imaging—a review of the past, the present, and the future.  Progress in Biomedical Engineering, 2022.
  * [125] Richard S Sutton.  Adapting bias by gradient descent: An incremental version of delta-bar-delta.  In AAAI, volume 92, pages 171–176. Citeseer, 1992.
  * [126] Yann LeCun, Yoshua Bengio, and Geoffrey Hinton.  Deep learning.  nature, 521(7553):436–444, 2015.
  * [127] Alex Krizhevsky, Ilya Sutskever, and Geoffrey E Hinton.  Imagenet classification with deep convolutional neural networks.  In Advances in neural information processing systems, pages 1097–1105, 2012.
  * [128] Michael M. Bronstein, Joan Bruna, Taco Cohen, and Petar Veličković.  Geometric deep learning: Grids, groups, graphs, geodesics, and gauges, 2021.
  * [129] Raghavendra Selvan and Erik B Dam.  Tensor networks for medical image classification.  In Medical Imaging with Deep Learning, pages 721–732. PMLR, 2020\.
  * [130] Chelsea Finn, Pieter Abbeel, and Sergey Levine.  Model-agnostic meta-learning for fast adaptation of deep networks.  In Doina Precup and Yee Whye Teh, editors, Proceedings of the 34th International Conference on Machine Learning, volume 70 of Proceedings of Machine Learning Research, pages 1126–1135. PMLR, 06–11 Aug 2017\.
  * [131] Allan Zhou, Tom Knowles, and Chelsea Finn.  Meta-learning symmetries by reparameterization, 2021.
  * [132] Jake Snell, Kevin Swersky, and Richard S. Zemel.  Prototypical networks for few-shot learning, 2017.
  * [133] Barret Zoph and Quoc V. Le.  Neural architecture search with reinforcement learning, 2017.
  * [134] Flood Sung, Yongxin Yang, Li Zhang, Tao Xiang, Philip H.S. Torr, and Timothy M. Hospedales.  Learning to compare: Relation network for few-shot learning.  In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), June 2018.
  * [135] David Silver, Aja Huang, Chris J Maddison, Arthur Guez, Laurent Sifre, George Van Den Driessche, Julian Schrittwieser, Ioannis Antonoglou, Veda Panneershelvam, Marc Lanctot, et al.  Mastering the game of go with deep neural networks and tree search.  nature, 529(7587):484, 2016.
  * [136] Florin Cristian Ghesu, Bogdan Georgescu, Yefeng Zheng, Sasa Grbic, Andreas Maier, Joachim Hornegger, and Dorin Comaniciu.  Multi-scale deep reinforcement learning for real-time 3d-landmark detection in ct scans.  IEEE Transactions on Pattern Analysis and Machine Intelligence, 2017\.
  * [137] Daniel Filan, Shlomi Hod, Cody Wild, Andrew Critch, and Stuart Russell.  Pruned neural networks are surprisingly modular.  arXiv preprint arXiv:2003.04881, 2020.
  * [138] Andreas K Maier, Christopher Syben, Bernhard Stimpel, Tobias Würfl, Mathis Hoffmann, Frank Schebesch, Weilin Fu, Leonid Mill, Lasse Kling, and Silke Christiansen.  Learning with known operators reduces maximum error bounds.  Nature machine intelligence, 1(8):373–380, 2019.
  * [139] Christopher Syben, Bernhard Stimpel, Jonathan Lommen, Tobias Würfl, Arnd Dörfler, and Andreas Maier.  Deriving neural network architectures using precision learning: Parallel-to-fan beam conversion.  In German Conference on Pattern Recognition (GCPR), 2018.
  * [140] George Em Karniadakis, Ioannis G Kevrekidis, Lu Lu, Paris Perdikaris, Sifan Wang, and Liu Yang.  Physics-informed machine learning.  Nature Reviews Physics, 3(6):422–440, 2021.
  * [141] Zong-Yi Li, Hongkai Zheng, Nikola B. Kovachki, David Jin, Haoxuan Chen, Burigede Liu, K. Azizzadenesheli, and Anima Anandkumar.  Physics-informed neural operator for learning partial differential equations.  ArXiv, abs/2111.03794, 2021.
  * [142] Li Li, Stephan Hoyer, Ryan Pederson, Ruoxi Sun, Ekin D Cubuk, Patrick Riley, Kieron Burke, et al.  Kohn-sham equations as regularizer: Building prior knowledge into machine-learned physics.  Physical review letters, 126(3):036401, 2021.
  * [143] Miles D Cranmer, Rui Xu, Peter Battaglia, and Shirley Ho.  Learning symbolic physics with graph networks.  arXiv preprint arXiv:1909.05862, 2019.
  * [144] Yinbao Cheng, Xiao huai Chen, Hong li Li, Z. Cheng, Rui Jiang, Jing Lü, and Huadong Fu.  Analysis and comparison of bayesian methods for measurement uncertainty evaluation.  Mathematical Problems in Engineering, 2018.
  * [145] M. Boumans.  Model-based type b uncertainty evaluations of measurement towards more objective evaluation strategies.  Measurement, 46:3775–3777, 2013.
  * [146] E. Crawley.  Intelligent structures for aerospace - a technology overview and assessment.  AIAA Journal, 32:1689–1699, 1994.
  * [147] Javier Garcıa and Fernando Fernández.  A comprehensive survey on safe reinforcement learning.  Journal of Machine Learning Research, 16(1):1437–1480, 2015.
  * [148] Dan Paulsson, Robert Gustavsson, and Carl-Fredrik Mandenius.  A soft sensor for bioprocess control based on sequential filtering of metabolic heat signals.  Sensors (Basel, Switzerland), 14:17864–17882, 10 2014.
  * [149] Weilin Fu, Katharina Breininger, Tobias Würfl, Nishant Ravikumar, Roman Schaffert, and Andreas Maier.  Frangi-Net: A Neural Network Approach to Vessel Segmentation.  Technical report, November 2017.  arXiv:1711.03345 [cs] type: article.
  * [150] Alejandro F. Frangi, Wiro J. Niessen, Koen L. Vincken, and Max A. Viergever.  Multiscale vessel enhancement filtering.  In William M. Wells, Alan Colchester, and Scott Delp, editors, Medical Image Computing and Computer-Assisted Intervention — MICCAI’98, Lecture Notes in Computer Science, pages 130–137, Berlin, Heidelberg, 1998. Springer.
  * [151] Oleksiy Rybakov, Daniel Stromer, Irina Mischewski, and Andreas Maier.  Segmentation of Fat and Fascias in Canine Ultrasound Images.  In Andreas Maier, Thomas M. Deserno, Heinz Handels, Klaus Hermann Maier-Hein, Christoph Palm, and Thomas Tolxdorff, editors, Bildverarbeitung für die Medizin 2018, Informatik aktuell, pages 263–268, Berlin, Heidelberg, 2018. Springer.
  * [152] Daniel Stromer, Vincent Christlein, Christine Martindale, Patrick Zippert, Eric Haltenberger, Tino Hausotte, and Andreas Maier.  Browsing through sealed historical manuscripts by using 3-D computed tomography with low-brilliance X-ray sources.  Scientific Reports, 8(1):15335, October 2018.
  * [153] Antonia Longo, S. Morscher, Jaber Malekzadeh Najafababdi, D. Jüstel, C. Zakian, and V. Ntziachristos.  Assessment of hessian-based frangi vesselness filter in optoacoustic imaging.  Photoacoustics, 20, 2020.
  * [154] Mengnan Liu, Fang Shuiliang, Huiyue Dong, and Cunzhi Xu.  Review of digital twin about concepts, technologies, and industrial applications.  Journal of Manufacturing Systems, 58, July 2020.
  * [155] Michael Grieves.  Digital Twin: Manufacturing Excellence through Virtual Factory Replication.  March 2015.
  * [156] David Jones, Chris Snider, Aydin Nassehi, Jason Yon, and Ben Hicks.  Characterising the Digital Twin: A systematic literature review.  CIRP Journal of Manufacturing Science and Technology, 29:36–52, May 2020.
  * [157] Y. Paquot, F. Duport, A. Smerieri, J. Dambre, B. Schrauwen, M. Haelterman, and S. Massar.  Optoelectronic reservoir computing.  Scientific Reports, 2, 2011.
  * [158] Dawid Przyczyna, Sébastien Pecqueur, D. Vuillaume, and K. Szaciłowski.  Reservoir computing for sensing - an experimental approach.  ArXiv, abs/2001.04342, 2020.
  * [159] Ye Xia, Weiming Guo, and Z. Miao.  Time-varying engineered reservoir for the improved estimation of atom-cavity coupling strength.  Photonics, 2023.
  * [160] K. Nakajima, K. Fujii, M. Negoro, K. Mitarai, and M. Kitagawa.  Boosting computational power through spatial multiplexing in quantum reservoir computing.  Physical Review Applied, 2018.
  * [161] Jiayin Chen, H. Nurdin, and N. Yamamoto.  Temporal information processing on noisy quantum computers.  ArXiv, abs/2001.09498, 2020.
  * [162] Mergen Kor, Ibrahim Yitmen, and Sepehr Alizadehsalehi.  An investigation for integration of deep learning and digital twins towards construction 4.0.  Smart and Sustainable Built Environment, ahead-of-print, 03 2022\.