2022

[1]Chao Hu

[9]Zhen Hu

1]Department of Mechanical Engineering, Iowa State University, Ames, 50011,
IA, USA

2]Department of Industrial and Systems Engineering, The Hong Kong Polytechnic
University, Kowloon, Hong Kong

3]Intelligent Maintenance and Operations Systems, EPFL, Lausanne, 12309, NY,
Switzerland

4]Information Modeling and Testing Group, National Institute of Standards and
Technology, Gaithersburg, 20877, MD, USA

5]Probabilistic Design and Optimization group, GE Research, Niskayuna, 12309,
NY, USA

6]Department of Mechanical Engineering, Seoul National University, Gwanak-gu,
151-742, Seoul, Republic of Korea

7]Department of Structural Engineering, University of California, San Diego,
La Jolla, 92093, CA, USA

8]Department of Civil and Environmental Engineering, Vanderbilt University,
Nashville, 37235, TN, USA

9]Department of Industrial and Manufacturing Systems Engineering, University
of Michigan-Dearborn, Dearborn, 48128, MI, USA

# A Comprehensive Review of Digital Twin - Part 1: Modeling and Twinning
Enabling Technologies

Adam Thelen<EMAIL_ADDRESS>Xiaoge Zhang<EMAIL_ADDRESS>Olga
Fink<EMAIL_ADDRESS>Yan Lu<EMAIL_ADDRESS>Sayan Ghosh
<EMAIL_ADDRESS>Byeng D. Youn<EMAIL_ADDRESS>Michael D. Todd
<EMAIL_ADDRESS>Sankaran Mahadevan<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>[  [  [  [  [  [  [  [  [

###### Abstract

As an emerging technology in the era of Industry 4.0, digital twin is gaining
unprecedented attention because of its promise to further optimize process
design, quality control, health monitoring, decision and policy making, and
more, by comprehensively modeling the physical world as a group of
interconnected digital models. In a two-part series of papers, we examine the
fundamental role of different modeling techniques, twinning enabling
technologies, and uncertainty quantification and optimization methods commonly
used in digital twins. This first paper presents a thorough literature review
of digital twin trends across many disciplines currently pursuing this area of
research. Then, digital twin modeling and twinning enabling technologies are
further analyzed by classifying them into two main categories: physical-to-
virtual, and virtual-to-physical, based on the direction in which data flows.
Finally, this paper provides perspectives on the trajectory of digital twin
technology over the next decade, and introduces a few emerging areas of
research which will likely be of great use in future digital twin research. In
part two of this review, the role of uncertainty quantification and
optimization are discussed, a battery digital twin is demonstrated, and more
perspectives on the future of digital twin are shared.

###### keywords:

Digital twin; Optimization; Machine learning; Enabling technology;
Perspective; Industry 4.0, Review

##  1 Introduction

This paper is the first in a series of two that analyze the roles of modeling
and twinning enabling technologies, uncertainty quantification (UQ), and
optimization in digital twins. Modeling and twinning enabling technologies are
fundamental methods used to bridge the information gap between a physical
system and its digital counterpart. In this paper, we review and analyze many
methods and modeling techniques currently used in digital twins, and classify
them into groups based on the direction in which data flows.

We begin by reviewing an existing definition of digital twin. A digital twin,
as originally proposed by Grieves (2014), is a virtual representation of a
complex physical asset in the digital space for the purpose of closely
characterizing the operations of the original physical process or system. The
precise representation of the physical system in cyberspace is enabled by
continuous data synchronization and information exchange between the digital
twin and the physical counterpart. Having a digital replica of the physical
entity of interest leads to enormous benefits spanning its entire life cycle,
including the development phase (e.g., product design, resource planning,
manufacturing process design), manufacturing phase (e.g., production process
planning, manufacturing process control, maintenance of manufacturing
equipment), service phase (e.g., performance and health monitoring,
maintenance and control of fielded products, path planning), and disposal
phase (e.g., end-of-life reuse, remanufacturing, and recycling). With the
ability to conduct what-if analyses in the digital space, the benefits of
digital twin technology are already materializing in a wide range of
applications ranging from high-value manufacturing industry (Ayerbe et al.,
2021) to personalized medicine (Corral-Acero et al., 2020), oil refinery
management (Min et al., 2019), risk identification, and city planning (Lu et
al., 2020a).

The digital twin concept centers around “individualized” digital models that
capture the unique characteristics of individual products or process units.
These models allow decision making to be optimized for each product or process
unit, rather than based on the average characteristics of the entire
population. This emerging technology poses new and challenging optimization
problems at the forefront of model-based design, smart manufacturing,
Industrial Internet of Things (IIoT), machine learning (ML), and predictive
maintenance. The industry-scale adoption of the digital twin concept entails
creating novel optimization solutions that use data collected from sensors and
inspections (physical to digital) to provide decision makers with actionable
information (digital to physical), thereby closing the digitalization loop.
Major benefits include the ability to optimize control/maintenance actions to
individual units and the potential to optimize the design of next-generation
products.

The promise of digital twin concept and the need to guide the effective
development of digital twin technology have motivated researchers to put
together many review papers on digital twin. For example, motivated by the
need to consolidate the many definitions and characteristics of digital twin,
several review papers have been reported in recent years, such as Barricelli
et al. (2019), Jones et al. (2020), VanDerHorn and Mahadevan (2021), Liu et
al. (2021b). These papers summarized various definitions of digital twin and
suggested generalized definitions and characteristics to distinguish digital
twin from other models. Lim et al. (2020a) presented a comprehensive review of
digital twin by considering its benefits to engineering product life cycle
management and business innovation. Qi et al. (2021) surveyed enabling
technologies and tools for digital twins and proposed a five-dimensional
digital twin model. Fuller et al. (2020) analyzed enabling technologies for
artificial intelligence, Internet of Things (IoT), and digital twins. Rasheed
et al. (2020) reviewed the latest methodologies and techniques related to the
construction of digital twins from the modeling perspective.

There are also numerous application-specific literature review papers. For
instance, Jiang et al. (2021) reviewed the implementation of digital twins in
civil engineering and discussed several problems that need to be addressed.
Xie et al. (2021) discussed applications of digital twins at different states
of a cutting tool’s life cycle. Errandonea et al. (2020) reviewed the
applications of digital twins for maintenance. Kritzinger et al. (2018)
provided a categorical review of digital twins in manufacturing and grouped
applications of digital twins into different categories according to the level
of integration. Focusing on a similar topic, Cimino et al. (2019) reviewed the
applications of digital twins in manufacturing. Along the same line, Ayerbe et
al. (2021) discussed recent advances in the digitalization of lithium-ion
battery cell manufacturing processes and proposed a three-layer structure of a
digital twin of large-scale battery manufacturing. Ozturk (2021) conducted a
literature survey over the applications of digital twins in the architectural,
engineering, construction, operation, and facility management industries. Wagg
et al. (2020) summarized the state-of-the-art and discussed future research
directions of digital twin, focusing on applications in engineering dynamics.
A survey of digital twins for verification and validation of industrial
automation systems is available in Löcklin et al. (2020). Tao et al. (2018)
presented a comprehensive review of digital twins in industry. He and Bai
(2021) reviewed applications of digital twins for intelligent and sustainable
manufacturing. Boje et al. (2020) discussed the future research directions of
digital twin in construction engineering. Uhlemann et al. (2017) discussed the
importance of digital twins in realizing cyber-physical production systems for
Industry 4.0.

While current review papers cover a good amount of literature on different
aspects of digital twin, such as definitions, enabling technologies, and
applications in different domains, not all important aspects of digital twin
have been covered in the current literature. Digital twin technology is an
emerging technology that exploits the synergy between several key enabling
technologies, including multiphysics simulations, sensing, ML, UQ, data
analytics, to name a few. Modeling and twinning methods are key to integrating
different technologies in a digital twin. However, a comprehensive review of
the modeling and twinning methods and the associated UQ and optimization
methods enabling digital twins is still missing. Such a review is essential
because it may guide effective, industry-scale implementations of digital
twins for different types of physical systems. Based on a literature review of
over 230 research papers on digital twins, this paper aims to fill this void
by (1) categorizing the modeling and twinning methods according to their roles
in digital twins, and (2) providing a detailed discussion on the importance of
UQ and optimization in the modeling and twinning methods of digital twins. The
challenges and future research directions on digital twin will also be
discussed based on the literature survey and lessons learned from case
studies.

Figure 1: Overview of topics covered in Part 1

In consideration of the large number of subjects covered, we split the review
paper into two parts to make it easier to read. The first paper is an
introduction to various definitions of digital twin in the literature and our
proposed five-dimensional definition of digital twin. In addition, the first
paper also covers commonly used modeling and twinning technologies to map a
physical system to its digital counterpart (P2V) and the actions/decisions
sent back by the digital twin that can be taken in the physical system (V2P).
Figure 1 depicts the overall structure of Part 1 of the review paper.

Part 2 of the review paper (Thelen et al., 2022b) centers around describing
the roles of UQ and optimization in digital twins so as to support a better
characterization of physical systems in the digital space. This second part of
the review also presents a case study on battery health management to showcase
the specific implementation of a battery digital twin and the power of UQ and
optimization in the digital twin in an end-to-end fashion. Additionally, Part
2 reviews the applications of digital twins at an industrial scale and
available open-source tools and datasets related to digital twins, followed by
a discussion on challenges and future research directions.

The remainder of this paper is organized as follows. Sec. 2 first discusses a
variety of digital twin definitions reported in the literature and then
conducts a comprehensive analysis on the trend of digital twin research. Sec.
3 categorizes the modeling enabling technologies in digital twins. Sec. 4
summarizes commonly used approaches for physical-to-virtual twinning. Sec. 5
elaborates the actions informed by virtual-to-physical connection that can be
carried out at various stages of a physical system. Sec. 6 presents
perspectives on modeling and twinning in digital twins. Sec. 7 concludes the
first part of the two-part review paper.

##  2 Definitions of digital twin and literature overview

###  2.1 What is a digital twin?

According to the literature and other documents, the concept of “digital twin”
was first implemented in the NASA Apollo 13 program in the 1960s. A digital
twin model of Apollo 13 was created on earth to allow engineers on the ground
to test possible solutions for a rescue mission in space (Barricelli et al.,
2019, Nguyen et al., 2021). Michael Grieves made the first documented
definition of digital twin in his presentation in the context of product life
cycle management in 2003 and later in a white paper (Grieves, 2014). Since
Grieves’s presentation on digital twin in 2003, various digital twin concepts
have been proposed. In a 2012 conference paper, Edward H. Glaessgen at NASA’s
Langley Research Center and David S. Stargel at the U.S. Air Force Office of
Scientific Research gave the first specific definition of digital twin in the
aerospace domain as:

_“A Digital Twin is an integrated multiphysics, multiscale, probabilistic
simulation of an as-built vehicle or system that uses the best available
physical models, sensor updates, fleet history, etc., to mirror the life of
its corresponding flying twin(Glaessgen and Stargel, 2012).”_

Since the first definition in the aerospace domain, digital twin has received
increasing attention in the past decade in various application domains, such
as manufacturing, automobile, energy, and civil engineering. Due to the broad
applications of digital twin in different domains, numerous application-
specific digital twin definitions have been proposed to capture the nature of
digital twin in these domains. Aiming to reduce the risk of diluting the
digital twin concept by a large number of definitions, efforts have been made
in recent years to consolidate digital twin definitions and terminologies in
several review papers (Barricelli et al., 2019, Jones et al., 2020, VanDerHorn
and Mahadevan, 2021, Liu et al., 2021b). In particular, VanDerHorn and
Mahadevan (2021) proposed a consolidated and generalized definition of digital
twin as _“a virtual representation of a physical system (and its associated
environment and processes) that is updated through the exchange of information
between the physical and virtual systems”_. Amongst the many definitions of
digital twin, the one specified in Glaessgen and Stargel (2012) is considered
one of the most broadly accepted definitions, even though the definition may
vary from application to application.

The distinction between a digital model, a digital shadow, and a digital twin
is unclear in many cases. Further, having a large number of vague and
inconsistent definitions of digital twin circulating in the literature may
adversely affect industry interest in the adoption of this technology,
creating a barrier to unleashing the maximum potential of the digital twin
technology (Wright and Davidson, 2020). In what follows, we attempt to
distinguish these three terms. It is argued that, for a digital model, data
flow between the physical space and virtual space is optional, or at the very
most, achieved manually as shown in Fig. 2. For a digital shadow, data flow is
unidirectional from physical to digital. But for digital twin, the data flow
has to be bidirectional (see Fig. 2) (Kritzinger et al., 2018). When digital
twin is used in control-related applications, the bidirectional data flow
needs to be automatic, often enabled by monitoring and control software. For
the application of digital twin to support decision making, such as predictive
maintenance, which will be reviewed in Sec. 5.2, however, the data flow from
virtual to physical involves humans in the loop who carry out maintenance
actions and, therefore, is not fully automatic but should be handled on time
to avoid unexpected breakdown. Moreover, Zhang et al. (2021a) pointed out that
since a physical entity usually has many different aspects that can be
modeled, it is always necessary to clarify for what aspect the digital twin is
constructed. Take a simple mechanical shaft as an example. The shaft may fail
due to fatigue, extreme torque, etc. Since it is difficult for the digital
space to mirror all aspects of the physical space, a digital twin is always
constructed for specific aspect(s) relevant to the engineering problem the
digital twin is used to solve, and these aspect(s) need to be specified before
the construction of the digital twin.

Figure 2: Digital model, digital shadow, and digital twin concept for a
lithium-ion battery cell within a larger pack Figure 3: Comparison of
different digital twin (DT) models and the proposed model

In addition, various digital twin models have been proposed in the past few
decades. Two highly cited digital twin models in the literature are the three-
dimensional digital twin model proposed by Grieves (2014) and the five-
dimensional digital twin model suggested by Tao et al. (2018). As shown in
Fig. 3, the three-dimensional digital twin model consists of a physical space,
a virtual space, and connections between them. Tao et al. (2018) expands the
three-dimensional digital twin model into a five-dimensional digital twin
model for manufacturing, which includes a real machine, a virtual machine, a
services component, digital twin data, and the connections as depicted in Fig.
3.

Based on the definition of digital twin given in (Glaessgen and Stargel, 2012)
as mentioned above, we suggest a new five-dimensional digital twin model as
illustrated in Fig. 3. The proposed five-dimensional digital twin model can be
considered an extended version of the three-dimensional model and is defined
as follows

${\rm{DT}}=\mathbb{F}({\rm{PS}},\;{\rm{DS}},\;{\rm{P2V}},\;{\rm{V2P}},\;{\rm{OPT}})$
(1)

As indicated in the equation above, the proposed five-dimensional digital twin
model consists of _a physical system (PS)_ , _a digital system (DS)_ , _an
updating engine (P2V)_ , _a prediction engine (V2P)_ , and _an optimization
dimension (OPT)_ , and $\mathbb{F}(\cdot)$ integrates all five dimensions
together to be an effective digital twin. The physical system (PS) with
sensing capability gathers data from multiple sources using various sensing
and data acquisition techniques. The updating engine (P2V) updates the state
of the digital model based on the sensor data. The updated digital models (DS)
are then employed to predict the future state of the physical system using the
prediction engine (V2P), thereby enabling predictive decision making feeding
back to the physical system through actuators (control, maintenance, path
planning, etc.). Finally, an optimization dimension (OPT, i.e., the fifth
dimension) supports the functionalities of the other four dimensions in the
digital twin by optimizing the data collection, modeling, state estimation,
decision making, etc. The optimization dimension consists of two elements,
namely offline optimization and online optimization. These two optimization
elements will be explained in detail in Part 2 of the review paper (Thelen et
al., 2022b). The seamless integration of the five dimensions (PS, DS, P2V,
V2P, and OPT) enables digital twins to fulfill the need for a real-time mirror
of the life cycle of a physical system, thereby supporting optimal decision
making.

To highlight how a five-dimensional digital twin might operate in practice, we
can look at the example of a digital twin for a lithium-ion battery cell
within a battery pack, shown in Fig. 2. When designing a lithium-ion battery
pack, engineers are most concerned with managing the rate and amount of
degradation of each cell in the pack driven by the cell’s usage (e.g.,
charging and discharging rates, depth of discharge, and temperature). In this
scenario, a digital twin model of each cell within a pack can be constructed
to track and forecast the capacity degradation trajectory of the cell and
ultimately drive decision making around when the pack needs to be retired from
service. Sensor measurements of the cell’s state of health (PS) can be used to
update (P2V, OPT) the digital model (DS) of the cell. Then, the digital model
(DS) can be used to predict the cell’s future state of health and degradation
trajectory (V2P). Engineers can then use the predicted future degradation
trajectory to inform decision making about the optimal time when the pack may
need to be retired from service (V2P, OPT). The complete flow of data through
all five digital twin dimensions outlined in Fig. 3 is what ultimately
characterizes this type of workflow as a true digital twin. This digital twin
application for forecasting the future health of a lithium-ion cell and using
the forecast health for predictive maintenance is extensively investigated as
a case study in Part 2 of this review.

With a focus on the modeling and twinning technologies (i.e., the updating and
prediction engines in Fig. 3), in the subsequent sections, we first perform an
extensive literature review of the state-of-the-art and then classify the
enabling technologies into distinct categories. The role of the optimization
dimension in digital twin and future research directions will also be
discussed.

Figure 4: Distribution of reviewed papers by domain (“Civil” includes civil
engineering, building engineering, smart city, and transportation;
“Mechanical” includes mechanical engineering, automotive, robotics, and
marine; “Others” includes supply chain, optical communication system, and
agricultural engineering.

###  2.2 Literature overview

Our literature review starts with a Google scholar search using the keyword
“digital twin”. In addition to the review papers summarized in Sec. 1, 230
research papers, excluding papers from predatory journals, were reviewed. The
majority of these 230 research papers were published between 2017 and 2022,
with some published between 2011 and 2012. They are classified into different
groups by application domain as shown in Fig. 4. This figure indicates that
manufacturing, followed by mechanical, civil, and structures, accounts for
most of the digital twin papers. 107 out of 230 papers come from the
manufacturing domain, which implies a leading role of manufacturing in the
digital twin field.

The following information is then extracted from the literature survey:

  * •

Top-cited papers, which are papers with an average number of citations per
year greater than 25 (the average number of citations per year to a paper is
its total number of citations divided by the number of years elapsed since the
publication of the paper).

  * •

Major journals, which are the journals where research papers on digital twin
are frequently published based on the 230 research papers reviewed.

  * •

Publicly available tools/datasets: tools and datasets related to digital twin
from the 230 papers and associated websites.

  * •

Modeling methods: methods used in the papers to create the virtual space.

  * •

Twinning methods: methods used to establish physical-to-virtual and virtual-
to-physical connections.

  * •

Optimization: roles of optimization in the digital twin models presented in
the papers.

Next, we will summarize the major findings related to the first two items
listed above. The methods of modeling, twinning (i.e., physical-to-virtual and
virtual-to-physical), and optimization will be summarized and classified into
different categories in Sec. 3. The publicly available tools/datasets will be
provided in Sec. 4.2 of Part 2 of the review paper.

####  2.2.1 Top-cited papers and trends in different domains

From the 230 research papers, we first identify papers whose average number of
citations per year is greater than 25. It is found that many papers in the
manufacturing domain satisfy this criterion. It is not entirely surprising
since digital twin has been a hot topic in smart manufacturing. We, therefore,
list the top-cited papers in the manufacturing domain separately from those in
the other domains.

Table 1: Top ten top-cited papers in manufacturing domain (Based on yearly average number of Google scholar citations as of April 2022) Title |  Author and Publication year |  Citations |  Application
---|---|---|---
Digital twin-driven product design, manufacturing and service with big data |  Tao et al. (2018) |  1435 |  Design and Manufacturing
Shaping the digital twin for design and production engineering |  Schleich et al. (2017) |  719 |  Production
Digital twin and big data towards smart manufacturing and industry 4.0: 360 degree comparison |  Qi and Tao (2018) |  789 |  Smart manufacturing
Digital twin shop-floor: a new shop-floor paradigm towards smart manufacturing |  Tao and Zhang (2017) |  585 |  Shop floor
Digital twin-based smart production management and control framework for the complex product assembly shop-floor |  Zhuang et al. (2018) |  313 |  Shop floor
Toward a Digital Twin for real-time geometry assurance in individualized production |  Söderberg et al. (2017) |  372 |  Geometry assurance
Digital twin-driven manufacturing cyber-physical system for parallel controlling of smart workshop |  Leng et al. (2019) |  235 |  Shop floor
Defining a digital twin-based cyber-physical production system for autonomous manufacturing in smart shop floors |  Ding et al. (2019) |  225 |  Shop floor
Digital twin-based designing of the configuration, motion, control, and optimization model of a flow-type smart manufacturing system |  Liu et al. (2021c) |  100 |  Configuration optimization
Digital twin service towards smart manufacturing |  Qi et al. (2018) |  231 |  Service in Manufacturing

Table 1 summarizes the top ten most-cited papers in the manufacturing domain
from the 230 reviewed papers, based on the average number of citations per
year. Qi and Tao are the the most highly-cited researchers in this field at
this time (Tao et al., 2018, Qi and Tao, 2018, Tao and Zhang, 2017, Qi et al.,
2018). Most of the top-cited papers in the manufacturing domain focus on
digital twins of manufacturing shop-floors, such as assembly, production
management and control, and configuration optimization. There is one paper
that focuses on geometry assurance in individualized production. Table 2 lists
the top-cited papers from the civil, aerospace, energy, and healthcare
domains, which reflect the trends of digital twin outside the manufacturing
sector. As shown in this table, aerospace, civil, and mechanical are leading
the trend of digital twin outside the manufacturing domain. Five out of the 12
top-cited papers concentrate on fault diagnostics and prognostics of physical
assets using digital twin (Xu et al., 2019, Li et al., 2017, Glaessgen and
Stargel, 2012, Jain et al., 2019, Tuegel et al., 2011). The others focus on
optimization, management, and service.

Following that, Fig. 5 shows the cumulative number of papers over the past
five years in each different application domain. It is clear that the number
of papers in the manufacturing domain is much higher than the other domains.
This agrees with the earlier analysis of the most highly cited papers which
found that a majority of them were in the manufacturing domain. Note that the
trends abruptly flatten from 2021 to 2022 because this review paper was
written in early 2022 and did not account for papers published later in the
year. It is expected that the increasing trend continue into 2022.

Figure 5: Cumulative number of papers of different application domains

To better analyze the trends in the other domains, we remove the manufacturing
data, and re-scale the plot. Fig. 6 shows the cumulative numbers of papers in
the past five years, excluding manufacturing. In recent years, the cumulative
number of papers in the other domains, especially civil, mechanical, and
energy, have been growing rapidly.

Figure 6: Cumulative number of papers of different application domains excluding manufacturing Table 2: Top-cited papers in each application domain excluding manufacturing (based on yearly average number of Google Scholar citations as of April 2022) Title |  Publication year |  Citations |  Application domain
---|---|---|---
A digital-twin-assisted fault diagnosis using deep transfer learning |  Xu et al. (2019) |  154 |  Mechanical (Automotive)
The digital twin paradigm for future NASA and U.S. Air Force vehicles |  Glaessgen and Stargel (2012) |  1226 |  Aerospace
Reengineering aircraft structural life prediction using a digital twin |  Tuegel et al. (2011) |  639 |  Aerospace
Dynamic Bayesian network for aircraft wing health monitoring digital twin |  Li et al. (2017) |  171 |  Aerospace
Machine learning based digital twin framework for production optimization in petrochemical industry |  Min et al. (2019) |  141 |  Energy (Oil industry)
A digital twin approach for fault diagnosis in distributed photovoltaic systems |  Jain et al. (2019) |  106 |  Energy
A BIM-data mining integrated digital twin framework for advanced project management |  Pan and Zhang (2021) |  60 |  Civil (Construction)
Developing a digital twin at building and city levels: Case study of West Cambridge campus |  Lu et al. (2020a) |  82 |  Civil (Smart city)
Digital twin–Proof of concept |  Haag and Anderl (2018) |  288 |  Mechanical (Structural Health Monitoring)
An application framework of digital twin and its case study |  Zheng et al. (2019) |  206 |  Others (Service)
How to tell the difference between a model and a digital twin |  Wright and Davidson (2020) |  104 |  Others (Service)
A novel cloud-based framework for the elderly healthcare services using digital twin |  Liu et al. (2019) |  171 |  Healthcare

####  2.2.2 Major journals

Table 3 presents major journals where the 230 reviewed papers were published.
Only journals with more than three published papers are listed. Shown in the
table, manufacturing-related journals, namely _Journal of Manufacturing
Systems_ , _International Journal of Advanced Manufacturing Technology_ , and
_Robotics and Computer Integrated Manufacturing_ , are leading journals. Other
leading journals include _Journal of Ambient Intelligence and Humanized
Computing_ , _IEEE Access_ , _Engineering Fracture Mechanics_ , and
_International Journal of Production Research_.

Table 3: Major journals on digital twin based on the reviewed papers Journal Name |  No of Papers
---|---
Journal of Manufacturing Systems |  25
IEEE Access |  10
International Journal of Advanced Manufacturing Technology |  9
Robotics and Computer Integrated Manufacturing |  6
International Journal of Computer Integrated Manufacturing |  6
Procedia CIRP |  6
CIRP Annuals - Manufacturing Technology |  5
IFAC Papers Online |  5
Journal of Intelligent Manufacturing |  4
Journal of Ambient Intelligence and Humanized Computing |  4
Engineering Fracture Mechanics |  4
International Journal of Production Research |  3
Advanced Engineering Informatics |  3
Automation in Construction |  3
Computer and Structures |  3

##  3 Modeling enabling technologies

Modeling plays an essential role in digital twins by creating a digital
replica of the physical entity of interest. According to the purpose and
complexity of different modeling methods, we classify them into the following
five categories.

###  3.1 Geometric modeling

##### (a) Solid modeling

Computer-aided design (CAD) is a virtual solid modeling technique where 3D
objects are created from a number of 2D drawings using computer software. Many
individual components can be combined in an assembly to create a 3D solid
model of a larger system. Solid models help designers and engineers see how a
product will look and function in the real world. There are a number of free
and paid-for 3D solid modeling software on the market. Each software package
differs in its intended audience and scope of available features. For
engineering, popular software includes Autodesk Inventor and AutoCAD, PTC
Creo, SolidWorks, and Catia (Cai et al., 2020). Free software packages include
Unity3D, Unreal Engine, and Blender (Leng et al., 2020, Matulis and Harvey,
2021, Liu et al., 2021e).

A simple but useful application of solid modeling for digital twin is to
create models of all the equipment for a production line. Then, the modeled
equipment can be moved around, checked for timing, or simulated consistent
with a range of different goals. For example, Liu et al. (2021f) created a
virtual workshop with various machining tools so that the order in which
different parts were machined could be optimized. Similar work by Yi et al.
(2021) investigated ideas for using 3D models of the equipment in an assembly
line to optimize the layout for reduced travel distance between steps.
Vathoopan et al. (2018) proposed creating a solid model of a robotic arm to
detect and alarm when the robot’s joints may have moved further than
physically possible as predicted by the solid model. Ayani et al. (2018)
created a complete solid model of an old analog-controlled production machine
so that its movements, timings, and controls could be uploaded to the cloud
and further studied. Huang et al. (2022a) modeled a CNC tool changing system
and all the tools inside of it for the purpose of improving tool change
speeds. Creating solid models for optimization problems which focus on the
physical size and location of manufacturing equipment and tools is relatively
easy because minor details like material properties and movable joints can be
ignored. If, however, the solid model will be used for simulation of stresses,
movements, or interference using physics-based simulations (see Sec. 3.2), the
modeling process requires all details about the equipment be defined,
increasing the modeling time substantially.

Another application of solid modeling in digital twin is simulating and
studying human-environment interaction. Solid modeling enables researchers to
investigate how humans fit into the digital twin environment. Vatankhah
Barenji et al. (2021) investigated concepts of human-robot interaction and
ways to improve energy savings. Pairet et al. (2019) proposed modeling an
entire offshore oil drilling platform to study human-robot interaction in the
dense 3D space. Similarly, Lohtander et al. (2018) created a 3D model of a
human assembly area to study ergonomics and optimize the placement of part
bins. This area of research is of great importance, especially in a
manufacturing setting, where humans and robots coexist in a constrained
environment. It is envisioned that solid modeling of environments will play a
large role in designing smarter robots which can better interact safely in
proximity to humans.

Other applications of solid modeling to digital twins include tolerancing,
geometry variance control, and defect control in manufacturing settings.
Digital twin models of a production process can be used to improve quality
control in various ways. Liu et al. (2022) investigated integrating solid
modeling and surface roughness prediction for adaptive manufacturing control.
Schleich et al. (2017) proposed computer aided tolerancing and geometry
variation control via a solid modeling approach. Zambal et al. (2018)
discussed how one might make a solid model system of a carbon fiber production
process so that deviations in the equipment or manufacturing process could be
simulated. Through simulation, the effects of equipment deviations on the
final finished carbon fiber product can be quantified and understood.

Solid modeling has many promising uses in digital twin applications. With the
ability to model the geometry of any object, solid modeling enables additional
optimization dimensions not captured through sensor measurements and other
techniques. One of the key problems using solid models in digital twins is the
excessive amount of time required to create models which can be used in
accurate physics-based simulations, such as finite element analysis (FEA) or
motion simulations in Sec. 3.2. In many cases, the solid models of the parts
and machines researchers aim to use in digital twin simulations have already
been created by the original equipment manufacturers (OEMs). Similar to ideas
discussed in Grieves (2014) white paper, it is entirely conceivable that
future high value assets (e.g., aircraft, spacecraft, industrial equipment,
and even entire production factories) could be delivered to the customer
accompanied by a complete digital solid model for use in a digital twin
framework. In turn, the extensive time commitment the OEM has sunk into
creating the initial 3D solid model of the product could now be recouped by
supporting the end user with a long-term reliability and predictive
maintenance contract. With lasting communication between the OEM and the end
user through the online solid model, safety issues can be managed quicker,
replacement parts can be identified more easily, and future upgrades can be
tested and checked virtually for compatibility with a given customer’s asset
before committing to manufacturing the parts. Ultimately, the sharing of solid
models between OEMs and end users is necessary to accelerate the adoption of
digital twin technology.

##### (b) Laser scanning

Laser scanning, also commonly referred to as LiDAR (light detection and
ranging), is a geometry modeling technique which uses laser light to
repeatedly sample the distance to different points on an object’s surface. The
recorded measurements are virtually assembled into a 3D point cloud of (X, Y,
Z) coordinate data to describe the surface morphology of the scanned object in
great detail. Unlike solid modeling techniques which can model internal
components, material changes, and geometries, laser scanning can only model
surface-level features. However, laser scanning makes up for its inability to
model internal geometries by generally being much faster at modeling objects.
When it is paired with the appropriate software, laser scanning can create a
point cloud of data points on a computer in near real-time.

Laser scanning has been used in many applications for digital twin. In the
manufacturing domain, laser scanning has been proposed for quickly checking
the geometry of traditionally manufactured or 3D printed parts (Almalki et
al., 2022, Wärmefjord et al., 2017, White et al., 2021). Also in
manufacturing, Dröder et al. (2018) showed a proof of concept for a digital
twin considering human-robot interaction. The researchers used a Microsoft
Kinect 2.0 LiDAR sensor to detect humans in a robotic arm cell. Based on the
proximity of the human to the robotic arm, the robotic arm’s path and speed
were automatically optimized in real-time to create a safe environment for the
human while not ceasing operation entirely. In contrast to other systems which
use laser trip-wires that completely stop the robot when a human enters the
area, the LiDAR-based digital twin system significantly improved manufacturing
performance while maintaining a safe working environment. In a different
application of laser scanning and robots, Wang et al. (2020a) laterally
mounted an industrial camera on a robotic welder to take images of the process
for improved real-time quality control via augmented reality data streaming.
LiDAR and other laser scanning techniques will be extremely useful for the
development of more complex manufacturing sensor networks connected to a
digital twin. Real-time data streaming to and from the physical and virtual
dimensions will enable more productive manufacturing lines without
compromising worker safety.

The ability of laser scanning to quickly map and model large structures,
landscapes, and buildings makes it particularly useful for digital twin
modeling in the civil engineering domain (Lu et al., 2020b, Pan and Zhang,
2021). One of the most common uses of LiDAR and laser scanning for digital
twin is for scanning the surface of large buildings (Deng et al., 2021),
bridges (Shim et al., 2019), or other structures (Lim et al., 2020b).
Specifically in Shim et al. (2019), the authors discussed how LiDAR laser
scanning could be used to periodically model the surface features of different
bridge components for preventative maintenance. In this system, the raw
scanned point cloud data is fed in batches to the cloud where structure
recognition deep learning (DL) algorithms sort the scanned images according to
the type of structure it is. Then, within the digital twin network, the images
are mapped into their respective ontological structure file (see Section 3.5
(b)) where they are further analyzed for cracks, pitting, shifting, or other
undesirable degradation indicators. Any positively identified defects would
then be loaded into a global solid model simulation for physics-based stress
analysis. In the next section, we discuss such physics-based simulations, and
their role in digital twins. Altogether, applications of LiDAR and laser
scanning for digital twin modeling will become extremely important in the
coming years as cities struggle to maintain the ever increasing number of
bridges and other critical infrastructure.

##### (c) VR, AR, and MR technologies

In addition to solid modeling and laser scanning, other techniques, such as
virtual reality (VR), augmented reality (AR), mixed reality (MR), have also
been explored as viable methods of constructing digital replicas of various
physical entities. These visualization and interaction techniques differ in
how they display relevant information. The most simple method, AR, overlays
relevant contextual data to the user by projecting over assets in the physical
world. Typically, the projections are done using light projection onto
surfaces, or in some cases, through a wearable headset with transparent
lenses. MR technologies further extend the capabilities of AR by allowing
interactions between the digital objects and the physical world. This
technology is closely followed by VR, which is a completely immersive digital
environment where interactions happen exclusively in the virtual world. These
data visualization and interaction technologies are promising enablers of
digital twin modeling because of their ability to connect users, assets, and
data streaming in real-time. VR, AR, and MR technology research is an active
field. For example, Tadeja et al. (2020) developed an immersive aerospace
design environment built upon VR to aid the design process of aerodynamic
components by interactively visualizing performance and geometry. Other work
by Cai et al. (2020) leveraged AR to link the layout information between a
physical and digital asset in a reconfigurable additive manufacturing system
made of robotic arms for tool path planning and simulation. Choi et al. (2022)
integrated MR in a digital twin to achieve safety-aware human-robot
collaboration, where MR was combined with safety-related monitoring to track
the shared workplace and real-time safety distance calculation. Research into
VR, AR, and MR technology remains a hot and active topic, as the role of these
powerful data visualization and interaction techniques is not yet fully
understood. In the future, it is expected that these technologies will enable
high-fidelity interactive digital twin models that can be used to visualize,
control, and optimize various processes and designs in real-time.

###  3.2 Physics-based modeling

Scientists observe the basic principles of how our world works and translate
them into physical laws. These laws define the rules for the motion of matter
and other physical phenomena through space and time. It is well-known that
partial differential equations (PDEs) can mathematically describe certain
physical laws. Two well-known examples of PDEs are the Navier–Stokes equations
describing fluid motion Chorin et al. (1990) and the heat equation describing
heat diffusion Widder (1976). Combining PDEs into a mathematical model and
solving this model allows engineers and scientists to simulate and observe the
physical phenomena governing a system or process. In computer-aided
engineering (CAE), engineers and researchers use software packages to
formulate mathematical models consisting of PDEs that describe the underlying
physics, initial conditions, and boundary conditions. After defining the
models, the software solves the PDEs to simulate the physical effects so that
engineers and researchers can better understand the physical interactions.
Simulating the solution approximates the input-output relationship of the
physical system, where the input consists of the physical system properties,
parameterized in the mathematical model, as well as the operating and
environmental conditions, and the output is the performance of interest.
Physics-based modeling software is diverse and complex in the range of
applications it can be used for. Below we outline some of the most common
types of physics engineers and researchers aim to simulate.

  * •

Types of physics: Physics-based modeling and simulation for use in digital
twins covers a wide scope. Below, we highlight and summarize some of the most
common types of physics and physical modeling techniques used for digital
twin.

    1. 1.

Solid body structural analysis of stress and strain is typically conducted
using finite element analysis (FEA) software where the objects are modeled as
a tessellation mesh of triangles, and the forces are calculated at each node
in the mesh. FEA is flexible in that the resolution of the mesh can be tuned
to balance the trade-off between simulation accuracy and computational costs.
FEA has so far been an important modeling technique for digital twin (Li et
al., 2017, Angjeliu et al., 2020, Millwater et al., 2019, Moi et al., 2020,
Karve et al., 2020, Ye et al., 2020, Liu et al., 2021c, Bellalouna, 2021, Wang
et al., 2022a). Specific examples of FEA for digital twin include stress
analysis of a ground vehicle suspension (Hu and Youn, 2011), wear analysis of
cutting tool in manufacturing (Zhang et al., 2021b), strain analysis (Revetria
et al., 2019), and dynamic structural analysis of an aircraft wing (Seshadri
and Krishnamurthy, 2017). As an initial digital twin proof of concept, Haag
and Anderl (2018) demonstrated simultaneous simulation and measurement of
forces in a beam bending test bench.

    2. 2.

Thermal and fluid flow analyses are typically conducted using computational
fluid dynamics (CFD) software (Zhou et al., 2021). Fluid physics are
especially relevant in aircraft and spacecraft simulation, where digital twin
was first conceived (Grieves, 2014). In many cases, fluids are closely tied to
heat transfer, as flowing fluids are exceptionally effective at exchanging
heat. Fan et al. (2013) performed thermal analysis for a battery cooling
system of an electric vehicle. In the future, thermal and fluid flow analyses
will prove exceptionally useful in digital twins.

    3. 3.

Kinematic and dynamic analyses of mechanisms using multi-body dynamics (MBD)
models are another type of physics commonly modeled. Ha et al. (2018) analyzed
the motion of robots using MBD; Xia et al. (2021a) used MBD to simulate and
test the interference and collisions of the components in their digital twin
model.

    4. 4.

Multiphysics simulations simultaneously simulate multiple coupled physical
phenomena by sharing information between individual physics-based simulations
(DeLaurentis and Mavris, 2000, Hu and Mahadevan, 2018a). Examples include
coupled FEA and CFD simulations in fluid-structure interaction for aircraft
wings (Tezduyar and Osawa, 2001) and wind turbines (Hsu and Bazilevs, 2012),
and coupled electrochemical-mechanical simulations for modeling Li-ion battery
cells (Allu et al., 2014, Wang et al., 2022b). Another example of coupled
multiphysics interaction is the hypersonic vehicle shown in Fig. 7. This
multiphysics simulation features coupled aerodynamic and aeroelastic analyses
to predict the complex vehicle response under hypersonic airflow conditions
(Culler and McNamara, 2010).

Figure 7: Coupled multiphysics simulation of a hypersonic vehicle panel (Hu
and Mahadevan, 2018a)

  * •

Levels of fidelity: When building a physics-based model, it is important to
consider the minimum level of detail (fidelity) required to approximate the
physical input-output relationship in order to optimize the balance between
computational costs and model accuracy. Take for example a fluid dynamics
problem which can be solved using different CFD models. With the highest
fidelity, direct numerical simulation can provide an accurate high-fidelity
solution to model turbulence flow. With slightly lower fidelity, large eddy
simulation and the Reynolds-averaged Navier–Stokes equations can also be used
to solve the same problems more quickly, albeit with a slightly lower
accuracy. Another example of multi-fidelity modeling arises in the battery
modeling community. There is a well-established hierarchy of electrochemical-
thermal models that consists of (from high- to low-fidelity) the 3D
Doyle–Fuller–Newman (DFN) model, 2+1D DFN model, the DFN model (or pseudo-two-
dimensional), and the single particle model Marquis et al. (2020). The field
of multi-fidelity modeling is rich with literature, and interested readers
should read several of the dedicated review papers (Fernández-Godino et al.,
2016, Peherstorfer et al., 2018, Giselle Fernández-Godino et al., 2019). One
final note regarding model fidelity is that the data-driven models presented
in Sec. 3.3 fall into the broad category of low-fidelity models, and great
effort is being done to find ways to combine physics-based and data-driven
models for improved fidelity (see Sec. 3.4)

When defining a relevant physics-based modeling problem, the engineering
problem at hand determines which types of physics should be modeled. Many
commercial CAE software tools streamline the execution of three basic steps:
(1) loading in 3D geometry data from a geometric model created in CAD software
or via laser scanning as reviewed in Sec. 3.1, (2) defining the initial
conditions and determining the geometric and physical bounds of the
simulation, and (3) solving the model to approximate the performance of
interest. Examples of software tools used to solve physics-based simulations
are ANSYS Mechanical, Abaqus FEA, and Altair HyperMesh for FEA, ANSYS Fluent,
Autodesk CFD, and SimScale for CFD, MATLAB & Simulink (Simscape Multibody),
MSC ADAMS, and RecurDyn for MBD, and COMSOL Multiphysics and MSC Nastran for
coupled multiphysics simulation. A few noteworthy open-source packages
relevant to digital twin are OpenFOAM and Stanford University’s Unstructured
(SU2) package for CFD, MBDyn for MBD, and for multiphysics simulation,
Advanced Simulation Library (ASL) and Chrono open-source options.

It is important to strike a balance between accuracy and computational cost
when choosing the level of fidelity for a physics-based simulation. No matter
how high the fidelity of a physics-based model might be, it is still only an
approximation of reality because of missing physics, flawed assumptions,
finite mesh discretization, etc. If the resulting model bias is large (likely
for a low-fidelity model), it is important to quantify the bias offline and
compensate for it online (see Sec. 2.1.2 in Part 2 of the review paper for a
detailed discussion on the offline calibration of dynamic system models).
Furthermore, it may be counter-intuitive to say that high-fidelity models are
not always preferred even when computational costs incurred in DS, P2V, and
V2P are not a concern. Practitioners need to keep in mind that high-fidelity
models contain larger numbers of parameters that need to be calibrated
offline, and during online deployment, a subset of the parameters need to be
further tuned. Offline validation and online fine tuning requires the use of
previously collected data and measurements. However, most of the time, very
limited data is available for both offline model parameter estimation and
online fine tuning. Practioners who deploy a high-fidelity model but have
inadequate data to properly tune the parameters may find that model
underperforms, and exhibits poor accuracy. As a result, practical
implementations of physics-based models are (1) a lower-fidelity model that
offers the desired accuracy at a lower cost and with a smaller number of
parameters for offline calibration and (2) a small and critical subset of the
model parameters are identified for online updating, reducing the amount of
data required (see also a discussion in Sec. 4.2 (a)).

###  3.3 Data-driven modeling

Data-driven models are needed for modeling physics in a digital twin under
either of two main scenarios:

  1. 1.

the underlying physics is too complicated or is not fully understood, and as a
result, building an accurate physics-based model is impossible; or

  2. 2.

the physics is well understood and can be modeled using available software,
but the simulation is too computationally expensive or time consuming to be
useful in a digital twin, especially when many model runs are required (such
as with UQ tasks).

Figure 8: Overview of data-driven models, where _italic_ font represents
statistical-modeling methods; and _non-italic_ font indicates ML methods.

In either scenario, data-driven models must be constructed to represent the
underlying physical input/output behavior, enabling efficient prediction of
digital states for real-time control and optimization in a digital twin. Shown
in Fig. 8, data-driven models can be grouped into three classes according to
their application domain, namely (1) data-driven models for _degradation
modeling_ , (2) data-driven _surrogate models_ which can entirely replace the
physics-based models discussed in Sec. 3.2, and (3) data-driven models for
_dynamic system identification_ using sensor measurements from a physical
system (PS in Eq. (1)). Moreover, it is worth mentioning that system
identification is fundamentally very similar to surrogate modeling. System
identification is generally the determination of a model’s characteristic
parameters (regardless of model class) that minimize some predictive error
metric, and one may note that essentially a surrogate model is trying to mimic
a system’s behavior also with a goal towards some minimized prediction error
metric. The main difference between the two is that surrogate modeling is
typically constructed and optimized using computer simulation data for any
type of physical simulation at any time scale, while system identification
mainly uses online sensor monitoring data and/or offline experimental data
collected to tune the specific model’s parameter(s). The close similarity is
why many of the surrogate modeling methods and system identification methods
can be used interchangeably. This relationship is further explained in Fig. 8.

Data-driven models for use in digital twins can be classified into two
categories according to the fundamental theory and construction (e.g.
numerical or analytical), namely _statistical models_ and _ML models_.
Statistical models and ML models have been used for both prediction and
inference (Danilo et al., 2018). “ _Statistical methods have a long-standing
focus on inference, which is achieved through the creation and fitting of a
project-specific probability model, while ML concentrates on prediction by
using general-purpose learning algorithms to find patterns in often rich and
unwieldy data_ ” (Danilo et al., 2018, Bzdok et al., 2017). In what follows,
we discuss the latest data-driven modeling trends and provide insights on how
each method can or has been used in a digital twin framework.

####  3.3.1 Statistical models

Statistical models play a vital role in establishing both P2V and V2P
connections in digital twins. For P2V connection, statistical models have been
used in the context of dynamic system identification to build state-space
models based on measurement data. For V2P connection, statistical models have
been extensively used to build degradation models for failure prognostics and
predictive maintenance of physical systems.

  * •

_Statistical models for dynamic system identification (P2V)._ System
identification is the process of learning a mathematical representation of a
dynamic system based on pairs of measured input/output data (Ljung, 1998).
System identification has been extensively studied using statistical methods
in the past decade, especially for linear dynamic systems.

Many approaches have been developed for linear system identification in both
the time domain and frequency domain (Ljung and Glover, 1981). In the time
domain, simple models, like auto-regressive moving average models (ARMA), have
been studied for the identification of dynamic systems with stationary
responses (Rojo-Álvarez et al., 2004, Martínez-Ramón et al., 2006). To
overcome the stationary limitation of ARMA, auto-regressive _integrated_
moving average (ARIMA) models have been developed for non-stationary dynamic
systems (Xuemei et al., 2010). We highlight the _I_ in ARIMA as it denotes the
model’s ability to model trending data as a series of differences, making it a
much more versatile statistical model. Other widely used approaches for time-
domain system identification include stochastic subspace identification (SSI)
methods (Peeters and De Roeck, 1999) and eigensystem realization algorithms
(Juang and Pappa, 1985). In the frequency domain, commonly used modeling
methods include frequency domain decomposition (FDD) (Brincker et al., 2001)
and the least squares complex frequency domain method (Guillaume et al.,
2003). Among the mentioned linear system time-domain and frequency-domain
methods, SSI, a parametric identification method based on stochastic state-
space models, is one of the most widely used approaches. Many variants of the
SSI method have been proposed in the past decade, such as reference-based SSI
(Peeters and De Roeck, 1999) and covariance-driven SSI (Döhler et al., 2011).
A detailed explanation of SSI is available in (Peeters and De Roeck, 1999,
Brincker and Andersen, 2006).

Numerous efforts have also been made to develop statistical methods for
nonlinear system identification (Billings, 2013, Piroddi and Spinelli, 2003,
Worden et al., 2018). The most widely used methods include the nonlinear
autoregressive exogenous (NARX) model (Piroddi and Spinelli, 2003), and the
nonlinear autoregressive moving average model with exogenous inputs (NARMAX)
model (Billings, 2013). In a NARX model, the value of the current output
$y_{k}$ is represented as a nonlinear function of past values of the output
(i.e., ${y_{k-1}},\;{y_{k-2,}},\;\cdots,{y_{k-p}}$) and current and past
values of exogenous inputs (i.e., ${u_{k}},\;{u_{k-1}},\;\cdots,\;{u_{k-q}}$)
as follows

$\begin{split}&{y_{k}}=g({y_{k-1}},\;{y_{k-2,}},\;\cdots,{y_{k-p}},\;\cr&\;\;\;\;\;\;\;\;{u_{k}},\;{u_{k-1}},\;\cdots,\;{u_{k-q}})+{\varepsilon_{k}},\cr\end{split}$
(2)

where $g(\cdot)$ is a nonlinear function, such as a nonlinear polynomial
function, $p$ and $q$ are respectively the number of lags of the output and
the exogenous inputs included in $g(\cdot)$, and ${\varepsilon_{k}}$ is a
Gaussian random variable with zero mean and unknown standard deviation
estimated based on observations along with parameters of $g(\cdot)$. In a
NARMAX model, the noise sequences (i.e.,
${\varepsilon_{k}},{\varepsilon_{k-1}},\cdots,{\varepsilon_{k-e}}$) are also
included in $g(\cdot)$ in addition to past values of exogenous inputs and the
response (Billings, 2013). When an ML method is employed to construct the
nonlinear function $g(\cdot)$, the nonlinear system identification methods
fall into the intersection of statistical methods and ML methods (Worden et
al., 2018).

There are also several review papers summarizing various system identification
approaches. Namely, Åström and Eykhoff (1971) conducted a detailed review of
various system identification methods of dynamic systems, Ljung (2010) shared
his perspectives on system identification, Sirca Jr and Adeli (2012) reviewed
approaches for system identification in structural engineering, and Kerschen
et al. (2006) conducted a comprehensive review on developments in system
identification of nonlinear dynamical structures and illustrated them using
numerical and experimental examples.

The dynamic system models identified using the system identification methods
reviewed above can enable real-time state estimation and control, a highly
desirable ability of a digital twin. For instance, He et al. (2019) used a
discrete-time state-space model identified from data to model various
component-level dynamics in constructing a virtual system in their digital
twin model. Similarly, Xia et al. (2021b) used the MATLAB system
identification toolbox (see Table 1 in Part 2 of the review paper) to build
digital models for their digital twin.

  * •

_Statistical methods for degradation modeling in predictive maintenance
(V2P)._ One of the most important elements in digital twins is degradation
modeling. Degradation modeling is vital for predicting the end of life of a
physical system to enable predictive maintenance (i.e., V2P). In some cases,
degradation mechanisms of the physical system involve complicated physics
which may not be fully understood. In these scenarios, a physics-based
degradation model cannot be constructed, and an empirical statistical model
must be used instead. Empirical modeling leverages historical data collected
from failed units to construct a degradation model. Shown in Fig. 8, many
statistical models have been proposed for degradation modeling of both single
and multiple failure mechanisms. For example, numerous stochastic process
models, including Markov models (Giorgio et al., 2011, Lee et al., 2010, Vega
et al., 2021), semi-Markov models (Kharoufeh et al., 2010, Liu et al., 2018a,
Compare et al., 2016), Wiener process, Poisson process, inverse Gaussian
process, Gamma process, and accelerated life testing (ALT) models (Hu and
Mourelatos, 2018), have been developed for degradation modeling of single
degradation mechanisms (Ye and Xie, 2015, Gorjian et al., 2010). For
degradation modeling with multiple degradation mechanisms, multi-phase
stochastic process models (Liao et al., 2021), piece-wise Markov process
models (Vega et al., 2020), copula functions (Peng et al., 2016), and system-
level ALT models (Moustafa et al., 2021), have been proposed.

While statistical degradation models allow for effective prediction of the end
of life of physical systems, there are several challenges that need to be
addressed in order for some of the models to be used in a digital twin. For
example, a Markov degradation model usually describes the degradation as a set
of highly abstract discrete states, such as ratings given in letters (e.g., A,
B, C, D). It is often difficult to directly use online sensor data, collected
from the physical system continuously and in real-time, to update the highly
abstract discrete states. This poses a real challenge in seamlessly
establishing the P2V connection in a digital twin system (see Fig. 3). To
tackle this challenge, researchers have proposed approaches integrating
Bayesian model updating methods reviewed in Sec. 4.2 with Markov chain model
for model updating and failure prognostics in Chiachío et al. (2020), Vega et
al. (2021). In addition, the uncertainty of a degradation model needs to be
properly quantified in order for such models to be fully integrated into a
maintenance decision making framework. Accurately quantifying the uncertainty
of prognostic models is essential to the effective integration of these models
into digital twins for predictive maintenance, especially for high-value
physical assets and in safety-critical applications, as will be discussed in
Sec. 5.2. A detailed discussion on the role of UQ in digital twins will be
discussed in Section 2.1 of Part 2 of the review paper.

####  3.3.2 ML models

With the increase in IIoT connected devices, big datasets have become
commonplace, and advanced data analysis methods have been developed to take
advantage of the collected knowledge. ML models, which can learn from a
dataset and be used to predict on new, unseen data, have become a popular
method for modeling physical systems in a digital twin (Min et al., 2019,
Ibrahim et al., 2020, Liu et al., 2020a, Priyanka et al., 2022, Elayan et al.,
2021, Ladj et al., 2021, Hu et al., 2021a). As shown in Fig. 8, ML methods can
be further categorized into conventional ML methods and newer, emerging DL
methods.

_Conventional ML methods_ , as indicated in Fig. 8, include feed-forward
neural networks (Chen and Billings, 1992), support vector machines (Zhang et
al., 2006), random forests (Chai and Zhao, 2019), and Gaussian process
regression (Worden et al., 2018). All of these methods have been used in
digital twins for (1) degradation modeling, (2) surrogate modeling to replace
first-principles models, and (3) dynamic system identification from data. For
example, for surrogate modeling, Chakraborty and Adhikari (2021), Molinaro et
al. (2021) used Gaussian process regression models to replace computational
simulations in digital twins, and thereby enable real-time prediction; Zhu et
al. (2021) trained ML models to build a digital workpiece model for a
manufacturing system digital twin; Zhou et al. (2022b) investigated using
multiple ML approaches to characterize the relationship between crack
size/shape and crack-front stress intensity factors in a helicopter. For the
identification of dynamic systems, particularly nonlinear systems, various ML-
based approaches have been proposed by following the NARX format given in Eq.
(2). When a Gaussian process regression model is used as $g(\cdot)$ in Eq.
(2), it is called a GP-NARX model (Doerr et al., 2018). When feed-forward
neural networks are used, it is called a NARX-net (Lin et al., 1996).

As a subarea of ML, _DL_ is a relatively new modeling technique which uses
large multi-connected networks with many parameters to approximate physical
process or functions (LeCun et al., 2015). Deep learning models are easily
reconfigured to have any number of parameters or architecture, and for this
reason, they have a significant advantage over traditional ML models when
dealing with massive high-dimensional heterogeneous datasets (Tang et al.,
2020, Rassõlkin et al., 2019, Hu et al., 2022a). As shown in Fig. 8 and
similar to conventional ML models, DL models have been applied in various
digital twin applications for degradation modeling, surrogate modeling, and
system identification. In particular, the application of DL models for
_degradation modeling_ to establish a V2P connection in a digital twin has
gained much attention recently. Applications of ML and DL models for
degradation modeling in failure prognostics and predictive maintenance will be
reviewed and discussed in detail in Secs. 4.4.4 and 5.2. Here, we briefly
summarize the application of DL models for surrogate modeling and dynamic
system identification.

  * •

In recent years, a large number of DL methods have been proposed for
_surrogate modeling_. DL models can replace many expensive physics-based
simulations by approximating the underlying physics using the highly-
interconnected model parameters. For example, Guo et al. (2022) developed a
surrogate model for thermal signature prediction in laser metal deposition
using DL models. Tang et al. (2020) proposed a DL-based surrogate modeling
method to predict dynamic subsurface flow in channelized geological models.
Similarly, Weber et al. (2020) developed a DL surrogate model for earth system
simulation models. In deep neural network-based surrogate modeling methods, it
is worth highlighting a particular type of neural network – long short-term
memory (LSTM) networks, which are a type of recurrent neural network. Unlike
regular feed-forward neural networks, LSTM has a feedback loop inside each
cell which allows information to persist in time throughout the network
(Hochreiter and Schmidhuber, 1997). As a result, it can capture long-term
dependencies in time series data. Because of this appealing feature, LSTM
networks have been extensively leveraged to build surrogate models for
sequence-to-sequence prediction with different number of inputs and outputs
(e.g., one to many, many to one, and many to many), such as in aerodynamics
(Zhang et al., 2021c), elasto-plastic finite element simulation (Im et al.,
2021), and microstructure evolution applications (Montes de Oca Zapiain et
al., 2021), among others. Moreover, numerous hybrid DL architectures, such as
convolutional neural network (CNN)-LSTM (Zhao et al., 2022) and CNN-
Bidirectional LSTM (Zhang and Yin, 2021), have been proposed in recent years
for surrogate modeling of various physical systems.

  * •

For _system identification_ , DL models are becoming more popular due to their
advantage over conventional ML and statistics-based models in handling high-
dimensional and heterogeneous data sources, such as videos and images. As
mentioned above, the main difference between system identification and
surrogate modeling is that system identification is mainly done using sensor
data collected during online operation, whereas surrogate modeling may be done
with either physics-based simulation data or sensor measurement data.

There are two main approaches for system identification using DL. In the first
approach, DL models are used to learn the _input-output_ relationship directly
in a nonlinear autoregressive exogenous scheme. In this way, the DL model
learns from data to act as a black-box state-space model of the system. The
learned states and state transition functions are encoded in the DL model’s
parameters and are typically not easy to visualize because the model’s contain
many thousands of parameters. Zhang and Mahadevan (2020) used an LSTM DL model
to learn the relationship between the measured aircraft velocity and flight
trajectory to facilitate the safety assessment of en-route flight. Ma et al.
(2021) used a probabilistic LSTM to learn the relationship between geometric
sensor data and thermal elongation. Hu et al. (2022b) developed a grasps
generation-and-selection CNN to facilitate performance assessment of robotic
grasps in the context of digital twin, where grasping state (i.e., position,
rotation angle, and gripper width) was captured and characterized in the form
of red–green-blue-depth images. Wu and Jahanshahi (2019) presented a deep
convolution neural network-based method for system identification of
structural dynamic systems. They showed that CNN-based system identification
methods are more robust than conventional multilayer perception neural
networks-based methods when the signals are contaminated by large noise.

In the second approach to system identification using DL, the state-space
model of a system is learned in a more easily interpreted, low-dimensional
form. The most popular DL model for this is the autoencoder (Masti and
Bemporad, 2021, Li and Yang, 2021, Otto and Rowley, 2019). Shown in Fig. 9, an
autoencoder is given the name _encoder_ because it takes high dimensional
inputs $I_{k-1}$ and encodes them as a low-dimensional state vector $x_{k}$.
The low-dimensional state vector can easily be viewed and visualized since it
is of much lower dimensionality than the system inputs and outputs. To
estimate the system outputs $O_{k}$, the state vector $x_{k}$ is decoded in
the decoder. In a recurrent fashion, the encoded state vector $x_{k}$ is
combined with control system inputs $u_{k}$ using another small neural network
which learns the state transition function $x_{k+1}=f(x_{k},u_{k})$ to update
the state estimate, $x_{k+1}$, at the next time step. The specifically
designed high- and low-dimensional parts of recurrent autoencoder networks
offer practitoners the ability to work with high-dimensional inputs and
outputs while still maintaining a better level of clarity and visualization by
observing the low-dimensional states $x$. Autoencoders and similar DL models
will play a vital role in understanding the many complex input/output
relationships in future digital twin frameworks.

Figure 9: A recurrent autoencoder-based DL model for system identification of
a nonlinear dynamic system (Masti and Bemporad, 2021).

We conclude our discussion on data-driven modeling by listing a few topics
that are crucial for the successful deployment of data-driven models in a
digital twin.

  1. 1.

Model selection: Even though both statistical models and ML models are getting
unprecedented attention in recent years, they have their own advantages and
disadvantages. For example, statistical models are highly explainable while
complex DL models are difficult to interpret and are often thus used in a
black-box way. There are so many data-driven models available, however,
selection of an appropriate data-driven model should always follow the
“Occam’s Razor” principle. For example, if a conventional ML model, such as a
Gaussian process regression model, can be employed to build an accurate data-
driven model, there is no need to construct a very complicated DL model for
the same task. The selection of an appropriate data-driven model for a
specific application of digital twin is an important optimization problem.

  2. 2.

UQ of ML models: George E.P. Box famously said “All models are wrong, but some
are useful,” as recorded in Box (1976). This is also true for all data-driven
models including statistical models, conventional ML models, and DL models. It
is therefore important to quantify the uncertainty in data-driven models,
especially ML models. UQ of ML models allows us to quantify the confidence
level of ML predictions. It plays a vital role in determining when the model
prediction will go wrong and how bad the prediction is if it is wrong. A
detailed discussion of such an important topic is given in Section 2.1.1 in
Part 2 of the review paper.

  3. 3.

Data collection: Data is essential for the learning of data-driven ML models.
The quantity and quality of available data significantly affect the
performance of data-driven models, and eventually the effectiveness of digital
twins. In engineering practice, high-fidelity physics-based simulations may be
computationally very expensive and physical experiments may be very time- and
resource-consuming. In those situations, it is difficult to collect large
volumes of high quality training data to build data-driven models. Ways to
tackle this challenge include (1) physics-informed ML, reviewed in Sec. 3.4;
(2) active learning where data are collected adaptively to refine data-driven
models in a data-efficient manner (Cohn et al., 1996), with Bichon et al.
(2008) being the first application of active learning to reliability analysis
under uncertainty; and (3) online model updating, as will be reviewed in Sec.
4.3, where an ML model is first trained using a small amount of data offline
and then is updated using data collected online to improve the prediction
accuracy.

###  3.4 Physics-informed ML

For some aspects of the physics in digital twins, both data from first-
principle (or physics-based) simulations and data collected from a physical
system may be available. When both physics and data are present, methods
presented in Sec. 3.2 and Sec. 3.3 can be integrated, resulting in hybrid
physics-ML approaches for modeling physical systems. For instance, Luo et al.
(2020) fused a first-principle model with operational data to build a multi-
domain model in a digital twin. Similar approaches have been attempted,
sometimes labeled as physics-informed ML, for physical system modeling in
digital twins (Liu et al., 2018b, Yu et al., 2020, Ritto and Rochinha, 2021,
Broo et al., 2022). Incorporating physical knowledge into data-driven ML
models can offer several attractive benefits, including better generalization
of ML models to unseen “out-of-distribution” test data, as will be discussed
in detail in 4.3, the ability to explain the underlying physics of the
predictions from the ML models, and speeding up the training process because
some physical laws can restrict the feasible region of solutions and limit the
values the model parameters can take. According to the way that a first-
principle model is integrated with a data-driven ML model, existing hybrid
approaches for physical system modeling can be broadly classified into six
approaches, shown in Fig. 10 and explained in what follows.

  * •

Approach 1: Physics-Informed Loss Function This approach modifies the loss
function of a data-driven ML model by adding a physics-informed loss term that
penalizes model predictions not compliant with first principles (or physics),
thus constraining the training of the ML model toward solutions that comply
with physics (Raissi et al., 2019a, Jagtap et al., 2020a, Lu et al., 2021).
This approach is the most common way to build hybrid physics-ML models and has
found success in many digital twin applications such as solving PDEs and
surrogate modeling (Raissi et al., 2019a, Zhu et al., 2019b, Haghighat et al.,
2021, Gao et al., 2021, Zhou et al., 2022a), inverse modeling (Raissi et al.,
2019b, Jagtap et al., 2020b), fault diagnostics (Shen et al., 2021), and
degradation modeling (Yan et al., 2022). This method of physics-informed ML is
different than the other approaches in that it does not necessarily require
any training data from a physics-based simulation, a significant advantage
over other methods.

  * •

Approach 2: Data Augmentation This second approach first runs first-principle
simulations to generate data at various states an operating conditions of a
physical system. Examples of states include healthy, degraded/damaged, and
failed for predictive maintenance applications. It then combines the generated
data (or synthetic data) with actual data (e.g., experimental data) to create
an augmented training dataset for training data-driven ML models. An example
that falls into this category is the digital twin model presented by Ritto and
Rochinha (2021). The authors generated different damage scenarios using first-
principle simulations to augment a training dataset for an ML classifier used
for damage detection of a bar structure. Another example is the application of
physics-informed ML to degradation diagnostics of lithium-ion batteries, where
a simple physics-based half-cell model was used to generate high-degradation
simulated data to allow training ML models on small, early-life experimental
data (Thelen et al., 2021). Using physics-based synthetic data to train an ML
model partially constrains the ML model’s output with the physics carried by
the first-principle model, thereby improving the generalization performance of
the ML model. Additionally, this data augmentation approach can reduce the
data size required for training large-scale DL models. A necessary condition
for the success of this approach is that the first-principle model captures
physics highly relevant to the problem at hand. However, the model does not
need to be highly accurate.

  * •

Approach 3: Transfer Learning This physics-informed ML approach is inspired by
the concept of transfer learning, which has been explored in various digital
twin applications, including multi-fidelity modeling, such as Huang et al.
(2022b) for melt pool modeling in additive manufacturing, process quality
monitoring, such as Kapusuzoglu and Mahadevan (2020) (the third type of
strategy) for in-line quality monitoring of additive manufacturing processes,
and data-driven prognostics such as Zhang et al. (2018) for turbofan engine
prognostics, Shen et al. (2020) for battery capacity estimation, and Wang et
al. (2021b) for bearing fault diagnosis. It first pre-trains an ML model using
a large quantity of data from first-principle simulations. It then fine-tunes
the model using a small quantity of real data (e.g., data from physical system
experiments). Similar to the data augmentation approach, this transfer
learning approach can improve the generalization of ML models and alleviate
the data size requirement. Its success also highly depends on how relevant the
physics captured by the first-principle model is to the
classification/regression problem that needs to be solved.

  * •

Approach 4: Delta Learning (Missing Physics) The basic idea of this delta
learning approach is to add a data-driven ML model, co-existing with a first-
principle simulation model, that learns to recover the unmodeled physics. This
hybrid modeling approach is commonly used in multi-fidelity modeling and
quantification of simulation model discrepancy. For example, Jiang et al.
(2022) added a polynomial chaos expansion model on top of a simplified
physics-based degradation model for failure prognostics of a structural system
called miter gates. The role of the polynomial chaos expansion model was to
represent the discrepancy of the simplified physics-based degradation model
due to unmodeled dynamics. Yucesan and Viana (2020) built a data-driven LSTM
model to augment a physics-based fatigue model, which created a hybrid model
for fatigue prognostics of wind turbines.

  * •

Approach 5: Delta Learning (ML Prediction) In this approach, an ML model
learns residuals on top of initial predictions by another ML model, trained
using data generated by a physics-based model. The final predictions are the
sum of the initial predictions and residuals. The initial predictions can also
be made directly by a physics-based model (Zeng et al., 2020). The logic
behind this hybrid physics-ML approach is two-fold: (1) the first-principle
model use baseline physics to produce initial, population-based predictions
that may not be accurate for individual system units but can generalize well
to different operating conditions; and (2) adding the data-driven residuals to
compensate for missing physics and unit-to-unit variability. The resulting
performance gains are improved generalization over pure data-driven ML and
improved prediction accuracy over purely physics-based modeling. Just like the
above physics-informed ML approaches, this delta learning approach may not
work well if the first-principle model does not capture relevant physics.
Issues may also arise when the first-principle model cannot capture the
effects of testing or operating conditions on experimental data.

  * •

Approach 6: ML-Assisted Prediction This sixth approach to physics-informed ML
uses a data-driven ML model to predict the input $\mathbf{x}$ or parameters
$\boldsymbol{\uptheta}$ of a first-principle model. When implementing this
approach in practice, the first step is identifying the input or a subset of
the parameters as variables, $\mathbf{{x}}_{\mathrm{FPM}}$ or
$\boldsymbol{{\uptheta}}_{\mathrm{FPM}}$, to be predicted by the ML model.
Then, the ML model estimates or, in some cases, forecasts
$\mathbf{{x}}_{\mathrm{FPM}}$ or $\boldsymbol{{\uptheta}}_{\mathrm{FPM}}$.
Finally, the first-principle model uses the ML estimates or forecasts to
predict the response of interest $y$ at current (estimation) or future
(forecasting) time steps. A clear benefit of this approach is the physical
interpretability, coming from (1) using physically meaningful variables as
intermediate $y$, predicted by an ML model, and (2) using a first-principle
model to predict the final $y$. Another advantage is the improved
generalization performance, attributable to the use of the first-principle
model. A typical challenge with this approach is the low identifiability of
the first-principle model’s input or parameters from observational data. In
extreme cases, large errors in the ML model’s estimates or forecasts propagate
through the first-principle model, causing the final prediction performance to
be worse than pure ML. Examples of Approach 6 can be found in grease
degradation forecasting (ML) for bearing fatigue life prediction (first-
principle modeling) (Yucesan and Viana, 2022) and degradation parameter
forecasting (ML) for battery capacity fade prediction (first-principle
modeling) Ramadesigan et al. (2011), Downey et al. (2019), Lui et al. (2021).

Figure 10: Six approaches to construct a hybrid of a physics-based and data-
driven ML model: (1) Modifying the loss function of an ML model by adding a
physics-informed penalty term (Karniadakis et al., 2021); (2) Generating
synthetic data using first-principle simulations (Ritto and Rochinha, 2021);
(3) Pre-training an ML model on physics-based synthetic data and then fine-
tuning it on experimental data (Shen et al., 2020, Kapusuzoglu and Mahadevan,
2020, Huang et al., 2022b); (4) Correcting a first-principle model by learning
unmodeled physics from data (Jiang et al., 2022); (5) Correcting an ML model,
trained with physics-based synthetic data, by learning its prediction residual
from experimental data (Thelen et al., 2022a); and (6) Learning to predict the
input of a first-principle model (Yucesan and Viana, 2022). The icon with “G”
inside a black circle indicates improvement in generalization over pure ML;
the icon with “I” inside indicates improvement in interpretability.

We note that the above list is not exhaustive but rather an incomplete
collection of representative approaches to physics-informed ML. For example,
an approach that could, in general, be categorized as Approach 6 is that the
output and parameters of a first-principle model are used to augment the input
of an ML model, as demonstrated for in-line quality monitoring of additive
manufacturing processes in (Kapusuzoglu and Mahadevan, 2020) (the second type
of strategy), and turbofan engine diagnostics in Chao et al. (2019) and
prognostics in Chao et al. (2022). The only difference is that the information
flow is in the opposite direction (i.e., the first-principle model now feeds
an additional input to the ML model). A seventh approach could be defining
physics-based kernel functions for ML models, such as the FEA-Net (Yao et al.,
2020) and physics-informed kernel functions for Gaussian process regression
(Yang et al., 2018). Interested readers can find more comprehensive and
detailed reviews in review papers more dedicated to the topic of physics-
informed ML, such as Karniadakis et al. (2021) for computational physics,
Aykol et al. (2021) for battery lifetime prediction, and Willard et al. (2020)
for a comprehensive collection of engineering applications. Despite tremendous
advances in combining physics and data for physical system modeling, much more
research is needed to address practical challenges impeding industry-scale
adoption of hybrid physics-ML modeling approaches. Among those challenges are
(1) a lack of large, realistic datasets relevant to different digital twin
applications for training, validating, and testing hybrid models and
benchmarking their generalization and interpretation performance (see also the
discussion in Section 4.2 of Part 2 of the review paper), (2) a shortage of
digital twin benchmark problems that allow researchers and practitioners to
compare prediction accuracy and computational efficiency of different hybrid
approaches and determine their suitability for real-time physical system
modeling, and (3) a lack of academia-industry collaborative effort in building
open-access, well-maintained platforms that can streamline data sharing, model
building, and benchmarking, such as many existing efforts to streamline ML
workflows.

###  3.5 System modeling

A physical system usually consists of many subsystems and components that
interact with each other in a variety of means. It is, therefore, important to
model the high-level interactions and the overall interconnection structure
among different components in the physical system. This section briefly
reviews two widely used techniques for modeling interactions in physical
systems.

##### (a) Unified Modeling Language (UML) and Systems Modeling Language
(SysML)

Towards systems modeling, the Unified Modeling Language (UML) is a general-
purpose language that is commonly used for software development in model-based
systems engineering (MBSE) (Ramos et al., 2011). In brief, UML is a graphical
modeling language that provides a standardized way to specify the structure of
a software system and visualize the interactions of different models in the
system (Glatt et al., 2021, Weilkiens, 2011). In engineered systems, UML has
been increasingly utilized to characterize the design of a system with regard
to its high-level structure, behavior, and interactions. To this end,
different diagrams are created to reflect the model in line with the intended
purpose from a certain perspective, and these diagrams can be grouped into two
categories:

  1. 1.

Structure diagram: It describes the static structure of elements in a system,
which is often adopted to document the architecture of software systems.
Structure diagram includes class diagram, component diagram, object diagram,
deployment diagram, etc. Take the class diagram as an example, it is often
used to describe a set of classes, interfaces, and collaborations and their
relationships in object-oriented software systems. For example, both classes
students and professors belong to a super class person, and the relationship
between professors and students is that professors supervise students.

  2. 2.

Behavior diagram: Differing from the structure diagram, behavior diagram
emphasizes on characterizing and visualizing the dynamic aspects of the
system. It is composed of activity diagram, use case diagram, sequence
diagram, state chart. Take the same students versus professors as an example,
in the case of course taking, the sequence diagram illustrates the very basic
steps in this task ordered by the timeline, starting from the very beginning
course registration to homework assignment/submission and homework grading and
report. In short, the sequence diagram summarizes the dynamic interactions
between different users (i.e., students, professors, administrators) and
applications (i.e., course registration, homework submission, homework grading
and report) over time.

The promise of UML in modeling complex system interactions has been
demonstrated in multiple studies. For example, Luo et al. (2019) leveraged UML
to characterize the connections among multiple physical subsystems (i.e.,
mechanical, electrical, hydraulic) in the digital twin of a CNC machine tool.
Note that UML is intensively used for interaction modeling, focusing heavily
on the software side.

Differing from UML, the Systems Modeling Language (SysML) is independent of
the specific discipline (e.g., software, hardware, personnel, or facilities),
and it is a general-purpose architecture modeling language in MBSE through
extending and omitting certain UML elements (Weilkiens, 2011). As driven by
user requirements, SysML consists of four key pillars: structure (e.g., system
hierarchies), behavior (e.g., state machine), properties (e.g., attributes of
time variables), and requirements (e.g., requirements’ hierarchies and
traceability). Using SysML facilitates the design, specification, analysis,
verification, and validation of engineering products and processes across a
wide range of systems and systems-of-systems. Thanks to SysML’s rich features
in cross-system extensibility and traceability, there is an increasing number
of studies investigating how SysML is adopted in digital twins (Sun et al.,
2020b). For example, Bachelor et al. (2019) leveraged the traceability of the
allocation of requirements to system function definitions in SysML, and used a
digital twin model of an ice protection system to deal with changes in system
design through data traceability across the abstracted lifecycle. Zhang et al.
(2022a) used SysML as a model fusion tool to establish the relationship among
various domain models in a shop-floor digital twin. A similar approach is also
developed by Liu et al. (2021a) to model the interaction of different models
in Unity3D for their shop-floor digital twin model.

In digital twins, as software elements (i.e., simulation models, database,
machine learning models, decision support modules) play increasingly important
roles, UML and SysML have been frequently used to manage the growing software
complexity, where a unified interface is provided to describe the structure,
behavior, and interactions among software modules. Such architecture-level
view of all elements and their interdependencies in the overall system brings
multiple benefits, such as ensuring the planned functionality and the
envisioned benefits, preventing delays of the actual implementation due to
planning errors, easing project transfer due to employee leave. The state-of-
the-art literature has witnessed prominent adoptions of UML/SysML in the
manufacturing industry. A representative usage of UML is showcased by Glatt et
al. (2021), where they considered the digital twin of cyber-physical
production systems. As the digital system contained a significant number of
elements and complex relationships between those elements, Glatt et al. (2021)
adopted a UML-based approach to model system structure and component
interactions associated with different system functions (i.e., monitoring,
diagnostics, prediction). The unified representation of class relationships
and system workflow substantially eases the understanding of the object-
oriented software systems in digital twins.

##### (b) Ontology

Figure 11: A simple ontology model for digital twins. Adopted from Steinmetz
et al. (2018).

In computer science and information science, ontology is a prevailing
knowledge engineering and representation paradigm to describe the properties
of a subject area and define how they are related to each other with explicit
descriptions and specifications. Towards this end, ontology encompasses a set
of domain-specific concepts, entities, and categories as well as their
definitions, attributes, and interrelationships to represent the subject
matter of concern. Suppose we are interested in building an ontology map for
an IoT system, and we have the basic information related to each device and
its digital counterpart. Then a tentative ontology map can be constructed like
Fig. 11. In this figure, rectangles denote the distinct types of entities
(i.e., device, sensor, and attribute), and arrows represent the relationships
between entities (e.g., each sensor has a sensing device).

Note that ontology maps are generalized data models in the sense that they
only account for general types of entities and their abstractions with certain
properties in common. But they do not include any specific information about
an entity in the domain. For example, rather than focus on the characteristics
of a specific device, an ontology map concentrates on the general concept of
devices by capturing common characteristics that most devices have. The
construction of such a unifying conceptual framework significantly fosters the
communication and sharing of the structure of information specific to each
domain that is understandable by both humans and machines (Castells et al.,
2006, Compton et al., 2012). The advantage of ontology in enabling the sharing
and reuse of domain knowledge has been one of the key drivers in the recent
surge in ontology applications, which has been broadly used in machine
learning and information science as a popular means of knowledge
representation (Ji et al., 2021). For example, Akroyd et al. (2021)
demonstrated a digital twin built with a general-purpose dynamic knowledge
graph in the World Avatar project, where ontology-based knowledge
representation was repeatedly exploited to facilitate a unifiable interface
for data queries.

Knowledge graphs, being a popular way to instantiate ontology, are commonly
used to connect concepts in ontologies with specific data for knowledge
engineering and information retrieval (Wang et al., 2017, Lin et al., 2015).
In a knowledge graph, nodes and edges represent the instances of concepts and
the relationships between instantiated entities, respectively. Attributable to
knowledge graphs’ power in querying, inferring, and generating knowledge in
the semantic space, they have been adopted as a key enabling technology to
achieve semantic interoperability of heterogeneous data and information in
engineered systems (Noy et al., 2019, Psarommatis, 2021). For example,
Banerjee et al. (2017) exploited the reasoning power in a knowledge graph and
developed a query language to extract and infer knowledge from large-scale
production line data to support manufacturing process management in digital
twins.

In the context of digital twin, ontology and knowledge graphs offer a common
language to uniquely refer to each possible object (i.e., component or sensor)
in the physical asset. Doing this greatly enhances our flexibility in
describing the evolution of digital twin when it is subject to
systems/subsystems/components changes (e.g., retrofitting) or modifications in
the asset management process (e.g., monitoring) over the entire life-cycle.
For example, Erkoyuncu et al. (2020) demonstrated the benefits of an ontology-
based framework explicitly, where ontology was used to enable the co-evolution
of the digital twin with its physical asset.

##  4 Physical-to-virtual (P2V) twinning enabling technologies

In this section, we summarize five categories of widely employed physical-to-
virtual twinning methods. The P2V twinning methods vary depending on the
modeling approach used. Fig. 12 presents the relationship between the
different modeling methods reviewed in Sec. 3, the P2V methods reviewed in
this section, and the V2P methods that will be discussed in Sec. 5. Shown in
the Fig. 12, some of the P2V twinning methods (e.g., probabilistic model
updating) are widely used across many different modeling methods, while some
other P2V methods (e.g., ontology-based reasoning) are designed for specific
modeling techniques. Next, we will explain the five P2V twinning techniques in
detail.

Figure 12: Relationship between modeling methods, P2V methods, and V2P
methods.

###  4.1 Physical measurements as input to the virtual space

A straightforward way of connecting the physical space with the virtual space
(P2V) is to use physical measurements as inputs (Lydon et al., 2019). Doing so
enables the digital model in the virtual space to predict the response of
physical system in real-time, thus supporting timely risk assessment and
decision making (V2P). The current approaches for this type of P2V connection
can be categorized into two main groups as follows:

  1. 1.

_Using physical measurements to update the movements or position of digital
models._ This group of P2V updating strategy is mainly for the modeling
methods described in Sec. 3.1. For instance, in the manufacturing domain,
real-time position data of various components in a manufacturing system
collected by the IoT system have been used to generate behavior models in a
digital twin (Ricondo et al., 2021), detect disturbances by comparing the
position measurements with prediction (Glatt et al., 2021), drive the movement
of digital models on the virtual shop floor and thereby optimize manufacturing
system configurations or shop scheduling (Leng et al., 2020, Liu et al.,
2021a, Fang et al., 2019, Guo et al., 2021, Zhang et al., 2020a, 2021b). In
the smart city domain, data on mobility (i.e., movement data of human,
vehicles, and other objects), infrastructure, and building conditions are
collected in real-time, and are used as inputs for simulations in the virtual
space for digital twin-based city development and planning (White et al.,
2021, Schrotter and Hürzeler, 2020).

  2. 2.

_Using physical system monitoring data as input for physics-based analysis or
design._ This group of updating strategy is mainly for the modeling methods
summarized in Sec. 3.2. For example, in the aerospace domain, Millwater et al.
(2019) and Guivarch et al. (2019) employed the physical flight condition data
collected from sensors as inputs of a digital model for risk assessment of
aircraft structures. In the design domain, Bellalouna (2021) utilized
operational data collected from sensors and transmitted the data to a digital
environment in the PTC Creo platform for physics-based topology optimization,
demonstrating a digital twin concept. In the manufacturing domain, Liu et al.
(2021c) performed physics-based hardware-in-the-loop simulation using real-
time data from a physical system to replace physical tests, thereby reducing
the verification time and cost.

When discussing P2V connections using sensor measurements, it is imperative to
also consider the available methods of data transfer, as they should be
carefully selected for the application at hand. Digital twin simulations can
require substantial speeds and rates of data transfer in order to enable real-
time optimization and control (see Section 5.3). In general, data transfer
methods can be categorized into two classes: wired and wireless. Wired
connections include Ethernet cables (twisted pair cables), coaxial cables, and
optical fiber cables, where each cable is rated for different data transfer
distances and rates. Wired data transfer is generally preferred, as it is
easier to prevent data loss and is generally quicker as the connections are
direct from source to receiver. Wireless communication methods include
Bluetooth, Wi-Fi, ultra-wide band, and near-field communication (NFC) (Hu et
al., 2021b). More recently, the 5G wireless communication standard has come to
market offering higher data throughput with lower latency. It is envisioned
that 5G technology will enable deployment of more complex and expansive sensor
networks to collect and transmit data in real-time for use in digital twins
(Cheng et al., 2018). These connected sensor and machinery networks, known
more commonly as the industrial internet of things (IIoT), will be responsible
for the data collection and transmission that will enable digital twins.
Ultimately, each of the aforementioned data transmission techniques will play
a critical role in enabling real-time simulation, process control, and
optimization for digital twins.

This straightforward P2V connection is only applicable to cases where digital
states (e.g., position, mobility, traffic flow) can be directly updated using
monitoring data. However, for many engineering problems, digital states cannot
be directly updated and are affected by various uncertainty sources. In those
cases, more advanced P2V twinning methods are required and will be discussed
in the subsequent sections.

###  4.2 Probabilistic model updating

####  4.2.1 Introduction to digital state

As shown in Fig. 3, updating a digital system with a time-varying state is an
essential characteristic of a digital twin. This state can be called a digital
state, defined as a set of digital state variables that characterize the
digital model(s) in a digital twin. Examples of digital states enabling
predictive maintenance include the health of rotating equipment (e.g., motors,
pumps, and compressors) (Wang et al., 2019) and machine tools (Luo et al.,
2020) on a production floor, the material properties and structural health
parameters of an unmanned aerial vehicle (UAV) (Kapteyn et al., 2020), and the
state of charge (SOC) and state of health parameters (e.g., capacity and
internal resistance) of a lithium-ion battery cell (Plett, 2004, Hu et al.,
2012, Li et al., 2020b). A digital state often cannot be directly measured (or
observed) but can be estimated through noisy measurements that depend on the
digital state. The digital state of a physical system in operation changes
over time and can be estimated as new information about the physical system
becomes available. The time evolution of the digital state can be modeled as a
dynamic system perturbed by a certain process noise.

Before we talk about the specifics of probabilistic model updating, let us
look at a few examples of digital states. A recent paper demonstrated a
structural digital twin of a UAV and provided a distinction between a physical
state and a digital state (Kapteyn et al., 2021). A key argument in this
distinction is that the physical state space should allow capturing variation
in the physical system’s state (e.g., the structural state of the UAV) and,
therefore, could have high complexity and dimensions, while the digital state
space should be designed to be simple enough to make the online estimation of
the digital state feasible and tractable. As a result, digital models
composing a digital twin should provide a digital state space that strikes a
balance between complexity (variation in the physical asset captured by the
digital state space is sufficient for performing meaningful diagnostics,
prognostics, and decision making) and tractability (the number of digital
state variables is sufficiently small to allow for good observability for a
fixed number of sensor types and quantity of sensors). Fig. 13 illustrates the
physical and digital states of the structure in a self-aware UAV application.
The digital state encompasses geometric parameters, structural parameters
(e.g., material properties and structural health parameters), and boundary
conditions. Most of these digital state parameters are calibrated offline, and
only a small subset of the structural parameters is dynamically updated online
with sensor data, likely to ensure good model identifiability (Arendt et al.,
2012). It is worth mentioning that the observational data (control inputs)
flow from the physical (digital) to digital (physical) space, which shares
similarities with the data and control elements in our proposed digital twin
model in Fig. 3.

Figure 13: An abstraction of a digital twin enabling a self-aware UAV, shown
as a physical asset-digital twin system. In this example, the digital state
could be a small subset of the physical state, allowing for tractable digital-
state estimation. Reproduced with permission (Kapteyn et al., 2021). Copyright
2021, Springer Nature.

Another example of a digital state is in an industrial application of digital
twin for smart manufacturing (Wang et al., 2019). Although not clearly
defined, the digital state of a rotor system seems to refer to parameters of a
geometric model, parameters of a rotor dynamics model, and material
properties. In the online phase, two parameters of rotor dynamics, namely the
critical speed and vibration amplitude under unbalance, are dynamically
estimated based on vibration data collected from the rotor system. Similar to
Kapteyn et al. (2021), a very small number of digital state parameters are
updated online, suggesting the importance of identifying a small yet critical
set of parameters for online updating.

The third and last example comes from a review paper on battery digital twin
(Li et al., 2020b). Similar to the digital state definitions in Wang et al.
(2019) and Kapteyn et al. (2021), the digital state of a lithium-ion battery
cell/module/pack can be defined as parameters fully characterizing a digital
model of the cell/module/pack. Of particular interest to battery state
estimation is the real-time SOC and state of health, the estimation of which
are essential functions of a battery management system. Since each battery
system in the field is equipped with a battery management system actively
estimating and controlling battery SOC and state of health, one could thus
argue that the digital twin concept is implemented on every battery system.

####  4.2.2 State estimation and Bayesian filters

Let us consider a sequence of unknown digital state vectors,
${{\bf{x}}_{0:k}}=\\{{{\bf{x}}_{0}},\;{{\bf{x}}_{1}},\;\cdots,\;{{\bf{x}}_{k}}\\}$
, and a sequence of noisy measurement vectors,
${{\bf{y}}_{1:k}}=\\{{{\bf{y}}_{1}},\;\cdots,\;{{\bf{y}}_{k}}\\}$. We can
formulate a discrete-time state-space model that takes the following form at
time step $k$ (i.e., time $t_{k}$)

$\begin{split}&{\rm{State}}\;{\rm{transition}}:\;{{\bf{x}}_{k}}={\bf{f}}({{\bf{x}}_{k-1}},\;{{\bf{u}}_{k-1}})+{{\boldsymbol{\upomega}}_{k}},\cr&{\rm{Measurement:}}\;\;\;\;{{\bf{y}}_{k}}={\bf{g}}({{\bf{x}}_{k}})+{{\bf{v}}_{k}},\cr\end{split}$
(3)

where ${\bf{x}}_{k}$ is the vector of state variables at the $k^{\mathrm{th}}$
time step, ${\bf{u}}_{k}$ is the vector of measured exogenous inputs and often
control inputs, ${{\boldsymbol{\upomega}}_{k}}$ is the vector of process noise
variables for the state, ${\bf{y}}_{k}$ is the vector of system observations
(or measurements), ${\bf{v}}_{k}$ is the vector of measurement noise
variables, and ${\bf{f}}(\cdot)$ and ${\bf{g}}(\cdot)$ are the state
transition and measurement functions, respectively. The state transition
equation in Eq. (3) models the time evolution of the state as a dynamic
system, perfectly defined by ${\bf{f}}(\cdot)$, perturbed by a process noise
${{\boldsymbol{\upomega}}_{k}}$; it can be rewritten as the transition
probability density $p({{\bf{x}}_{k}}|{{\bf{x}}_{k-1}})$. The measurement
equation depicts the dependence of the current measurement ${\bf{y}}_{k}$ on
the current state ${\bf{x}}_{k}$; it can be rewritten as the conditional
density of the measurement ${\bf{y}}_{k}$ given the state ${\bf{x}}_{k}$,
$p({{\bf{y}}_{k}}|{{\bf{x}}_{k}})$. At this point, the state-space model in
Eq. (3) can be graphically represented as a hidden Markov model shown in Fig.
14.

Figure 14: An illustrative schematic of a hidden Markov model. This model
consists of: (1) the hidden state ${\bf{x}}_{k}$ that evolves according to the
state transition equation in Eq. (3), and (2) the measurement ${\bf{y}}_{k}$
directly related to the hidden state by the measurement equation in Eq. (3). A
prior probability density $p({{\bf{x}}_{0}})$, usually obtained from an
offline calibration of the digital model (see Sec. 2.2.2 of Part 2 of the
review paper), specifies the initial probability distribution of the hidden
state at the $0^{th}$ time step (i.e., $k=0$)

The objective of the state estimation task in Eq. (3) and Fig. 14 is to
estimate, in a recursive way, the hidden state at the current time step,
${\bf{x}}_{k}$, from the current and past noisy measurements (Särkkä, 2013).
Mathematically, we want to compute the marginal conditional distribution of
${\bf{x}}_{k}$ given on all available measurements up to the current step
($k$), ${{\bf{y}}_{1:k}}$. This marginal conditional of the digital state can
be denoted as $p({{\bf{x}}_{k}}|{{\bf{y}}_{1:k}})$ and be computed using
Bayes’ theorem as follows (Leser et al., 2020, Vega et al., 2022)

$\begin{split}&p({{\bf{x}}_{k}}|{{\bf{y}}_{1:k}})={{p({{\bf{y}}_{k}}|{{\bf{x}}_{k}})p({{\bf{x}}_{k}}|{{\bf{y}}_{1:k-1}})}\over{\int{p({{\bf{y}}_{k}}|{{\bf{x}}_{k}})p({{\bf{x}}_{k}}|{{\bf{y}}_{1:k-1}}){d}{{\bf{x}}_{k}}}}},\cr&\propto
p({{\bf{y}}_{k}}|{{\bf{x}}_{k}})p({{\bf{x}}_{k}}|{{\bf{y}}_{1:k-1}}),\end{split}$
(4)

where ${{\bf{y}}_{1:k}}$ are observations from $t_{1}$ to $t_{k}$, “$\propto$”
stands for “proportional to”, and as described earlier, $p(\cdot|\cdot)$ is a
conditional probability density function.

The above equation is analytically intractable. Recursive Bayesian filtering
approximates the marginal conditional by recursively executing two steps: a
transition step and an update step. Alternative, non-filtering approaches such
as Markov chain Monte Carlo and importance sampling approximate the full
conditional, $p({{\bf{x}}_{1:k}}|{{\bf{y}}_{1:k}})$. These alternative
approaches are not as attractive in the context of digital twin because (1)
they could be computationally much more costly than recursive Bayesian
filtering, especially when $k$ becomes large, and (2) approximating the full
conditional is unnecessary for most digital twin applications.

Table 4 compares four (recursive) Bayesian filters regarding system
nonlinearity, state distribution constraints, and computational efficiency
(Särkkä, 2013). If the state-space model in Eq. (3) is perfectly linear and
Gaussian (i.e., ${\bf{f}}(\cdot)$ and ${\bf{g}}(\cdot)$ are linear functions
and ${{\boldsymbol{\upomega}}_{k}}$ and ${{\bf{v}}_{k}}$ follow Gaussian
distributions), the Kalman filter, one of the simplest and most efficient
Bayesian filters, finds the exact solution to the recursive filtering problem,
i.e., the solution to the conditional marginal is exact. Suppose
${\bf{f}}(\cdot)$ or ${\bf{g}}(\cdot)$ is nonlinear and or ${{\bf{v}}_{k}}$ is
non-Gaussian. In that case, the extended Kalman filter provides a first-order
approximation to the marginal posterior by (1) linearizing the state-space
model, and (2) assuming Gaussian noise (Gustafsson, 2010). This approximation
may be accurate for state-space models whose ${\bf{f}}(\cdot)$ or
${\bf{g}}(\cdot)$ has mild nonlinearity and where true marginal posteriors are
unimodal and symmetric. For state-space models with higher function
nonlinearity, the unscented Kalman filter propagates a small set of points
(called sigma points), sampled based on the prior mean and covariance of
${\bf{x}}_{k}$, through the nonlinear ${\bf{f}}(\cdot)$ and ${\bf{g}}(\cdot)$
functions, from which a posterior mean and covariance of ${\bf{x}}_{k}$ are
estimated while assuming Gaussian. It can accommodate quadratic (second-order)
terms in general and even cubic (third-order) terms in some cases when
${{\boldsymbol{\upomega}}_{k}}$ and ${{\bf{v}}_{k}}$ are Gaussian.

Table 4: A comparison of four different recursive Bayesian filtering methods Quantity of interest |  Kalman filter |  Extended Kalman filter |  Unscented Kalman filter |  Particle filter
---|---|---|---|---
Ability to handle nonlinear functions (${\bf{f}}(\cdot)$ and ${\bf{g}}(\cdot)$) |  Linear |  Mildly nonlinear (linearized with first- or second-order Taylor expansion) |  Nonlinear (can handle quadratic and even cubic terms) |  Nonlinear
Process noise and measurement noise (${{\boldsymbol{\upomega}}_{k}}$ and ${{\bf{v}}_{k}}$) |  Gaussian |  Gaussian |  Gaussian |  Gaussian and non-Gaussian (no limitation)
Posterior state distribution ($p({{\bf{x}}_{k}}|{{\bf{y}}_{1:k}})$) |  Gaussian |  Unimodal (only one peak) and symmetric |  Unimodal (only one peak) |  Unimodal and multimodal (no limitation)
Computational efficiency |  High |  Medium |  Medium-low |  Low

Particle filters differ drastically from the family of Kalman filters in that
it does not assume state and noise are Gaussian and characterize uncertainties
using Gaussian mean and variance. Instead, particle filters draw a large set
of weighted samples (or particles) from the marginal posterior (or a
distribution proportional to the marginal posterior) at each time step and use
these particles as an approximation to the posterior (Särkkä, 2013,
Gustafsson, 2010). An example of pseudocode for a generic particle filter
algorithm is given in Sec. Appendix A: A generic particle filter algorithm
(see Figure 28). As shown in the pseudocode, particle filtering consists of
three key steps: state transition, weight evaluation and normalization, and
resampling. An easy-to-understand explanation of a particle filter can be
found in an animated video on YouTube (Svensson, 2022b). This video gives a
toy example of positioning an aircraft based on measurements of the distance
to the ground and flight elevation. The MATLAB source code for this
illustration can be accessed openly on GitHub (Svensson, 2022a). As a result
of using the particle approximation, particle filters work well on any non-
Gaussian state (${{\bf{x}}_{k}}$) and noise ($\boldsymbol{\upomega}_{k}$ and
${\bf{v}}_{k}$) distributions and highly nonlinear transition and measurement
functions (${\bf{f}}(\cdot)$ and ${\bf{g}}(\cdot)$). These attractive
advantages come with a price: higher computational costs. For example, the
unscented Kalman filter, the most computationally complex member in the Kalman
family, needs to propagate a set of 7 sigma points to estimate the SOC,
capacity, and internal resistance position velocity of a lithium-ion battery
(capacity and internal resistance are two state-of-health parameters that
affect the energy and power capability of a battery). However, a particle
filter may propagate hundreds of weighted particles at each step to achieve
good convergence speed and estimation accuracy. Also, particle filters do not
scale easily to high-dimensional state spaces, where we may have no particles
near the true state, also known as particle deprivation, and the numbers of
particles required to achieve reasonable estimation become prohibitively high.
Therefore, applications of particle filters for state estimation in the
digital twin context have been mostly limited to a small number of state
dimensions (typically $<$ 5), consistent with what we found in Li et al.
(2020b), Wang et al. (2019), Kapteyn et al. (2021).

It is worth mentioning that approaches have been developed that use cheap-to-
evaluate surrogate models to replace the state transition function
(${\bf{f}}(\cdot)$), measurement function (${\bf{g}}(\cdot)$), or both when
these functions are computationally expensive. These approaches alleviate, to
some degree, the computational burden imposed by particle filers by allowing a
more efficient propagation of thousands or even millions of weighted particles
with a lower computational cost. These efforts make particle filter a popular
technique for state estimation in many digital twin applications as reported
in the literature Li et al. (2017), Kapteyn et al. (2021), Ye et al. (2020).

Two other state estimation problems relevant to digital twin are called
smoothing and prediction. These two problems aim to estimate past (smoothing)
and future (prediction) states, respectively. Figure 15 illustrates the time
requirements for all three state estimation methods. Each of the three methods
play a different role in digital twins because they focus on different time
frames. Smoothing methods operate on previously collected data and generally
reduce the size of the data in the process. For example, smoothing methods can
be used to understand a physical system’s state history over time after the
data has been collected. On the contrary, prediction methods are concerned
with estimating future states using historical data. Shown in Fig. 15, the
_prediction_ state estimation method is estimating the state of the system at
time $k$ using historical data. While smoothing was used to understand
historical system state, prediction is typically used for understanding future
system state. Prediction in digital twin is typically used for planning and
control; two use cases which require some degree of understanding about the
trajectory of system states.

Figure 15: A schematic distinguishing among three different state estimation
problems. The differences lie in the period of the available measurements
relative to the state estimation time.

In digital twin applications, state estimation results are used to inform
decision making algorithms. The timeliness required for effective decision
making using digital twin influences the choice of state estimation
methodology. For example, fault diagnosis of machinery requires that the
machine’s health status be estimated at the present time. In this case, the
ideal state estimation methodology to use in a fault diagnosis digital twin
would be filtering, as it provides state estimates at the present time. On the
other hand, health forecasting and remaining useful life (RUL) prediction
require state estimates of system health at future times, and are thus only
enabled by prediction methods which predict state at future time steps (See
Fig. 15) (Wang et al., 2019, Tian et al., 2011, Bangalore and Patriksson,
2018). Another example is that a structural digital twin of a UAV with an
updated damage state can be used to predict (by simulation) the probability of
failure for an upcoming flight maneuver and thus informing flight path
planning and control (Kapteyn et al., 2021). The idea is to allow a
structurally healthy UAV to perform aggressive maneuvers to reduce flight
distance and time and constrain a structurally damaged UAV to act more
conservatively to incur minimal damage in future maneuvers and prevent
unexpected structural failure. The last example is that the state-of-charge
and state-of-health estimate of a lithium-ion battery can be used to optimize
the current profile (i.e., charging C-rate vs. SOC) during fast charging (Li
et al., 2020b, 2021, Wei et al., 2021).

####  4.2.3 Dynamic Bayesian network

A Bayesian network (BN), which is also called a Bayes net, is a probabilistic
graphical model that represents a set of random variables and their
probabilistic relationships as a directed acyclic graph (Murphy, 2002,
Friedman et al., 1997). A dynamic Bayesian network (DBN) extends a standard BN
by considering the time evolution of variables and is used to model dynamic
systems. It is a generalization of the hidden Markov model given in Fig. 14
(see Sec. 4.2 (b)). The difference between a hidden Markov model (HMM) and a
DBN is that hidden Markov model uses a single hidden state variable to
represent the entire state space whereas the DBN represents the hidden state
as a set of random variables connected in a graph (Murphy, 2002). A DBN allows
for the modeling of nonlinear dynamic systems with arbitrary nonlinearities
and distributions.

Fig. 16 presents an illustrative example of a DBN with five nodes. At each
time instant $t_{k}$, the DBN has a static BN. The static BNs at two adjacent
time instants $t_{k}$ and $t_{k+1}$ are connected through a transition BN. As
a directed acyclic graph, a DBN consists of two essential elements, namely
vertices (also called nodes) and edges. In a DBN, the vertices or nodes are
random variables, such as ${x_{1,k}},{x_{2,k}},{x_{3,k}},{x_{4,k}},\textrm{and
}{y_{1,k}}$ in Fig. 16. The edges directed from one node to another, such as
${x_{3,k}}\rightarrow{y_{1,k}}$ and ${x_{1,k}},{x_{2,k}}\rightarrow{x_{3,k}}$,
represent the probabilistic causality relationship between the nodes.

Figure 16: Illustration of a DBN, which is a generalization of the hidden
Markov model given in Fig. 14.

_Nodes_ in a DBN could represent discrete random variables or continuous
random variables. For instance, as part of the U.S. Air Force funded digital
twin effort, Li et al. (2017) represented bolt looseness and a damage state
(i.e., elastic/plastic zone) as discrete random variables and a crack length
as a continuous random variable in their DBN-based digital twin that modeled
the fatigue crack growth on an aircraft wing. Moreover, the nodes that an edge
is directed from and directed to are called the parent node and the child
node, respectively. For example, for edge
${x_{1,k}},{x_{2,k}}\rightarrow{x_{3,k}}$ in Fig. 16, ${x_{1,k}},{x_{2,k}}$
are parent nodes and ${x_{3,k}}$ is a child node. Nodes without parent nodes
are called root nodes (e.g., ${x_{1,k}},{x_{2,k}}$ in Fig. 16).

_Edges_ connecting parent nodes with the corresponding child node(s) in a DBN,
such as ${x_{3,k}}\rightarrow{y_{1,k}}$ and ${x_{4,k}}\rightarrow{y_{1,k}}$ in
Fig. 16, represent the causality relationship(s) between the parent node(s)
and the child node(s). The probabilistic causal relationships between a group
of parent nodes connected to a common child node are modeled as a conditional
probability table (CPT) for discrete nodes and a conditional probability
density (CPD) function for continuous nodes. The CPT contains the conditional
probability mass function of a child node covering all possible realizations
of its parent nodes. For example, the probability of having a bolt looseness
or not for a given realization of load condition of an aircraft is described
as a CPT in Li et al. (2017). The CPD could be a simple, analytical
probability density function or a numerical probabilistic predictive model,
such as a Gaussian process regression model (Karve et al., 2020), a data-
driven stochastic process model (Yu et al., 2021), or a reduced-order model
with noise (Kapteyn et al., 2021). For instance, if Gaussian process
regression is employed to construct the CPD of $x_{3,k}$ given in Fig. 16, a
Gaussian process regression model ${x_{3,k}}=\hat{G}({x_{1,k}},\;{x_{2,k}})$
is first built according to the input-output dependence depicted by the edges
${x_{1,k}},{x_{2,k}}\rightarrow{x_{3,k}}$. After that, we have the CPD of
$x_{3,k}$ as

$p({x_{3,k}}|{x_{1,k}},{x_{2,k}})=\phi\left({{{{x_{3,k}}-{\mu_{\hat{G}}}({x_{1,k}},{x_{2,k}})}\over{{\sigma_{\hat{G}}}({x_{1,k}},{x_{2,k}})}}}\right),$
(5)

in which ${{\mu_{\hat{G}}}({x_{1,k}},{x_{2,k}})}$ and
${{\sigma_{\hat{G}}}({x_{1,k}},{x_{2,k}})}$ are respectively the mean and
standard deviation of the Gaussian process regressor-predicted $x_{3,k}$ for
given values of the parent nodes $x_{1,k}$ and $x_{2,k}$, and
$\phi\left(\cdot\right)$ is the probability density function of a standard
normal random variable.

The physic-based models described in Sec. 3.2, such as FEA models, can also be
used to construct CPDs by accounting for uncertainty sources in the models.
Since digital twins require real-time model updating and each updating run may
need to evaluate CPTs or CPDs thousands of times, high-fidelity physics-based
models are usually replaced with computationally cheaper data-driven surrogate
models (see Sec. 3.3) in the practical implementation of DBNs. In addition to
the _nodes_ and _edges_ , as mentioned above, a transition BN is used to
connect two BNs at two adjacent time instants. This is essential to making the
BN _dynamic_. Taking the DBN given in Fig. 16 as an example, the transition of
each state variable between two adjacent time instants $t_{k}$ and $t_{k+1}$
is given by

$p({x_{i,k+1}}|{x_{i,k}}),\forall i=1,\;2,\;3,\;4,$ (6)

where $p({x_{i,k+1}}|{x_{i,k}})$ is the transition probability of state
variable $x_{i}$ from $t_{k}$ to $t_{k+1}$, which is defined by a CPT or CPD
and shown as a dashed edge in Fig. 16 (i.e.,
${x_{i,k}}\rightarrow{x_{i,k+1}},i=1,2,3,4$).

An important property of BNs and DBNs is the _conditional independence_
property, which means that the nodes are independent from each other if they
are not connected by directed edges, given the values of their parent nodes.
For instance, for given values of $x_{1,k}$ and $x_{2,k}$ in Fig. 16,
$x_{3,k}$ and $x_{4,k}$ are independent from each other. Thanks to this
property and based on the chain rule of conditional probabilities, the
likelihood function of a DBN can be expressed as a product of individual CPTs
and/or CPDs conditioned on the parent nodes (see an example given in Appendix
B). The decomposition of the joint probability density into a product of
individual CPTs and CPDs as illustrated in Eq. (14) in Appendix B really
facilitates model updating and the construction of complex DBNs for digital
twins. One just needs to properly define the CPTs and CPDs for individual
nodes and connect them using a directed acyclic graph. After that, the
inference can be performed using generic Bayesian inference methods including
the commonly used Markov chain Monte Carlo simulation (Brooks, 1998) and
sequential Monte Carlo methods such as particle filtering as described in Sec.
4.2 (b), and given in Appendix A. As has been pointed out by Murphy (2002),
the flexibility of DBNs in incorporating various CPTs and CPDs using directed
acyclic graphs makes DBNs advantageous over various hidden Markov models
including classical hidden Markov models and hierarchical hidden Markov
models. The fact that the CPDs in a DBN could be any type of probabilistic
model with any nonlinearities and distribution types makes DBNs a flexible
tool capable of fusing heterogeneous data sources (e.g., images, text, audio)