# Material-Response-Informed DeepONet and its Application to Polycrystal
Stress-strain Prediction in Crystal Plasticity

Junyan He1<EMAIL_ADDRESS>Deepankar Pal1 Ali Najafi1 Diab Abueidda3 Seid
Koric2,3 Iwona Jasiuk2 1 Ansys Inc., Canonsburg, PA, USA
2 Department of Mechanical Science and Engineering, University of Illinois at
Urbana-Champaign, Urbana, IL, USA
3 National Center for Supercomputing Applications, University of Illinois at
Urbana-Champaign, Urbana, IL, USA

###### Abstract

Crystal plasticity (CP) simulations are a tool for understanding how
microstructure morphology and texture affect mechanical properties and are an
essential component of elucidating the structure-property relations. However,
it can be computationally expensive. Hence, data-driven machine learning
models have been applied to predict the mean-field response of a polycrystal
representative volume element to reduce computation time. In this work, we
proposed a novel Deep Operator Network (DeepONet) architecture for predicting
microstructure stress-strain response. It employs a convolutional neural
network in the trunk to encode the microstructure. To account for different
material properties, boundary conditions, and loading, we proposed using
single crystal stress-strain curves as inputs to the branch network,
furnishing a material-response-informed DeepONet. Using four numerical
examples, we demonstrate that the current DeepONet can be trained on a single
material and loading and then generalized to new conditions via transfer
learning. Results show that using single crystal responses as input
outperforms a similar model using material properties as inputs and overcomes
limitations with changing boundary conditions and temporal resolution. In all
cases, the new model achieved a $R^{2}$ value of above 0.99, and over 95% of
predicted stresses have a relative error of $\leq$ 5%, indicating superior
accuracy. With as few as 20 new data points and under 1min training time, the
trained DeepONet can be fine-tuned to generate accurate predictions on
different materials and loading. Once trained, the prediction speed is almost
$1\times 10^{4}$ times faster the CP simulations. The efficiency and high
generalizability of our DeepONet render it a powerful data-driven surrogate
model for CP simulations in multi-scale analyses.

###### keywords:

Deep Operator Network (DeepONet) , Crystal Plasticity (CP) , Representative
Volume Element (RVE) , Stress-strain Curve , Homogenization ,

*[inlinelist,1]label=(),

##  1 Introduction

It is well-known that the microstructure of polycrystalline materials
significantly influences the material response (Zhao et al., 2021; Keller et
al., 2011). Recent advancements in modern additive manufacturing (AM)
techniques have allowed accurate control of the manufacturing process
parameters, and the microstructure can be tailored (Liu et al., 2022a) to
achieve desired mechanical properties. Effective adjustment of process
parameters to optimize mechanical properties relies on a thorough
understanding of the process-structure-property relations for the AM process.
To elucidate how process parameters such as laser power and scan speed affect
the microstructure, process simulation models were developed to obtain the
thermal history during the print. Cellular automata were used to simulate the
development of microstructure during solidification (Herriott et al., 2019;
Zhang and Zhang, 2019; Zhang and He, 2023). To understand how the
microstructure affects mechanical properties and evolves in subsequent
mechanical loading, prior research has relied on crystal plasticity (CP)
models (Zhang et al., 2007; Bridier et al., 2009; Joshi et al., 2022; He and
Admal, 2021; Cheng and Ghosh, 2015) to capture the local influence on the
grain microstructures. However, CP simulations can be quite computationally
expensive and even prohibitive for large models. Therefore, fast-Fourier-
transform-based CP models have been utilized (Paramatmuni and Kanjarla, 2019;
Cocke et al., 2023), and adaptive remeshing schemes for complex polycrystals
have been used in finite-element (FE) methods (Phung et al., 2021; Phung and
Spear, 2019) to improve CP simulation speeds. In the meantime, outside of
classical computational mechanics, the recent improvement of GPU-accelerated
high-performance computing and machine learning (ML) techniques has led to
significant strides in neural network (NN) applications in various fields,
including plasticity simulations. Physics-informed NNs have been successfully
applied to solve path-dependent plasticity problems (He et al., 2023a;
Abueidda et al., 2021a; Niu et al., 2023). Data-driven NNs have been used to
predict the elastic-plastic stress-strain curves (Liu et al., 2022b; He et
al., 2023b; Roy and Guha, 2023; Gorji et al., 2020; Abueidda et al., 2021b),
and the contours of the plastic strain during plastic loading (He et al.,
2024). Similarly, ML techniques have been applied to CP, such as to predict
the stress distribution within the representative volume element (RVE)
(Ibragimova et al., 2022; Frankel et al., 2020), the mean stress-strain
response of the polycrystal RVE (Bonatti et al., 2022; Martinitz and Hartmann,
2023; Ali et al., 2019; Miyazawa et al., 2019; Bishara and Li, 2023;
Sadeghpour and Nonn, 2022), and the yield locus of CP model with different
grain textures (Fuhg et al., 2022; Nascimento et al., 2023). However, most of
these works rely on simple feed-forward neural networks (FNN) or convolutional
neural networks, and they use the material property values, Euler angles, and
texture of the microstructure as inputs. Furthermore, most models were trained
for a single set of material properties.

Recently, the Deep Operator Network (DeepONet) was pioneered by Lu et al.
(2021a). It is an architecture designed to learn the underlying operator of
the governing equation but not a specific solution, making it more
generalizable to parameterized input functions, boundary conditions, and
varying materials. The classical DeepONet architecture is characterized by a
branch and a trunk network. The branch network encodes the parameterized input
variables/functions, and the trunk network encodes the underlying geometry.
The outputs of the branch and trunk were then combined via a dot product,
producing the final NN output. In its original form, the branch and trunk
networks are FNNs (Lu et al., 2021a). In contrast, others have proposed
different variations of it, such as using a convolutional neural network to
extract image-based inputs (Wu et al., 2024; He et al., 2023c; Oommen et al.,
2022), a recurrent neural network to predict time-dependent fields (He et al.,
2024, 2023d), and adding additional data communication links to improve
performance (Wang et al., 2022). Moreover, DeepONets have been employed to
solve computational mechanics problems in heat transfer (Koric and Abueidda,
2023), elasticity (Goswami et al., 2022a), plasticity (He et al., 2023c) as
well as fracture (Goswami et al., 2022b). Previous studies have also explored
transfer learning on DeepONets to generalize the predictions to unseen inputs
(Goswami et al., 2022a; Xu et al., 2023).

With the proven track record of DeepONet and its excellent generalization
capability, it is of high research interest to apply this novel NN
architecture to polycrystals in CP simulations. Specifically, this current
work seeks to develop a versatile and highly generalizable framework that can
be applied to different material properties, microstructure textures, as well
as loading conditions, and it should be data-efficient without requiring many
CP simulations as training data. This current work also looks beyond using
material properties as direct inputs to the NN and explores a way to input
material response and boundary conditions in a more mechanically meaningful
manner by utilizing the response of single crystals. Since CP simulations are
widely used in multi-scale analyses to generate homogenized stress-strain
curves and yield functions needed by macro-scale models, this work focuses
only on predicting the mean-field stress-strain curves of RVEs instead of
trying to predict the local stress distribution. This paper is organized as
follows: Section 2 provides an overview of the crystal plasticity model,
single crystal responses, as well as the proposed neural network architecture.
Section 3 presents and discusses the performance of the NN model with four
different numerical examples. Section 4 summarizes the outcomes and
limitations and highlights future works.

##  2 Methods

###  2.1 Crystal plasticity model

In this work, a CP model is used to capture the response of polycrystal RVEs.
We adopted a finite deformation formulation, where the multiplicative
decomposition of the deformation gradient $\bm{F}$ is employed (Reina and
Conti, 2014):

$\bm{F}=\frac{\partial\bm{u}}{\partial\bm{X}}+\bm{I}=\bm{F}^{e}\bm{F}^{p},$
(1)

where $\bm{F}^{e}$ and $\bm{F}^{p}$ denote the elastic and plastic deformation
gradients, respectively. Only plasticity due to dislocation slip is
considered; hence the plastic velocity gradient $\bm{L}^{p}$ is defined from
Schmid’s law as (Clayton, 2010):

$\bm{L}^{p}=\dot{\bm{F}^{p}}(\bm{F}^{p})^{-1}=\sum^{N_{ss}}_{\alpha=1}\dot{\gamma}^{\alpha}\bm{P}^{\alpha},$
(2)

where $N_{ss}$ and $\dot{\gamma}$ denote the number of slip systems and slip
rate. For face-centered cubic crystals, the slip rate can be expressed as:

$\dot{\gamma}^{\alpha}=\dot{\gamma}_{0}\,{\rm{exp}}\left\\{\frac{-\Delta
F}{k_{B}T}\left[1-\left(\frac{|\tau^{\alpha}|-g_{a}}{g^{\alpha}-g_{a}}\right)^{p}\right]^{q}\right\\}{\rm{sgn}}(1+{\rm{sgn}}(\tau^{\alpha})),$
(3)

where $\dot{\gamma}_{0}$ is a reference slip rate, $k_{B}$ is the Boltzmann
constant, $T$ is the temperature, $g^{\alpha}$ denotes the slip resistance of
slip system $\alpha$, and sgn denotes the sign function. To model the material
hardening behavior during plastic loading, the slip resistance can evolve
according to a hardening law:

$\dot{g}^{\alpha}=q^{\alpha\beta}h^{\beta}|\dot{\gamma}^{\beta}|,$ (4)

where $q^{\alpha\beta}$ is a matrix relating the self- and latent-hardening of
different slip systems and $h^{\beta}$ is the hardening modulus. Strain
hardening behavior can saturate, and $g^{\alpha}$ is bounded between its
initial value $g_{0}^{\alpha}$ and saturation value $g_{max}^{\alpha}$. Due to
material nonlinearity, Newton-Raphson iterations are employed to iteratively
solve for the current $\bm{F}^{p}$ at each element integration point. Once the
plastic strain is known, the second Piola-Kirchhoff stress can be determined
as:

$\bm{T}^{PK2}=\frac{1}{2}\mathbb{C}(\bm{F}^{eT}\bm{F}^{e}-\bm{I}),$ (5)

where $\mathbb{C}$ denotes the fourth-order elasticity tensor. The elastic
behavior of the grains is assumed to be anisotropic, modeled by a cubic
elastic material model, and can be uniquely defined by three constants
$C_{11}$, $C_{12}$, and $C_{44}$. The Cauchy stress $\bm{\sigma}$ is
calculated from $\bm{T}^{PK2}$ as:

$\bm{\sigma}=\frac{1}{det(\bm{F})}\bm{F}\bm{T}^{PK2}\bm{F}^{T}.$ (6)

In this work, we limit ourselves to oligocrystalline polycrystals under plane-
strain conditions. It is further assumed that all the grain orientations are
in-plane and can be described by a single Euler angle in the range
[-180°,180°].

###  2.2 Single crystal behaviors and material-response-informed DeepONet

By design, a CP model captures the influence of local microstructure (e.g.,
grain size, grain shape, and preferred grain orientations) on the elastic-
plastic material behavior. Therefore, CP simulation results depend not only on
the material properties of the single crystal and boundary conditions but also
on the underlying microstructure, leading to significant variation in
response. In this section, we aim to construct a novel DeepONet architecture
to separate the influence of microstructure from the influence of material
properties and boundary conditions and do so in a mechanically meaningful way.

To motivate the architecture of the DeepONet, first consider the elastic-
plastic response of single crystals. Let a single crystal in the shape of a
unit cube be discretized by _one_ hexahedral finite element. Each single
crystal is a sample point in the full crystal orientation space. We employed
36 different single crystal sample points whose orientations are [4.86°,
9.73°, 14.59°, $\dots$, 175.13°], respectively. Note that only the positive
half of the orientation space is sampled due to cubic material symmetry,
although further simplification is possible using the crystal’s four-fold
symmetry. We also emphasize that this sampling is fixed and does not change
with the microstructures of the different polycrystal RVEs considered in this
work. Each single-crystal model is subject to the same boundary conditions and
loading as the polycrystal RVEs of interest. In this work, we focus on the
volume-averaged mean-field response, defined as:

$\hat{\sigma}=\frac{1}{V_{\Omega}}\int_{\Omega}\sigma
dV=\frac{1}{N^{e}}\sum_{i=1}^{N^{e}}\sigma^{i},$ (7)

where $\,\hat{}\,$ denotes a volume-averaged quantity, $V_{\Omega}$ is the
volume of the domain, and $\sigma$ can be the scalar von-Mises stress or any
stress tensor components. For a uniform mesh, the integral simplifies to a
summation, and $N^{e}$ denotes the number of elements in the mesh. Taking
tension as an example, the tensile stress-strain response of the single
crystals is shown in Fig. 1a, while the global stress-strain response of a
typical RVE is shown in Fig. 1b. Two well-known rules of mixture, the Voigt
(1889) and Reuß (1929) models are shown in Fig. 1b by considering the
individual grains as different constituents in the mixture. Note that although
the individual single crystal response differs drastically from that of the
RVE, the approximations constructed from the two simple rules of mixture
become close to the RVE response, hinting that a more complex rule of mixture
may be determined to refine the approximation. This resemblance of the global
RVE behavior with the two simple approximations is rooted in the fact that the
average strain state between the RVE and the single crystals are similar
(since identical boundary conditions are applied); see Fig. 1c.

(a)

(b)

(c)

Figure 1: LABEL:sub@basis Single crystal responses under tension.
LABEL:sub@basis_avg A typical polycrystal RVE response in tension compared to
the Voigt and Reuss approximations computed from all single crystal responses.
LABEL:sub@avg_strain Mean strain response of a typical polycrystal RVE and
single crystals.

From the observation above, we see that the single crystal responses (each
computed from _one_ finite element) can be manipulated to resemble global RVE
responses (computed from _many_ finite elements). From this perspective, we
can view the single crystal stress-strain curves as ”basis functions” (not
necessarily orthogonal), and the global stress-strain curves of polycrystal
RVEs under similar mean strain states can be approximated by a certain
nonlinear rule of mixture to be determined by a NN. Most importantly, we
emphasize that the single crystal responses contain information for both
material properties _and_ boundary conditions. With that information
implicitly represented in the form of single-crystal stress-strain curves,
capturing the influence of local microstructure is necessary. The DeepONet
architecture is well-suited for this problem structure. A typical DeepONet
contains a trunk network that encodes the geometry of interest and a branch
network that encodes various input functions that affect the network output.
For the case of polycrystal RVE response prediction, the trunk network will
encode the local microstructure measured by grain orientations to capture its
influence, and the branch network will encode the single crystal responses
(i.e., ”basis functions” of the RVE response). Specifically, since
microstructure can be described by a 2D array of grain orientations and the
influence of the neighboring grains matters, a convolution neural network, the
residual U-Net (ResUNet) (Diakogiannis et al., 2020), was used in the trunk.
The trunk ResUNet takes 2D crystal orientation arrays in shape $[64\times
64\times 1]$ and generates its embedding of shape $[64\times 64\times HD]$
where $HD$ denotes the hidden dimension of the DeepONet. The ResUNet used in
this work has 3 encoding residual block levels and 24622 trainable parameters.
We employed a simple feed-forward neural network (FNN) in the branch network
to encode the 36 single-crystal responses in shape $[T\times 36]$, where $T$
denotes the number of time steps in the single crystal stress-strain curves.
The branch network then encodes the single crystal responses into shape
$[T\times HD]$. $HD=16$ was used for all examples in this work. The FNN
contains three layers of neurons: $[72,36,16]$ and has 5884 trainable
parameters. A tensor-product operation combines the encoded outputs from the
branch and trunk networks, followed by spatial averaging (similar to how the
average of the stress at all finite elements yields the mean stress of the
RVE, see Eq. (7)). This can be expressed as:

$\hat{\sigma}_{t}=\frac{1}{64^{2}}\sum^{64}_{i=1}\sum^{64}_{j=1}\sum^{HD}_{k=1}B_{tk}T_{ijk}+\beta,$
(8)

where $\hat{\sigma}_{t}$, $\bm{B}$, $\bm{T}$, and $\beta$ denote the predicted
mean stress at time step $t$, branch output, trunk output, and a trainable
scalar bias, respectively. A schematic of the proposed DeepONet is shown in
Fig. 2.

Figure 2: Schematic of the proposed DeepONet.

All NNs were implemented in the DeepXDE framework (Lu et al., 2021b) with a
TensorFlow backend (Abadi et al., 2015). The Python implementation of ResUNet
was modified from the original code in Palasek (2022). All models were trained
for 80000 epochs with a batch size of 4 and a learning rate of $1\times
10^{-3}$. The Adam optimizer (Kingma and Ba, 2014) was used, and the scaled
mean squared error (MSE) was used as the loss function, which is defined as:

${\rm{MSE}}=\frac{1}{N}\sum^{N}_{i=1}(f_{FE}-f_{Pred})^{2},$ (9)

where $N$, $f_{FE}$, and $f_{Pred}$ denote the number of data points, the FE-
simulated values, and the NN-predicted values, respectively.

The proposed DeepONet can be trained on only _one_ set of material properties
under _one_ loading condition to save training time and the required training
data. This trained network is denoted as the baseline network. Transfer
learning is used with minimal additional training data to extend the
prediction to different material properties and/or different loading
conditions. In addition to new RVE training data for the new material
properties and boundary conditions, the corresponding single-crystal responses
are also required. Since the convolutional layers are generally accepted as
universal (Liu et al., 2021), only the task-specific branch FNN is fine-tuned
during transfer learning. The branch network is fine-tuned for 20000 epochs
with a lower learning rate of $5\times 10^{-4}$.

The most novel proposal in this work is the use of single crystal responses as
a means to carry information on both the material properties and boundary
conditions into the DeepONet instead of directly feeding the material property
values. Koric et al. (2023) previously investigated the use of DeepONet to
capture stress contours for different material properties, where the material
properties (i.e., Young’s modulus, yield stress, and hardening modulus) are
inputs to the branch network (see Figure 1 therein). This idea is adopted here
to construct a baseline DeepONet model for performance comparison with our
proposed architecture. To maintain a fair comparison, we only change the
branch network in the baseline DeepONet, which is now an FNN taking 9
different scalar inputs and embedding them to an output shape of $[T\times
HD]$ with a single layer of neurons. Doing so results in an FNN with 8000
trainable parameters, slightly more than that in the proposed DeepONet. The 9
input parameters consist of 8 material property parameters [$C_{11}$ ,
$C_{12}$ , $C_{44}$ , $h^{\beta}$ , $g_{0}^{\alpha}$ , $g_{max}^{\alpha}$ ,
$\dot{\gamma}_{0}$ , $\Delta F$] as well as 1 parameter on the loading
condition, which is the magnitude of the applied strain. Since only a single
scalar input is used to capture the different levels of applied strain, the
baseline model cannot be extended toward different types of loading (e.g.,
from tension to shear). An identical training procedure was applied to train
this baseline DeepONet. For easier reference, the baseline DeepONet taking
material properties as inputs will be denoted as MP-DeepONet, while the
proposed architecture taking single crystal responses will be called SC-
DeepONet in subsequent discussions.

###  2.3 Training data generation

To generate diverse training data containing different grain morphologies, the
synthetic microstructure generation tool DREAM.3D (Groeber and Jackson, 2014)
was used, and 1000 microstructures were generated with random Euler angles.
Four polycrystal microstructures were selected and displayed in Fig. 3. The
microstructures were represented by a uniform mesh with $64\times 64$ bi-
linear hexahedral elements (Solid 185 in Ansys), and the edge length of the
microstructures was set to 1 $mm$. The 3D finite elements were constrained in
the thickness direction (out of the page, geometry was one element thick) to
create a plane strain condition (the crystal plasticity model was implemented
for 3D solid elements only; hence, the use of 2D plane-strain elements was not
possible). The single crystal was modeled by a single hexahedral finite
element (except for the shear case, where a $10\times 10$ uniform mesh was
used), and identical displacement boundary conditions were applied.

(a)

(b)

(c)

(d)

Figure 3: Four typical synthetic microstructures in the training data set,
contour color is assigned based on Euler angle.

To test the extent of generalizability of the proposed network, especially
towards different material properties and loading, two different materials and
four different loading cases were considered. The first material was 2024-T3
aluminum alloy, and the second material was pure copper. The CP material
parameters for aluminum were calibrated to experimental data provided in the
work of Esmaeili et al. (2015). Selected CP material properties are shown in
Table 1.

Table 1:  Material properties used in the crystal plasticity simulation222Unit in MPa unless otherwise noted | $C_{11}$ | $C_{12}$ | $C_{44}$ | $h^{\beta}$ | $g_{0}^{\alpha}$ | $g_{max}^{\alpha}$ |  $\dot{\gamma}_{0}$ [$s^{-1}$] |  $\Delta F$ [J]
---|---|---|---|---|---|---|---|---
Al | 131038 | 80314 | 30942 | 1750 | 220 | 400 | 1.732$\times 10^{6}$ | 2.5$\times 10^{-19}$
Cu | 166957 | 120900 | 75000 | 187 | 22 | 153 | 1.52$\times 10^{7}$ | 2.85$\times 10^{-19}$

Four different loading conditions and material combinations were considered in
this work:

  1. 1.

Tension in the Y direction, 1% engineering strain, aluminum

  2. 2.

Tension in the Y direction, 0.125% engineering strain, copper

  3. 3.

Shear in the X-Y plane, 2% engineering strain, aluminum

  4. 4.

Sinusoidal fully-reversible cyclic loading for 1.5 cycles, 0.125% maximum
strain, copper

Ansys Mechanical 2023R2 was used to conduct the CP simulations on different
microstructures, material properties, and loading conditions. An in-house
python code was developed to convert the microstructure data in .vtk format
(generated from DREAM.3D) to material assignment commands in Ansys parametric
design language and pass the crystal properties and grain orientation to the
solver. To account for the material behavior nonlinearity and path-dependent
plasticity, all CP simulations were solved using a full Newton solver with
automatic time stepping. The minimum and maximum step sizes used were $2\times
10^{-3}s$ and $1\times 10^{-2}s$, respectively. All external loads were
applied as displacement boundary conditions and were ramped linearly over the
step time (1s for cases 1-3, 3s for case 4). For load cases 1-3, the mean von-
Mises stress $\sigma_{VM}$ of the RVE was stored in 50 uniform time steps, and
it is defined as:

$\sigma_{VM}=\sqrt{\frac{3}{2}\bm{S}:\bm{S}},$ (10)

where $\bm{S}=\bm{\sigma}-\frac{1}{3}tr(\bm{\sigma})\bm{I}$ is the deviatoric
part of the stress tensor. For the cyclic loading case, the axial stress
component $\sigma_{y}$ was stored in 240 time steps. It is best to scale the
input and output data to suitable ranges for better NN training and accuracy.
The input Euler angles were scaled linearly to the range $[0,1]$. At each time
step, the RVE stress response was scaled using a min-max scaler with the data
ranges determined from the corresponding single crystal responses at that time
step.

##  3 Results and discussion

All simulations were conducted with two high-end AMD EPYC 7763 Milan CPU
cores. All NN training and inference were conducted using a single Nvidia A100
GPU card on Delta, an HPC cluster hosted at the National Center for
Supercomputing Applications (NCSA). To evaluate the model performance in the
test set, three quantitative metrics were used. They are the relative error,
mean absolute error, and $R^{2}$ value:

$\displaystyle{\rm{Relative\;error}}=\left|\frac{f_{FE}-f_{Pred}}{f_{FE}}\right|\times
100\%,$ (11)
$\displaystyle{\rm{MAE}}=\frac{1}{N_{T}}\sum_{i=1}^{N_{T}}\left|f_{FE}-f_{Pred}\right|,$
$\displaystyle
R^{2}=1-\frac{\sum_{i=1}^{N_{T}}\left(f_{FE}-f_{Pred}\right)^{2}}{\sum_{i=1}^{N_{T}}\left(f_{FE}-\bar{f}_{FE}\right)^{2}},$

where $f_{FE}$, $f_{Pred}$, $N_{T}$, $\bar{f}_{FE}$ denote the finite element
(FE) simulated field value, NN-predicted field value, number of test cases,
and the mean value of the FE-simulated field values, respectively.

###  3.1 Comparison between single crystal and RVE responses

The idea of using single crystal responses as ”basis functions” to obtain the
RVE responses via certain nonlinear rule of mixture relies on the assumption
that the different RVE responses are bounded within the variation of the
single crystal responses given a similar strain state. To check the validity
of this assumption, we plotted the single crystal responses against all the
polycrystal RVE responses for the respective cases in Fig. 4.

(a)

(b)

(c)

(d)

Figure 4: LABEL:sub@c1 Aluminum under 1% tension. LABEL:sub@c3 Copper under
0.125% tension. LABEL:sub@c2 Aluminum under 2% shear. LABEL:sub@c4 Copper
under cyclic loading.

From the curves shown in Fig. 4, it is evident that there are significant
variations in polycrystal RVE responses for each case due to local
microstructure changes. It is also observed that the stress-strain curves from
different cases drastically differ in shape and magnitude due to changing
material properties and/or loading conditions. However, despite the
significant variations, all polycrystal RVE responses fall entirely within the
variations of the single crystal responses (i.e., black lines in Fig. 4),
therefore showing the validity of our assumption. It is also emphasized that
the single crystals and the RVEs should be subject to identical displacement
BCs that lead to a similar mean strain state, since strain (as a function of
time) was not an direct input to the DeepONet, and only the stress values at
different time steps were fed into the DeepONet. Hence, it is required that
the data points at the same time step share an similar strain state.

###  3.2 DeepONet performance for a single material and loading

Before conducting transfer learning, both DeepONets were first trained on a
set of aluminum data, which was under 1% tensile strain. 1000 simulations were
conducted, and 80% of the data was used in training while 20% was reserved for
testing. To test the repeatability of the results, each model was
independently trained 4 times with randomly generated training/testing data
split. Out of the 4 repetitions, the baseline MP-DeepONet achieved an averaged
relative error of 3.02%, while that of the proposed SC-DeepONet was 1.45%.
Training times for both networks were similar, with an average training time
of 429s and 445s, respectively. The inference time per case was similar for
both models, at around $8\times 10^{-3}$s. On average, running each CP
simulation took 196.7s, making the inference 24586 times faster than the
direct numerical simulation. Detailed performance metrics for the best-
performing run out of the four are shown in Table 2.

Table 2: Performance metrics for the aluminum under 1% tension Model | Relative error [%] | Mean absolute error [MPa] |  $R^{2}$ value
---|---|---|---
MP-DeepONet | 2.61 | 9.97 | 0.978
SC-DeepONet | 1.06 | 3.62 | 0.997

To further compare the distribution of prediction errors of the MP- and SC-
DeepONets, histogram and scatter plots are shown in Fig. 5. Selected stress-
strain curves are shown in Fig. 6. They are ranked by the percentile of mean
relative error in the predictions, and the 0th (best case), 50th (median
case), 85th and 100th (worst case) percentiles are shown for statistical
representation.

(a)

(b)

(c)

Figure 5: LABEL:sub@hist Histogram of absolute stress error. Scatter plot comparing simulated and predicted stresses for: LABEL:sub@mp_corr MP-DeepONet, LABEL:sub@sc_corr SC-DeepONet.  (a) MP-DeepONet, best |  (b) MP-DeepONet, 50th pct |  (c) MP-DeepONet, 85th pct |  (d) MP-DeepONet, worst
---|---|---|---
(e) SC-DeepONet, best |  (f) SC-DeepONet, 50th pct |  (g) SC-DeepONet, 85th pct |  (h) SC-DeepONet, worst
Figure 6: Testing results for the two models at different error percentiles,
for aluminum under 1% tension.

From the results, we observe that the SC-DeepONet could provide a higher
prediction accuracy while keeping a training time similar to that of the MP-
DeepONet. Both networks could efficiently predict the mean stress-strain curve
at a rate 5 orders of magnitude faster than FE simulations. It is also worth
mentioning that the 36 single crystal responses were obtained in merely 5s
(total time), which was only 2.5% the time of one RVE simulation. Note that
the single crystal responses are only generated _once_ for this material and
loading and do not change with the RVE microstructure and the number of
training data points. Therefore, the computational expense associated with
using the SC-DeepONet is negligible. For MP-DeepONet, 85.6% of the time step
predictions have a relative error $\leq 5\%$, while that for SC-DeepONet is
98.1%. The stress-strain plots in Fig. 6 show that, in the best case, both
networks can faithfully predict the mean stress-strain curve of the RVE.
However, SC-DeepONet continues to deliver accurate predictions with minimal
deviation up to that 85th percentile case, while the MP-DeepONet already shows
significant over-prediction. In the worst case, both networks accurately
predicted the elastic response and the hardening slope but under-predicted the
initial yield stress.

###  3.3 Generalization to different materials and strains

In Section 3.2, we demonstrated that different DeepONets could be conditioned
to a single material property/loading state and provide reasonable
predictions. We now investigate how MP- and SC-DeepONets can be extended to
different materials and strain levels. In this section, we considered the case
of copper under 0.125% tension, for which case 400 simulation results were
generated. To investigate the effect of the number of copper data points on
the transfer learning accuracy, five different numbers of data were used in
the fine-tuning: [5, 10, 20, 50, 100]. For repeatability, each training was
repeated 4 times. Training times with 20 new data points were similar for MP-
and SC-DeepONets and were close to 45s. Once trained, the prediction time for
both models remained similar to the baseline case. The average CP simulation
time was 75.5s, making the DeepONet predictions 9436 times faster. Generating
the single crystal responses took merely 4s, which is quick compared to the
time of a single RVE simulation. The transfer learning results with different
data points are shown in Fig. 7a and Fig. 7b. The models fine-tuned using 20
new data points were chosen for further inspection, and the histogram of the
prediction errors is shown in Fig. 7c. Key accuracy metrics are presented in
Table 3. The predicted stress-strain curves from the two models at different
error percentiles are depicted in Fig. 8.

Table 3: Performance metrics for the copper under 0.125% tension Model | Rel. err. (before) [%] | Rel. err. (after) [%] | Mean abs. err. [MPa] | $R^{2}$
---|---|---|---|---
MP-DeepONet | 2.92$\times 10^{6}$ | 3.82 | 1.22 | 0.975
SC-DeepONet | 2.87 | 1.11 | 0.36 | 0.997

(a)

(b)

(c)

Figure 7: Final prediction error for different number of transfer learning data points: LABEL:sub@tr_mp_cu MP-DeepONet and LABEL:sub@tr_sc_cu SC-DeepONet. LABEL:sub@tr_cu_hist Error histogram for both DeepONets after transfer learning with 20 data points. (a) MP-DeepONet, best |  (b) MP-DeepONet, 50th pct |  (c) MP-DeepONet, 85th pct |  (d) MP-DeepONet, worst
---|---|---|---
(e) SC-DeepONet, best |  (f) SC-DeepONet, 50th pct |  (g) SC-DeepONet, 85th pct |  (h) SC-DeepONet, worst
Figure 8: Testing results for the two models at different error percentiles
after transfer learning with 20 data points, for copper under 0.125% tension.

The transfer learning results in Table 3 reveal more differences in the MP-
and SC-DeepONets. We see that for MP-DeepONet, the prediction error on copper
data before any transfer learning is high, indicating that the information the
network learned from the previous aluminum data is not directly generalizable.
However, in the case of SC-DeepONet, with the help of the new single crystal
basis functions for copper under tension, the prediction error is already less
than 3% before any transfer learning, indicating that the combination of local
microstructure information (extracted from the ResUNet trunk network) and
material response information (provided to the branch as new single crystal
responses) is effective and highly generalizable. For both networks,
prediction performance generally increased with more copper data added to the
training. However, the SC-DeepONet provided higher accuracy than MP-DeepONet
for all numbers of copper training data. With merely 20 copper RVE data used
in fine-tuning, the SC-DeepONet achieved close to 1% relative error, and the
mean absolute error in stress was less than 0.5 MPa, showing extreme data
efficiency and generalizability of the proposed architecture. When inspecting
the predicted stress-strain curves in Fig. 8, we see that the SC-DeepONet
predicted behavior matched closely with the CP simulations up to 85th
percentile. Another vital advantage of the SC-DeepONet can be seen by
comparing the worst-case predictions of the two networks. Comparing the
overall RVE response for aluminum (Fig. 4a) and copper (Fig. 4c), it is
immediately apparent that the copper material displays more elastic
anisotropy. From the last column of Fig. 8, we see that the SC-DeepONet, with
the help of the anisotropic single crystal responses, can accurately capture
the elastic slope even in the worst case. At the same time, the MP-DeepONet
struggles to capture the initial modulus when the elastic responses vary
significantly in the test set due to material anisotropy. These observations
prove that using single crystal stress-strain curves as inputs to the branch
network is a more mechanically meaningful way to capture material behavior
than directly feeding the network with material property values.

###  3.4 Generalization to different boundary conditions

In this section, we move away from tensile loading and investigate SC-DeepONet
generalizability for aluminum under shear loading. 400 RVE simulations were
conducted, and new single-crystal responses were generated. Contrary to the
tensile cases, where those responses were generated from a single finite
element, for shear, a uniform mesh of 10$\times$10 elements was used to model
the single crystals and obtained their responses 333If the simple shear
boundary conditions are applied on a single finite element, it will lead to a
state of uniform shear strain since all the degrees of freedom are prescribed.
This is not the case for an inhomogeneous polycrystal consisting of multiple
elements since it has additional unconstrained degrees of freedom that can
deform, and the strain state will not be similar. Therefore, multiple elements
are needed to model single crystals to approximate the nonuniform strain state
in the RVE.. Doing so incurred additional time for generating the single
crystal ”basis functions”, but the total time of 504s is still short compared
to running the full RVE simulation at 122s each. The time required to generate
the single crystal responses amounts to about 1% of the time to generate the
400 RVE simulations. Transfer learning of SC-DeepONet was attempted with a
different number of new data, and results are shown in Fig. 9. Performance
metrics and predicted stress-strain curves for the model fine-tuned with 20
new data points are shown in Table 4 and Fig. 10, respectively. The transfer
learning process took 46s and the prediction time was $8\times 10^{-3}$s per
case, 15248 times faster than the FE simulation.

Table 4: Performance metrics for the aluminum under 2% shear Model | Rel. err. (before) [%] | Rel. err. (after) [%] | Mean abs. err. [MPa] | $R^{2}$
---|---|---|---|---
SC-DeepONet | 23.73 | 1.81 | 5.50 | 0.994

(a)

(b)

Figure 9: LABEL:sub@tr_sh Final SC-DeepONet prediction error for different numbers of transfer learning data points. LABEL:sub@tr_sh_corr Correlation of predicted and FE simulated stress data. (a) SC-DeepONet, best |  (b) SC-DeepONet, 50th pct |  (c) SC-DeepONet, 85th pct |  (d) SC-DeepONet, worst
---|---|---|---
Figure 10: Testing results for SC-DeepONet at different error percentiles
after transfer learning with 20 data points, for aluminum under 2% shear.

In this section, we only tested the SC-DeepONet and did not use the MP-
DeepONet. MP-DeepONet cannot be fine-tuned via transfer learning (without
modifying the branch FNN architecture) for this case as it does not have the
means to distinguish between tensile (what it was initially trained on) and
shear loading: the type of boundary condition is challenging to encode due to
its discreteness and therefore was not included as input to MP-DeepONet (only
the tensile strain magnitude was included). From this perspective, the single
crystal stress-strain curves under identical boundary conditions provide a
natural and mechanically meaningful method to implicitly represent only the
_effects_ of the boundary conditions rather than trying to represent and
encode the boundary conditions themselves. After all, when predicting the
homogenized RVE responses, we are only interested in understanding how
changing the boundary conditions affects the stress-strain curves. As seen in
Table 4, the SC-DeepONet trained on aluminum tensile data did not provide good
prediction in shear before transfer learning, having a significant relative
error of nearly 26%. However, the mean error reduced drastically to below 3%
after exposure to only 5 new data points and continued to improve with more
training data added. This again speaks to the data efficiency of the proposed
architecture. From the scatter plot of predicted stress points, we see that
the prediction correlated well with the CP simulation with a $R^{2}$ value of
0.994, and 95% of the predictions had a relative error $\leq 5\%$. From Fig.
10, we see that the SC-DeepONet captures the shape of the global stress-strain
curves thanks to the help of the single crystal responses and accurately
predicts the elastic slope and hardening behavior. For the cases with
relatively large prediction errors, it is observed that SC-DeepONet tends to
under-predict the flow stress.

###  3.5 Generalization to cyclic loading

In the last example, we investigate how a DeepONet trained on monotonic
loading data of aluminum can be extended to predict the cyclic loading
behavior of copper with the help of the corresponding single crystal
responses. 186 RVE simulations were generated, and the average run time for
each CP simulation was 2256.8s. Generating the single crystal responses
(modeled by a single finite element) took 10s, significantly shorter than
running a full RVE simulation. A sinusoidal, fully reversible displacement was
applied in the Y-direction for a maximum engineering strain of 0.125% (see
Fig. 11a). The mean axial stress $\sigma_{y}$ of the RVE was recorded at 240
uniformly spaced time steps (while all previous examples only had 50 time
steps). Transfer learning results with different new data points are shown in
Fig. 11b. Performance metrics and predicted stress-strain curves for the model
fine-tuned with 20 new data points are shown in Table 5 and Fig. 12,
respectively. The transfer learning process took 47s, and the DeepONet
prediction was over $2.8\times 10^{5}$ times faster than the FE simulation.

Table 5: Performance metrics for the copper under cyclic loading Model | Rel. err. (before) [%] | Rel. err. (after) [%] | Mean abs. err. [MPa] | $R^{2}$
---|---|---|---|---
SC-DeepONet | 154.90 | 7.34 | 0.54 | 0.999

(a)

(b)

(c)

Figure 11: LABEL:sub@cyc_bc Applied axial strain as a function of time. LABEL:sub@tr_cyc Final SC-DeepONet prediction error for different numbers of transfer learning data points. LABEL:sub@tr_cyc_corr Correlation of predicted and FE simulated stress data. (a) SC-DeepONet, best |  (b) SC-DeepONet, 50th pct |  (c) SC-DeepONet, 85th pct |  (d) SC-DeepONet, worst
---|---|---|---
Figure 12: Testing results for SC-DeepONet at different error percentiles
after transfer learning with 20 data points, for copper under cyclic loading.

Similar to Section 3.4, only SC-DeepONet was tested but not the MP-DeepONet.
Two reasons for that are: (1) in MP-DeepONet, only the magnitude of the
applied strain is used as input to encode the loading, this works well for
monotonic loading but does not capture the time-dependent applied displacement
in a fully reversible cyclic loading; (2) MP-DeepONet was trained to output 50
time steps only, its branch FNN has a fixed output shape of $[50\times HD]$.
Therefore, it is not possible to change the number of output time steps
without adding new layers to the branch FNN of MP-DeepONet (while in the
current transfer learning scheme, we are only fine-tuning the weights and bias
of the FNN without changing its architecture). Neither of those is a
limitation of the SC-DeepONet using single crystal responses as inputs. First,
the single crystal stress-strain curves carry information on the time-
dependent axial displacement, removing the need for explicit information on
the applied boundary conditions. Secondly, the number of output time steps for
the SC-DeepONet follows directly from the input functions, as each time step
is treated as an independent sample with 36 features (stress from the 36
single crystal samples at that time). Therefore, no changes to the branch FNN
architecture are warranted when predicting stress-strain curves with different
temporal resolutions. These two observations show the versatility of the SC-
DeepONet compared to using material properties and loading conditions as
direct inputs to the MP-DeepONet. From Table 5, we see that the SC-DeepONet
before any transfer learning lead to significant prediction errors, but this
is expected due to the profound differences in the RVE responses (comparing
Fig. 4a with Fig. 4d) caused by a change of both material properties and
loading conditions. With only 10 new data points used in transfer learning,
the relative error dropped to below 10%. It is also worth pointing out that
the use of relative error might be biased in this case since the axial stress
can become 0 during cyclic loading, and relative error for a small stress
quantity close to 0 can be quite large. Nonetheless, with 20 data points in
transfer learning, 96.9% of the predicted stresses had a relative error below
5%, and the mean absolute stress error was only around 0.5 MPa. Different
stress-strain curves in Fig. 12 also show satisfactory agreement with the
corresponding FE simulation results, indicating that the SC-DeepONet can
effectively capture the cyclic response of polycrystal RVEs given information
on the single crystal responses.

##  4 Conclusions, limitations, and future work

The present work introduces a novel material response-informed DeepONet
architecture to predict the mean-field stress-strain responses for
polycrystalline RVEs. It leverages a ResUNet in the trunk (similar to that of
He et al. (2023c)) to encode the grain orientations in the polycrystal RVE and
learn the interactions of a voxel in the domain with its neighbors. The most
novel contribution of the current work is the use of single crystal stress-
strain responses as inputs to the branch network of the DeepONet. The single
crystal response curves contain information on the material properties,
boundary conditions, and loading, effectively serving as ”basis functions” to
reconstruct the RVE response. A dot product operation combines the encoded
output from the branch and trunk networks to produce the prediction. In this
view, the SC-DeepONet combines the information on grain-to-grain interaction
(extracted from the ResUNet trunk network) and information on the individual
grain behavior (inputs to the FNN branch network) to produce a nonlinear,
microstructure-dependent rule of mixture as an estimation of the RVE response.
It captures all key elements that influence the global response in crystal
plasticity simulations. The proposed DeepONet architecture is compared to a
more traditional approach, where the material properties and applied strain
values are used as direct inputs to the DeepONet branch network. When trained
on data involving a single material and loading, both DeepONets can deliver
satisfactory predictions of global stress-strain curves. However, when asked
to generalize to different material and tensile strain levels, the MP-DeepONet
that takes material properties as inputs suffers from large prediction errors
and cannot capture the more anisotropic elastic behavior in the new data. On
the other hand, SC-DeepONet gives less than 3% relative error even before any
transfer learning and reduces to about 1% with only 20 new data points. SC-
DeepONet accurately predicts the RVE stress-strain curve up to about 85th
percentile error and captures the correct elastic and hardening behavior even
in the worst case. SC- DeepONet is further tested on different loading
conditions, such as shear and cyclic tensile loading. The SC-DeepONet can be
fine-tuned to produce a mean relative error of 1.8% and 7.3%, respectively,
with only 20 new data points. The predicted stress-strain curves in both cases
agree well with the CP FE simulations. Training of the SC-DeepONet is
efficient, the baseline training on 800 data points took less than 5mins,
while all transfer learning procedures finished within 1min. Once fine-tuned,
the prediction speed is nearly 10000 times faster than FE simulations.
Generating the single crystal responses is also computationally cheap, taking
less than 10s for single element cases and about 1% of the RVE data generation
time if the single crystal has to be modeled with multiple finite elements. It
is therefore concluded that the proposed SC-DeepONet is highly data efficient.
It overcomes the limitations of using material properties in the branch
network, such as being unable to encode different types of boundary
conditions, and not being able to predict different number of time steps
without modifying the branch network architecture. These results are evidence
that the new architecture is versatile and highly generalizable.

The current DeepONet can be a valuable tool in efficient multi-scale analyses.
Due to the high computational expense, it is often prohibitive to run CP
simulations on engineering structures. Yet the ability of CP simulations to
capture the effect of local microstructure on mechanical response can be vital
for predicting the performance of additively manufactured components, where
the local microstructure can vary significantly due to print conditions and
geometry. In this sense, it is sufficient to take a homogenization approach,
where the microstructure-informed global response of a small representative
volume is calculated from CP, and continuum plasticity models can be used to
capture the location-dependent elastic-plastic behavior. This process
inherently demands many CP simulations on RVEs with different microstructures
and can change if the part is manufactured with a different material. The
current SC-DeepONet provides an efficient surrogate model for evaluating the
mean-field responses given a new microstructure, material, and loading
condition. Although not mentioned in the current work, the SC-DeepONet can be
easily extended to treat 3D polycrystal RVEs using a 3D analog of the ResUNet
and more single-crystal responses that uniformly sample the full grain
orientation space. Once properly trained, the SC-DeepONet can efficiently
generate location-dependent homogenized elastic-plastic material properties
for additively manufactured components, where the underlying microstructure
can come from part-scale process simulations. It furnishes a seamless and
efficient integrated computational materials engineering (ICME) workflow
(Panchal et al., 2013; Liu et al., 2018), and the differentiable NN is well-
suited for gradient-based optimization to optimize mechanical properties.

Finally, it is worth reiterating the assumptions and limitations of the
current work. The central idea of this work is to use the single crystal
responses under identical boundary conditions as input ”basis functions” and
approximate the RVE response by constructing a nonlinear rule of mixture. It
requires the assumption that the mean strain states between the single crystal
samples and the polycrystal RVE are similar, since strain is not directly
passed into the SC-DeepONet. It is emphasized that this is only valid in an
average sense, and the local strain state at individual finite elements within
the RVE might be drastically different from that experienced by the single
crystals. Therefore, the current SC-DeepONet architecture cannot be used to
predict the full contour of the stress field since strain similarity does not
hold on a local level. It should only be used in a homogenization setting
where the local response is unimportant and only the mean-field response is of
interest. In future work, we will extend the proposed DeepONet to account for
3D polycrystal RVEs and expose the network to more realistic microstructures
generated using cellular automata simulation of additive manufacturing.

## Replication of results

The data and source code that support the findings of this study can be found
at https://github.com/Jasiuk-Research-Group. Note to editor and reviewers: the
link above will be made public upon the publication of this manuscript. During
the review period, the data and source code can be made available upon request
to the corresponding author.

## Conflict of interest

The authors declare that they have no conflict of interest.

## Acknowledgements

The authors would like to thank the National Center for Supercomputing
Applications (NCSA) at the University of Illinois, and particularly its
Research Consulting Directorate, the Industry Program, and the Center for
Artificial Intelligence Innovation (CAII) for their support and hardware
resources. This research is a part of the Delta research computing project,
which is supported by the National Science Foundation (award OCI 2005572) and
the State of Illinois, as well as the Illinois Computes program supported by
the University of Illinois Urbana-Champaign and the University of Illinois
System. Finally, the authors would like to thank Professors George
Karniadakis, Lu Lu, and the Crunch team at Brown, whose original work with
DeepONets inspired this research.

## CRediT author contributions

Junyan He: Methodology, Formal analysis, Investigation, Writing - Original
Draft. Deepankar Pal: Supervision, Writing - Review & Editing. Ali Najafi:
Supervision, Writing - Review & Editing. Diab Abueidda: Supervision, Writing -
Review & Editing. Seid Koric: Supervision, Resources, Writing - Review &
Editing, Funding Acquisition. Iwona Jasiuk: Supervision, Writing - Review &
Editing.

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