# Predicting Mechanically Driven Full-Field Quantities of Interest with Deep
Learning-Based Metamodels

Saeed Mohammadzadeh
Division of Systems Engineering
Boston University
Boston, MA 02215
<EMAIL_ADDRESS>
&Emma Lejeune
Department of Mechanical Engineering
Boston University
Boston, MA 02215
<EMAIL_ADDRESS>

###### Abstract

Using simulation to predict the mechanical behavior of heterogeneous materials
has applications ranging from topology optimization to multi-scale structural
analysis. However, full-fidelity simulation techniques such as Finite Element
Analysis, while effective, can be prohibitively computationally expensive when
they are used to explore the massive input parameter space of heterogeneous
materials. Therefore, there has been significant recent interest in machine
learning-based models that, once trained, can predict mechanical behavior at a
fraction of the computational cost compared to full fidelity simulations. Over
the past several years, research in this area has been focused mainly on
predicting single Quantities of Interest (QoIs). However, there has recently
been an increased interest in a more challenging problem: predicting full-
field QoI (e.g., displacement/strain fields, damage fields) for mechanical
problems. Due to the added complexity of full-field information, network
architectures that perform well on single QoI problems may perform relatively
poorly in the full-field QoI problem setting. This problem is also challenging
because, even outside the Mechanics research community, deep learning
approaches to image-to-image mapping and full-field image analysis remain
poorly understood. The work presented in this paper is twofold. First, we made
a significant extension to the Mechanical MNIST dataset designed to enable the
investigation of full field QoI prediction. Specifically, we added Finite
Element simulation results of quasi-static brittle fracture in a heterogeneous
material captured with the phase-field method. This problem was chosen as a
broadly relevant "challenge problem" for full-field QoI prediction. Second, we
investigated multiple Deep Neural Network architectures and subsequently
established strong baseline performance for predicting full-field QoI. We
found that a MultiRes-WNet architecture with straightforward data augmentation
achieves $0.80\%$ and $0.34\%$ Mean Absolute Percentage Error on full-field
displacement prediction for Equibiaxial Extension and Uniaxial Extension
datasets in the Mechanical MNIST Fashion dataset, respectively. In addition,
we found that our MultiRes-WNet architecture combined with a basic
Convolutional Autoencoder achieves a mean $F_{1}$ score of $0.87$ on the newly
added Mechanical MNIST Crack Path dataset. In addition to presenting the
results in this paper, we have released our model implementation and the
Mechanical MNIST Crack Path dataset under open-source licenses. We anticipate
that future researchers will directly use our model architecture on related
datasets and potentially design models that exceed the baseline performance
for predicting full-field QoI established in this paper.

##  1 Introduction

Understanding and predicting the behavior of heterogeneous materials is a
critical focus of mechanics researchers [47]. Over the past several decades,
theoretical and computational advances have enabled high-accuracy simulation
of heterogeneous materials behavior with applications ranging from patient-
specific soft tissue simulation [31, 60], to material microstructure design
[6, 25, 83], to large scale structural analysis [16]. Full-fidelity modeling
techniques such as the Finite Element Method [28] are typically quite
effective, but they come with a major shortcoming: they are extremely
computationally expensive [58], especially for advanced material models
implemented on highly resolved meshes [5]. High computational cost often
limits both our capability for thorough state-space exploration and prediction
[69]. On the other hand, metamodels, also referred to as surrogate models, are
“models of models” that can replicate a portion of the results of the original
high fidelity models through a portion of the state space with some (ideally
low) associated error [17]. When metamodels are constructed with supervised
machine learning methods, they are trained on data that is initially obtained
from the outcomes of high fidelity models (e.g., Finite Element simulation)
[32, 88, 67]. Once trained, metamodels can then be used to make predictions on
unseen data [43, 46]. Machine learning methods such as Gaussian Process
Regression [72] and Convolutional Neural Networks [21] have been used in
mechanics-based metamodeling [34, 85]. Critically, these metamodels can be
constructed to be computationally cheap [70], which is massively enabling for
applications such as topology optimization [4], multi-scale modeling [56, 40,
65], and uncertainty quantification [7]. However, despite the surge of
interest in metamodeling for solid mechanics problems [44], most work has
focused on predicting single Quantities of Interest (QoI) from descriptions of
a material domain [80, 20, 39]. In this work, we will address the more
challenging problem of designing metamodels to predict full field QoI such as
full field displacement, and full field damage [18, 19]. Notably, this has
been a recently growing area of interest in the solid mechanics research field
[15, 63, 81]. And, designing effective methods for full field QoI prediction
is compelling because these methods: (1) leverage data that is typically
already being generated by running full fidelity simulations for simulating
single QoI problems, and (2) will lead to much richer predictions than single
QoIs.

In this work, our goal is to increase the impact of machine learning based
metamodels in solid mechanics by designing methods for full field QoI
prediction. Full field QoI prediction exhibit some similarities with image-to-
image translation tasks from the computer vision research field [30]. For
example, image segmentation [9] and optical flow analysis [14] are both
computer vision tasks that can require full field predictions. Despite the
proliferation of methods for full field image analysis, it is only recently
that researchers have begun adapting these methods for problems in mechanics
[49]. And, unlike in the computer vision field [68], there is a severe lack of
open source datasets that can be used to explore problems in solid mechanics
[61, 53]. Therefore, the focus of this work is twofold: (1) on advancing full
field QoI prediction in solid mechanics, and (2) on addressing the problem of
lack of benchmark data for machine learning approaches to solid mechanics
problems [13] by curating and disseminating open source datasets for full
field QoI prediction. In recent previous work, we introduced the Mechanical
MNIST dataset collection [35, 37, 38, 36, 41], a series of open-source
datasets, that includes Finite Element simulation results of applying a fixed
set of loading and boundary conditions (e.g., uniaxial extension and
equibiaxial extension) to a two-dimensional heterogeneous material domain. In
this work, we extend the Mechanical MNIST dataset collection by adding
“Mechanical MNIST Crack Path,” a dataset of Finite Element simulation results
of quasi-static brittle fracture in a heterogeneous material domain captured
via the phase-field method [75, 2]. This significant extension is specifically
designed as an interesting and challenging full field prediction problem
because: 1) Computational models of fracture are typically quite complex and
computationally expensive, and thus metamodeling approaches are a promising
way to gain insight; and 2) Crack path prediction brings unique challenges
that computer scientists and the machine learning community do not inherently
address such as the requirement for high performance prediction of a small
region of the image (i.e., the crack) that is not present in the original
image. With our open source Mechanical MNIST dataset collection, we tackle the
problem of full-field QoI prediction by proposing a Convolution-based deep
learning framework, the “MultiRes-WNet.” This work presents not only a
metamodeling approach that other researchers can readily implement, but also a
framework for others to build directly on this work.

The remainder of the paper is organized as follows. In Section 2, we first
briefly discuss the previously published Mechanical MNIST and Mechanical MNIST
Fashion datasets, and then introduce our new dataset, Mechanical MNIST Crack
Path. Notably, our new dataset has two versions, a “lite” version designed for
accessibility [51], and an “extended” version designed so that other
researchers can build on our work in a less constrained manner [52]. In
Section 2, we also present the network architecture of the MultiRes-WNet,
along with additional details of our metamodel implementation. Then, in
Section 3, we present the results of training our proposed metamodeling
framework for both the displacement prediction task on the Mechanical MNIST
and Mechanical MNIST Fashion datasets and the crack path prediction task on
the Mechanical MNIST Crack Path dataset. Finally, in Section 4, we present
concluding remarks and summarize possibilities for future work.

##  2 Methods

As stated in 1, there are two main contributions presented in this paper. In
Section 2.1, we describe our contribution to the “Mechanical MNIST” dataset
collection: “Mechanical MNIST Crack Path.” Then, in Section 2.2, we present
the details of our proposed novel metamodeling approach. We briefly note that
additional information on the phase-field model used for dataset generation is
presented in Appendix B, and additional details of our metamodel
implementation are presented in Appendix C.

###  2.1 Datasets and Data Generation

To test the efficacy of our proposed metamodeling technique, we need datasets
to perform computational experiments. In Section 2.1.1, we briefly describe
the full field QoI problem present in the original Mechanical MNIST dataset
and the recently added Mechanical MNIST Fashion dataset. Then, in Section
2.1.2, we introduce our challenging addition to this collection, the
“Mechanical MNIST Crack Path” dataset. Visualizations of the different input
parameter spaces in the Mechanical MNIST dataset collection are available in
Appendix A.

####  2.1.1 Mechanical MNIST and Mechanical MNIST Fashion

Inspired by the classic MNIST dataset used by the computer vision research
community [33], the original Mechanical MNIST and the Mechanical MNIST Fashion
datasets [35, 37, 38] are benchmark datasets for comparing different
metamodeling strategies on mechanical data. Each dataset contains Finite
Element simulation results where a fixed set of loading and boundary
conditions (e.g., uniaxial extension and equibiaxial extension) are applied to
a two-dimensional heterogeneous material domain. The heterogeneous material
domain (a $28\times 28$ unit square) in both datasets corresponds directly to
bitmaps in the original MNIST and the Fashion MNIST datasets [77]. The domain
is modeled as a compressible Neo-Hookean material, and bitmap pixel intensity
determines the elastic modulus of the corresponding region within the material
domain. Therefore, each material block consists of a soft background matrix
with a significantly stiffer region within. Both datasets contain the total
change in the strain energy, total $x$ and $y$ reaction force, and full-field
$x$ and $y$ displacements sampled on a $28\times 28$ unit grid to match the
size of the input material domain. In previous studies, we have investigated
metamodels that predict single QoI (e.g., change in strain energy). In this
study, we will focus exclusively on the full-field QoI prediction problem
illustrated in Fig. 1a and 1b.

####  2.1.2 Mechanical MNIST Crack Path

Motivated by recent interest in designing machine learning models to predict
material failure [63, 18, 19, 71, 23], we decided to add a dataset to the
Mechanical MNIST collection that captures crack propagation in a heterogeneous
material. In particular, we were interested in designing a dataset where the
goal of full field QoI prediction would be crack path prediction because this
is a substantially different challenge than the full-field displacement
prediction problem present in the original Mechanical MNIST dataset. For our
new dataset, “Mechanical MNIST Crack Path,” we model quasi-static brittle
fracture in a two dimensional heterogeneous material with inclusions whose
positions are dictated by the Fashion MNIST bitmap using the phase-field
method.

As a brief background, the phase-field method is a state-of-the-art approach
to fracture modeling [75, 84, 2, 50]. The core idea of the phase-field method
is to approximate the discontinuity caused by a fracture with a damage field
variable $\phi$. In this work, we follow the unified phase field method
introduced by Wu in 2017 [73]. This approach contains a general form for the
crack density function $\gamma(\phi,\nabla\phi)$ which approximates the crack
by a regularized functional $\Gamma_{d}(\phi)$, and is written as follows:

$\Gamma\approx\Gamma_{d}(\phi):=\int_{\Omega}\gamma(\phi,\nabla\phi)\,\mathrm{d}\textbf{x}=\int_{\Omega}\frac{1}{c_{0}}\left(\frac{1}{l_{0}}\alpha(\phi)+l_{0}|\nabla\phi|^{2}\right)\mathrm{d}\textbf{x}\quad\mathrm{with}\quad
c_{0}=4\int_{0}^{1}\sqrt{\alpha(\phi)}\,\mathrm{d}\phi,$ (1)

where $\alpha(\phi)$ is the geometric crack function and $l_{0}$ is an
internal length scale parameter[73]. We refer to Appendix B for further
details regarding the phase field theory and Finite Element implementation. We
note briefly that, despite the benefits of the phase field method, it is quite
computationally expensive because accurately capturing the crack geometry
typically requires a very fine mesh surrounding the zone where the crack path
could potentially propagate.

In the Mechanical MNIST Crack Path dataset, illustrated in Fig. 1c, we define
the material domain as a square with a side length of $1$. There is an initial
crack of fixed length ($0.25$) on the left edge of the domain. The bottom edge
of the domain is fixed in $x$ (horizontal) and $y$ (vertical), the right edge
of the domain is fixed in $x$ and free in $y$, and the left edge is free in
both $x$ and $y$. The top edge is free in $x$, and in $y$ it is displaced such
that, at each step, the displacement increases linearly from zero at the top
right corner to the maximum displacement on the top left corner. Maximum
displacement starts at $0.0$ and increases to $0.02$ by increments of $0.0001$
($200$ simulation steps in total). The heterogeneous material distribution is
obtained by adding very stiff circular inclusions to the domain using the
Fashion MNIST bitmaps as the reference location for the center of the
inclusions. Specifically, each center point location is generated randomly
inside a square region (a fraction of the size of the domain) defined by the
corresponding Fashion MNIST pixel when the pixel has an intensity value higher
than $10$. In addition, a minimum center-to-center distance limit of $0.0525$
is applied while generating these center points for each sample. The values of
Young’s Modulus $E$, Fracture Toughness $G_{f}$, and Failure Strength $f_{t}$
near each inclusion are increased with respect to the background domain by a
variable rigidity ratio $r$. The background values for $E$, $G_{f}$, and
$f_{t}$ are $210000$, $2.7$, and $2445.42$ respectively. The rigidity ratio
throughout the domain depends on position with respect to all inclusion
centers such that the closer a point is to the inclusion center the higher the
rigidity ratio will be. The rigidity ratio is defined as:

$r(x)=1+\left\\{\begin{array}[]{cr}3&d<d_{min}\\\ \frac{3d_{min}^{2}}{\beta
d_{min}^{2}+(1-\beta)d^{2}}&d\geq d_{min}\end{array}\right.\quad with\quad
d=S_{\alpha}(||x-P_{1}||,\dots,||x-P_{n}||)$ (2)

where $\alpha=-100$, $\beta=0.9$, $d_{min}=0.0075$, and $P_{i}$ is the center
point of inclusion $i$. Function $S_{\alpha}$ is the smooth minimum function,
and is defined as:

$S_{\alpha}(x_{1},\dots,x_{n})=\frac{\sum_{i=1}^{n}x_{i}e^{\alpha
x_{i}}}{\sum_{i=1}^{n}e^{\alpha x_{i}}}\quad where\quad
S_{\alpha}\xrightarrow[]{}\min\;as\;\alpha\xrightarrow[]{}-\infty\,.$ (3)

We note that the full algorithm for constructing the heterogeneous material
property distribution is included in the simulations scripts shared on GitHub,
with access details given in Section 5.

Figure 1: Summary of the Mechanical MNIST datasets: a) and b) show the results
of finite element simulations of uniaxial and equibiaxial extension for sample
materials in the Mechanical MNIST and Mechanical MNIST Fashion datasets
respectively; c) shows a schematic of the domain and boundary conditions
defined for the Mechanical MNIST Crack Path dataset, as well as material
distributions, damage fields and force vs. displacement curves for three
sample cases.  A visualization of the three different input parameter domains
in the Mechanical MNIST dataset collection is available in Appendix A

####  2.1.3 Note on the two versions of the Mechanical MNIST Crack Path
dataset

With this manuscript, we provide two separate versions of the Mechanical MNIST
Crack Path dataset. We have designed a lite version of the dataset, available
through the OpenBU Institutional Repository, such that others can download the
dataset and directly use it to create metamodels without any additional
modification. In addition, a full version of the dataset, available through
the Dryad Repository, contains more information but will require preprocessing
before it can be used to train metamodels. Notably, the full version of the
dataset allows for more flexibility in metamodeling, and is intended to be
used by researchers who are familiar with the underlying mechanical problem.
The OpenBU version of the dataset contains:

  * •

An array corresponding to the rigidity ratio to capture heterogeneous material
distribution reported over a uniform $64\times 64$ grid.

  * •

The binary damage field at the final level of applied displacement reported
over a uniform $256\times 256$ grid.

  * •

The force-displacement curves for each simulation (which are not addressed in
this manuscript).

The larger version of the dataset contains:

  * •

The locations of the center of each inclusion, where the script to
subsequently extract rigidity ratio matrices with a desired resolution is
available on GitHub.

  * •

The displacement and damage fields every ten simulation steps reported over a
uniform $256\times 256$ grid.

  * •

The full mesh resolution displacements and damage fields at both the last step
of the simulation and at the early stage of damage initiation (when the
reaction force is maximum).

  * •

The force-displacement curves for each simulation.

###  2.2 Metamodeling

In this Section, we briefly summarize the metamodel architecture that we
developed for predicting full field QoI for mechanics based problems. In
Section 2.2.1, we introduce the MultiRes-WNet architecture, in Section 2.2.2
we describe our autoencoder based approach to managing the high domain
resolution required for crack path prediction, and in Section 2.2.3 we briefly
note alternative approaches that were (in our experience) less effective than
our final metamodel design. In Appendix C, we provide a quantitative
comparison between our proposed MultiRes-WNet and three representative
alternatives.

####  2.2.1 MultiRes-WNet Network Architecture

The U-Net [59] architecture is a type of Convolutional Neural Network (CNN)
developed mainly for image segmentation tasks [79, 87, 3]. The U-Net has been
widely used in medical image analysis [78, 42, 22]. Furthermore, in recent
years, the U-Net architecture has been adopted in several areas of solid
mechanics research for full-field QoI prediction, such as modeling the non-
linear relation between a contact force and the displacement field in
hyperelastic material domains [49], and full-field stress and strain
predictions in composite geometries [82]. In fluid mechanics, the U-Net
architecture has also been used for velocity and pressure field prediction
[8], optimal mesh generation [27], and correcting numerical error induced by a
coarse-grid simulation of turbulent flows [55]. The U-Net architecture
consists of a downsampling (encoder) section to extract high-level features
from the input image, and an upsampling (decoder) section to construct the
desired output. In a typical implementation, the encoder takes the input array
(image or otherwise), and applies a series of convolutions followed by ReLU
activation functions, followed by max pool operations for each level of
downsampling. The decoder section of the U-Net takes inputs at each level that
are constructed by concatenating the encoder output of the same level with a
transpose convolution of the lower level encoder output (referred to in short
as an “UpConv”). This basic “U” motif can be seen in Fig. 2a.

In this work, after our initial dissatisfaction with the efficacy of the
standard U-Net, we propose a modified version of the U-Net architecture for
predicting full field mechanics based QoI. The key modifications that we
applied to the UNet architecture are summarized as follows: 1) we stack two
UNets in series, 2) we use depthwise separable convolutions, 3) we replace
double convolutions at each layer with a block of serial convolutions with
internal residual paths, and 4) we add several convolutional filters to the
residual path connecting encoder output to the decoder input. Our first type
of major modification the the UNet, the “MultiRes” architecture, was initially
proposed as a MultiRes-UNet [29] with two key modifications to the standard
U-Net architecture: 1) replacing the sequence of regular convolutions by a
multi-resolution convolutional block (MultiRes-Conv-Block), and 2) passing the
encoder output at each level through a convolutional residual path (ResPath)
before concatenation with the upsampled tensor. These modifications are
schematically illustrated in Fig. 2b and Fig. 2c. The addition of the
MultiRes-Conv-Block enables feature extraction at different zoom levels, and
the intuition behind the ResPath modification is that it avoids direct
concatenation of low-level features in the encoder section with rich features
in the decoder section of the network. In addition to the initial major
modification, we replaced the regular convolutions in the MultiRes-Conv-Block
with Depthwise Separable Convolutions [11]. The idea of these is to first
apply separate convolutional filters on each channel, and then apply a point-
wise filter that combines values of a pixel over all channels. Finally,
inspired by [76] and [12], we stack two MultiRes-UNets with depthwise
separable convolutions in series. We found through experiments that doubling
the number of parameters with this approach is more effective than doubling
the number of channels in a single UNet. Because of this addition, we refer to
our network as a "MultiRes-WNet," illustrated in Fig. 2a.

Figure 2: We combined the U-Net with Multi-resolution Convolutional Blocks,
ResPath, and Depthwise Separable Convolutions to form MultiRes-WNet, which
consists of two identical MultiRes-UNet models put together in series. We
schematically illustrate: a) the MultiRes-WNet Network Architecture; b) the
Multi-resolution Convolutional Block that replaces regular convolutions in the
U-Net to enable feature extraction at different zoom levels; and (c) the
ResPath that applies a few convolutions on the input with low-level features
from the encoder to match the output with rich features in the decoder
section.

####  2.2.2 An autoencoder based approach to handling the high output domain
resolution requirements for accurate crack path prediction

For the Mechanical MNIST and MNIST Fashion datasets, the full field QoI
prediction challenge involves mapping the $28\times 28$ material property
input domain to the full-field displacement output on a $28\times 28$ grid
(see Section 2.1.1 for details). For this scenario, the MultiRes-WNet
described in Section 2.2.1 requires no additional modification. However, for
the Mechanical MNIST Crack Path dataset, a $256\times 256$ array is required
for the damage field output domain to sufficiently capture the crack path
geometry. Also, based on the network architecture we use, the resolution of
the material distribution input domain must match the resolution of the output
domain. In this scenario, we run into a potential limitation of the MultiRes-
WNet architecture: an unacceptably high training time. Quadrupling the
resolution of the input images from $64\times 64$ to $256\times 256$ results
in an approximately ten to fourteen times greater training time when using
consistent computational resources.  With the resources presently available to
us (two NVIDIA TESLA V100 GPUs), training time for a $64\times 64$ input
domain is $\approx 1$ day, therefore an order of magnitude increase in
training time would be highly detrimental. In addition, because the output
domain damage variable is equal to zero over most of the problem domain
(everywhere except near the crack), using Mean Squared Error (MSE) loss will
often result in poor prediction performance where the network ultimately
predicts a severely smeared, rather than binary, damage field. To address
these challenges, we combine our MultiRes-WNet architecture with a basic
Convolutional Autoencoder [86] to reduce the size of the damage field arrays
for both faster and more accurate training of the MultiRes-WNet, illustrated
in Fig. 3. Specifically, we first train the convolutional autoencoder on a
subset of the training data to effectively transform a ($256\times 256$)
damage field into a ($64\times 64$) encoded damage field, and then reconstruct
the true ($256\times 256$) damage field with a negligible error, illustrated
in Fig. 3. Once the autoencoder is trained, we first encode the damage field
for the entire training set, and then train the MultiRes-WNet to predict the
encoded damage field. As illustrated in Fig. 3, the MultiRes-WNet will map the
material property distribution to the encoded damage field. Then, we
reconstruct the predicted high-resolution damage field with the previously
trained decoder section of the autoencoder.  This simple to implement
modification provided us with a pragmatic way of dealing with the
aforementioned drawbacks of training a MultiRes-WNet directly on the unencoded
representation of the binary damage field. We anticipate that alternative
approaches could be similarly effective and we make no claim that the use of
an autoencoder is the optimal approach to this problem.

Figure 3: A schematic illustration of the MultiRes-WNet combined with an
autoencoder for crack path prediction. The autoencoder encodes the high-
resolution damage field to a low-resolution array, and the MultiRes-WNet
learns to map the low-resolution material distribution to low-resolution
encoded damage field. The decoder of the trained autoencoder takes the
predicted low-resolution encoded damage field and outputs the predicted high-
resolution damage field.

####  2.2.3 Notes on metamodel design process and alternative approaches
investigated

As stated briefly in 2.2.1, we used the UNet model in our first attempt at
designing an effective metamodel for full-field QoI prediction problems. This
initial exploration focused on displacement field prediction for the
Mechanical MNIST datasets. Specifically, we tested the original UNet and
several modified versions of the UNet: the Channel-Unet [10], the Attention
U-Net [54], the parallel and coupled UNet architectures [8], the W-Net [76],
and multiple combinations of these modifications. Although there is certainly
more room for exploration, we found that the MultiRes-WNet was the best
performing metamodel in our experiments. Once we had settled on the Multires-
Wnet architecture as a promising method for predicting displacement fields, we
tested the MultiRes-WNet on an inherently different task of crack path
prediction using the Mechanical MNIST Crack Path dataset. Although it was
necessary to use an autoencoder to handle the high-resolution data
requirements outlined in 2.2.2, the MultiRes-WNet performed well on the crack
path prediction task. However, despite our promising results, we believe that
metamodeling techniques in this field are still far from perfect, and we
expect future studies to focus on designing metamodels that exceed the current
state of the art performance established in this paper for full-field QoI
prediction on the Mechanical MNIST, Mechanical MNIST Fashion, and Mechanical
MNIST Crack Path benchmark datasets.

##  3 Results and Discussion

###  3.1 Predicting Full Field Displacement for the Mechanical MNIST and
Mechanical MNIST Fashion Datasets

In order to evaluate the performance of our proposed MultiRes-WNet
architecture, illustrated in Fig. 2, we trained the MultiRes-Wnet on all
$60000$ samples provided in the Mechanical MNIST Uniaxial Extension,
Mechanical MNIST Equibiaxial Extension, Mechanical MNIST Fashion Uniaxial
Extension, and Mechanical MNIST Fashion Equibiaxial Extension datasets. As
stated in Section 2.1.1, these datasets map $28\times 28$ material property
distribution arrays to $28\times 28$ $x$ and $y$ displacement fields. For all
metamodel training, we set the learning rate to $10^{-2}$ with a decay rate of
$0.5$ at every $50$ epochs. We set the batch size to $32$, and the number of
channels in the first layer to $64$, which corresponds to approximately $2$
million total metamodel parameters. In addition, we trained each model for
$200$ epochs using Adam optimizer. Throughout this Section, we evaluate each
trained metamodel on all $10000$ available test cases, and report the absolute
error in displacement prediction for each test case averaged over the
$28\times 28$ grid. In addition, we divide the absolute error value by the
mean nodal displacement averaged over all samples in each dataset to obtain a
dimensionless Absolute Percentage Error ($APE$) metric for fair performance
comparison between the different datasets that may have different levels of
applied displacement.

Figure 4: Metamodel performance evaluation for displacement prediction: a)
MAPE vs. training set size for models trained on subsets of the Mechanical
MNIST Fashion Equibiaxial Extension Dataset (with and without data
augmentation) and tested on the Mechanical MNIST Fashion test set; b) Sample
visualizations of the predicted vs. true displacements for test cases with
$APE\approx 1.0\%$; c) Histograms of the frequency of the predicted
displacement field absolute percentage error for Mechanical MNIST Uniaxial
Extension and Equibiaxial Extension test datasets; and d) Mechanical MNIST
Fashion Uniaxial Extension and Equibiaxial Extension test datasets. In c) and
d) the illustrations of the predicted vs. true displacement field are the
worst performing example from each test dataset.

In Fig. 4a, we show the effect of both training set size and data augmentation
on metamodel test performance for the Mechanical MNIST Fashion Equibiaxial
Extension dataset. We note briefly that for both Mechanical MNIST Fashion
datasets, we performed both horizontal and vertical flips (separately and
combined) to augment the size of the initial training set to $4\times 60000$
training points. Because the original Mechanical MNIST datasets were “easier”
(i.e., lower metamodel test error) we did not perform data augmentation for
either of them. We note that the results presented in Fig. 4a show that data
augmentation consistently improves the model performance, and that at least
$10000$ training cases are required for our model to effectively learn the
distribution of the test set. For the metamodels trained with all $60000$
training points (including data augmentation for the Mechanical MNIST Fashion
datasets), metamodel performance is summarized in Table 1. Briefly, we
observed a Mean Absolute Percentage Error (MAPE) of $0.27\%$ and $0.34\%$ for
Uniaxial Extension and $0.50\%$ and $0.80\%$ for Equibiaxial Extension in the
Mechanical MNIST and Mechanical MNIST Fashion datasets, respectively.
Qualitatively, as shown in Fig. 4b, for a single test case with $APE<1\%$, the
true and predicted displacement fields are visually indistinguishable. If
$APE=1\%$ is considered the cutoff for “high quality” predictions, our
metamodels achieve $99.87\%$ and $97.50\%$ of high quality predictions on the
Mechanical MNIST Uniaxial and Equibiaxial Extension datasets and $99.25\%$ and
$78.84\%$ high quality predictions on the Mechanical MNIST Fashion Uniaxial
and Equibiaxial Extension datasets respectively. The distributions of test
error for all four datasets are shown in Fig. 4c and Fig. 4d. In addition to
showing the distribution of test error, we added visualization of the worst
prediction (i.e., highest test error) from each of the four datasets. These
results show that the MultiRes-WNet architecture is quite effective for
predicting full field displacement on all four datasets, and is thus likely a
good choice for related problems in mechanics. In Appendix D.1, we compare
three alternative Neural Network architectures with the MultiRes-WNet, and, as
shown in Table 3, the MultiRes-WNet outperforms all of these alternatives. In
addition, we analyze the performance of the MultiRes-WNet in strain field
prediction in Appendix E. In short, the MultiRes-WNet is an effective approach
for predicting strain fields in addition to displacement fields.

Table 1: MultiRes-WNet performance predicting full field displacement on the
Mechanical MNIST and Mechanical MNIST Fashion datasets. All values reported
are test error.

|  | Mechanical MNIST | Mechanical MNSIT Fashion
---|---|---|---
|  | Uniaxial Extension | Equibiaxial Extension | Uniaxial Extension | Equibiaxial Extension
X Displacements | MAE | 0.0101 | 0.0128 | 0.0132 | 0.0229
Y Displacements | 0.0129 | 0.130 | 0.0149 | 0.0204
Total Displacements | 0.0181 | 0.0205 | 0.0222 | 0.0344
MAPE | 0.27% | 0.50% | 0.34% | 0.80%

Table 2: MultiRes-WNet performance predicting crack path on the Mechanical
MNIST Crack Path. All values reported are for the test.

| Mechanical MNIST Crack Path
---|---
| Continuous Path | Discontinuous Path | All Samples
Number of Samples | 8861 | 1139 | 10000
F1 Score | 0.893 | 0.693 | 0.870
Avg. Wrong Pixel Count (FP+FN) | 147 | 411 | 177

###  3.2 Predicting Full Field Damage for the Mechanical MNIST Crack Path
Dataset

As stated in Section 2.1.2, we created the Mechanical MNIST Crack Path dataset
as an additional challenge for our proposed MultiRes-WNet metamodel. In
addition to the MultRes-WNet, we also used a convolutional autoencoder to
reduce the resolution of the $256\times 256$ array required to resolve the
crack path captured by the phase field model damage field to a $64\times 64$
array (see Section 2.2.2 for details). Therefore, we trained our MultiRes-WNet
to predict a $64\times 64$ encoded array from a $64\times 64$ material
property distribution input array. Before training our MultiRes-WNet, we train
the autoencoder on $10,000$ samples randomly selected from the training set of
the Mechanical MNIST Crack Path dataset. Note that in Appendix D.2, we
evaluate an alternative method for crack prediction based on a 1D
representation of the crack path and found that it was less effective than our
proposed autoencoder approach.

Our trained autoencoder is able to regenerate true damage fields with an
average accuracy of $99.995\%$ (i.e., $2.74$ incorrect pixels out of $65536$
per sample). With the trained autoencoder, we generated encoded damage fields
for $60000$ samples in the training set and all $10000$ samples in the test
set. Then, after we mapped the $64\times 64$ material property distribution
arrays to $64\times 64$ encoded damage fields with our MultiRes-WNet, we used
the trained decoder to generate the $256\times 256$ predicted damage fields.
To evaluate our model, we set a damage field threshold $(\phi>0.5)$ and thus
treat the predicted damage field as a binary image, in which pixel value $=1$
denotes the crack path (i.e., the representation of the crack path).

We train our MultiRes-WNet on all $60000$ training samples provided in the
Mechanical MNIST Crack Path dataset. We set the learning rate to $10^{-2}$
with a decay rate of $0.5$ at every $25$ epochs and trained the model for
$100$ epochs in total using Adam optimizer. We set the batch size to be $32$
and the number of channels in the first layer to be $64$. We evaluated the
trained metamodel on the $10000$ test cases provided in the dataset. To
quantitatively evaluate the model performance, we used the Sørensen–Dice
index, referred to as the $F_{1}$ score and defined as:

$F_{1}=\frac{2TP}{2TP+FP+FN},$ (4)

where $TP$, $FP$, and $FN$ are the total number of true positive, false
positive and false negative pixels in a predicted sample. We note that the
number of true negative pixels does not affect this score which is desirable
because the number of true negative pixels is two orders of magnitude larger
than $TP$, $FP$, and $FN$ pixels combined.

Figure 5:  Metamodel performance evaluation on the Mechanical MNIST Crack Path
dataset. The histograms show the frequency of test set results with respect to
$F_{1}$ score interval (see Eqn. 4). The predicted samples are grouped into
continuous (dark blue) and discontinuous (light blue) paths. To contextualize
these scores, we show: a) Three discontinuous paths with different scores
(referred to as “Incorrect Predictions”); b) Three continuous paths with
different scores (referred to as “Plausible Alternative Paths”); and c) A
sample continuous path with a relatively high F1 score (referred to as a
“Correct Prediction”). To contextualize model performance, we consider a
continuous paths with a F1 score higher than $0.85$ a “Correct Prediction” for
this dataset. d) Three samples of correctly predicted crack paths with $F_{1}$
scores of $0.85$, $0.90$, and $0.95$.

As illustrated in Fig. 5, we categorize the predicted crack path patterns into
continuous and discontinuous paths. Notably, for the Mechanical MNIST Crack
Path dataset, all predicted crack paths should be continuous. In Fig. 5, we
show histograms of the $F_{1}$ score of the predicted damage field for the
test data separated by category. To help qualitatively interpret these
results, we consider damage field predictions as “Correct” if the predicted
path is continuous, and if the corresponding $F_{1}$ score is above a
threshold of $0.85$. We show an example of “Correct” behavior in Fig. 5c. For
continuous predictions with a low $F_{1}$ score (below $0.85$), we observed
that the crack path tends to follow a “Plausible Alternative Path,” where the
crack propagates around inclusions in an alternative manner to the true
behavior. We show three examples of this behavior in Fig. 5b.  In Appendix F,
we discuss this qualitative categorization in more detail and show the
sensitivity of our results to the chosen “Plausible Alternative Path”
threshold. Notably, a possible explanation for the existence of plausible
alternative paths is the information loss due to the low-resolution
representation of the material domain. If the predicted path is discontinuous,
we describe it as “Incorrect” regardless of the $F_{1}$ score. We show three
examples of this behavior in Fig. 5a. Our trained model achieves a mean
$F_{1}$ score of $0.893$ on the continuous path group, a $0.693$ on the
discontinuous path group, and a $0.870$ overall. Qualitatively, we consider
$78.77\%$ of the predictions as “Correct,” $9.84\%$ as “Plausible Alternative
Paths,” and $11.39\%$ as “Incorrect.” Table 2 summarizes prediction results on
the Mechanical MNIST Crack Path dataset. In contrast to excellent recent work
on Physics-informed Neural Networks (PINNs) developed for fracture prediction
tasks [18, 19], our approach has the benefit of achieving accurate predictions
on heterogeneous materials with varying material and micro-structural
distributions. Looking forward, we anticipate that the evaluation framework
presented in this Section could be used to compare our proposed MultiRes-WNet
to related alternative approaches [63, 26]. Specifically, we hope that our
open source benchmark dataset combined with our well-defined quantitative and
qualitative score metrics will enable direct comparison between our approach
and the approaches developed by others.

##  4 Conclusion

In this paper, our goal was to advance the state of the art in predicting
mechanically driven full field quantities of interest in a supervised learning
framework. To this end, we propose a MultiRes-WNet, a novel type of
Convolutional Neural Network, for mechanics-based full field quantity of
interest prediction. We first showcase the capabilities of the MultiRes-WNet
in accurately predicting full-field displacements in simulations of
heterogeneous hyperelastic material domains from the previously created
Mechanical MNIST [36] and Mechanical MNIST Fashion [38] datasets. After this
initial success, we realized that it was necessary to introduce a more
challenging dataset to evaluate our proposed method. Therefore, we created the
Mechanical MNIST Crack Path dataset [51] through Finite Element simulations of
a heterogeneous material domain undergoing quasi-static brittle fracture
captured through the phase-field method. This significant extension to the
Mechanical MNIST dataset collection [36] allows us to study a different type
of full-field QoI prediction – full field crack path prediction. With this
dataset, we show that by combining the MultiRes-WNet with an autoencoder, the
MultiRes-Wnet is capable of making reasonably accurate crack path predictions
on the Mechanical MNIST Crack Path dataset. Critically, we also introduce both
an interpretable approach to reporting error on Mechanical MNIST Crack Path
and an extended version of the dataset [52] that will allow other researchers
to readily evaluate alternative metamodeling strategies against the baseline
performance that we report here.

Despite the remarkable performance of MultiRes-WNet in full-field displacement
prediction, there are multiple potential directions for future work. For
example, future research could explore methods to integrate boundary
conditions as a model input [62], or include information about the underlying
physics of the problem [57]. In addition, there is a significant need for
methods that rely on smaller training set sizes [41]. Furthermore, our results
for the crack path prediction task can likely be improved upon. Critically, we
anticipate that information loss in using low-resolution representations of
the material distribution domain is a major contributor to prediction error.
One promising avenue for future work would involve developing alternative
methods that would function well with higher resolution input domains. In
addition, future studies should explore alternative data curation and
metamodeling strategies for crack path prediction that represent the crack
path as a one dimensional entity [24] rather than as a two dimensional damage
field. In the extended version of the Mechanical MNIST Crack Path dataset [52]
we provide $256\times 256$ resolution information of displacements and damage
fields for 20 steps throughout the simulation and full mesh resolution of
displacements and damage fields at early stage of damage initiation and at the
final displacement such that other researchers can explore these potential
alternatives and make a direct comparison to the work presented in this paper.
We also briefly note that there are multiple additional supervised and
unsupervised learning problems beyond the scope of this paper that could be
formulated with these datasets. Although this work emphasizes the
effectiveness of the MultiRes-WNet in the end-to-end mapping of a
heterogeneous material to desired full-field QoIs, metamodeling in this area
is still far from perfect. Therefore, we anticipate that by publishing both
our dataset and our metamodel implementation under open-source licenses we
will enable future research interested in mechanics-based metamodeling to
exceed the baseline performance established here.  And, perhaps even more
critically, we hope that there will soon be many more open access mechanics
based datasets available to further challenge the efficacy of our MultiRes-
WNet and inspire new methodological approaches.

##  5 Additional Information

The lite version of the Mechanical MNIST Crack Path Dataset used in this
manuscript is available through the OpenBU Institutional Repository at
https://open.bu.edu/handle/2144/42757 [51]. The full version of the dataset is
available through the Dryad Digital Repository
https://doi.org/10.5061/dryad.rv15dv486 [52]. The Mechanical MNIST and
Mechanical MNIST Fashion Datasets are also available through the OpenBU
Institutional Repository https://open.bu.edu/handle/2144/39371. All code used
to generate and process Mechanical MNIST Crack Path is available on GitHub at
https://github.com/saeedmhz/Mechanical-MNIST-Crack-Path. All code used to
construct the MultiRes-WNet is also available on GitHub
https://github.com/saeedmhz/MultiRes-WNet.

##  6 Declaration of competing interest

The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.

##  7 Acknowledgements

We would like to thank the staff of the Boston University Research Computing
Services and the OpenBU Institutional Repository (in particular Eleni Castro)
for their invaluable assistance with generating and disseminating Mechanical
MNIST Crack Path. This work was made possible through start up funds from the
Boston University Department of Mechanical Engineering, funds from the
Division of Systems Engineering, the David R. Dalton Career Development
Professorship, and the Hariri Institute Junior Faculty Fellowship.

##  Appendix A Visualization of the Mechanical MNIST Collection Input
Parameter Spaces

In this Section, we provide supplementary information on the different input
patterns in the Mechanical MNIST dataset collection. At present, there are
three different input parameter distributions within the collection: (1) the
MNIST handwritten digits [33]; (2) the Fashion MNIST clothing images [77]; and
(3) the Mechanical MNIST Crack Path pattern which contains inclusions with
center positions that are a random function of the Fashion MNIST bitmap
pattern (see eqn. 2). To better visualize how these input parameter spaces are
structured, we use the Uniform Manifold Approximation and Projection (UMAP)
dimension reduction technique to project these high dimensional spaces into
two dimensions [48]. All three input parameter spaces are shown in Fig. 6.
From this visualization, we can see that the MNIST, Fashion MNIST, and MNIST
Crack Path datasets have distinctly organized input parameter spaces. Notably,
the MNIST dataset is organized into $10$ quite distinct clusters corresponding
to the $10$ classes in the original dataset [33], while the Fashion MNIST and
MNIST Crack Path datasets are not, despite also being based on a dataset with
$10$ classes [77]. Looking forward, it will be important to further evaluate
the predictive ability of the MultiRes-WNet for datasets with alternative
material property distribution input spaces. For example, it would be broadly
useful to the mechanics research community to develop multiple datasets that
each address specific challenges in architected material design beyond the
scope of the current Mechanical MNIST dataset.

Figure 6:  Visualization of the input parameter space for the Mechanical MNIST
dataset [36], the Mechanical MNIST Fashion dataset [38], and the Mechanical
MNIST Crack Path dataset [51] using the UMAP algorithm with default parameters
(n_neighbors=15, min_dist=0.1, n_components=2, metric="euclidean") [48].

##  Appendix B Details of Phase Field Model Implementation

In this Section, we summarize the process of data generation through Finite
Element simulations of phase field fracture. In Section B.0.1, we briefly
introduce the phase-field method for quasi-static brittle fracture in general
form. In Section B.0.2, we provide details of the specific form of the phase
field method that we use in this work.

####  B.0.1 Phase Field Modeling of Quasi-static Brittle Fracture

The core idea of the phase-field method in fracture modeling [75] is to
approximate the discontinuity caused by fracture with a damage field variable.
In Fig. 7a, we show a domain $\Omega\subset\mathbb{R}^{n}$ $(n=1,2,3)$ with a
crack occupying the region defined by $\Gamma\subset\mathbb{R}^{n-1}$ in which
$\partial\Omega_{u}$ and $\partial\Omega_{t}$ define the Dirichlet and Neumann
boundary conditions respectively. In Fig. 7a, we show an approximation of the
crack via a continuous damage parameter $\phi$ where $\phi=1$ represents the
crack and $\phi=0$ denotes the undamaged region. In Fig. 7b, we show a one-
dimensional illustration of three common choices for $\phi(\textbf{x})$.
Following the paradigm established by Miehe et al. [50], Wu [73] introduced a
unified method to approximate the crack by a regularized functional
$\Gamma_{d}(\phi)$, Equ. 5, for quasi-static brittle fracture using the crack
surface density function $\gamma(\phi,\nabla\phi)$:

$\Gamma\approx\Gamma_{d}(\phi):=\int_{\mathcal{B}}\gamma(\phi,\nabla\phi)\;\mathrm{d}\textbf{x}=\int_{\mathcal{B}}\frac{1}{c_{0}}\left(\frac{1}{l_{0}}\alpha(\phi)+l_{0}|\nabla\phi|^{2}\right)\mathrm{d}\textbf{x}\quad
with\quad c_{0}=4\int_{0}^{1}\sqrt{\alpha(\phi)}d\phi,$ (5)

where $\alpha(\phi)$ is the geometric crack function and $l_{0}$ is an
internal length scale parameter, and $\mathcal{B}\subset\Omega$ is the
localization band over which the crack is smeared.

Figure 7: Phase-field fracture modeling: a) shows a domain
$\Omega\subset\mathbb{R}^{d}$ $(d=1,2,3)$ with a fracture occupying the region
defined by $\Gamma\subset\mathbb{R}^{d-1}$ on the right in which
$\partial\Omega_{u}$ and $\partial\Omega_{t}$ determine the Dirichlet and
Neumann boundary conditions respectively and an approximation of the fracture
with a continuous damage parameter $\phi$ where $\phi=1$ represents the crack
and $\phi=0$ denotes the undamaged region on the left; b) one dimensional
plots of three common choices for $\phi(x)$; c) force vs displacement curve
for the problem domain of the Mechanical MNIST Crack Path Fig. 1c for a fixed
$l_{0}=0.020$ and different $\frac{l_{0}}{h}$ ($l_{0}$ is the length scale
parameter and $h$ is the minimum mesh size); $\frac{l_{0}}{h}\geq 5$ produces
reliable simulation results; d) force vs displacement curve for the problem
domain of the Mechanical MNIST Crack Path Fig. 1c for a fixed
$\frac{l_{0}}{h}=5$ and different length scale parameters. Note that for these
illustrative plots, we simulate a homogeneous domain. In the Mechanical MNIST
Crack Path dataset, $l_{0}=0.015$ and $h=0.003$ are chosen for generating the
dataset.

The weak form of the governing equations for quasi-static brittle fracture,
where we must find $\mathbf{u}\in\mathcal{U}_{u}$ and
$\phi\in\mathcal{U}_{\phi}$, can be written as follows:

$\left\\{\begin{array}[]{lr}\int_{\Omega}\left(\bm{\sigma}:\nabla_{sym}\delta\mathbf{u}\right)\mathrm{d}\mathbf{x}=\delta\mathcal{P}&\forall\delta\mathbf{u}\in\mathcal{V}_{u}\\\
&\\\
\int_{\mathcal{B}}\left(-Y\delta\phi+G_{f}\delta_{\phi}\gamma\right)\mathrm{d}\mathbf{x}=0&\forall\delta\phi\in\mathcal{V}_{\phi}\end{array}\right.$
(6)

where $\delta\mathcal{P}$ denotes the power of the external body and surface
forces, $G_{f}$ is the fracture toughness, and the stress tensor $\bm{\sigma}$
and the crack driving force $Y$ are defined as follows:

$\left\\{\begin{array}[]{l}\bm{\sigma}:=\frac{\partial\psi}{\partial\bm{\epsilon}}=\omega(\phi)\frac{\partial\psi_{0}}{\partial\bm{\epsilon}}\\\
\\\
Y:=-\frac{\partial\psi}{\partial\phi}=-\omega^{\prime}(\phi)\frac{\partial\psi}{\partial\omega}=-\omega^{\prime}(\phi)\bar{Y}\end{array}\right.\quad\mathrm{with}\quad\psi_{0}(\bm{\epsilon})=\frac{1}{2}\bm{\epsilon}:\mathbb{E}_{0}:\bm{\epsilon}$
(7)

where $\psi_{0}(\bm{\epsilon})$, $\psi(\bm{\epsilon},\phi)$, and
$\omega(\phi)$ are the initial strain energy function, the free energy density
function, and the energetic degradation function respectively, and
$\mathbb{E}_{0}$ is the elasticity tensor. Also, test and trial spaces are
defined as follows:

$\begin{array}[]{ll}\mathcal{U}_{u}:=\Big{\\{}\mathbf{u}\;|\;\mathbf{u}(\mathbf{x})=\mathbf{u}^{*},\;\forall\mathbf{x}\in\partial\Omega_{u}\Big{\\}},&\mathcal{V}_{u}:=\Big{\\{}\delta\mathbf{u}\;|\;\delta\mathbf{u}=\mathbf{0},\;\forall\mathbf{x}\in\partial\Omega_{u}\Big{\\}}\\\
&\\\
\mathcal{U}_{\phi}:=\Big{\\{}\phi\;|\;\phi(\mathbf{x})\in[0,1],\;\dot{\phi}(\mathbf{x})\geq
0,\;\forall\mathbf{x}\in\mathcal{B}\Big{\\}},&\mathcal{V}_{\phi}:=\Big{\\{}\delta\phi\;|\;\delta\phi(\mathbf{x})\geq
0,\;\forall\mathbf{x}\in\mathcal{B}\Big{\\}}\end{array}$ (8)

Finally, we note that following [73] we consider the following generic
expressions for $\alpha(\phi)$ and $\omega(\phi)$:

$\begin{array}[]{c}\alpha(\phi)=\xi\phi+(1-\xi)\phi^{2}\\\ \\\
\omega(\phi)=\frac{(1-\phi)^{2}}{(1-\phi)^{2}+a_{1}\phi(1+a_{2}\phi)},\end{array}$
(9)

where $\xi$, $a_{1}$, and $a_{2}$ are constants.

Figure 8: The figure shows detailed architecture of MultiRes-UNet, Multi-Res-
Conv-Block, and ResPath. Padding for each convolution is chosen such that the
input and output arrays for that convolution have the same dimensions. D-Conv
is used to refer to depthwise convolution. In Multi-Res-Conv-Block*, depthwise
convolutions are replaced by regular convolutios.

####  B.0.2 Details of the Phase Field Method Formulation

The model introduced in Section B.0.1 by the crack driving force defined in
eqn. 7 results in an isotropic phase-field damage model with identical tensile
and compressive fracture behaviour. To address this issue, we follow the
hybrid model described in [2, 73]. As proposed in [74], we replace the crack
driving force formulation in eqn. 7 by:

$Y:=-\frac{\partial\bar{\psi}}{\partial\phi}=-\omega^{\prime}(\phi)\bar{Y}\quad\mathrm{with}\quad\bar{\psi}(\bm{\epsilon},\phi)=\omega(\phi)\bar{\psi}_{0}(\bm{\epsilon})=\omega(\phi)\left(\frac{1}{2E_{0}}\left(max\\{0,\bar{\sigma}_{1}\\}\right)^{2}\right)$
(10)

where $E_{0}$ is the Young’s Modulus and $\bar{\sigma}_{1}$ is the major
principal stress. Furthermore, to ensure crack irreversibility, we use the
local history filed $\mathcal{H}(\mathbf{x},t)$ proposed in [50]:

$\bar{Y}=\mathcal{H}(\mathbf{x},t):=\max_{t^{\prime}}\bar{\psi}_{0}(\bm{\epsilon},t^{\prime})\,.$
(11)

Finally, we follow the optimal characteristic functions for length scale
independent phase-field model proposed in [74] and choose the following values
for parameters $\xi$, $a_{1}$ and $a_{2}$:

$\left\\{\xi=2,\;a_{1}=\frac{4l_{ch}}{c_{0}l_{0}},\;a_{2}=-\frac{1}{2}\right\\}\quad\mathrm{with}\quad
c_{0}=\pi\quad\mathrm{and}\quad l_{ch}=\frac{E_{0}G_{f}}{f_{t}^{2}}$ (12)

where $f_{t}$ is the failure strength. We note briefly that our full FEniCS
[1, 45] finite element implementation is available on GitHub, with a link
provided in Section 5.

##  Appendix C Details of our MultiRes-WNet implementation

To supplement the description of our MultiRes-WNet in Sections 2.2 and 3, we
have included additional detail on our implementation of the MultiRes-WNet
network in Fig. 8. Critically, note that the padding value for each
convolution is such that the input and output arrays for that convolution have
the same dimensions. In addition to this schematic illustration, please note
that we have uploaded our code to GitHub under an open source license, see
Section 5 for access information.

##  Appendix D Comparing the MultiRes-WNet to Alternative Model Architectures

In this Section, we report a summary of our investigation comparing the
performance of the MultiRes-WNet to alternative model architectures commonly
used in the field. Essentially, we show the performance of other approaches
that were not designed specifically for these mechanics-based datasets to make
the relative benefits of the MultiRes-WNet clearer. In Section D.1, we compare
the performance of four different neural network structures including
MultiRes-WNet for displacement prediction. Then, in Section D.2, we summarize
the results of an alternative approach for crack prediction based on a 1D
representation of the crack path and a feed forward neural network
architecture.

###  D.1 Full-field Displacement Prediction Comparison

For this comparison, we trained and tested all machine learning models on the
Mechanical MNIST Fashion Equibiaxial Extension dataset, as it is the most
challenging displacement-based dataset in our collection. The four Neural
Networks trained for direct comparison are as follows:

  * •

FNN Feed-Forward Neural Network ($3$ convolutional, $2$ fully connected
layers) with $5,832,240$ parameters

  * •

UNet ($2$-level) with $1,863,489$ parameters

  * •

WNet ($2$-level) with $3,726,978$ parameters

  * •

MultiRes-WNet (see Section 2.2.1 for details) with $1,941,594$ parameters

All four networks were trained using the Adam Optimizer for $200$ epochs with
a learning rate of $10^{-2}$ for the first $100$ epochs and $10^{-3}$ for the
last $100$ epochs. In contrast to the Mechanical MNIST Fashion results
presented in Fig. 4d, we did not use data augmentation for the results
reported in this Section. The results of the investigation are reported in
Fig. 10 and Fig. 9. As Fig. 9 shows, for this dataset, the performance of the
MultiRes-WNet is superior to all other alternatives considered for prediction
in both the “x” and “y” directions. However, it is worth pointing out that the
WNet, which has twice as many parameters as MultiRes-WNet, is the second best
model and performs only slightly worse than MultiRes-WNet. Notably, a standard
$UNet$ architecture has quite poor performance on these data, as shown in both
the training/test error with respect to number of epochs curves in Fig. 9, and
the histogram of test error with worst case sample prediction visualizations
in Fig. 10. Summarized performance evaluation metrics are reported in Table 3
for a clear quantitative comparison of the different models. All code used to
construct these models and reproduce our results is available on GitHub
https://github.com/saeedmhz/MultiRes-WNet.

Figure 9:  Training (left) and test (right) error comparison for the FNN,
UNet, WNet, and MultiRes-WNet models trained on the Mechanical MNIST Fashion -
Equibiaxial Extension dataset for full-field displacement prediction in both
the X (top) and Y (bottom) directions.  Table 3:  Performance comparison
between the MultiRes-WNet and the three alternative Neural Networks
investigated in this Section. All values reported correspond to test error.
Please note that in contrast to the MultiRes-WNet results reported in Table 1,
these are results without data augmentation.

| Mechanical MNIST Fashion - Equibiaxial Extension
---|---
| MSE X Displacement | MSE Y Displacement | MAE | MAPE
FNN | 0.00629 | 0.00750 | 0.08657 | 2.027
UNet | 0.02438 | 0.11873 | 0.19251 | 4.507
WNet | 0.00279 | 0.00233 | 0.05035 | 1.179
MultiRes-WNet | 0.00270 | 0.00194 | 0.04867 | 1.139

Figure 10:  Test error histograms for the FNN, UNet, WNet, and MultiRes-WNet
models trained on the Mechanical MNIST Fashion - Equibiaxial Extension
dataset. Each histogram is annotated with a visualization of the four worst
predictions in the test set. Average test performance is summarized in Table
3.

###  D.2 Crack Path Prediction Comparison

For the results reported in Section 3.1, we used our MultiRes-WNet to predict
the two dimensional damage field in the Mechanical MNIST Crack Path dataset
[51]. The mean $F_{1}$ score with this approach was 0.87, and, qualitatively,
we determined that 78.77 % of predictions were “Correct” (defined as
predicting a continuous path with an $F_{1}$ score greater than $0.85$). To
give these results some additional context, we also explored an alternative
crack path prediction approach that relies on a simpler method for crack
representation (1D vector vs. 2D autoencoder) and a standard feed-forward
neural architecture rather than the MultiRes-WNet. To achieve this, we first
preprocessed the outputs of our Mechanical MNIST Crack Path dataset (2D damage
fields) to obtain vectors that approximate 1D crack paths with the crack path
$y$ coordinates sampled uniformly over the width of the domain. We note that
for our Mechanical MNIST crack path dataset, the crack can be reasonably
represented through this approach. However, implementing a similar naive
approach would be nontrivial in more complex scenarios such as crack branching
or multiple cracks. Then, we trained a feed-forward neural network (FNN)
similar to the VGG-Net [64] (a set of CNNs for feature extraction followed by
fully connected layers with $3,535,264$ parameters in total) on this processed
dataset. We interpret the performance of this trained FNN via the two
histograms illustrated in Fig. 11. In Fig. 11, the left histogram shows the
$l_{2}$ distance between the predicted and true vectors representing the crack
path in the test set. The right histogram is obtained by reconstructing the 2D
damage fields from both the predicted and true 1D crack paths in the test set,
and calculating the corresponding $F_{1}$ score (see eqn. 4). We briefly note
that for 2D damage field reconstruction, we set the value of each pixel in the
$256\times 256$ domain to $1$ if the 1D crack curve crosses that pixel, and
then offset the crack path obtained through this process by one pixel upward
and downward in “y” direction while the rest of the pixels are zero. We chose
to present the test error in two ways to better facilitate a fair comparison
between the FNN vector prediction approach and our MultiRes-WNet damage field
prediction. We also annotated each histograms with best, worst, and average
predictions to aid in qualitative comparison. Comparing this alternative
approach with our proposed method, we note that the predicted paths using this
alternative approach are overly smooth, and even in the best cases predictions
they fail to follow the complex irregularities that naturally exist in our
crack path dataset. In addition, though it is difficult to perform a fair
comparison between our proposed approach and this straightforward alternative,
we note that our proposed approach achieves a much higher average $F_{1}$
score than the alternative approach ($0.87$ vs. $0.49$), and overall a much
better predictive ability is seen in the representative examples in Fig. 5 and
Fig. 11. Since it would also be non-trivial to predict either crack
bifurcation or multiple cracks using the vector approximation approach, we
believe that our proposed approach will be more broadly applicable to others
in the future. All code used to construct this alternative model and reproduce
our results is available on GitHub https://github.com/saeedmhz/MultiRes-WNet.

Figure 11:  Test error histograms for crack path prediction using the 1D
vectorized representation of the crack path and a FNN model architecture. The
left histogram measures the $l_{2}$ error based on true and predicted vector
representations of the crack and the right histogram measures the $F_{1}$
error based on a reconstruction of the 2D damage field from both the true and
predicted crack path vectors. The histograms are annotated with visualizations
of the true and predicted crack paths for best, worst, and average
predictions.

##  Appendix E Predicting Full Field Strain with the MultiRes-WNet

The results presented in Section 3.1 show that the MultiRes-WNet is able to
successfully predict displacement fields with high accuracy on the Mechanical
MNIST [36] and the Mechanical MNIST Fashion [38] datasets. To investigate the
capability of the MultiRes-WNet in predicting strain fields in addition to
displacement fields, we have extended our Mechanical MNIST Fashion dataset
[38] to contain strain data. Specifically, these datasets now contain the
deformation gradient tensors for each example sampled over the same uniform
$28\times 28$ grid as the displacement field. To demonstrate the efficacy of
the MultiRes-WNet at predicting strain fields, we trained the MultiRes-WNet to
predict the first component of the deformation gradient tensor
($\mathbf{F}_{11}$). For additional comparison, we also calculated
$\mathbf{F}_{11}$ by differentiating both the true and predicted displacement
fields sampled on the uniform $28\times 28$ grid and compared the results of
these three different versions of calculated $\mathbf{F}_{11}$ with the ground
truth values of $\mathbf{F}_{11}$ obtained directly from the Finite Element
simulations. Briefly, we used the CubicSpline(x,y,axis) function from the
scipy.interpolate module to interpolate the displacement field from the given
$28\times 28$ displacement array sampled along the $x$ axis (axis=0) and used
CubicSpline.derivative(nu=1) to compute the displacement gradient [66]. We
report the results of our investigations in Fig. 12. The left histogram
compares the test set error when differentiating the true displacement field
to the test set error when differentiating the MultiRes-WNet predicted
displacement field. The right histogram compares the test set error for direct
prediction of $\mathbf{F}_{11}$ to the test set error when differentiating the
MultiRes-WNet predicted displacement field. In all cases, the error is with
respect to the ground truth computed directly from the Finite Element
simulations. Our investigation shows that the test error for $\mathbf{F}_{11}$
prediction is significantly lower when we directly predict $\mathbf{F}_{11}$
compared to when we derive $\mathbf{F}_{11}$ from the predicted ($28\times
28$) displacement field. However, the left histograms in Fig. 12 shows that
this difference predominantly stems from differentiating an interpolation over
a low-resolution displacement field rather than from error in the displacement
predictions themselves. In brief, we found that direct prediction of
$\mathbf{F}_{11}$ using the MultiRes-WNet architecture described in Section
2.2.1 without data augmentation was excellent, with $0.84$ mean percent error.
And, we found that the prediction of $\mathbf{F}_{11}$ obtained from
differentiating the MultiRes-WNet predicted displacement field had similar
error to the prediction of $\mathbf{F}_{11}$ obtained from differentiating the
true displacement field, with $3.93$ and $3.38$ mean percent error
respectively.

Figure 12:  Test error histograms for predicting $\mathbf{F}_{11}$ through
both differentiating the displacement field and direct prediction. The left
histogram compares the test set error when differentiating the true
displacement field to the test set error when differentiating the MultiRes-
WNet predicted displacement field. The right histogram compares the test set
error for direct prediction of $\mathbf{F}_{11}$ to the test set error when
differentiating the MultiRes-WNet predicted displacement field. In all cases,
the error is with respect to the ground truth value of $\mathbf{F}_{11}$
computed directly from the Finite Element simulations.

##  Appendix F Additional Information on the Definition of Qualitative Error
Metrics for Crack Path Prediction

In Section 3.2, in addition to using the $F_{1}$ score to quantitatively
evaluate our model performance, we categorizes predictions into three
qualitative groups: “Correct”; “Plausible Alternative Path”; and “Incorrect.”
Illustrative examples are shown in Fig. 5. To do this, we first labeled all
predictions that lead to a discontinuous path as “Incorrect” predictions.
Then, for the remaining continuous path predictions, we chose a cutoff
threshold for the $F_{1}$ score where any prediction with a $F_{1}$ score less
than the threshold was labeled as a “Plausible Alternative Path.” The
remainder of the predictions with both a continuous path and a $F_{1}$ score
higher than the cutoff threshold were labeled as “Correct” predictions. In
Fig. 13, we provide a plot that shows how sensitive our qualitative results
are to the choice of the cutoff threshold. We selected $0.85$ for the
threshold, and, as shown in Fig. 13, our selection of $0.85$ occurs just
before a steep transition (this is also reflected in the histogram in Fig. 5).
Selection of a $0.85$ threshold was informed by both the curve shown in Fig.
13, and manual inspection of the data (see examples in Fig. 5) which revealed
that $0.85$ was a slightly conservative threshold for isolating crack paths
that adhere to the ground truth and only have slight errors around their
margin (see the zoomed in segments of Fig. 5). With $0.85$ as the $F_{1}$
score cutoff we categorized $78.77$% of the test set predictions as “Correct.”

Figure 13:  Influence of the $F_{1}$ score cutoff in delineating between
“Plausible Alternative Path” and “Correct” test predictions of our MultiRes-
WNet on the Mechanical MNIST Crack Path dataset. In this work, we have
selected $0.85$ as the minimum threshold for “Correct” prediction.

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