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The term material structure is used here to describe the spatial arrangement of structural and chemical heterogeneities, which constitute a material at a specified instant of time and govern its properties at that instant of time. For a given chemical composition, thermodynamics predicts an equilibrium crystallographic phase (or a multiphase mixture), and at finite temperature, an equilibrium vacancy concentration. Yet materials are rarely in their thermodynamic ground state. Essentially, an overwhelming subset of the material structural features represent metastable or unstable defects created throughout the process history. Conventionally, material structure defects have been classified based on their dimensionality as planar grain boundaries, linear dislocations, and point-wise atomic impurities; these are but the simplest examples of a myriad of complex microstructural features (see Figure 1). The material structure is not usually static but evolves when stimulated by exposure to energy (thermal, mechanical, chemical, etc.). Through state-of-the-art processing, the most perfect undoped, isotopically pure silicon single crystals have been produced to purity levels of >99.9999%. On the other hand, the most sophisticated structural alloys benefit from their complex, multiscale arrangement of the lower length scale structural features, reminiscent of the hierarchical nature of biological systems. Hence, the challenge for a digital twin of a material system is to represent the necessary complexity of the inherently high-dimensional material structure features with sufficient fidelity to capture the relevant subset that controls the material response of interest. Complicating matters, no single experimental technique is capable of comprehensively digitizing the material’s complete internal structure.

A digital twin of a material system should be able to instantiate a representative volume of the material with sufficient statistical sampling of all the relevant lower length-scale structural features and their spatial arrangements. Given the roughly eight orders of magnitude in length scales (from ∼Å to ∼ cms) involved, it should become clear that such instantiations cannot be deterministic or unique. Therefore, what is required here is the ability to produce multiple instantiations that reflect as accurately as possible the inherent stochasticity of the material structure for a given nominal composition and process history. Laplace conjectured that by knowing every atom, its position and momentum, we could anticipate the behavior of the material (marquis de Laplace, 1814). While this statement reflects accurately the expected causal relationship between the material structure and its associated properties, it reflects a practically impossible pursuit. Therefore, we take the viewpoint that the digital twin of a material system is intended to be a minimally sufficient reduced-order representation of Laplace’s “demon.” A tractable digital twin of a material system should therefore utilize a versatile (broadly applicable to all material classes and length scales) low-dimensional representation of the material structure that would allow efficient learning of the functional response of the material system. Based on the earlier discussion, it is also clear that the low-dimensional representation of the material structure can only reflect suitably defined statistical measures at different material length scales; henceforth, such salient statistical measures of the material structure will be referred as features. Because of our interest in instantiating the material structure in our digital twins, it is important that the selected feature set should produce realistic, sufficiently accurate, instantiations of the material structure that can be subsequently correlated with its functional response. This is not a trivial requirement. For example, most of the conventionally used statistical measures of the material structure, such as the overall alloy composition, phase volume fractions, and the averaged grain sizes are woefully inadequate for producing the required instantiations of the multiscale material structure for our digital twins. More advanced approaches involving a richer set of microstructure statistics (e.g., orientation and mis-orientation distributions, grain aspect ratio distributions) have led to concepts such as statistically equivalent representative volume elements (McDowell and LeSar, 2016; Ghosh and Groeber, 2020). Some of these concepts have also been implemented in open-source codes such as DREAM.3D (Groeber and Jackson, 2014; Ghosh and Groeber, 2020).

A comprehensive and systematic framework available today that is capable of providing the requisite feature engineering capabilities for the material structure is the framework of n-point spatial correlations (Torquato and Stell, 1982; Torquato and Haslach, 2002; Fullwood et al., 2010; Niezgoda et al., 2011; Adams et al., 2012; Niezgoda et al., 2013). In recent work, Kalidindi and co-workers (e.g., Kalidindi, 2015) have developed and demonstrated an efficient and broadly applicable computational framework and toolsets for addressing this task. Broadly referred as Materials Knowledge Systems (MKS), this framework takes advantage of the computational efficiency of voxelated representations and Fast Fourier Transform (FFT) algorithms to implement the theoretical framework of n-point spatial correlations. The feasibility and benefits of this approach have been demonstrated on a wide variety of material classes and material structures at different length scales [from the atomic (Gomberg et al., 2017; Kaundinya et al., 2021) to dislocation length scales (Robertson and Kalidindi, 2021a) to microscale (Generale and Kalidindi, 2021)].

At its core, MKS defines and utilizes a material structure function (Kalidindi, 2015) that maps a selected combination of spatial position   x ∈ Ω   (the physical volume of the material domain) and a local material state h ∈ H (includes information such as phase identifiers, chemical compositions, lattice orientations, defect densities) to suitable measures (e.g., density) that reflect the intensity of h at x . Mathematically, one can express this function as m ( h , x ) . Implicit in this definition is the expectation that H needs to be identified suitably to capture the complete set of material states of interest at the different material structure length scales. Features of the material structure can then be defined as expectations of suitably scaled moments of m ( h , x ) . For example, the expected value of m ( h , x ) over Ω can provide a set of 1-point features that can be interpreted as the volume fractions of h in Ω (Kalidindi, 2015). Similarly, the expected value of m ( h , x ) m ( h ′ , x + r ) over Ω can produce a set of 2-point features that can be interpreted as the joint probability of realizing h at x and h′ at x + r , where r denotes a specified vector separating the two spatial points randomly selected from Ω . Although, one can define higher-order features (e.g., 3-point features), one often finds a sufficiently large number of features in the 2-point feature set, as it includes all permutations of ( h , h′ ) over a very large domain of r (this domain includes all distinct set of all vectors of interest that can be placed in Ω ). The adequacy of the set of 2-point features in capturing the salient features of the material structure (including the set of features identified in conventional practices in materials science and engineering) has been established for a broad range of material classes (Latypov et al., 2019; Generale and Kalidindi, 2021) as well as the different structure length scales (Fullwood et al., 2010; Robertson and Kalidindi, 2021a; Kaundinya et al., 2021) encountered in them.

The MKS framework described in Figure 4 produces a very large number of features, even when using only the 2-point feature set. For extracting a low-dimensional feature set, one needs to use a suitable dimensionality reduction technique. Of the various options for this task, principal component analysis (PCA) has been found to be particularly attractive. First, it allows for an unsupervised learning of the salient low-dimensional features based on maximization of captured variance. Therefore, it identifies a consistent set of features that can be used across multiple PSP surrogate models, allowing for full interoperability among collections of such models. In other words, since the salient features are identified without the knowledge of the specific targets (i.e., outputs) of the surrogate model, they can be used for different targets (for example, in the predictions of very different properties of a given material system). Second, the PCA basis can be inspected and interpreted to a limited extent, allowing for the low-dimensional features to be associated with some (limited) physical meaning. Third, since PCA essentially involves a rotational transform of the original space, it preserves distances between datapoints in the original space. Finally, the orthogonal decomposition involved in the PCA allows for practically useful reconstructions of the full feature list, i.e., a reconstruction of the high-dimensional feature list from the low-dimensional feature list. Of course, these reconstructions are approximate because of PC truncation. However, since the PC representations are maximized to capture variance, it is possible to make sure that the approximation introduced by the truncation is within acceptable tolerance. The PC scores obtained from the application of PCA on an ensemble of 2-point feature sets (one set corresponds to one material structure) serve as a highly effective low-dimensional feature set for the material structure in our digital twins. There exist a multitude of other options for dimensionality reduction of the feature space, such as isomap or kernel PCA. However, the nonlinear embeddings employed in these techniques can introduce distortions into the latent space that negate the benefits of PCA identified above (Hu et al., 2022).

As stated earlier in Section 2.1, the physical twin is not defined as a single instantiation of a material structure, but rather as the outcome of a stochastic generative process that yields instantiations that we then observe. The MKS framework described above provides a mathematically compact representation using computationally efficient tools. However, many tools (e.g., phase-field simulations, micromechanical finite element models) only take specific instantiations of the material structure as inputs. Therefore, successful creation of digital twins for materials requires the ability to move between statistical representations of material structure and their three-dimensional physical instantiations at low computational cost. While the computation of 2-point spatial correlations from instantiations is relatively easy (Cecen et al., 2016), the inverse computation is not trivial. Very recently, it has been shown that the three-dimensional material structures can be instantiated from their 2-point feature sets with minimal computational cost (Robertson and Kalidindi, 2021b). As a result of the many advantages described above, the MKS framework along with its open-source code repository PyMKS (Brough et al., 2017) offers a powerful, currently available, toolset for addressing the challenges of building digital twins of materials systems.

It is also noted that there are a number of other options based on neural networks that allow one to combine feature engineering and property prediction into a single-step framework. These approaches offer attractive avenues when one is interested in a limited number of properties as targets. If one insists on de-coupling the form and function of the digital twins (as we have argued here), then it is imperative to pursue feature engineering of the material structure independently from establishing property predictors (discussed in the next section). In this context, it should be recognized that the autoencoder-decoder networks (Herr et al., 2019) offer an interesting option. These networks do address the unsupervised feature engineering of the material structure. Therefore, the features identified from such networks can then be input into other neural networks for property predictions. This idea represents an open research avenue that merits further exploration.

Reliable prediction of the effective properties of a given material sample is a challenging task. At a high level, the main options are to either measure experimentally the properties of interest or to leverage known physics (often delivered in physics-based simulation packages) to estimate their values. Both approaches face hurdles when one desires to produce a multiscale, digital twin for materials. On the experimental front, the effort and cost involved in measuring all of the properties of interest along with the related information (e.g., anisotropy, variances) over the multiple material length scales of interest are often prohibitive. On the modelling front, there is substantial uncertainty in the model forms and/or parameter values used in the physics-based models. It is therefore clear that neither approach by itself is optimal in getting us the requisite information. In this regard, the recent emergence and successful application of materials data analytics tools has opened up new avenues for addressing these gaps.

Recently (Kalidindi, 2015; Kalidindi, 2020), it has been argued that process-structure-property (PSP) linkages can be defined over different material-structure length scales to capture the core knowledge needed to study multiscale material responses. It is argued here that the same PSP linkages can be utilized to predict the functional response of the material digital twin. This is because the PSP linkages can be used to update both the changes in the multiscale material structure due to the imposed service conditions (using suitably defined process-structure evolution linkages) but also their associated properties (using structure-property linkages). The required PSP linkages need to be formulated using available data that might often be disparate, incomplete and/or uncertain. Most importantly, the framework for predicting the function of the material digital twin should allow easy (and possibly frequent) updates as new data becomes available. It is also likely that one needs to chain together multiple PSP models in order to make the predictions of the function of the material digital twin.

A Bayesian framework has the potential to address scale-bridging with uncertain physics. The proposed Bayesian framework will be described next using the structure-property (SP) linkages as an example. However, they will be formulated such that they can also be easily applied to capturing process-structure linkages (PS). Typically, SP linkages are formulated to take structure variables as inputs and predict property values as output. The mapping implied in these linkages can be expressed as P   =  ℱ ( μ ) ,   where P is a property variable and μ denotes a vector of structure features (e.g., the PC scores of the 2-point feature set described in Section 2.2.1). Both P and μ should be treated as stochastic variables. This naïve definition makes the governing physics implicit in the formulation of ℱ . It would be much more desirable for SP linkages to explicitly treat the governing physics as additional input variables to the mapping, i.e., to refine the desired mapping as P   =  ℱ ( μ , φ ) , where φ denotes the governing physics. In practice, φ would represent a vector of parameters defining the physical mechanisms controlling the response of the material physical twin (e.g., parameters used in constitutive modeling of the material response). This refinement is advantageous in two ways. Firstly, it allows one to treat φ as a stochastic vector variable, which often exhibits a significant amount of uncertainty for a selected material physical twin. Secondly, it allows for the uncertain physics to be passed between linkages. This is particularly useful for the multiscale phenomenon that occur in material systems, as the uncertain physics learned in one length scale can still be utilized at another length scale. An example of this scale-bridging is depicted schematically in Figure 5. The first linkage estimates the indentation yield strength (effective property) of a single grain given the grain’s orientation (structure variable) and critically resolved shear strengths (physics variables). The second linkage estimates the bulk yield strength (property) given the two-point statistics of the grain orientations (structure variables) and the same critically resolved shear strengths (physics variables). Since the physics variables in these two linkages are the same, the uncertain knowledge of the physics variables extracted in the grain-scale data (could come from experiments and/or simulations) can be upscaled and utilized in making predictions of the effective properties at the polycrystal scale.

In establishing the material physics parameters, one has to exploit all of the available data, collected from disparate sources (e.g., experiments and physics-based simulations). Machine learning of φ for a selected material physical twin can be accomplished using a Bayesian update strategy expressed as: p ( φ | E ) ∝ p ( E | φ )   p ( φ ) (1) where E denotes the set of available experimental observations, p ( φ ) is the prior (reflecting our best initial guess), p ( E | φ ) is the likelihood of realizing the observations in E , and p ( φ | E ) denotes the updated posterior on φ . Although Eq. 1 looks very simple, its practical usage for learning the controlling physics in multiscale material phenomena has been hindered by several factors. First, only the physics-based simulation tools that faithfully mimic the experiments performed to obtain E can allow for the computation of the likelihood term in Eq. 1. This is because only these tools allow arbitrary specification of the governing physics φ . However, a brute-force application of physics-based tools for computing the likelihood is prohibitively expensive because of the extremely large number of simulations one needs to perform to accomplish this task since it entails performing simulations covering a large domain of likely governing physics for all of the experimental observations in E . Second, the proportionality in Eq. 1 implies that one needs to develop and implement a suitable strategy for establishing the proportionality factor. Recent work (Castillo et al., 2021) has demonstrated that it is possible to train AI/ML models on simulation results produced by physics-based models, which can then be used to compute the likelihood term in Eq. 1. Furthermore, they would also allow for the implementation of Markov-Chain Metropolis-Hastings (MCMH) approaches for sampling the posterior in Eq. 1 without explicitly computing the proportionality factor. It is important to note that the posterior estimate of φ is not restricted to come from any single source of data. As an example, let us consider the situation where the data becomes available from different test modes (these could be indentation tests and micro-pillar tests for grain-scale mechanical measurements). In such situations, we need to establish different surrogate models for each test mode. Let P 1 =  ℱ ( μ , φ ) and P 2 =  ℱ ( μ , φ ) represent such surrogate models. Since the underlying microstructure and physics variables have the exact same meanings in both models, one can use both models with their respective experimental datasets for sampling a consistent posterior for φ . Once the posterior is established, one can establish the desired SP linkage in a stochastic framework by marginalizing as: p ( μ | E ) = ∫ f ( μ , φ ) p ( φ | E ) d φ (2)

As noted above, the practical implementation of Eqs 1, 2 needs the establishment of suitable AI/ML surrogates. These usually take the form ℱ ( μ , φ ) , and can be accomplished using Gaussian Process Regression (GPR). The central advantage of the proposed strategy here is that the formulation of the needed AI/ML models is generally a one-time effort. In other words, when these are properly designed to cover large input domains in the space of the controlling physical parameters and the space of relevant material structures, they only need to be performed once [examples can be seen in prior work (Castillo and Kalidindi, 2019; Castillo et al., 2021)]. This feature allows for a relatively low-computational cost update of the surrogate model as new experimental data becomes available. It is also possible to suggest new experiments that maximize the potential for improving the accuracy of the predictions (i.e., reducing the prediction uncertainty). This is most efficiently accomplished using established concepts of information gain such as the posterior predictive variance (Castillo and Kalidindi, 2019; Castillo et al., 2021; Castillo and Kalidindi, 2021), expected improvement (Solomou et al., 2018; Takhtaganov and Müller, 2018; Talapatra et al., 2018; Ghoreishi and Allaire, 2019), and expected information gain (Pandita et al., 2019).

An example application of the proposed Bayesian approach methodology is depicted in Figure 6, taken from the work of Castillo et al. (2021). In this example, the information from spherical indentation measurements on individual grains in a polycrystalline sample and the corresponding simulations using crystal plasticity finite element models are combined to establish distributions on the unknown values of the critical resolved shear strengths of four different families of potentially active slip systems in a selected Ti alloy. The approach described in this study resulted in at least one order of magnitude savings in both the overall cost and effort expended, when compared to the conventional approaches that employed small-scale testing to obtain the same information.

Material structure measurements capture the state of the material before, during, and after evolution, and material property measurements quantify various characteristics of evolution (e.g., resistance to evolution, evolution rates). The constellation of methods used to measure material structure and properties is extensive, and here we only mention two general trends. First, the digital data stream is becoming more entrenched in the instruments used to measure material properties. Just a generation ago, material structures were documented on film and quantification was performed by manual measurements; lab instruments utilized strip-chart recorders that created an analog graphical representation of the data. Now, not only have data streams become digitized, but increasingly, the data collection instruments are networked and remotely accessible. Yet significant concerns remain regarding the cyber vulnerability of both the data and the instrument, and institutional regulations regarding interconnectivity are highly disparate. Second, with the continuing advances in measurement sensors, data transfer, and data storage, the data streams are becoming increasingly dense, requiring thoughtful strategies for intelligent data reduction. Additionally, unconventional datasets, collected with alternative low-cost methods are proving to have utility. Previous trends in measurement science have focused on increases in precision and accuracy of data. Now, the focus is shifting to affordable high-density data streams that can provide similar or complementary information content to the existing suite of ultra-precise measurements.

The external stimuli (e.g., thermo-chemo-mechanical loading) driving material structure evolution need to be tracked through the use of suitable sensors. Sensors generally transduce various forms of energy (Table 1) into electrical signals that can be transformed into digital data. The transduction can also involve intermediate forms of energy, e.g., magnetic or optical. All forms of sensing have limits in resolution, range, accuracy, and precision. The fidelity of the digitized resolution of the external stimulus captured by the sensor is limited by the accuracy of the correlation of the electrical signal to the intensity of the imposed stimulus, and the bit-depth of the stored information. The fidelity of an environmental measurement can also be limited by the temporal and spatial resolution of the sensor. Sensor arrays allow for spatial mapping of a field (e.g., temperature field on a sample surface) of interest, with the spatial resolution limited by the spacing between individual sensors in the array. Alternately, one can acquire such information using a single sensor and rapidly scanning a region of interest; this strategy will lead to some degree of temporal disregistry between individual measurements.

The high volume and high variety of materials data quickly outpaces rudimentary data organization techniques typically used by humans (project specific folder structures, ad hoc organization or note taking). We therefore require more sophisticated data management tools to manage the storage and organization of the materials data relevant to the digital twin. In their most basic forms data management tools act as simple data repositories, centralized locations where data is held and made accessible to others. However, simple data repositories do not necessarily provide a systematic scheme for the organization of the data or metadata therein. Digital twins require the establishment of standards and protocols to catalogue, vet, compare, and use data reliably and credibly in automated (and possibly autonomous) protocols (Kalidindi and De Graef, 2015; Sorkin et al., 2020). Consequently, data management solutions for digital twins should aim to at least meet FAIR data principles: Findability, Accessibility, Interoperability, and Reusability (Wilkinson et al., 2016). FAIR data should have: (1) globally assigned, rich, searchable metadata with a unique persistent identifier and clear provenance; (2) standardized communication protocols for data storage and retrieval; (3) consistent, widely utilized, non-proprietary standards employed for data formatting. Data repositories generally only meet the most basic aspects of FAIR—namely accessibility. Materials databases progress further towards FAIR principles by providing greater searchability. Databases allow users to construct and carry out complex queries to search for information, and therefore improve searchability. However, their searchability is generally limited to tabular data. Furthermore, databases are also generally limited in their interoperability and reusability. In particular, they are not well suited for the materials data needed for digital twins as there is no natural way to describe the relational connections between disparate materials data (e.g., temporal variations along process paths, nested composition relationships, multimodal data describing single sample).

In order to truly realize FAIR data principles for materials data, we need to adopt emerging software tools in ontologies and linked data. Ontologies for data management are an open-world framework where we construct a standardized language to connect and describe objects. There currently exists many standardized languages used to describe ontologies such as OWL (McGuinness and Van Harmelen, 2004), RDF (Lassila and Swick, 1998), or JSONLD (Sporny et al., 2014). These languages all describe data in subject-predicate-object triples where we link the subject and the object through some rule (the predicate). One way to capture this information is through the formation of knowledge graph consisting of nodes (subjects, objects) and edges (predicates). Knowledge graphs allow for easily understood visual depictions of metadata, and for the application of emergent graph-based AI toolsets for the automated identification of new connections between aggregated elements of a complex heterogeneous dataset.

A recently proposed materials ontology (Voigt and Kalidindi, 2021) shown in Figure 7A can prove valuable in our effort to collect and curate the data needed for a materials digital twin. This ontology consists of four primary classes of entities (denoted by circles) that can serve as subjects or objects: Process, Material, Tool, and Data. A total of nine predicates (denoted by arrows) have been defined to link these objects. Process nodes hold information about process parameters, tool nodes describe the settings and characteristics of machines, and data nodes hold the payloads of interest (images, tabular data, etc). A material node describes the state of the material along a nominal process. Therefore, every time an action is taken on a material, we produce a new material node. This allows us to easily associate data with a point along a process path. As an example, a given steel (Material) produced after a specified thermo-mechanical processing route (Process) can be studied in a microscope (Tool); the results of the study are captured in a file (Data). Figure 7B depicts an example knowledge graph for a steel. The process begins with a generic low carbon steel node (seen at the bottom of the knowledge graph). It then undergoes a standard annealing step to get a uniform starting material, and proceeds through a specialized intercritical annealing and quench steps to its final form (labelled as 750-00-000 in the knowledge graph). Along the processing route shown, we are able to connect the various data/metadata collected. For example, it is seen that both the starting material and the final material have associated SEM images. The final material also has a datasheet generated using a known software package (defined by a Tool node) which took a known load-displacement curve (defined by a Data node) as input. Ontologies allow us to systematically capture interconnected materials data and allow for the context of a dataset to be robustly described and communicated, thus enhancing the reusability of the data.

There currently exist several software packages than can be used to support the mathematical framework proposed in Section 2.2. For structure quantification PyMKS (Brough et al.) offers computationally efficient tools for the feature engineering of material internal structures. PyMKS supports various data transformations needed to capture information on a wide range of material local states encountered in different material classes at different material structure length scales. PyMKS utilizes Dask, a distributed framework for developing python applications, to facilitate computations involving large datasets on supercomputers and large clusters (Rocklin, 2015). Subsequent to feature engineering, surrogate model building can be accomplished via a wide variety of popular python packages; examples include Statsmodels (Seabold and Perktold, 2010) for basic statistical models, SKLearn (Pedregosa et al., 2011) for machine learning tools, PyTorch (Paszke et al., 2019) and TensorFlow (Abadi et al., 2016) for neural networks/deep learning tools.

AI tools support digital twins beyond the needs of the mathematical framework alone. AI based segmentation strategies have gained traction, and Bayesian CNNs have recently been used to characterize the segmentation uncertainty in materials images (LaBonte et al., 2020). AI tools have also been effective in fusing multimodal materials data. Multi-input NNs have proven effective in combining data from multiple sources and different data types. For example, numeric and categorical data, assessed via multi-layer perceptron algorithms can be directly combined with image-based convolutional NNs (Azim and Aggarwal, 2014). While data streams are typically experimental, it can sometimes be beneficial to integrate high-fidelity simulation data from traditional high-performance computing approaches (e.g., atomistic modeling, phase-field, finite element) to augment “missing” experimental data or to represent functional dependencies/sensitivities that were not exposed in the experimental datasets. For instance, well-established experimental methods such as diffraction measurements are being implemented into computational models as a complement of the interpretation of experimental results (Coleman et al., 2014; Kunka et al., 2021). Alternatively, researchers have recently used generative machine learning algorithms such as generative adversarial network (GAN) to generate large materials and process libraries (Banko et al., 2020).