3

Virtual Representation: Foundational Research Needs and Opportunities

The digital twin virtual representation comprises a computational model or set of coupled models. This chapter identifies research needs and opportunities associated with creating, scaling, validating, and deploying models in the context of a digital twin. The chapter emphasizes the importance of the virtual representation being fit for purpose and the associated needs for data-centric and model-centric formulations. The chapter discusses multiscale modeling needs and opportunities, including the importance of hybrid modeling that combines mechanistic models and machine learning (ML). This chapter also discusses the challenges of integrating component and subsystem digital twins into the virtual representation. Surrogate modeling needs and opportunities for digital twins are also discussed, including surrogate modeling for high-dimensional, complex multidisciplinary systems and the essential data assimilation, dynamic updating, and adaptation of surrogate models.

FIT-FOR-PURPOSE VIRTUAL REPRESENTATIONS FOR DIGITAL TWINS

As discussed in Chapter 2, the computational models underlying the digital twin virtual representation can take many mathematical forms (including dynamical systems, differential equations, and statistical models) and need to be “fit for purpose” (meaning that model types, fidelity, resolution, parameterization, and quantities of interest must be chosen and potentially dynamically adapted to fit the particular decision task and computational constraints). The success of a digital twin hinges critically on the availability of models that can represent the physical counterpart with fidelity that is fit for purpose, and that can be used to

issue predictions with known confidence, possibly in extrapolatory regimes, all while satisfying computational resource constraints.

As the foundational research needs and opportunities for modeling in support of digital twins are outlined, it is important to emphasize that there is no one-size-fits-all approach. The vast range of domain applications and use cases that are envisioned for digital twins requires a similarly vast range of models: first-principles, mechanistic, and empirical models all have a role to play.

There are several areas in which the state of the art in modeling is currently a barrier to achieving the impact of digital twins, due to the challenges of modeling complex multiphysics systems across multiple scales. In some cases, the mathematical models are well understood, and these barriers relate to our inability to bridge scales in a computationally tractable way. In other cases, the mathematical models are lacking, and discovery of new models that explain observed phenomena is needed. In yet other cases, mathematical models may be well understood and computationally tractable to solve at the component level, but foundational questions remain around stability and accuracy when multiple models are coupled at a full system or system-of-systems level. There are other areas in which the state of the art in modeling provides potential enablers for digital twins. The fields of statistics, ML, and surrogate modeling have advanced considerably in recent years, but a gap remains between the class of problems that has been addressed and the modeling needs for digital twins.

Some communities focus on high-fidelity models in the development of digital twins while others define digital twins using simplified and/or surrogate models. Some literature states that a digital twin must be a high-resolution, high-fidelity replica of the physical system (Bauer et al. 2021; NASEM 2023a). An early definition of a digital twin proposed “a set of virtual information constructs that fully describes a potential or actual physical manufactured product from the micro atomic level to the macro geometrical level. At its optimum, any information that could be obtained from inspecting a physical manufactured product can be obtained from its Digital Twin” (Grieves 2014). Other literature proposes surrogate modeling as a key enabler for digital twins (Hartmann et al. 2018; NASEM 2023c), particularly recognizing the dynamic (possibly real-time) nature of many digital twin calculations.

Conclusion 3-1: A digital twin should be defined at a level of fidelity and resolution that makes it fit for purpose. Important considerations are the required level of fidelity for prediction of the quantities of interest, the available computational resources, and the acceptable cost. This may lead to the digital twin including high-fidelity, simplified, or surrogate models, as well as a mixture thereof. Furthermore, a digital twin may include the ability to represent and query the virtual models at variable levels of resolution and fidelity depending on the particular task at hand and the available resources (e.g., time, computing, bandwidth, data).

Determining whether a virtual representation is fit for purpose is itself a mathematical gap when it comes to the complexity of situations that arise with digital twins. For a model to be fit for purpose, it must balance the fidelity of predictions of quantities of interest with computational constraints, factoring in acceptable levels of uncertainty to drive decisions. If there is a human in the digital twin loop, fitness for purpose must also account for human–digital twin interaction needs such as visualization and communication of uncertainty. Furthermore, since a digital twin’s purpose may change over time, the requirements for it to be fit for purpose may also evolve. Historically, computational mathematics has addressed accuracy requirements for numerical solution of partial differential equations using rigorous approaches such as a posteriori error estimation combined with numerical adaptivity (Ainsworth and Oden 1997). These kinds of analyses are an important ingredient of assessing fitness for purpose; however, the needs for digital twins go far beyond this, particularly given the range of model types that digital twins will employ and the likelihood that a digital twin will couple multiple models of differing fidelity. A key feature for determining fitness for purpose is assessing whether the fusion of a mathematical model, potentially corrected via a discrepancy function, and observational data provides relevant information for decision-making. Another key aspect of determining digital twin fitness for purpose is assessment of the integrity of the physical system’s observational data, as discussed in Chapter 4.

Finding 3-1: Approaches to assess modeling fidelity are mathematically mature for some classes of models, such as partial differential equations that represent one discipline or one component of a complex system; however, theory and methods are less mature for assessing the fidelity of other classes of models (particularly empirical models) and coupled multiphysics, multi-component systems.

An additional consideration in determining model fitness for purpose is the complementary role of models and data—a digital twin is distinguished from traditional modeling and simulation in the way that models and data work together to drive decision-making. Thus, it is important to analyze the entire digital twin ecosystem when assessing modeling needs and the trade-offs between data-driven and model-driven approaches (Ferrari 2023).

In some cases, there is an abundance of data, and the decisions to be made fall largely within the realm of conditions represented by the data. In these cases, a data-centric view of a digital twin (Figure 3-1) is appropriate—the data form the core of the digital twin, the numerical model is likely heavily empirical (e.g., obtained via statistical or ML methods), and analytics and decision-making wrap around this numerical model. An example of such a setting is the digital twin of an aircraft engine, trained on a large database of sensor data and flight logs col-

lected across a fleet of engines (Aviation Week Network 2019; Sieger 2019). Other cases are data-poor, and the digital twin will be called on to issue predictions in extrapolatory regimes that go well beyond the available data. In these cases, a model-centric view of a digital twin (Figure 3-1) is appropriate—a mathematical model and its associated numerical model form the core of the digital twin, and data are assimilated through the lens of these models. Examples include climate digital twins, where observations are typically spatially sparse and predictions may extend decades into the future (NASEM 2023a), and cancer patient digital twins, where observations are typically temporally sparse and the increasingly patient-specific and complex nature of diseases and therapies requires predictions of patient responses that go beyond available data (Yankeelov 2023). In these data-poor situations, the models play a greater role in determining digital twin fidelity. As discussed in the next section, an important need is to advance hybrid modeling approaches that leverage the synergistic strengths of data-driven and model-driven digital twin formulations.

MULTISCALE MODELING NEEDS AND OPPORTUNITIES FOR DIGITAL TWINS

A fundamental challenge for digital twins is the vast range of spatial and temporal scales that the virtual representation may need to address. The following section describes research opportunities for modeling across scales in support of digital twins and the need to integrate empirical and mechanistic methods for

hybrid approaches to leverage the best of both data-driven and model-driven digital twin formulations.

The Predictive Power of Digital Twins Requires Modeling Across Scales

For many applications, the models that underlie the digital twin virtual representation must represent the behavior of the system across a wide range of spatial and temporal scales. For systems with a wide range of scales on which there are significant nonlinear scale interactions, it may be impossible to represent explicitly in a digital model the full richness of behavior at all scales and including all interactions. For example, the Earth’s atmosphere and oceans are components of the Earth system, and their instantaneous and statistical behaviors are described respectively as weather and climate. These behaviors exhibit a wide range of variability on both spatial scales (from millimeters to tens of thousands of kilometers) and temporal scales (from seconds to centuries). Similarly, relevant dynamics in biological systems range from nanometers to meters in spatial scales and from milliseconds to years in temporal scales. In biomedical systems, modeling requirements range across scales from the molecular to the whole-body physiology and pathophysiology to populations. Temporal ranges in nanoseconds represent biochemical reactions, signaling pathways, gene expression, and cellular processes such as redox reactions or transient protein modifications. These events underpin the larger-scale interactions between cells, tissues, and organs; multiple organs and systems converge to address disease and non-disease states.

Numerical models of many engineering systems in energy, transportation, and aerospace sectors also span a range of temporal and spatial resolutions, and complexity owing to multiphysics phenomena (e.g., chemical reactions, heat transfer, phase change, unsteady flow/structure interactions) and resolution of intricate geometrical features. In weather and climate simulations, as well as in many engineered and biomedical systems, system behavior is explicitly modeled across a limited range of scales—typically, from the largest scale to an arbitrary cutoff scale determined by available modeling resources—and the remaining (small) scales are represented in a parameterized form. Fortunately, in many applications, the smaller unresolved scales are known to be more universal than the large-scale features and thus more amenable to phenomenological parameterization. Even so, a gap remains between the scales that can be simulated and actionable scales.

An additional challenge is that as finer scales are resolved and a given model achieves greater fidelity to the physical counterpart it simulates, the computational and data storage/analysis requirements increase. This limits the applicability of the model for some purposes, such as uncertainty quantification, probabilistic prediction, scenario testing, and visualization. As a result, the demarcation between resolved and unresolved scales is often determined by computational constraints

rather than a priori scientific considerations. Another challenge to increasing resolution is that the scale interactions may enter a different regime as scales change. For example, in atmospheric models, turbulence is largely two-dimensional at scales larger than 10 km and largely three-dimensional at scales smaller than 10 km; the behavior of fluid-scale interactions fundamentally changes as the model grid is refined.

Thus, there are incentives to drive modeling for digital twins in two directions: toward resolution of finer scales to achieve greater realism and fidelity on the one hand, and toward simplifications to achieve computational tractability on the other. There is a motivation to do both by increasing model resolution to acquire data from the most realistic possible model that can then be mined to extract a more tractable model that can be used as appropriate.

Finding 3-2: Different applications of digital twins drive different requirements for modeling fidelity, data, precision, accuracy, visualization, and time-to-solution, yet many of the potential uses of digital twins are currently intractable to realize with existing computational resources.

Finding 3-3: Often, there is a gap between the scales that can be simulated and actionable scales. It is necessary to identify the intersection of simulated and actionable scales in order to support optimizing decisions. The demarcation between resolved and unresolved scales is often determined by available computing resources, not by a priori scientific considerations.

Recommendation 3: In crafting research programs to advance the foundations and applications of digital twins, federal agencies should create mechanisms to provide digital twin researchers with computational resources, recognizing the large existing gap between simulated and actionable scales and the differing levels of maturity of high-performance computing across communities.

Finding 3-4: Advancing mathematical theory and algorithms in both data-driven and multiscale physics-based modeling to reduce computational needs for digital twins is an important complement to increased computing resources.

Hybrid Modeling Combining Mechanistic Models and Machine Learning

Hybrid modeling approaches—synergistic combinations of empirical and mechanistic modeling approaches that leverage the best of both data-driven and model-driven formulations—were repeatedly emphasized during this study’s information gathering (NASEM 2023a,b,c). This section provides some examples of how hybrid modeling approaches can address digital twin modeling challenges.

In biology, modeling organic living matter requires the integration of biological, chemical, and even electrical influences that stimulate or inhibit the living material response. For many biomedical applications, this requires the incorporation of smaller-scale biological phenomena that influence the dynamics of the larger-scale system and results in the need for multiphysics, multiscale modeling. Incorporating multiple smaller-scale phenomena allows modelers to observe the impact of these underlying mechanisms at a larger scale, but resolving the substantial number of unknown parameters to support such an approach is challenging. Data-driven modeling presents the ability to utilize the growing volume of biological and biomedical data to identify correlations and generate inferences about the behavior of these biological systems that can be tested experimentally. This synergistic use of data-driven and multiscale modeling approaches in biomedical and related fields is illustrated in Figure 3-2.

Advances in hybrid modeling in the Earth sciences are following similar lines. Models for weather prediction or climate simulation must solve multiscale and multiphysics problems that are computationally intractable at the necessary level of fidelity, as described above. Over the past several decades of work in developing atmospheric, oceanic, and Earth system models, the unresolved scales have been represented by parameterizations that are based on conceptual models of the relevant unresolved processes. With the explosion of Earth system observations from remote sensing platforms in recent years, this approach has been modified to incorporate ML methods to relate the behavior of unresolved processes to that of resolved processes. There are also experiments in replacing entire Earth system components with empirical artificial intelligence (AI) components. Furthermore, the use of ensemble modeling to approximate probability distributions invites the use of ML techniques, often in a Bayesian framework, to cull ensemble members that are less accurate or to define clusters of solutions that simplify the application to decision-making.

In climate and engineering applications, the potential for hybrid modeling to underpin digital twins is significant. In addition to modeling across scales as described above, hybrid models can help provide understandability and explainability. Often, a purely data-driven model can identify a problem or potential opportunity without offering an understanding of the root cause. Without this understanding, decisions related to the outcome may be less useful. The combination of data and mechanistic models comprising a hybrid model can help mitigate this problem. The aerospace industry has developed hybrid digital twin solutions that can analyze large, diverse data sets associated with part failures in aircraft engines using the data-driven capabilities of the hybrid model (Deshmukh 2022). Additionally, these digital twin solutions can provide root cause analysis indicators using the mechanistic-driven capabilities of the hybrid model.

However, there are several gaps in hybrid modeling approaches that need to be addressed to realize the full potential value of these digital twin solutions. These gaps exist in five major areas: (1) data quality, availability, and affordabil-

ity; (2) model coupling and integration; (3) model validation and calibration; (4) uncertainty quantification and model interpretability; and (5) model scalability and management.

Data quality, availability, and affordability can be challenging in biomedical, climate, and engineering applications as obtaining accurate and representative data for model training and validation at an affordable price is difficult. Prior data collected may have been specific to certain tasks, limited by the cost of capture and storage, or deemed unsuitable for current use due to evolving environments and new knowledge. Addressing data gaps based on the fit-for-purpose requirements of the digital twin and an analysis of current available data is crucial. Minimizing the need for large sample sizes and designing methodologies to learn robustly from data sets with few samples would also help overcome these barriers. AI methods might be developed to predict a priori what amount and type of data are needed to support the virtual counterpart.

Combining data-driven models with mechanistic models requires effective coupling techniques to facilitate the flow of information (data, variables, etc.) between the models while understanding the inherent constraints and assumptions of each model. Coupling is complex in many cases, and model integration is even more so as it involves creating a single comprehensive model that represents the features and behaviors of both the data-driven and the mechanistic-driven model within a coherent framework. Both integration and coupling techniques require harmonizing different scales, assumptions, constraints, and equations, and understanding their implications on the uncertainty associated with the outcome. Matching well-known, model-driven digital twin representations with uncharacterized data-driven models requires attention to how the various levels of fidelity comprised in these models interact with each other in ways that may result in unanticipated overall digital twin behavior and inaccurate representation at the macro level. Another gap lies in the challenge of choosing the specific data collection points to adequately represent the effects of the less-characterized elements and augment the model-driven elements without oversampling the behavior already represented in the model-driven representations. Finally, one can have simulations that produce a large data set (e.g., a space-time field where each solution field is of high dimension) but only relatively few ensembles. In such cases, a more structured statistical model may be required to combine simulations and observations.

Model validation is another evident gap that needs to be overcome given the diverse nature of the involved data-driven and mechanistic models and their underlying assumptions. Validating data-driven models heavily relies on having sufficient and representative validation data for training as well as evaluating the accuracy of the outcome and the model’s generalizability to new data. On the other hand, mechanistic-driven models heavily rely on calibration and parameter estimation to accurately reproduce against experimental and independent data. The validation and calibration processes for these hybrid models must be harmonized to ensure the accuracy and reliability required in these solutions.

Uncertainty quantification and model explainability and interpretability are significant gaps associated with hybrid systems. These systems must accurately account for uncertainties arising from both the data-driven and mechanistic-driven components of the model. Uncertainties can arise from various factors related to both components, including data limitations and quality, model assumptions, and parameter estimation. Addressing how these uncertainties are quantified and propagated through the hybrid model is another gap that must be tackled for robust predictions. Furthermore, interpreting and explaining the outcomes may pose a significant challenge, particularly in complex systems.

Finally, many hybrid models associated with biomedical, climate, and engineering problems can be computationally demanding and require unique skill sets. Striking a balance between techniques that manage the computational complexity of mechanistic models (e.g., parallelization and model simplification) and techniques used in data-driven models (e.g., graphics processing unit coding, pruning, and model compression) is essential. Furthermore, hybrid approaches require that domain scientists either learn details of computational complexity and data-driven techniques or partner with additional researchers to experiment with hybrid digital twins. Resolving how to achieve this combination and balance at a feasible and affordable level is a gap that needs to be addressed. Additionally, the model will need to be monitored and updated as time and conditions change and errors in the system arise, requiring the development of model management capabilities.

While hybrid modeling provides an attractive path forward to address digital twin modeling needs, simply crafting new hybrid models that better match available data is insufficient. The development of hybrid modeling approaches for digital twins requires rigorous verification, validation, and uncertainty quantification (VVUQ), including the quantification of uncertainty in extrapolatory conditions. If the hybrid modeling is done in a way that the data-driven components of the model are continually updated, then these updating methods also require associated VVUQ. Another challenge is that in many high-value contexts, digital twins need to represent both typical operating conditions and anomalous operating conditions, where the latter may entail rare or extreme events. As noted in Conclusion 2-2, a gap exists between the class of problems that has been considered in VVUQ for traditional modeling and simulation settings and the VVUQ problems that will arise for digital twins. Hybrid models—in particular those that infuse some form of black-box deep learning—represent a particular gap in this regard.

Finding 3-5: Hybrid modeling approaches that combine data-driven and mechanistic modeling approaches are a productive path forward for meeting the modeling needs of digital twins, but their effectiveness and practical use are limited by key gaps in theory and methods.

INTEGRATING COMPONENT AND SUBSYSTEM DIGITAL TWINS

The extent to which the virtual representation will integrate component and subsystem models is an important consideration in modeling digital twins. A digital twin of a system of systems will likely couple multiple constituent digital twins. Integration of models and data to this extent goes beyond what is done routinely and entails a number of foundational mathematical and computational challenges. In addition to the software challenge of coupling models and solvers, VVUQ tasks and the determination of fitness for purpose become much more challenging in the coupled setting.

Modeling of a complex system often requires coupling models of different components/subsystems of the system, which presents additional challenges beyond modeling of the individual components/subsystems. For example, Earth system models couple models of atmosphere, land surface, river, ocean, sea ice, and land ice to represent interactions among these subsystems that determine the internal variability of the system and its response to external forcing. Component models that are calibrated individually to be fit for purpose when provided with observed boundary conditions of the other components may behave differently when the component models are coupled together due to error propagation and nonlinear feedback between the subsystems. This is particularly the case when models representing the different components/subsystems have different fidelity or mathematical forms, necessitating the need for additional mathematical operations such as spatiotemporal filtering, which adds uncertainty in the coupled model.

Another example is the coupling of human system models with Earth system models, which often differ in model fidelity as well as in mathematical forms. Furthermore, in the context of digital twins, some technical challenges remain in coupled model data assimilation, such as properly initializing each component model. Additional examples of the integration of components are shown in Box 3-1.

Interoperability of software and data are a challenge across domains and pose a particular challenge when integrating component and subsystem digital twins. Semantic and syntactic interoperability, in which data are exchanged between and understood by the different systems, can be challenging given the possible difference in the systems. Furthermore, assumptions made in one model can be distinct from the assumptions made in other models. Some communities have established approaches to reducing interoperability—for example, though the use of shared standards for data, software, and models, or through the use of software templates—and this is a critical aspect of integrating complex digital twin models.

Finding 3-6: Integration of component/subsystem digital twins is a pacing item for the digital twin representation of a complex system, especially if different fidelity models are used in the digital twin representation of its components/subsystems.

SURROGATE MODELING NEEDS AND OPPORTUNITIES FOR DIGITAL TWINS

Surrogate models play a key role in addressing the computational challenges of digital twins. Surrogate models can be categorized into three types: statistical data-fit models, reduced-order models, and simplified models.

• Statistical data-fit models use statistical methods to fit approximate input-output maps to training data, with the surrogate model employing a generic functional form that does not explicitly reflect the structure of the physical governing equations underlying the numerical simulations.
• Reduced-order models incorporate low-dimensional structure learned from training data into a structured form of the surrogate model that reflects the underlying physical governing equations.
• Simplified models are obtained in a variety of ways, such as coarser grids, simplified physical assumptions, and loosened residual tolerances.

Surrogate modeling is a broad topic, with many applications beyond digital twins. This section focuses on unique challenges that digital twins pose to surrogate modeling and the associated foundational gaps in surrogate modeling methods. A first challenge is the scale at which surrogate modeling will be needed. Digital twins by their nature may require modeling at the full system scale, with models involving multiple disciplines, covering multiple system components, and described by parameter spaces of high dimensions. A second challenge is the critical need for VVUQ of surrogate models, recognizing the uncertain conditions under which digital twins will be called on to make predictions, often in extrapolatory regimes. A third challenge relates to the dynamic updating and adaptation that is key to the digital twin concept. Each one of these challenges highlights gaps in the current state of the art in surrogate modeling, as the committee discusses in more detail in the following.

Surrogate modeling is an enabler for computationally efficient digital twins, but there is a limited understanding of trade-offs associated with collections of surrogate models operating in tandem in digital twins, the effects of multiphysics coupling on surrogate model accuracy, performance in high-dimensional settings, surrogate model VVUQ—especially in extrapolatory regimes—and, for data-driven surrogates, costs of generating training data and learning.

Surrogate Modeling for High-Dimensional, Complex Multidisciplinary Systems

State-of-the-art surrogate modeling has made considerable progress for simpler systems but remains an open challenge at the level of complexity needed for digital twins. Multiple interacting disciplines and nonlinear coupling among

disciplines, as needed in a digital twin, pose a particular challenge for surrogate modeling. The availability of accurate and computationally efficient surrogate models depends on the ability to identify and exploit structure that is amenable to approximation. For example, reduced-order modeling may exploit low-rank structure in a way that permits dynamics to be evolved in a low-dimensional manifold or coarse-graining of only a subset of features, while statistical data-fit methods exploit the computational efficiencies of representing complex dynamics with a surrogate input-output map, such as a Gaussian process model or deep neural network. A challenge with coupled multidisciplinary systems is that coupling is often a key driver of dynamics—that is, the essential system dynamics can change dramatically due to coupling effects.

One example of this is Earth system models that must represent the dynamics of the atmosphere, ocean, sea ice, land surface, and cryosphere, all of which interact with each other in complex, nonlinear ways that result in interactions of processes occurring across a wide range of spatial and temporal scales. The interactions involve fluxes of mass, energy (both heat and radiation), and momentum that are dependent on the states of the various system components. Yet in many cases, the surrogate models are derived for the individual model components separately, and then coupled.

Surrogate models for coupled systems—whether data-fit or reduced-order models— remain a challenge because even if the individual model components are highly accurate representations of the dynamics and processes in those components, they may lose much of their fidelity when additional degrees of freedom due to coupling with other system components are added. Another set of challenges encompass important mathematical questions around the consistency, stability, and property-preservation attributes of coupled surrogates. A further challenge is ensuring model fidelity and fitness for purpose when multiple physical processes interact.

Finding 3-7: State-of-the-art literature and practice show advances and successes in surrogate modeling for models that form one discipline or one component of a complex system, but theory and methods for surrogates of coupled multiphysics systems are less mature.

An additional further challenge in dealing with surrogate models for digital twins of complex multidisciplinary systems is that the dimensionality of the parameter spaces underlying the surrogates can become high. For example, a surrogate model of the structural health of an engineering structure (e.g., building, bridge, airplane wing) would need to be representative over many thousands of material and structural properties that capture variation over space and time. Similarly, a surrogate model of tumor evolution in a cancer patient digital twin would potentially have thousands of parameters representing patient anatomy, physiology, and mechanical properties, again capturing variation over space and

time. Deep neural networks have shown promise in representing input-output maps even when the input parameter dimension is large, yet generating sufficient training data for these complex problems remains a challenge. As discussed below, notable in the literature is that many apparent successes in surrogate modeling fail to report the cost of training, either for determining parameters in a neural network or in tuning the parameters in a reduced-order model.

It also remains a challenge to quantify the degree to which surrogate predictions may generalize in a high-dimensional setting. While mathematical advances are revealing rigorous insights into high-dimensional approximation (Cohen and DeVore 2015), this work is largely for a class of problems that exhibit smooth dynamics. Work is needed to bridge the gap between rigorous theory in high-dimensional approximation and the complex models that will underlie digital twins. Another promising set of approaches uses mathematical decompositions to break a high-dimensional problem into a set of coupled smaller-dimension problems. Again, recent advances have demonstrated significant benefits, including in the digital twin setting (Sharma et al. 2018), but these approaches have largely been limited to problems within structural modeling.

Finding 3-8: Digital twins will typically entail high-dimensional parameter spaces. This poses a significant challenge to state-of-the-art surrogate modeling methods.

Another challenge associated with surrogate models in digital twins is accounting for the data and computational resources needed to develop data-driven surrogates. While the surrogate modeling community has developed several compelling approaches in recent years, analyses of the speedups associated with these approaches in many cases do not account for the time and expense associated with generating training data or using complex numerical solvers at each iteration of the training process. A careful accounting of these elements is essential to understanding the cost–benefit trade-offs associated with surrogate models in digital twins. In tandem, advances in surrogate modeling methods for handling limited training data are needed.

Finding 3-9: One of the challenges of creating surrogate models for high-dimensional parameter spaces is the cost of generating sufficient training data. Many papers in the literature fail to properly acknowledge and report the excessively high costs (in terms of data, hardware, time, and energy consumption) of training.

Conclusion 3-2: In order for surrogate modeling methods to be viable and scalable for the complex modeling situations arising in digital twins, the cost of surrogate model training, including the cost of generating the training data, must be analyzed and reported when new methods are proposed.

Finally, the committee again emphasizes the importance of VVUQ. As noted above for hybrid modeling, development of new surrogate modeling methods must incorporate VVUQ as an integral component. While data-driven surrogate modeling methods are attractive because they reduce the computational intractability of complex modeling and require limited effort to implement, important questions remain about how well they generalize or extrapolate in realms beyond the experience of their training data. This is particularly relevant in the context of digital twins, where ideally the digital twin would explore “what if” scenarios, potentially far from the domain of the available training data—that is, where the digital twin must extrapolate to previously unseen settings. While incorporating physical models, constraints, and symmetries into data-driven surrogate models may facilitate better extrapolation performance than a generic data-driven approach, there is a lack of fundamental understanding of how to select a surrogate model approach to maximize extrapolation performance beyond empirical testing. Reduced-order models are supported by literature establishing their theoretical properties and developing error estimators for some classes of systems. Extending this kind of rigorous work may enable surrogates to be used for extrapolation with guarantees of confidence.

Data Assimilation, Dynamic Updating, and Adaptation of Surrogate Models

Dynamic updating and model adaptation are central to the digital twin concept. In many cases, this updating must be done on the fly under computational and time constraints. Surrogates play a role in making this updating computationally feasible. At the same time, the surrogate models themselves must be updated—and correspondingly validated—as the digital twin virtual representation evolves.

One set of research gaps is around the role of a surrogate model in accelerating digital twin state estimation (data assimilation) and parameter estimation (inverse problem). Challenges surrounding data assimilation and model updating in general are discussed further in Chapter 5. While data assimilation with surrogate models has been considered in some settings, it has not been extended to the scale and complexity required for the digital twin setting. Research at the intersection of data assimilation and surrogate models is an important gap. For example, data assimilation attempts to produce a state estimate by optimally combining observations and model simulation in a probabilistic framework. For data assimilation with a surrogate model to be effective, the surrogate model needs to simulate the state of the physical system accurately enough so that the difference between simulated and observed states is small. Often the parameters in the surrogate model itself are informed by data assimilation, which can introduce circularity of error propagation.

A second set of gaps is around adaptation of the surrogate models themselves. Data-fit surrogate models and reduced-order models can be updated as more data become available—an essential feature for digital twins. Entailing multiphysics coupling and high-dimensional parameter spaces as discussed above, the digital twin setting provides a particular challenge to achieving adaptation under computational constraints. Furthermore, the adaptation of a surrogate model will require an associated continual VVUQ workflow—which again must be conducted under computational constraints—so that the adapted surrogate may be used with confidence in the virtual-to-physical digital twin decision-making tasks.

KEY GAPS, NEEDS, AND OPPORTUNITIES

In Table 3-1, the committee highlights key gaps, needs, and opportunities for realizing the virtual representation of a digital twin. This is not meant to be an exhaustive list of all opportunities presented in the chapter. For the purposes of this report, prioritization of a gap is indicated by 1 or 2. While the committee believes all of the gaps listed are of high priority, gaps marked 1 may benefit from initial investment before moving on to gaps marked with a priority of 2.

TABLE 3-1 Key Gaps, Needs, and Opportunities for Realizing the Virtual Representation of a Digital Twin

REFERENCES