Predicting MicrostructureSensitive FatigueCrack Path in 3D Using a Machine Learning Framework
Abstract
The overarching aim of this paper is to explore the use of machine learning (ML) to predict the microstructuresensitive evolution of a threedimensional (3D) crack surface in a polycrystalline alloy. A convolutional neural network (CNN)based methodology is developed to establish spatial relationships between micromechanical/microstructural features in a cyclically loaded, uncracked microstructure and the 3D crack path, the latter quantified by the vertical deviation (i.e., zoffset) of the crack along a specified axis. The proposed methodology consists of (i) a feature selection and reduction scheme to identify a lowerdimensional representation of the experimentally measured microstructure and computed micromechanical fields, which allows for computational feasibility in predicting the zoffsets; (ii) a CNN model to compute the zoffset as a function of the local, lowerdimensional feature data; and (iii) a radial basis function smoothing spline to ensure spatial continuity between the independently predicted zoffsets. The proposed CNNbased methodology is shown to improve on the accuracies obtained using existing ML models such as XGBoost and to provide a definitive way of quantifying model uncertainty associated with CNN predictions. To further investigate the applicability of ML models, multiple prediction strategies with which to deploy ML algorithms are proposed and the relative performance of ML algorithms corresponding to each prediction strategy are analyzed. The presented work thus provides a framework to find an encoded representation of 3D microstructure and micromechanical data and develop methods to predict microstructuresensitive crack evolution based on this encoded representation, while quantifying associated prediction uncertainties.
Introduction
Fatigue cracking contributes to a majority of inservice failures of engineering structures, which are often driven by onset and accumulation of microstructurally small cracks.1 The initiation and earlystage growth of these small cracks are dependent on microstructural features that influence, for example, crack interactions at grain boundaries.2 Computational modeling of crack initiation and growth can be complex, partly because understanding the effects of a broad range of microstructural features on the constitutive response can be extremely difficult to achieve using conventional continuum approaches.2 While the growth rate of long cracks can be accurately predicted using the Paris–Erdogan law (or similar relationships),3, 4, 5 microstructurally small cracks can exhibit greater variability in their growth behavior compared with long cracks.4 This is evidenced by the larger scatter in the growth rate of small cracks with the same nominal value of stress intensity factor.4 Yet understanding the behavior of these small cracks is critical, as they can account for upwards of 80% to 90% of a component’s fatigue lifetime.6 As such, accurate predictions of crack behavior can improve the reliability of structural components, as well as lower associated production and maintenance costs, as described in Ref. 7.
In an attempt to model the behavior of microstructurally small cracks, both empirical and numerical approaches have been employed based on certain underlying principles and influencing factors. Fatemi and Yang8 and Hussain5 review early phenomenological theories on smallcrack behavior, in which plasticity effects, metallurgical effects, and crack closure are suggested as possible explanations for smallcrack behavior. Factors such as grain shape and crystal orientation, nearneighbor distances, grain fracture toughness, and intrinsic flaw size are also considered as factors contributing to crack nucleation and earlystage propagation.9,10 In some cases, finiteelement analysis using crystal plasticity models, as originally proposed by Asaro,11 are used in literature to investigate the effect of the spatial variability of microstructural features and micromechanical fields on smallcrack behavior (e.g., see Ref. 12). A more comprehensive review of empirical and numerical approaches investigating the behavior of small cracks can be found in other literature.5, 6, 7, 8,13
While the aforementioned approaches often rely on stress intensity factors12 and/or fatigue indicator parameters14,15 as representative mesoscopic surrogates for the driving force behind crack initiation and propagation, there is a need for a more comprehensive, general framework that accounts for the complex spatial, nonlinear relationships between the relevant features, i.e., microstructural features and micromechanical fields, and a given representation of the crack. Datadriven methods can potentially be a viable alternative approach to address this challenge, as they can leverage large, highdimensional datasets, obtained through experiments or simulations, to model these complex relationships.16, 17, 18 Machine learning (ML) models have already exhibited a wide range of applicability within the materials science community, including for materials discovery,19 optimal design of experiments,20 and imagebased materials characterization.21,22 In the context of predicting crack behavior, datadriven methods such as principal component analysis (PCA) have been employed to determine reducedorder representations that correlate with fatigue indicator parameters (FIPs).23,24 In similar work,25 a random forest learning algorithm was employed to predict stress hot spots that were computed from fullfield crystal plasticity simulations, where the algorithm was trained using features that encode the local crystallography, geometry, and connectivity of the microstructure. Probabilistic models such as Bayesian networks,7,26 which are nonparametric and can account for uncertainties in predictions, have also been used to compute fatiguerelated parameters such as residual life and equivalent stress intensity factors, respectively.
This paper expands previous work on datadriven methods in fatigue modeling and proposes a framework using a convolutional neural network (CNN) model to predict 3D crack paths based on microstructural and micromechanical features. While previous, relevant work using datadriven approaches has primarily focused on identifying micromechanical and microstructural variables that contribute to the direction and rate of crack propagation,26 or on mapping global variables (such as chemical composition, grain size, heat treatment, and cyclic stress intensity factor) to the onedimensional crack growth rate,27 the work presented herein concerns the use of CNN to quantitatively predict the local crack path, in 3D, as a function of local microstructural and micromechanical features. CNNs are particularly well suited for problems that require finding spatial, nonlinear relationships between input and a given response variable of interest and have been successfully used in related applications such as classification of microstructures based on scanning electron microscopy (SEM) images20 and determination of material properties based on microstructure.28,29 Prior to training the CNN model, the input features (i.e., microstructural features and micromechanical fields) are selected based on previous correlation analysis by Pierson et al.30 (described below). PCA analysis is performed to convert the relevant input features to unique lowdimensional descriptors, whose values are specific to a given location within a microstructure. The 3D map of these descriptor values is then introduced into the CNN model. As a postprocessing step, to retain spatial continuity, a smoothing operation is performed on the predictions of crack surface elevations obtained from the CNN model.
The next subsection provides background information on the experimental and simulation data used to train and evaluate the ML model.
Prior Work by the Authors
In previous work,30 the authors conducted a systematic correlation analysis between computed micromechanical fields in a 3D, uncracked polycrystal and the observed path of an eventual fatigue crack. Specifically, an experimentally measured volume of an AlMgSi alloy31 was modeled using a highfidelity, concurrentmultiscale, finiteelement mesh with a crystalplasticity constitutive model.32 Cyclic loading was simulated at a load ratio of R = 0.5 (consistent with experiment), and computed field variables (or derivatives thereof) based on stress, strain, and slip were parameterized to a regular 3D grid. A complete list of the 22 variables computed at each time step during the finiteelement simulation is presented in Table I. Additionally, the cyclic change in each of the variables was computed between the peak and minimum load for each of five simulated loading cycles. Figure 1 shows six of the parameterized variables (five cyclic damage metrics and the cyclic micromechanical Taylor factor33). Once the cyclic change in the computed field variables was shown to converge, the spatial gradients of the variables were calculated using a finitedifference approach. In total, 88 variables were considered in the correlation study, of which 44 were based on spatial gradients of the micromechanical fields. The parameterized variables were then systematically correlated with distance to the known crack surface. Correlation coefficients for all 88 variables are shown in Fig. 2. In general, the spatial gradients of the micromechanical field variables (Fig. 2c and d) exhibited a stronger correlation with the crack path than did the field variables themselves (Fig. 2a and b). To assess whether the correlation coefficient values shown in Fig. 2c and d were meaningful, correlation analyses were also performed between the 88 variables and alternative paths throughout the microstructure. The correlation coefficients for the alternative paths were consistently weaker than those for the actual crack surface, suggesting that micromechanical fields of the cyclically loaded, uncracked microstructure might provide some degree of predictiveness for the microstructurally small fatigue crack path. Results from the previous correlation analyses are leveraged in the current work involving physicsinformed ML to predict the crack surface evolution.
List of 22 variables computed during finiteelement simulation30
\(D_1\)  Maximum value of accumulated slip among 12 octahedral slip systems 
\(D_2\)  Maximum value of total accumulated slip over each slip plane 
\(D_3\)  Accumulated slip summed over all slip systems 
\(D_4\)  Maximum value of energy dissipated on a given slip plane during plastic deformation 
\(D_5\)  Modified Fatemi–Socie parameter 
\(\bar{\epsilon }\)  Symmetric strain tensor composed of \(\epsilon _{xx}\), \(\epsilon _{yy}\), \(\epsilon _{zz}\), \(\epsilon _{xy}\), \(\epsilon _{xz}\), and \(\epsilon _{yz}\) 
\(\epsilon _1\)  Principal eigenvalue of the strain tensor 
\(\epsilon _{\rm vM}\)  von Mises strain 
\(\bar{\sigma }\)  Symmetric stress tensor composed of \(\sigma _{xx}\), \(\sigma _{yy}\), \(\sigma _{zz}\), \(\sigma _{xy}\), \(\sigma _{xz}\), and \(\sigma _{yz}\) 
\(\sigma _1\)  Principal eigenvalue of the stress tensor 
\(\sigma _{\rm vM}\)  von Mises stress 
\(M^{\rm micro}\)  Micromechanical Taylor factor 
Methods
Selection and Extraction of Input Features
Based on the previously described correlation analysis, a new, lowdimensional representation of the data is computed from the original set of features, one that is amenable to spatial relationbased learning algorithms, without losing highvalue information contained within the raw features. To achieve this, the existing data are transformed into a new domain that has a low dimensionality for each point in the 3D microstructure. As an analogy, consider how an image is represented in a lowdimensional color space (such as RGB) and contains a vector with three entries at each of the pixels on a twodimensional (2D) plane. This representation allows for feasible convolutions and other operations during training. In this work, the dimensionality of each point within the microstructure is first reduced from 88 raw features to a descriptor vector containing three elements, while still retaining the highvalue information for predicting the crack path.
The correlation analysis described by Pierson et al.30 is used to select the subset of features likely to contribute to spatial deviations on the crack surface. This feature selection is imperative to the feasibility of efficient model training; using all 88 features at all \(1.7235 \times 10^8\) points in the volume would render the computation intractable, given hardware constraints. It was shown in Ref. 30 that the spatial gradients of the micromechanical features are more strongly correlated with the crack path than are the micromechanical features themselves, as shown in Fig. 2. Among the 44 spatial gradient features considered, the 22 features associated with the cyclic change between loading and unloading (Fig. 2d) have comparatively higher Pearson correlation coefficients. Thus, the feature set considered for input to the ML models is first downselected to these 22 features.
Principal component analysis (PCA) is subsequently performed on these 22 features to further reduce the dimensionality of the feature set. PCA is a commonly used dimensionality reduction tool that provides the orthogonal basis vectors in decreasing order of the amount of variance explained along their directions.34 The 22 features are normalized prior to performing PCA. The PCA analysis shows that the first three modes explain \(96\%\) of the total variance, and as such, those three principal components (\(\alpha _1, \alpha _2, \alpha _3\)) are selected as the basis for the new data. In other words, the 22 features at each point in the microstructure are projected onto this basis, resulting in a threeelement descriptor vector based on the micromechanical metrics at each point.
In addition to the features based on micromechanical fields, an additional feature is considered based purely on the geometrical configuration of the microstructure. Experimental evidence has long suggested that microstructurally small fatigue cracks behave very differently near grain boundaries than within a single grain, manifested as crack deceleration or deflection.35, 36, 37 To account for such a spatially dependent relationship, an additional feature called \(d_{\rm GB}\), which is the distance from a given point to the nearest grain boundary, is added to the descriptor set to produce a locationspecific, fourelement descriptor. Thus, the descriptor vector \(\mathbf {{x}_{i}}\) corresponding to each location i can be expressed as follows: \({\mathbf {{x}_{i}}} = [\alpha _{1i}, \alpha _{2i}, \alpha _{3i}, {d}_{{\rm GB},i}]\). For comparison, the ML models are also trained using a descriptor vector of only the experimentally measured Euler angles (represented within the fundamental zone, to account for crystal symmetry) corresponding to each location, to see how predictions based on a simpler, more rudimentary description of the microstructure perform in comparison with the feature vector \(\mathbf {{x}_{i}}\).
General Approach to Predict Crack Path
 1.
The local vertical deviation in the crack surface with respect to a “neighboring point,” as quantified by the zoffset (\(\varDelta z\)), is assumed to be dependent upon a region of influence \(\varDelta x_{v} \times \varDelta y_{v} \times \varDelta z_{v}\,\mu {\text{m}}^3\) centered around that point. In this analysis, we consider \(\varDelta x_{v} = \varDelta y_{v} = \varDelta z_{v} = 10\,\mu {\text{m}}\), meaning that the zoffset at a given point is assumed to be dependent upon microstructural and micromechanical features contained within a cube of dimension \(10\,\mu \text{m}\), as shown in Fig. 3. The microstructure is discretized onto a 3D grid with resolution of \(1\, \mu \text{m}\),30 with each point in the grid being defined by the entries of the descriptor vector \({\mathbf {{x}_{i}}}\). The selection of the neighboring point depends on the type of prediction strategy adopted, as explained later in this section.
 2.
The set of targets (i.e., zelevation values) paired with the corresponding features is partitioned into training and test sets. The features and the targets in the training set are used to train a given ML algorithm (e.g., CNN model). The trained ML model is used to predict the zoffset, and subsequently compute the zelevations at test locations using the features in the test set.
 3.
Optionally, once the predicted zelevations are obtained, a radial basis function (RBF) smoothing spline is applied on the zelevations as a postprocessing step to ensure spatial continuity between independently predicted zoffset values. The RBF package in the Python package scipy is used to perform this operation. Note that the RBF scheme is only applied in the translational prediction strategy, which is described later in this section. An averaging of multiple predictions is implemented within the radial prediction strategy (also described later), which serves as a postprocessing smoothing operation.
 4.
Finally, the accuracy of the ML algorithm is evaluated by comparing the predicted (and subsequently corrected using RBF) zelevations with the corresponding actual values using the mean squared error and the \(R^2\) metrics.
In the radial prediction strategy, the ML model is trained on a semicircular region emanating outward from the nucleation point of the crack. The test data (i.e., data for which the predictions are made) corresponds to a region that extends radially outward with respect to the training set, as shown in Fig. 4. This prediction is done by moving pointwise along a semicircular region of the crack surface and predicting the zoffset for three of the adjacent points on the xy plane. If the predictions of multiple “known” points overlap for a given “unknown” point, the mean of all the predictions is taken. This averaging operation serves as a smoothing operation, as mentioned earlier in this section. Once a prediction for the zcoordinate is produced, the point is added to the “known” region. This continues until the zcoordinates of all points within the domain of the final crack front have been predicted. Figure 4 also presents a schematic of this radial prediction approach. In the radial prediction scheme, 33,985 data points are used to train the CNN model before the trained model is used to predict on 17,754 data points.
The translational and radial strategies represent two different parameterizations of the crack surface. While cracks evolve physically as arbitrary 3D shapes, expressing and predicting such evolution requires a parametric description (i.e., the definition of a fundamental basis or direction along which the crack path is to be predicted).
Convolutional Neural Network Model to Predict Crack Surface
Convolutional neural networks (CNNs) are considered as the primary ML candidate to predict the crack surface using the general approach presented in “General Approach to Predict Crack Path” section, since CNNs account for the topology of the input data (i.e., allow extraction of higherlevel features by considering local correlations among spatially proximal lowerlevel features38). CNNs usually employ a hierarchical structure to determine how these local feature maps correlate with the response variable of interest. A CNN model is defined by weight tensors that indicate the relative influence that a spatial arrangement of features has on a given output. Aside from ensuring a lower number of free weight parameters, the spatial sharing of the weights also enforces a degree of shift invariance within the model.38 Through an optimization process, the CNN model “learns” how the spatial arrangement of the specific features is related to the output variable to be predicted. Once trained, the CNN model can then be used in a computationally efficient, forward sense to make predictions.
The application of CNNs is pertinent to the problem of predicting the zoffset presented here, as the zoffset at a given point is likely to be dependent on the relative spatial arrangement of the descriptors (as developed in “Selection and Extraction of Input Features” section) within a given region—in our case, a cubic volume surrounding that point. It is thus hypothesized that the proposed CNN model would perform comparatively better than other ML algorithms. XGBoost, a scalable tree boosting algorithm that has achieved stateoftheart results in several ML challenges39 and has been previously employed in materials sciencerelated applications,40 is selected as the primary ML model for comparison with the proposed CNN. Note that support vector regression (SVR) was also initially considered as an ML candidate; however, SVR exhibited significantly poor performance (e.g., \(R^2 < 0.6\) for the radial strategy), aside from suffering from poor scalability of computational time with increasing size of training data.41
Figure 5 presents a schematic of the CNN architecture implemented to predict the crack surface, as quantified by the zoffsets. For any given point on the crack surface, we consider as input a \(10\,\mu \mathrm{m} \times 10\,\mu \mathrm{m} \times 10\,\mu \mathrm{m}\) volume surrounding the point, where each voxel within the input volume occupies 1 \(\mu \mathrm{m}^3\) and has associated with it the feature vector \({\mathbf {{x}_{i}}}\) (or an input vector describing the Euler angles). A filter (which can be thought of as a scanning volume) of size \(3\,\mu \mathrm{m} \times 3\,\mu \mathrm{m} \times 3\,\mu \mathrm{m}\) then convolves (or scans) through the input volume. Each position within the scanning volume has an initially arbitrary coefficient, or weight value, associated with it, which, considering the entire scan volume, can be represented by a weight tensor, \(\bar{\omega} \). This weight tensor for a given filter is initialized at the beginning of the training phase. As the scanning volume convolves throughout the larger input volume, each element in the feature vector \({\mathbf {{x}_{i}}}\) at a given point in the scanning volume is multiplied by the corresponding weight value at that location. This convolution process continues until the scanning volume has scanned the entire input volume. The output from the convolution step is a matrix of weighted values representing the spatial arrangements of the \({\mathbf {{x}_{i}}}\) features. The next step shown in Fig. 5 is to apply a nonlinear transfer function to the elements of the matrix, whose effect (over subsequent iterations, or epochs) is to emphasize important features and deemphasize nonimportant features. In this work, we apply a rectified linear (or ReLu) transfer function. Once the matrix is updated to reflect the application of the nonlinear transfer function, a process known as “max pooling” is carried out, in which the updated matrix is subdivided into regions that correspond to neighborhoods from the original input volume. The maximum value from each subregion of the matrix is extracted and input to a new, lowerdimensional matrix that represents a new set of intermediate features for the volume (rather than the original set of \({\mathbf {{x}_{i}}}\) features). All of the above steps are carried out using multiple independent filters. In this work, four filters are applied, which results in four new, lowerdimensional feature matrices.
XGBoost as a Comparative ML Model
The performance of the CNN is compared against that of XGBoost, a type of gradientboosting algorithm39 that does not take into account the spatial arrangements of features. Gradientboosting algorithms make predictions by using an ensemble of predictive ML models, where these predictive models are added sequentially over multiple iterations to minimize a given loss function.43 In the case of gradient tree boosting algorithms, the predictive ML model takes the form of a decision tree, in which first and secondorder derivatives of the loss function with respect to the function value (i.e., prediction of each decision tree architecture) are used to determine the optimal decision tree architecture.
XGBoost is a variant of gradient tree boosting algorithm with multiple algorithm modifications and parallelization techniques to improve its computational efficiency, the details of which are available in other literature.39 The CNN is hypothesized to perform better than XGBoost for two main reasons: (i) the CNN inherently accounts for the spatial arrangement of local features within the microstructure (as represented by the descriptor vector or by the set of Euler angles), and (ii) the CNN model might be better suited to predicting a continuous variable such as zoffset compared with XGBoost, which constructs a piecewiseconstant model for each leaf in the decision tree. This hypothesis is explored in “Results and Discussion” section.
Computing Model Uncertainty Associated with CNN Predictions
Uncertainty quantification of parameters pertaining to fatigue phenomena is critical for reliability analysis and safety evaluation of structural components.44,45 The sources of such uncertainties can be diverse, such as variability in material properties, data uncertainty, and model uncertainty.46 In our study, quantification of model uncertainty is most relevant.46 Previous work by Rovinelli et al.7 has successfully explored Bayesian approaches for estimating the uncertainties associated with predictions of global measures (viz., residual life); however, it is of interest in this work to quantify the uncertainties associated with estimation of local parameters relating to small cracks. One of the contributions of this paper, therefore, is to demonstrate how we can leverage the CNN model presented in “Convolutional Neural Network Model to Predict Crack Surface” section to compute the model uncertainty associated with the predictions of local zelevation at given location (x, y), based on microstructural and micromechanical features in proximity to that location.
As such, the dropout operation mentioned in “Convolutional Neural Network Model to Predict Crack Surface” section is utilized to quantify the uncertainty in local zelevation prediction. As mentioned in “Convolutional Neural Network Model to Predict Crack Surface” section, the dropout layer can be used to stochastically regularize the CNN model against overfitting by aggregating over multiple neural network configurations.47 When dropout is applied between two layers l and \(l+1\) (see Fig. 5), the interconnection between a given intermediate feature (also known as a “node” in the multilayer perceptron) in layer l and another intermediate feature in layer \({l+1}\) is retained with probability p. This means that, in a given single iteration during training, the set of intermediate features in layer \(l+1\) is obtained by means of randomly subsampling from the set of intermediate features in the previous layer l with probability p.
Gal and Ghahramani48 suggested that, beyond simply preventing overfitting during training, the dropout layer can be further exploited during prediction to compute the uncertainty of a deep neural network model prediction, without needing any modifications to the model or the optimization objective function, and without compromising the prediction accuracy. They showed that the loss function for a neural network model with dropout is equivalent to a loss function that minimizes the Kullback–Leibler divergence between an approximate distribution and the posterior distribution of a deep Gaussian process.48,49 The approximate distribution of predictions in this case is obtained through multiple samplings of the zoffset using the CNN model with dropout. As the objective here is to obtain a distribution of predictions, the training configurations are retained for inference as well; i.e., rather than rescaling the predictions by p, the interconnections between nodes are each dropped with probability p during prediction, just as was done during the training phase.
In this analysis, therefore, we leverage this finding from Gal and Ghahramani to not only predict the crack path, but also to compute the uncertainty associated with the CNN predictions of the crack path at a given location. The prediction of the crack path at a given point can depend on prior predictions at neighboring points, and as such, the associated uncertainty propagation is accounted for using the uncertainty propagation rules, as presented in Ref. 50.
Results and Discussion
Table II presents the accuracies of the 3D crack path predictions using CNN and XGBoost, as obtained using the translational prediction strategy shown in Fig. 4. To indicate the relative benefits of using these ML algorithms, the accuracies obtained are compared with a baseline hypothetical crack surface, which is obtained by simply extruding the crack surface in the xdirection at the end of the training region through the entire prediction region. This hypothetical crack surface is treated as the naive prediction. The improvements in the \(R^2\) value relative to the naive prediction are presented in the final column of Table II. All of the models, with the exception of XGBoost trained using only the Euler angles, lead to better predictions of 3D crack path than simply taking the naive approach of extending the crack forward. Overall, the CNN with RBF smoothing outperforms the XGBoost model for a given training set.
The results in Table II show that both the CNN (with RBF smoothing) and XGBoost models exhibit improvement in accuracy when the descriptor vector (i.e., PCA values and \(d_{\rm GB}\)) developed in “Selection and Extraction of Input Features” section is used as the feature type, compared with cases where only Euler angles are used. However, note that the improvement in accuracy is significantly higher in the case of XGBoost (improvement in \(R^2\) from 0.669 to 0.88), compared with the relatively marginal improvement when CNN is selected as the ML algorithm (improvement in \(R^2\) from 0.844 to 0.891). While this observation supports our initial hypothesis that the encoded descriptor vector \({\mathbf {{x}_{i}}}\) can serve adequately as a local reducedorder representation of the microstructural and micromechanical features, it also indicates that, in comparison with the XGBoost model, the CNN gains less relative benefit from having access to the highfidelity, finiteelementbased inputs than simply having access to the crystal orientations at each point in the discretized domain. One possible reason for this is that the CNN’s ability to account for the spatial arrangement of crystal orientations allows it to learn the relevant relationships that would otherwise be learned from the finiteelement results. For a nonspatiallyaware ML model, such as XGBoost, it is imperative that the training dataset accounts, a priori, for the spatial relationships of crystal orientations, whereas this is not necessarily required for the CNN model. In the future, considering alternative features that represent crystal orientations more uniquely than Euler angles (see work by Mangal and Holm25) could help to generalize the CNN’s predictive capability across microstructural datasets.
Prediction results of crack surface elevation for different combinations of ML model and input feature type
Model  Feature type  \({R^2}\)  RMSE (\(\mu \mathrm{m}\))  % Increase in \(R^2\) 

Naive approach  N/A  0.788  14.17  – 
CNN (with RBF)  Descriptor vector, \(\mathbf {{x}_{i}}\)  0.891  10.16  +13.1 
CNN (with RBF)  Euler angles  0.844  12.16  +7.1 
CNN (without RBF)  Descriptor vector, \(\mathbf {{x}_{i}}\)  0.832  12.60  +5.73 
XGBoost  Descriptor vector, \(\mathbf {{x}_{i}}\)  0.888  10.31  +12.6 
XGBoost  Euler angles  0.669  17.69  −15.2 
The ML algorithms are now compared for the radial prediction strategy, focusing on the CNN and XGBoost models with the descriptor vector, \(\mathbf {{x}_{i}}\), used as input. Table III presents the accuracies of the ML algorithms corresponding to the radial prediction strategy, which is more likely to resemble crack front profiles than the translational strategy. Figure 9a presents the actual crack surface, with the boundary indicating the split between the training and test regions. Note that the outer profile of the predicted region is not perfectly semicircular. This is because the predictions were truncated at the last known crack front profile from experiment.31 The comparison in Table III shows that, for the radial prediction strategy, CNN performs marginally better than XGBoost. Inspection of Fig. 9b, c and comparison with 9a also reveals that CNN performs noticeably better than XGBoost in regions where the crack surface elevation is abnormally high (i.e., zelevation \(\ge 340\,\mu \mathrm{m}\)). Hence, note that, while global measures such as \(R^2\) and RMSE provide valuable, quantitative, information for an aggregated measure of accuracy of a given ML model, these measures are homogenized over the entire crack surface domain and do not necessarily convey information about how ML algorithms predict the crack surface in these anomalous, local regions. A qualitative comparison of the MLpredicted crack surface with the actual crack surface—particularly in such anomalous regions—can, therefore, provide additional insight into the performance of the ML model in predicting the crack path.
As stated in “General Approach to Predict Crack Path” section, the size of the training data corresponding to the radial prediction strategy is less than 25% of that corresponding to the translational case, so it is possible that the CNN is able to generalize comparatively better than XGBoost when fewer training data are available. One potential reason for the improved generalizability of the CNN could be due to its ability to account for the spatial distribution of descriptors, as discussed in “Convolutional Neural Network Model to Predict Crack Surface” section—for which XGBoost does not inherently account. In the context of a regression problem, XGBoost, which is a decision treebased algorithm, constructs a piecewiseconstant model with each leaf fit to a single value during training. When the size of the training data is comparatively smaller, it is possible that the leaf values are fit to a discrete value (corresponding to a zoffset) that is not sufficiently resolved.
Figure 9d shows the cumulative uncertainty maps corresponding to the predictions presented in Fig. 9b using the method presented in “Computing Model Uncertainty Associated with CNN Predictions” section. The figure shows that the uncertainties in CNN predictions are comparatively high in regions where the zelevations are high, which could be due to the relative lack of data points with high zelevations in the training region, as mentioned previously.
The implementation of the ML framework presented here is subject to several limitations. One of the limitations of the presented work is that only one set of experimental data of the crack surface, along with the corresponding raw microstructural and micromechanical features from the uncracked volume, is available to train the ML models. While the current crack surface does exhibit anomalous regions in crack path deviation, the same framework is anticipated to be applicable to (and perhaps even more useful for) more tortuous crack surfaces. In that regard, there is still a need to investigate the performance of the proposed ML framework on similar highresolution datasets beyond the one presented here. The proposed framework also does not explicitly handle crack initiation, but instead predicts the crack surface path once the initiation site is known. Finally, as mentioned at the end of “Prior Work by the Authors” section, one could view the lack of an explicit discontinuity in the training data as a limitation of the framework. However, the hypothesis explored here is that the micromechanical fields in the uncracked microstructure encode (within some spatial range of the initiation site) the path that a crack will eventually take. It is expected that, beyond this finite range, the fields will hold less predictive information, at which point an actual crack surface might need to be modeled. In fact, it is anticipated that a hybrid of the ML framework could be integrated with a computationally expensive, highfidelity model of an actual evolving discontinuity to help advance the crack in a more computationally efficient manner. This is an area for future investigation.
The proposed framework for predicting crack paths could prove useful in various materials science applications. For example, the same method for predicting the crack surface elevation could be implemented for predicting the local crack growth rate, which could lead to improved structuralprognosis capabilities. In addition to structural prognosis, the ML framework for predicting crack paths could also be applied in materials design applications. For example, rapid predictions of crack paths could be made for various microstructural instantiations under a given set of boundary conditions to identify or downselect optimal microstructural arrangements for controlled crack propagation.
Crack surface height prediction results for different ML models using the proposed descriptor vector (radial prediction strategy)
Model  Feature type  \({R^2}\)  RMSE (\(\mu \mathrm{m}\) ) 

CNN  Descriptor vector, \(\mathbf {{x}_{i}}\)  0.842  8.410 
XGBoost  Descriptor vector, \(\mathbf {{x}_{i}}\)  0.784  9.344 
Conclusion

The proposed descriptor vector, which is used as a reducedorder representation of local micromechanical and microstructural features, performs better (in terms of \(R^2\) and RMSE) than Euler angles in predicting the 3D crack surface when used as the feature set in a given ML algorithm. However, the improvement in accuracy is only significant for the XGBoost algorithm. In the case of CNN, the relative advantage in using the descriptor vector is marginal.

In the absence of micromechanical fields, i.e., with only crystal orientations as inputs, the CNN performs fairly well (\(R^2 = 0.844)\) in predicting the crack path. A likely reason for this is that the CNN’s ability to account for the spatial arrangement of crystal orientations allows it to learn the relevant relationships that would otherwise be learned from finiteelement simulations.

The CNN model also allows for computation of model uncertainties associated with its predictions by using the dropout operation, which can be used to compute (and visualize) the uncertainty maps corresponding to the predicted crack surface.
Notes
Acknowledgements
This material is based on research sponsored by the Air Force Office of Scientific Research Young Investigator Program under Agreement No. FA95501510172. The support and resources of the Center for High Performance Computing at the University of Utah are gratefully acknowledged.
References
 1.H. Mughrabi, Phil. Trans. R. Soc. A 373(2038), 20140132 (2015)CrossRefGoogle Scholar
 2.S. Kumar and W.A. Curtin, Mater. Today 10(9), 34 (2007)CrossRefGoogle Scholar
 3.P. Paris and F. Erdogan, J. Basic Eng. 85(4), 528 (1963)CrossRefGoogle Scholar
 4.D. Davidson, K. Chan, R. McClung and S. Hudak, Comprehensive Structural Integrity: Small Fatigue Cracks, ed. I. Milne, R.O. Ritchie, and B.L. Karihaloo (Elsevier, 2003), pp. 129–164.Google Scholar
 5.K. Hussain, Eng. Fract. Mech. 58(4), 327 (1997)CrossRefGoogle Scholar
 6.A.J. McEvily, J. Soc. Mater. Sci. 47(3Appendix), 3 (1998)Google Scholar
 7.A. Rovinelli, Y. Guilhem, H. Proudhon, R.A. Lebensohn, W. Ludwig, and M.D. Sangid, Modell. Simul. Mater. Sci. Eng. 25(4), 045010 (2017)CrossRefGoogle Scholar
 8.A. Fatemi and L. Yang, Int. J. Fatigue 20(1), 9 (1998)CrossRefGoogle Scholar
 9.J. Bozek, J. Hochhalter, M. Veilleux, M. Liu, G. Heber, S. Sintay, A. Rollett, D. Littlewood, A. Maniatty, and H. Weiland et al., Modell. Simul. Mater. Sci. Eng. 16(6), 065007 (2008)CrossRefGoogle Scholar
 10.M. Li, S. Ghosh, O. Richmond, H. Weiland, and T. Rouns, Mater. Sci. Eng. A 265(1–2), 153 (1999)CrossRefGoogle Scholar
 11.R.J. Asaro, J. Appl. Mech. 50(4b), 921 (1983)CrossRefGoogle Scholar
 12.G. Potirniche, S. Daniewicz, and J. Newman Jr., Fatigue Fract. Eng. Mater. Struct. 27(1), 59 (2004)CrossRefGoogle Scholar
 13.S. Suresh and R. Ritchie, Int. Metals Rev. 29(1), 445 (1984)Google Scholar
 14.W.D. Musinski and D.L. McDowell, Int. J. Fatigue 37, 41 (2012)CrossRefGoogle Scholar
 15.C.P. Przybyla, W.D. Musinski, G.M. Castelluccio, and D.L. McDowell, Int. J. Fatigue 57, 9 (2013)CrossRefGoogle Scholar
 16.A. Agrawal and A. Choudhary, APL Mater. 4(5), 053208 (2016)CrossRefGoogle Scholar
 17.Y. Liu, T. Zhao, W. Ju, and S. Shi, J. Mater. 3(3), 159 (2017)Google Scholar
 18.A.D. Spear, S.R. Kalidindi, B. Meredig, A. Kontsos, and J.B. le Graverend, JOM 70, 1143 (2018)CrossRefGoogle Scholar
 19.M.W. Gaultois, A.O. Oliynyk, A. Mar, T.D. Sparks, G.J. Mulholland, and B. Meredig, arXiv preprint arXiv:1502.07635 (2015)
 20.J. Ling, M. Hutchinson, E. Antono, B. DeCost, E.A. Holm, and B. Meredig, Mater. Discov. 10, 19 (2017)CrossRefGoogle Scholar
 21.B.L. DeCost and E.A. Holm, Comput. Mater. Sci. 110, 126 (2015)CrossRefGoogle Scholar
 22.J. Xu, X. Luo, G. Wang, H. Gilmore, and A. Madabhushi, Neurocomputing 191, 214 (2016)CrossRefGoogle Scholar
 23.S. Jha, R. Brockman, R. Hoffman, V. Sinha, A. Pilchak, W. Porter, D. Buchanan, J. Larsen, and R. John, JOM 70, 1147 (2018)CrossRefGoogle Scholar
 24.N.H. Paulson, M.W. Priddy, D.L. McDowell, and S.R. Kalidindi, Int. J. Fatigue 119, 1 (2019)CrossRefGoogle Scholar
 25.A. Mangal and E.A. Holm, Int. J. Plast. 111, 122 (2018). https://doi.org/10.1016/j.ijplas.2018.07.013 CrossRefGoogle Scholar
 26.A. Rovinelli, M.D. Sangid, H. Proudhon, and W. Ludwig, npj Comput. Mater. 4(1), 35 (2018)CrossRefGoogle Scholar
 27.H. Fujii, M. DJC, and B. HKDH, ISIJ Int. 36(11), 1373 (1996)Google Scholar
 28.A. Cecen, H. Dai, Y.C. Yabansu, S.R. Kalidindi, and L. Song, Acta Mater. 146, 76 (2018)CrossRefGoogle Scholar
 29.Z. Yang, Y.C. Yabansu, R. AlBahrani, W.K. Liao, A.N. Choudhary, S.R. Kalidindi, and A. Agrawal, Comput. Mater. Sci. 151, 278 (2018)CrossRefGoogle Scholar
 30.K.D. Pierson, J.D. Hochhalter, and A.D. Spear, JOM 70(7), 1159 (2018). https://doi.org/10.1007/s1183701828842 CrossRefGoogle Scholar
 31.A.D. Spear, S.F. Li, J.F. Lind, R.M. Suter, and A.R. Ingraffea, Acta Mater. 76, 413 (2014)CrossRefGoogle Scholar
 32.A.D. Spear, J.D. Hochhalter, A.R. Cerrone, S.F. Li, J.F. Lind, R.M. Suter, and A.R. Ingraffea, Fatigue Fract. Eng. Mater. Struct. 39(6), 737 (2016)CrossRefGoogle Scholar
 33.D. Raabe, M. Sachtleber, Z. Zhao, F. Roters, and S. Zaefferer, Acta Mater. 49(17), 3433 (2001). https://doi.org/10.1016/s13596454(01)002427 CrossRefGoogle Scholar
 34.I. Jolliffe, in International Encyclopedia of Statistical Science (Springer, Berlin, 2011), p. 1094Google Scholar
 35.J. Lankford, Fatigue Fract. Eng. Mater. Struct. 5(3), 233 (1982). https://doi.org/10.1111/j.14602695.1982.tb01251.x CrossRefGoogle Scholar
 36.T.G. Zhai, A. Wilkinson, and J. Martin, Acta Mater. 48(20), 4917 (2000)CrossRefGoogle Scholar
 37.K. Miller, Inst. Mech. Eng. Part C Mech. Eng. Sci. 205(5), 291 (1991)CrossRefGoogle Scholar
 38.Y. LeCun and Y. Bengio et al., Handb. Brain Theory Neural Netw. 3361(10), 1995 (1995)Google Scholar
 39.T. Chen and C. Guestrin, in Proceedings of the 22nd ACM Sigkdd International Conference on Knowledge Discovery and Data Mining (ACM, New York, 2016), p. 785Google Scholar
 40.L. Laugier, D. Bash, J. Recatala, H.K. Ng, S. Ramasamy, C.S. Foo, V.R. Chandrasekhar, and K. Hippalgaonkar, arXiv preprint arXiv:1811.06219 (2018)
 41.A. Bordes, S. Ertekin, J. Weston, and L. Bottou, J. Mach. Learn. Res. 6(Sep), 1579 (2005)MathSciNetGoogle Scholar
 42.D.P. Kingma and J. Ba, arXiv preprint arXiv:1412.6980 (2014)
 43.J.H. Friedman, Ann. Stat. 29, 1189 (2001)CrossRefGoogle Scholar
 44.M.S. Chowdhury, C. Song, and W. Gao, Eng. Fract. Mech. 78(12), 2369 (2011)CrossRefGoogle Scholar
 45.S.C. Kang, H.M. Koh, and J.F. Choo, Probab. Eng. Mech. 25(4), 365 (2010)CrossRefGoogle Scholar
 46.S. Sankararaman, Y. Ling, and S. Mahadevan, Eng. Fract. Mech. 78(7), 1487 (2011)CrossRefGoogle Scholar
 47.N. Srivastava, G. Hinton, A. Krizhevsky, I. Sutskever, and R. Salakhutdinov, J. Mach. Learn. Res. 15(1), 1929 (2014)MathSciNetGoogle Scholar
 48.Y. Gal and Z. Ghahramani, in International Conference on Machine Learning, p. 1050 (2016)Google Scholar
 49.Y. Gal and Z. Ghahramani, arXiv preprint arXiv:1506.02158 (2015)
 50.A summary of error propagation (2007). http://ipl.physics.harvard.edu/wpuploads/2013/03/PS3_Error_Propagation_sp13.pdf. Accessed 14 June 2019.