# Data-Driven Modeling of Dislocation Mobility from Atomistics using Physics-
Informed Machine Learning

Yifeng Tian  Information Sciences Group, Computer, Computational and
Statistical Sciences Division (CCS-3), Los Alamos National Laboratory, Los
Alamos, 87545, NM, USA  Soumendu Bagchi  Physics and Chemistry of Materials,
Theoretical Division (T-1), Los Alamos National Laboratory, Los Alamos, 87545,
NM, USA  Liam Myhill  Department of Materials Science and Engineering, Clemson
University, Clemson, 29623, SC, USA  Giacomo Po  Department of Mechanical and
Aerospace Engineering, University of Miami, Miami, 33146, FL, USA  Enrique
Martinez  Department of Materials Science and Engineering, Clemson University,
Clemson, 29623, SC, USA  Yen Ting Lin  Information Sciences Group, Computer,
Computational and Statistical Sciences Division (CCS-3), Los Alamos National
Laboratory, Los Alamos, 87545, NM, USA  Nithin Mathew  Physics and Chemistry
of Materials, Theoretical Division (T-1), Los Alamos National Laboratory, Los
Alamos, 87545, NM, USA  Danny Perez  Physics and Chemistry of Materials,
Theoretical Division (T-1), Los Alamos National Laboratory, Los Alamos, 87545,
NM, USA

###### Abstract

Dislocation mobility, which dictates the response of dislocations to an
applied stress, is a fundamental property of crystalline materials that
governs the evolution of plastic deformation. Traditional approaches for
deriving mobility laws rely on phenomenological models of the underlying
physics, whose free parameters are in turn fitted to a small number of
intuition-driven atomic scale simulations under varying conditions of
temperature and stress. This tedious and time-consuming approach becomes
particularly cumbersome for materials with complex dependencies on stress,
temperature, and local environment, such as body-centered cubic crystals (BCC)
metals and alloys. In this paper, we present a novel, uncertainty
quantification-driven active learning paradigm for learning dislocation
mobility laws from automated high-throughput large-scale molecular dynamics
simulations, using Graph Neural Networks (GNN) with a physics-informed
architecture. We demonstrate that this Physics-informed Graph Neural Network
(PI-GNN) framework captures the underlying physics more accurately compared to
existing phenomenological mobility laws in BCC metals.

##  1 Introduction

Almost a century ago [1, 2, 3], dislocations were identified as carriers of
plastic deformation in crystalline materials. A key construct of the theory is
the response of dislocations to an applied stress state, the _dislocation
mobility law_ , which describes the relationship between the force experienced
by the dislocation ($\mathbf{f}$) and the resultant velocity ($\mathbf{v}$).
In crystals with low lattice resistance and planar core structure, e.g., Face-
Centered Cubic (FCC) metals, the mobility law is usually simplified to a
viscous damping relationship $\mathbf{v}=\mathbf{M}(T)\cdot\mathbf{f}$, where
$\mathbf{M}$ is the matrix of mobility coefficients and $T$ is the temperature
[4], under the assumption of over-damped dynamics. In metals, $\mathbf{M}$
encodes dissipation mechanisms due to both electrons and phonons, with
dominant phonon contribution except at very low temperatures. Theoretical
calculations of phonon scattering predict a linear dependence of the mobility
coefficients on temperature [5] with a temperature-independent leading term
proposed recently [6, 7]. While this picture holds for edge dislocations in
Body-Centered Cubic (BCC) metals, the complicated core structure of the BCC
screw dislocations results in additional complexity. Screw dislocations in BCC
are known to have high lattice resistance. This in turn results in a strong
temperature dependence of the flow stress as the mobility of screw
dislocations in BCC are controlled by kink-pair nucleation and migration
enthalpy. In addition, screw dislocation motion is strongly influenced by
multiple stress components, in addition to the resolved shear stress in the
direction of the Burgers vector, resulting in non-Schmid effects [8, 9, 10].
These complexities are typically accounted for in the mobility law by
introducing functional dependencies on the local stress tensor $\bm{\sigma}$,
line orientation $\bm{\xi}$, temperature and possibly other local internal
variables, that is [11]

$\displaystyle\bm{v}=\bm{g}_{\alpha}(\bm{\sigma},\bm{\xi},T,\ldots)\,.$ (1)

The function $\bm{g}_{\alpha}$ is known as the (glide) _mobility law_ for slip
system $\alpha$ [12, 13, 14] The functional dependencies of the mobility law
are in turn informed by phenomenological approaches based on kink-pair
nucleation and kink diffusion. A review of the phenomenological approach for
deriving dislocation mobility laws can be found in [15]. State-of-the art
methods to develop such models for dislocation mobility (e.g., in BCC
materials) [16, 17, 18] have generally focused on screw and edge type
dislocations and smooth interpolation between these for other line
orientations. However, this approach presents multiple pitfalls. For example,
it has been shown previously that “singular” line orientations exist in BCC
metals [19, 20] which result in cusps in the mobility. Dislocations in these
singular orientations, and those vicinal to them, move by motion and
propagation of kink pairs. Therefore, their mobility cannot be accurately
represented using a smooth interpolation between screw and edge dislocations.
Moreover, the mobility becomes a complex function of local chemical
environments in the case of conventional and high entropy alloys (HEAs). The
atomically-rough energy landscape for dislocation motion, which is believed to
be an important characteristic of HEAs [21, 22], results in Arrhenius-type
mobility laws which are highly non-linear in stress and temperature [15]. It
has been demonstrated in BCC-based HEAs that the mobility of mixed
dislocations [23] and multiplicity of glide planes [24] can drive the overall
plastic behavior, in contrast to BCC metals. The traditional phenomenological
approach informed by a handful of intuition-driven data-points to derive the
mobility law can quickly become error-prone and cumbersome in such scenarios.

These challenges can in principle be addressed by using data-driven
approaches, whereby active learning techniques are used to efficiently explore
the parameter space for learning the dislocation mobility law with minimal
human intervention. Such approaches are especially relevant with the advent of
exascale computing platforms where hierarchical multi-scale methods, deployed
at scale with automated model refinement, can potentially be used to drive
materials design for targeted functionality. Moreover, the improved accuracy
of Machine-Learning (ML) interatomic potentials [25] promises affordable near-
quantum accurate predictions, which could enable upscaling of atomic-level
defect dynamics to predictive mesoscale models such as Discrete Dislocation
Dynamics (DDD) [26, 27, 28, 29, 30, 31]. To date, the exploration of these
ideas is in a relatively nascent stage. Previous studies have explored the
possibility of using graph-theoretical techniques for learning specific
aspects of dislocation glide. Moraes et al. [32] developed a graph-based
surrogate model for glide of edge dislocations in BCC Fe, trained to molecular
dynamics (MD) data. In this representation, sub-domains within the MD model
are represented using nodes on a ring graph. The dislocations are then
represented as random walkers which jump between neighboring nodes following a
Poisson process. More recently, Bertin and Zhou [33] developed a Graph Neural
Network (GNN) approach to model DDD, with the aim of replacing its time-
integration procedure. They demonstrated that this DDD-GNN approach can learn
ground-truth (GT) DDD data of a dislocation gliding in a field of obstacles.
While these studies show the potential for accelerating materials modeling
through machine learning, bridging mesoscale dynamics (i.e. DDD) with
information from higher fidelity (i.e. atomistic) simulations involving more
general defect characteristics (i.e. mixed orientations of dislocations)
remains to be explored.

Our objective in this study is to develop a generalizable framework for data-
driven modeling of dislocation dynamics mobility law. Our proposed framework
integrates high-throughput MD simulation datasets, physics-inspired machine
learning model structures, and uncertainty quantification, accelerating the
development of mesoscale models of dislocation mobility. A schematic diagram
with different components of our proposed modeling workflow is shown in Fig.
1. To demonstrate our approach, GT data was generated from classical MD
simulations of dislocation motion using an Embedded Atom Method (EAM) type
interatomic potential (Fig. 1a); the workflow would remain formally identical,
albeit more expensive, if ML interatomic potentials were used instead. The MD
simulation data consists of dislocation dipoles and shear loops moving under
different values of resolved shear stress (RSS) at various temperatures. Note
that, while Non-Schmid effects are important in BCC materials, our aim here is
to demonstrate the ML workflow supported by high-throughput large-scale
atomistic trajectory generation protocols for a relatively simple data set
with only RSS. Coarse-grained representations of dislocations were obtained
from this atomistic representation using the Dislocation Extraction Algorithm
(DXA) as implemented in the Open Visualisation Tool (OVITO)[34] software. The
DXA representation, which consists of nodes with line segments between them,
was converted into a numerical representation with nodes, quadrature points or
Gauss points (GPs), and line segments as shown in Fig. 1b. Relevant input
features of these dislocation configurations, which will serve as input for an
ML procedure, such as the Cauchy stress tensor ($\bf\sigma$, shown
schematically in Fig. 1c), segment tangent vector (${\bf s}$), and the
Burger’s vector (b) were computed using a recent version of the open-source
Mechanics of Defects Evolution Library (MoDELib) supporting periodic boundary
conditions [35]. We remark that the computation of stress tensor by MoDELib is
based on linear elasticity theory and the resulting stress can deviate from
the true stress experienced by the dislocation. Nevertheless, we hypothesize
that the stress tensor computed by MoDELib contains sufficient information for
us to make accurate predictions, and the error due to the linear assumption,
if any, will be corrected by our proposed GNN-based ML procedure below. We
also note that the physical features of the dislocation lines, such as the
Burger’s vector, segment tangent vector, and local stress, are defined on both
the GPs and nodes, while the displacements ($\Delta$${\bf x}$) are defined on
the nodes alone, with several GPs being present between any given pair of
nodes in general. Dislocation network, thus represented as a set of nodes and
Gauss points (GPs) in DDD can be naturally encoded as a graph. In Figures 1d
and 1e, we present the visualization of the dislocation network, showcasing
both its physical representation extracted from DDD for ML and its graph
representation. The physical network serves as an abstract depiction of the
dislocation network, which consists of nodes ($N_{i}$) and GPs ($GP_{i}$),
connected by the dislocation segments. Figure 1e illustrates the corresponding
heterogeneous graph, a type of graph containing vertices with different
features, for learning purposes. Within this heterogeneous graph, we introduce
two types of vertices, $v_{N}$ representing the nodes and $v_{\text{GP}}$
representing the GPs in the physical network. In the graph representation,
instead of connecting $v_{\text{GP}}$’s sequentially into a segment between
nodes $v_{N}$, we directly link GPs to their nearest nodes. This heterogeneous
graph enables the direct exchange of information between GPs and their
neighboring nodes, facilitating efficient information transfer.

To achieve the goal of the proposed work, we present a general physics-
informed data-driven approach for modeling dislocation mobility law. Our
approach incorporates a physics-inspired mathematical structure based on the
theoretical descriptions of dislocation mobility into the proposed GNN model.
Features are designed to respect physical constraints, e.g. rotational and
translational invariance. The proposed physics-informed GNN mobility law (PI-
GNN) is trained using a coarse-grained MD dataset, with a loss function
defined in Sec. 4.3. To quantify the predictive uncertainty and facilitate
accelerated learning, we employ an uncertainty quantification-driven active
learning framework [36] that is capable of providing on-the-fly uncertainty
measures and automatically query new MD simulations to improve the model’s
predictability and uncertainty. By studying the mobility of $\\{110\\}$
dislocations in BCC Fe, we demonstrate the effectiveness of this PI-GNN
framework to accurately capture the relevant physics of dislocations
systematically.

Figure 1: Schematic of ground truth data generation: a Atomistic
representation of a dislocation segment (green atoms represent the defect core
while the purple denotes pristine atoms). b Idealised nodal representation of
the dislocation segment obtained from Dislocation Extraction Algorithm (DXA).
Blue spheres represent dislocation nodes (denoted as $N_{1},N_{2},...$) and
yellow cubes represent the Cauchy stress tensor, one of the features, defined
at Gauss Points (denoted as $GP_{1},GP_{2},...$) using the linear elasticity
kernel in MoDELib Discrete Dislocation Dynamics (DDD) code. c Zoom-in of the
local stress state ($\bf\sigma$) at a given Gauss point, with the definition
of segment tangent vector (${\bf s}$) and the Burger’s vector (b). d General
representation of the dislocation configuration with multiple Gauss points
between nodes. e Graph representation of the dislocation configuration.

##  2 Results

Learning the mobility function using Graph Neural Networks [37] in the
Physics-informed Machine Learning (PIML) framework. A GNN is a type of neural
network designed to act on graph-structured data, where nodes represent
entities (e.g., the nodes and GPs on the dislocation) and edges represent
relationships (e.g., the spatial proximity of the nodes) between them. GNNs
learn powerful representations of graph data by propagating information
between neighboring nodes. Unlike convolutional neural networks on grids, GNNs
respect rotational and translational invariance, making them well-suited for
learning on irregular graph domains. As such, GNNs serve as an ideal neural
architecture for the coarse-grained representation of dislocation structures
generated by DXA. Furthermore, the design of the PI-GNN architecture was
informed by established physics in linear elasticity theory pertaining to
dislocation dynamics in continuum media [15], i.e.,

$\displaystyle\frac{{\rm d}{\bf x}}{{\rm d}t}=\frac{{\bf f}({\bf\sigma},{\bf
b},{\bf s})}{B(T,{\bf b},{\bf s})}\exp\left(-\frac{\epsilon({\bf\sigma},{\bf
b},{\bf s},T)}{k_{B}T}\right),$ (2)

where ${\bf f}$, $B$, and $\epsilon$ are separate networks modeling the
contribution of the Peach-Koehler force, drag, and thermally activated
processes in dislocation dynamics. These networks, referred to as F-net,
B-net, and E-net, are jointly trained on time-series data of the dislocation
dynamics, generated by MD and processed through the DXA-MoDELib pipeline.
Through message passing [38], the PI-GNN not only uses the local information
but also information on nearby nodes to learn higher-order corrections to
linear theories. More details of the PI-GNN model we adopted can be found in
Sec. 4 and in the Supplementary Sec. LABEL:sisec:PI-GNN.

Uncertainty quantification-driven active learning (UQ-AL) framework to
facilitate efficient learning. We propose a workflow, outlined in Fig. 2, for
training the PI-GNN while aiming to minimize its predictive uncertainty.
Rather than generating one large dataset all at once, the workflow begins by
learning from an initial dataset, and then progressively queries new datasets
where the trained models exhibit the most uncertainty. In each iteration, we
independently train $M_{en}$ identical GNN models with randomized
initialization of weights and biases. These models are then evaluated along
the trajectories of a set (8) of newly generated datasets with randomly
sampled simulation parameters $\Lambda=\\{T,|\bm{\sigma}|,...\\}$, and the
uncertainties of the models’ single-step predictions are quantified. This can
be considered as an ensemble learning [36] method. We integrate the top 50% of
new configurations that induce the highest predictive uncertainty into the
existing training dataset. The PI-GNN ensemble is retrained on this augmented
dataset, and the process repeats. We note that this method is an approximation
of more rigorous but computationally expensive UQ-AL frameworks (e.g.,
Bayesian Optimization [39]) for two reasons. First, the variability of the
$M_{en}$ model predictions is only a lower bound to the true uncertainty [36].
Second, we select the most uncertain candidates from randomly sampled
simulation parameters, rather than directly querying the single most uncertain
point across the entire simulation parameter space. More details of the
proposed UQ-AL framework can be found in Sec. 4.

Figure 2: Flow chart of UQ-driven active learning framework. The UQ-AL
framework is composed of two computational platforms, the high-throughput
microscopic MD simulations, and PIML-UQ platform. The UQ-AL workflow begins by
generating and learning from an initial dataset, and then progressively
queries new datasets where the trained models exhibit the most uncertainty. In
each iteration, we independently train an ensemble of identical GNN models
with randomized initialization of weights and biases. These models are then
evaluated on newly generated datasets with randomly sampled simulation
parameters, and the uncertainties of the models’ single-step predictions are
quantified. We integrate the top 50% of new configurations that induce the
highest predictive uncertainty into the existing training dataset. The PI-GNN
ensemble is retrained on this augmented dataset, and the process repeats.

We illustrate the advantages of the proposed UQ-AL framework in learning the
mobility law using the dipole dataset generated from MD simulations as
described in Sec. 4.1. In this numerical experiment, the PI-GNN models are
continuously retrained as more data are acquired and integrated into the
training set, and we compare the performance of the model when the new
datasets are either actively selected by the UQ-AL or passively queried by
randomization. We fix the total number of ensemble models $M_{en}$ at 20. The
initial dataset ${\bf D}(\Lambda)$ are generated on the set of simulation
parameters: temperature $T\in\\{300\text{ K},500\text{ K}\\}$, RSS
$|{\bf\sigma}|\in\\{0.25\text{ GPa},0.5\text{ GPa}\\}$, and the line
orientation (i.e., the angle between the tangent vector ${\bf s}$ and the
Burger’s vector ${\bf b}$) $\text{angle}({\bf s},{\bf
b})\in\\{45^{\circ},135^{\circ}\\}$. In each iteration, we maintain a
candidate pool of 8 new trajectories by sampling the simulation parameters,
$T\sim\text{Unif}(100\text{ K},1000\text{ K})$,
$|{\bf\sigma}|\sim\text{Unif}(0.1\text{ GPa},1.0\text{ GPa})$, and
$\text{angle}({\bf s},{\bf b})\in(0^{\circ},180^{\circ})$, to create new
trajectories of dislocation dynamics. With UQ-AL, the trained PI-GNN ensemble
is evaluated along the trajectories to quantify their averaged single-step
predictive uncertainty; we select and integrate the top 50% (4) trajectories
with the highest predictive uncertainties (i.e, the PI-GNN ensemble models
disagree with the most) into the existing training dataset. With passive
learning, we randomly select 4 new trajectories from the pool and integrate
them into the training dataset. For both methods, the selected trajectories
are removed from the candidate pool, which is then replenished by 4 newly
sampled parameter settings to generate the trajectories for the next
iteration.

Figure 3: Comparison of the proposed UQ-AL framework with a passive learning
method. The uncertainty measure of predictions for nodal displacement as a
function of the a active (AL) and b passive learning (PL) iterations.

Figure 3 illustrates how the model performance improves as more data are
acquired into the training set, for both active and passive selection of the
datasets. Evidently, with the proposed UQ-AL framework, the prediction
uncertainty is consistently reduced through the iterations. In comparison, the
predictive uncertainty with passive selection is higher and fluctuates more
between iterations, even though it eventually decreases in the large-data
limit. Suppose we set the uncertainty threshold to be $25\%$ of the mean nodal
displacement, $\sigma(\Delta{\bf x})=25\%\mu(\Delta{\bf x})$, the active and
passive acquisition of the new training dataset requires 8 and 15 iterations,
or equivalently 36 and 64 queries (initial 8 fixed, then 8 randomly sampled
ones to populate the candidate pool in the first iteration, and 4 randomly
sampled ones in each of the subsequent iterations) of new trajectories,
respectively. As such, UQ-AL saves more than $40\%$ of computational resources
for generating training dataset in this experiment. Because the proposed UQ-AL
can efficiently query computationally expensive high-fidelity datasets to
learn mesoscale models and thus accelerate the development of data-driven
mesosocpic models for DDD, we deploy this framework in the following analysis
as part of the data acquisition pipeline.

Figure 4: The predicted single-step evolution for various dislocation
configurations and learning method. a PI-GNN mobility law is trained on
coarse-grained dipole dislocation dataset with different RSS values,
temperatures, and angles between Burger’s vector and dipole tangent. The
predictions are also made on dipole configurations with different orientations
at 500K and 0.9GPa. b PI-GNN mobility law trained on dislocation loop
expansion data and prediction on dislocation loops at 500K and 0.9, 1.2, and
1.5GPa. c Testing the generalizability of the PI-GNN mobility law by training
on dipole dislocation data and predicting on loop expansion.

Predictability and generalizability of the PI-GNN mobility law. In the
following analysis, the proposed UQ-AL is used to train the PI-GNN model until
the predicted uncertainty is reduced to 25% of the mean nodal displacement.
The 50% top-performing models in the ensemble are selected for the following
validation analysis, in which we compare the single-step ($\Delta t\approx
1ps$) prediction of the trained models to a validation dataset. See Sec. 4 for
details on the train/test/validation dataset. We designed three different
experiments: (a) the PI-GNNs ensemble is trained on a dataset comprised of
nominally straight dislocation dipoles and compared against the dipole
validation dataset, (b) the PI-GNNs ensemble is trained on a dataset comprised
of expanding shear loops and tested on a loop validation dataset, and (c) the
PI-GNNs ensemble is trained on the dipole dataset and tested on the loop
dataset. Figure 4 shows selected examples of the single-step prediction; more
results can be found in Supplementary Sec. LABEL:sisec:pignnresults

We observe that the PI-GNN is capable of learning an effective mobility law
for various dislocation dipole configurations. Experiment (a) (Fig. 4a) shows
that the PI-GNN model trained on various conditions of temperature, RSS, and
line orientation of a dipole configuration, can learn an effective mobility
law to predict the evolution of the dislocation with an out-of-sample dipole
configuration. In contrast to experiment (a), where we explicitly sampled the
line orientation of the dislocation for training, the PI-GNN learns the
dependence of the line orientation directly from the angular distribution on
the evolving loops in experiment (b), and accurately predicts the evolution of
out-of-sample loop configurations. Experiment (c) shows that the PI-GNN is
able to learn the mobility law from the dipole dataset and generalize to the
out-of-distribution prediction task for loop configurations. Such
generalizability indicates that the PI-GNN is learning the physics of the
dislocation dynamics, instead of memorizing the training data.

Quantitative error analyses of the learned PI-GNN mobility law are reported in
Fig. 5, where the panels’ layout is analogous to Fig. 4. The prediction error
is quantified by both the percentage and absolute errors, the computation of
which is based on the proposed loss function (See Sec. 4.3 for more
information). For the dislocation dipole, the median prediction error is in
general very small for different orientations, temperatures and stress levels
(500K and 0.9GPa is shown in the figure). For screw-like dislocation where the
angle(${\bf b},{\bf s}$) is close to $0^{\circ}$ and $180^{\circ}$, the
percentage error is comparatively larger. This is mainly caused by an
imbalanced training dataset, due to the timescale separation between the
dynamics of screw dislocations and mixed/edge dislocations. At the timescales
of the dataset (several picoseconds) used in this study, the screw dislocation
dynamics are dominated by thermal fluctuations. Therefore, the signal-to-noise
ratio of the nodal displacement, which is the prediction goal of the PI-GNN
mobility law, is much smaller than that of the edge. The small signal-to-noise
ratio can only be seen for configurations near the pure screw configuration
(i.e., angle$\left(\mathbf{b},\mathbf{s}\right)=0^{\circ}$ and $180^{\circ}$),
while the majority of the data points do not have this signature.
Consequently, the trained PI-GNN model predicts the edge and mixed
configurations better than the pure screw configuration. Future work to
rectify the training data imbalance will be discussed in Sec. 3. When we
compare the absolute error in Fig. 5b, the absolute errors for different
angles are at approximately the same range. For dislocation loops, the single-
step prediction percentage error is also very small. The errors in
generalization are plotted in Fig. 5d. As explained above, we train the model
in this case using dislocation dipoles and make predictions on shear loops.
Compared to Fig. 5c, even though the percentage error is larger, it still
falls in an acceptable range, with the median at approximately $20\%$. This
shows that with the PIML structure and features, the PI-GNN model can still
capture roughly $80\%$ of the total motion even without any additional
training on the complex dataset. These numerical experiments demonstrate the
potential generalizability of the proposed UQ-AL PI-GNN framework for modeling
complex dislocation networks using simple datasets.

Figure 5: The predicted single-step error for various dislocation
configurations and learning method. PI-GNN mobility law is trained on coarse-
grained dipole dislocation datasets with different applied stress levels,
temperatures, and angles between Burger’s vector and dipole tangent. The
predictions are also made on dipole configurations with different orientations
at 500K and 0.9GPa. Both the a percentage error and b absolute unnormalized
error are shown. c PI-GNN mobility law trained on dislocation loop expansion
data and prediction on dislocation loops at 500K and 700K and various applied
stress levels. d Testing the generalizability of the PI-GNN mobility law by
training on dipole dislocation data and predicting on loop expansion.

Integrating PI-GNN mobility law with DDD for iterative multi-step predictions.
In Fig. 6, the predicted iterative multi-step evolution of both dipole and
loop configurations are visualized at different consecutive time steps. We
show results at 0.9GPa and 500K, and angle$({\bf b},{\bf s})\approx
80^{\circ}$ for the dislocation dipole. For each row in Fig. 6, the panels
from left to right show the forward evolution in time, where configurations
from preceding time steps are also visualized as translucent markers. As
observed in Fig. 6, the PI-GNN mobility law enables us to achieve
qualitatively accurate long-time predictions. The coarse-grained model
prediction will eventually deviate from the GT, but within the visualized
prediction horizon, the relative error remains less than 10%. Quantitative
analysis of the errors in multi-step prediction is provided in Fig.
LABEL:fig:MultistepError.

Figure 6: The predicted multi-step evolution for various dislocation
configurations and learning method. For a given snapshot, the GT is
represented using blue-filled circles, the PI-GNN prediction using red-filled
circles with translucency applied to configurations from all preceding time
steps. a A dislocation dipole(angle$({\bf b},{\bf s})\approx 80^{\circ}$) at
0.9GPa and 500K predicted using PI-GNN mobility law trained on dislocation
dipole data. b An expanding dislocation loop at 500K and 1.2GPa, predicted
using PI-GNN mobility law trained on dislocation loop expansion data.

Interpretability of the PI-GNN mobility law. The phenomenological model,
represented by Eq. 2, involves three distinct physical aspects of dislocation
dynamics, namely the force experienced by the dislocation given a stress
state, dissipative mechanisms resulting in drag, and thermal activation, each
of these making a unique contribution. We model each aspect using a separate
GNN as shown in Eq. 2 and train them altogether with a single loss function.
Even though quantitatively isolating each process from the trained PI-GNN
mobility law is challenging, the proposed framework still offers qualitative
insights into their sensitivity and dependence on the input features. For
instance, scaling both F-net and B-net by a constant factor does not alter the
output. To resolve the identifiability issue arising from the linear constant
factor, we compute the ratio of F-net over B-net, as well as other components
of the PI-GNN mobility law, against various input features.

In Fig. 7a, we plot the output of F-net/B-net at different angles and applied
stress levels. Despite the identifiability challenge, combining the outputs of
both networks allows us to study their collective contributions to the
prediction. With the increase of applied stress level, the predicted
contributions to nodal displacement become larger as expected. For different
angles representing edge, screw or mixed dislocations, the magnitude of the
output is larger around $90^{\circ}$ (i.e., edge dislocation), and small at
$0^{\circ}$ and $180^{\circ}$ ( i.e., screw dislocation), accurately capturing
the well-known disparities in the mobility of screw and edge dislocations in
BCC metals. An interesting observation is that the output is not symmetric
with respect to $90^{\circ}$ and is skewed toward larger angles. In Fig. 7b,
the dependence of the combined F-net/B-net on temperatures is plotted. We
observe that the increase in phonon drag with increasing temperature is also
captured by the GNNs. It is important to note that the temperature-dependence
of phonon drag is known to be stronger for edge-like dislocations [17] and the
GNN learns this behavior; as shown in Fig. 7b that the temperature-dependence
is stronger close to $90^{\circ}$ or edge dislocation as it is apparent from
the GT MD data (see Fig. LABEL:fig:Schew_EdgeM111Mobility in Supplementary
Information for dislocation velocities as a function of stress and temperature
for screw, edge, and mixed configurations).

Figure 7c illustrates the dependence of the contribution from E-net output on
the dislocation orientations. We note that for screw dislocations, with
angles$({\bf b},{\bf s})\approx 0^{\circ},180^{\circ}$, the PI-GNN mobility
law qualitatively captures the comparatively higher activation energy, leading
to lower output from the E-net. Conversely, at the mixed configurations, the
E-net output becomes larger. In addition, we observe a cusp at $110^{\circ}$,
predicting a reduced contribution from the E-net for that singular
orientation. The combined output from all three networks in Fig. 7d shows that
the PI-GNN mobility law is capable of predicting expected dependencies on
temperatures and angles. For instance, the increased phonon drag from high
temperatures causes the dislocation lines to move slower at some orientations.
In the case of screw dislocations with a low applied stress level, the nodal
displacement is dominated by thermal activation, which has a reversed order
compared to other angles (see the inset in Fig. 7d). This is also apparent for
the singular orientation at to $110^{\circ}$, where thermal activation
dominates at low temperatures. Such cusps in dislocation mobility is known to
be present in BCC metals [19, 20] due to singular dislocation orientations
which move by kink nucleation and propagation. The predicted trends are
aligned very well with what is observed in MD simulations (see Fig.
LABEL:fig:rawMDcomMobility0.25GPa), demonstrating the accuracy of the PI-GNN
mobility law and the ability of GNNs to learn complex dependence of
dislocation mobility on different parameters.

Figure 7: Physics-informed interpretation of the three components of the PI-
GNN mobility law. a Box plots of the $\frac{\mathbf{F}}{\mathbf{B}}$ for
different orientations, applied stress levels at 500K. b Box plots of the
$\frac{\mathbf{F}}{\mathbf{B}}$ for different orientations, and different
temperatures at 0.5GPa.c Box plots of the $\mathbf{E}$ for different
orientations, and different temperatures at 0.5GPa. d Box plots of the total
predicted displacement for different orientations, and different temperatures
at 0.25GPa.

##  3 Discussion

The central objective of this work is to provide a general, physics-informed
data-driven framework for modeling the mobility law in discrete dislocation
dynamics simulations autonomously from higher fidelity molecular dynamics
data. When constructing the modeling framework, we focused on the following
aspects: generalizability, interpretability, and uncertainty quantification.
To achieve generalizability for the model, we adopt the flexible graph-based
data structure to represent the dislocation network, which can be easily
extended to dislocation configurations beyond those studied in this paper. We
demonstrate that this framework is capable of learning many complex aspects of
dislocation motion in BCC metals, such as highly anisotropic dislocation
mobility between screw and edge configurations, cusps in dislocation mobility
at singular line orientations, and temperature dependence of phonon drag. The
interpretability of the proposed framework results from the physics-inspired
structure of the PI-GNN mobility law and physics-informed feature engineering.
We illustrate the interpretability of the proposed framework by analyzing
learned components of the PI-GNN, which showed qualitative agreement with
various physical aspects of dislocation mobility. Uncertainty quantification
is built into the proposed framework to inform data selection from MD
simulations for active learning, which is shown to significantly accelerate
the development of PI-GNN mobility laws. We also expect the proposed framework
to be beneficial when the dimensionality of the simulation parameters is even
larger, e.g. for learning Non-Schmid effects in BCC.

Further work focusing on improving the PI-GNN mobility law using the proposed
UQ-AL framework include (1) mixed modeling of the screw and mixed/edge
dislocations, trained by MD data generated at different timescales for
capturing both the fast-evolving mixed/edge dislocations and slow-evolving
screw dislocations, (2) time scale-aware feature engineering for mobility law
input, for enabling variable time-step integration, and (3) stochastic
modeling of screw-like dislocations. In addition to the mobility law, an
important component of DDD simulations is the description of discrete events
resulting in topological re-arrangements of the dislocation network. These
consist of cross-slip of screw dislocations, junction formation and
annihilation, and dislocation nucleation. Our PI-GNN mobility law is capable
of predicting a 3-D displacement and thus, it is well set up to predict cross-
slip events. Extension of the PI-GNN mobility law to describe dislocation
junctions and nucleation is possible using advanced methodologies like dynamic
graphs [40, 41], for example. We also note that the extension of our framework
to more complex systems like alloys is possible by using additional features
for local environment dependence. These advancements will be considered in
future studies.

We are aware of a pre-print by Bertin et al. [42], independently advocating a
GNN-based method for learning dislocation mobilities. While the basic concept
is similar, namely representing dislocations as graphs, there are multiple
distinctions between the two approaches as follows: (1) Training protocols: we
tested multiple, UQ-driven, training protocols, with relatively simple MD
simulations of straight dislocations and expanding shear loops in broad
parameter settings. In contrast, Bertin et al. used large-scale MD simulations
of high strain-rate deformation, using a narrow set of loading conditions; (2)
Our PI-GNN approach allows the development of a physically interpretable non-
linear mobility law, while such an interpretation is not straightforward in
the approach by Bertin et al.; (3) Features for our PI-GNN model are composed
of local stress tensor and line geometry (defined by Burgers vector and line
tangent) as opposed to nodal forces in Bertin et al.; (4) Loss function in our
approach is defined using a distance metric between nodal positions, as
opposed to a Nye tensor based approach in Bertin et al. While there are pros
and cons to the two approaches, a detailed evaluation of their relative merits
will be the subject of a future study.

##  4 Methods

In this section, we describe the generation of GT data from MD simulations,
calculation of relevant features using the linear elasticity kernel in MoDELib
Discrete Dislocation Dynamics (DDD) code, physics-informed graph neural
networks (PI-GNN) framework for modeling mobility law of discrete dislocation
dynamics, including the GNN architecture, physics-informed feature
engineering, generalized loss function, and uncertainty-driven active learning
(AL).

###  4.1 Automated on-the-fly generation of dislocation mobility data in MD

Generation of dislocation motion data from molecular dynamics simulations has
been widely addressed in past [43, 16, 18, 17]. It is a common practice to use
free surface boundary conditions in the direction normal to the glide plane
while building atomistic models with specific dislocation configurations
moving under externally applied stress and temperature. Evaluation of the
effect of free surfaces, on the stress experienced by the dislocation,
requires solution of a boundary value problem coupled with the DDD framework.
To circumvent this, we used periodic boundary conditions and devised an
automated model generation protocol where straight dislocation dipoles on a
slip system $[11\bar{1}](101)$ are created for an arbitrary line orientation
(defined by angle $\theta$) with the Burger’s vector ($\pm\textbf{b}$). The
simulation box is aligned as $b\sin{\theta}$ (x), $b\cos{\theta}$ (y) and
$[101]$ (z) with approximate lengths of 30 nm, 18 nm and 40 nm in these
directions, adjusted based on the crystallography of Fe BCC unit cell. The
dislocation line is oriented parallel to y axis. Extra half-plane of atoms are
removed along x with a thickness of $b\sin{\theta}$. Note, for screw dipoles
($\theta=0^{o}$) this results in no deletion, while for an edge
($\theta=90^{o}$) a complete half-plane with a thickness of $b$ is removed. We
use the periodic image correction superimposed on the Volterra displacement
field with 4 periodic images on both sides along Cartesian directions x and z
of the box, i.e., 63 [$(4\times 2)^{2}-1)$] images in total, following Cai et
al [44]. We report a few sample box-orientations in Table 1, chosen out of a
total 25 cases ranging from $0^{o}$ to $180^{o}$.

For simulating the motion of a shear loop, an initial loop of radius $15$ nm
is created on a $[11\bar{1}](101)$ system. The simulation box is then affinely
deformed according to the linear elastic solution corresponding to the applied
stress to prevent the loop from shrinking due to self interactions. To
generate high-throughput dislocation trajectories, we subject both the loop
and dipole configurations to a resolved shear stress range of
$\sigma\in\\{0.1,0.25,0.5,0.75,1.0\\}$ GPa and a temperature range of
$T\in\\{100,300,500,700,1000\\}$ K. All MD simulations were performed using
the EAM potential for iron (Fe) developed by Mendelev et al.[45].

Table 1: List of few sample atomistic supercell orientations out of a total of 25 cases. Glide-plane normal $(101)$ is along $z$-axis for all the dipole configurations. Line angles w.r.t. Burger’s vector ($\vec{b}$) |  $x$-axis orientation |  $y$-axis orientation (line direction)
---|---|---
$0^{o}$ | $[1\bar{2}\bar{1}]$ | $[11\bar{1}]$
$40.32^{o}$ | $[11\>\bar{4}\>\overline{11}]$ | $[2\,11\,\bar{2}]$
$81.07^{o}$ | $[10\,7\,\overline{10}]$ | $[\bar{7}\,20\,7]$
$110.41^{o}$ | $[28\,58\,\overline{28}]$ | $[\overline{29}\,28\,29]$
$163.14^{o}$ | $[\bar{8}\,34\,8]$ | $[\overline{17}\,\bar{8}\,17]$

To efficiently perform active learning, on-the-fly processing of million atom
MD trajectories is crucial. Our dipole models have 3-4 million atoms and MD
simulations were performed using the kokkos-implementation of LAMMPS [46]. We
apply a predefined global stress tensor and temperatures of the simulation box
by running simulations in the NPT ensemble for 40 ps with a timestep of 1 fs.
On NVIDIA A100 GPUs, 1 million EAM atoms/GPU can be simulated at a rate of up
to 10ns/day, excluding any input/output (I/O) overhead. To avoid reducing the
simulation rate due to I/O as well as the storage of large volumes of MD data,
we implement an in memory and on-the-fly analysis of dislocation trajectories
via the Dislocation Extraction Algorithm (DXA) from Ovito-API [47].
Simulations were analyzed every 0.1 ps, with nodal spacings of $1.5$ nm. This
is schematically shown in Figs. 1a and 1b.

###  4.2 Augmenting coarse-grained MD data with local features

Coarse-grained representations of dislocations present in the MD simulations
are generated using the DXA algorithm. Using the same nodes identified by DXA
and connecting the nodes via segments that are further discretized using Gauss
points (GPs) (as shown in Fig. 1b), an exact geometrical representation of the
MD dislocation can be replicated in MoDELib, which is used to compute the
local stress state at the GPs (Fig. 1c). The stress state along the
dislocation network is computed using a non-singular isotropic linear theory
of dislocations [48], and it includes dislocation-dislocation interactions as
well as the external stress [49, 50, 51, 52, 53, 54, 55, 56, 57, 58]. MoDELib
supports a periodic dislocation topology [35], and the stress field of the
periodic dislocation network is computed using an Ewald summation strategy
[59], which tapers the real-space long-range stress field by a user-defined
characteristic length. Convergence of the stress calculation is typically
realized using eight periodic images in each direction. Augmented coarse-
grained MD snapshots are finally obtained, which consist of discretized
dislocation lines using nodes, connectivities, and stress information at the
GPs including both dislocation-dislocation interactions and external
contributions. It is from these snapshots, containing both the geometry from
MD and the local stress state computed by DDD that we move forward to
construct the physical network and train the PI-GNN. More details about the
methodology can be found in Supplemental Information.

###  4.3 Modeling mobility law using Physics-informed Graph Neural Network

Representing Dislocations using Heterogeneous graphs. Graphs are a type of
data structure that describes a set of objects, often referred to as vertices,
and their relationships, or edges. Graphs are expressive and powerful at
representing complex non-Euclidean data structures in social science,
engineering, and physical science applications [60, 38]. In the context of
dislocation dynamics, we demonstrate the mapping between the physical and
computational representation of a dislocation network in Figs. 1d and 1e.
Physical representation consists of nodes and GPs, connected by the
dislocation segments, illustrated by Fig. 1d. The physical features of the
dislocation lines are defined on the GPs between two nodes, such as the
Burger’s vector, segment tangent vector, or local stress, while the features
of the nodes consist only of positions. The data structure of a dislocation
network can be easily mapped to a heterogeneous graph, where we introduce two
types of vertices: $v_{N}$ representing nodes and $v_{\text{GP}}$ representing
GPs in the physical network.

Figure 8: Overview of the PI-GNN architecture and training loss function. a
Physics-informed graph neural network structure for learning the mobility law
in Discrete Dislocation Dynamics (DDD), which consists of three different
networks: F-net, B-net, and E-net. Each network consists of hidden layers,
which are stacked HeteroSage GNNs, and physical layers, which are physically
inspired functions adopted from a phenomenological description of dislocation
mobility. b The proposed HeteroSage GNN structure and message passing diagram
for heterogeneous graphs. c,d Illustrations of the proposed generalized loss
function for measuring the distance between two dislocation configurations
with different total number of nodes resulting from different discretization.

Physics-informed graph neural network. Figure 8a presents the structure of the
graph neural network inspired by the physics-based phenomenological model of
dislocation dynamics [15] as shown in Eq. 2. We decompose the PI-GNN mobility
law into three components: F-net, B-net, and E-net, each with distinct input
features and network structures, depending on the physics-based mobility
function. As shown in figure 8a, we model the three network components with
two types of layers, hidden GNN layers, which consist of a series of black-box
layers for learning feature embedding from the graph data structure, and
physics layers, in which physical constraints for each component of the
mobility function can be injected. With these two types of layers, we aim to
combine the functional expressiveness of GNNs and the interpretability of
physical structure to construct a powerful, robust, and predictive model for
dislocation dynamics. For the hidden layers, we extend the widely used
GraphSAGE [61], which were designed to operate on homogeneous graphs to sample
and aggregate features from large graphs, to one that can operate on
heterogeneous graphs. We refer to this extended model as HeteroSAGE in this
work. Figure 8b illustrates the message-passing and feature operations of each
HeteroSAGE layer: the features defined on the neighboring vertices are
aggregated using an aggregator function, which is then concatenated with local
features and passed through a fully-connected network. This process is then
repeated in the opposite direction to generate the output of one HeteroSAGE
layer. By stacking multiple HeteroSAGE layers, the model can learn complex
feature embeddings from the graph. For the physics layers, we encode the
mathematical structure of physics-based mobility law using differentiable
functions. This design ensures that the output of each component adheres to
the corresponding constraints dictated by the physics-based model. More
details on the structure can be found in the Supporting Information.

Feature engineering. In modern ML applications, deep neural networks, coupled
with large datasets are commonly used to discover informative feature
embeddings for effective predictions. However, in scientific applications, a
set of carefully designed features can inject physics into the model, leading
to improved prediction robustness and reduced dataset size requirements. In
this work, we use a set of specially designed features from nodes and GPs as
the input of the PI-GNN mobility law. We leverage translational invariance and
rotational invariance, which are important in physical systems, for guiding
the feature engineering processes. The input features of the PI-GNN are
designed such that the aforementioned constraints are automatically satisfied
by the model, thus injecting crucial physics-based induction biases into the
model. Details on the selected features can be found in the Supplementary Sec.
LABEL:sisec:featureengineering.

Generalized loss function. After constructing the GNN model, we define a loss
function that quantifies the differences between the model-predicted evolution
of the dislocation and the ground truth, whose minimization will constrain the
parameters of the PI-GNN during learning. However, a challenge arises when
formulating the loss function from coarse-grained MD simulation snapshots, as
the total number of nodes in dislocations separated by the time step $\Delta
t$ may differ and there is no clear one-to-one mapping between nodes. This
hinders a straightforward computation of their differences using the commonly
used Mean-Squared-Error (MSE) in traditional ML tasks. To resolve this issue,
we propose a generalized loss function to measure the distances between two
dislocations. As illustrated in figures 8c and 8d, the dislocations can be
described as a collection of nodes and segments/edges connecting them. Instead
of using the traditional point-wise positional difference to construct the
loss function, we define a node-to-segment distance based on the geometric
projection as:

$\displaystyle l(N_{i}^{D_{1}},e_{j,k}^{D_{2}})=\begin{cases}\left|{\bf
x}^{\prime D_{1}}_{N_{i}}-{\bf x}^{D_{1}}_{N_{i}}\right|,{\bf x}^{\prime
D_{1}}_{N_{i}}\in e^{D_{2}}_{j,k},\\\ \text{min}\left\\{\left|{\bf
x}^{D_{1}}_{N_{i}}-{\bf x}^{D_{2}}_{N_{j}}\right|,\left|{\bf
x}^{D_{1}}_{N_{i}}-{\bf x}^{\prime D_{2}}_{N_{k}}\right|\right\\},{\bf
x}^{\prime D_{1}}_{N_{i}}\notin e^{D_{2}}_{j,k}.\end{cases}$ (3)

With the node-to-segment distance $l(N_{i}^{D_{1}},e_{j,k}^{D_{2}})$ defined,
we use the minimum value of the node $N^{D_{1}}_{i}$ to all the segments
$e^{D_{2}}_{j,k}\in D_{2}$ as the node-to-dislocation distance
$l(N_{i}^{D_{1}},D_{2})$, which can then be used to compute the dislocation-
to-dislocation distance as the sum of the node-to-dislocation distances
$l(D_{1},D_{2})=\sqrt{\sum_{i}l(N_{i}^{D_{1}},D_{2})^{2}}$. We notice that the
proposed distance measure based on equations 3, is not symmetric, i.e.
$l(D_{1},D_{2})\neq l(D_{2},D_{1})$. In this work, we adopt a loss function
based on a symmetric distance measure, which is achieved by averaging the
distances computed in both directions to avoid the potential issue due to the
smoothing effects (see Supplementary Sec. LABEL:sisec:generalloss for more
details):

$\displaystyle L(D_{1},D_{2})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[l(D_{1},D_{2})+l(D_{2},D_{1})\right],$ (4)
$\displaystyle Loss$ $\displaystyle=$
$\displaystyle\sum_{i}L(D^{\text{GT}}_{i},D^{\text{Pred}}_{i}).$ (5)

###  4.4 Uncertainty quantification-drive active learning

Ensemble learning for uncertainty quantification. In the context of machine
learning for scientific applications, particularly in cases where fully-
resolved simulation data is sparse, the neural network-based model
demonstrates competitive prediction accuracy. However, this may become
problematic when a single, over-parameterized NN provides predictions for new
datasets with unwarranted confidence. To address this issue, we adopt the
ensemble learning approach to properly quantify the prediction uncertainty for
modeling the mobility law using PI-GNN.

Ensemble learning is a machine learning training approach aimed at achieving
improved prediction performance by combining the predictions from multiple
independently-learned models. The objective of the ensemble learning approach
lies in generating multiple instances of model parameters that span the local
minima within the parameter-loss landscape of the NN-based model. We then
approximate the PDF of the predictions using a Gaussian distribution, whose
mean and variance are computed using the predictions from the top-performing
ensemble models:

$\displaystyle\frac{\widehat{d\bf{x}}}{dt}$ $\displaystyle\approx$
$\displaystyle\overline{\mu}\left(\frac{d\bf{x}}{dt}\right)=\frac{2}{M_{en}}\sum_{i=1}^{M_{en}/2}\left.\frac{d\bf{x}}{dt}\right|_{\theta_{i}},$
(6) $\displaystyle\sigma^{2}\left(\frac{d{\bf x}_{j}}{dt}\right)$
$\displaystyle\approx$
$\displaystyle\frac{2}{M_{en}}\sum_{i=1}^{M_{en}/2}\left(\left.\frac{d{\bf
x}_{j}}{dt}\right|_{\theta_{i}}-\frac{\widehat{d{\bf x}}_{j}}{dt}\right)^{2},$
(7)

where ${d\bf{x}}/{dt}|_{\theta_{i}}$ refer to the PI-GNN prediction of the
nodal displacement using the parameter $\theta_{i}$. Through the ensemble
learning procedure, we can obtain an estimate of the distribution of PI-GNN
outputs, which can inform the UQ of predictions.

Active learning. Active learning is a special procedure in ML that can
interactively query a new dataset based on a pre-determined strategy. The
active learning approach is commonly used in supervised learning setups where
it is very expensive to manually enumerate new data points for the query. When
a proper strategy for selecting new data points is used, the total number of
data points to learn a model can be dramatically reduced compared to a passive
learning strategy with a randomly queried dataset.

Figure 2 shows the workflow for the UQ-based active learning loop, which
includes data generation, model training, UQ and data point selection. The
details of the active learning procedure are described below:

  1. 1.

Generate initial MD dataset with a set of randomly selected simulation input
features $\Lambda=\left\\{\lambda_{i}\right\\}$, where $\lambda_{i}$ can be
different combinations of temperature, applied stress, orientations, Burger’s
vector, etc. The simulated MD trajectories are then coarse-grained and post-
processed to obtain snapshots of the dislocations nodes/GPs and their
features. We denote the MD dataset with parameters $\Lambda$ as ${\bf
D}(\Lambda)$.

  2. 2.

Train an ensemble of $M_{en}$ PI-GNNs on the dataset ${\bf D}(\Lambda)$ each
with random initialization of model parameters. We employ a
train/test/validation data split ratio of $70\%/20\%/10\%$.

  3. 3.

Randomly sample a new set of MD simulation input features $\Lambda^{\prime}$
and run new MD simulations with them and regroup with existing test dataset to
generate a test dataset ${\bf D}_{t}({\Lambda^{\prime}})$.

  4. 4.

Take the top $50\%$ ($M_{en}/2$) performing trained PI-GNN models and use the
ensemble variance of the nodal displacement prediction as a measure of
prediction uncertainty.

  5. 5.

If the highest uncertainty level is lower than a preset threshold, stop
training. Otherwise, combine the most uncertain set of the test data with the
training data to form a new set of training data, i.e. ${\bf
D}(\Lambda\cup\Lambda^{\prime})$, which is then used to re-train (using
previously converged model parameters as the initial weights) the ensemble
models.

  6. 6.

Move to step 3.

## Acknowledgements

The team is supported by the Laboratory Directed Research and Development
(LDRD) Project “Accelerated Dynamics Across Computational and Physical Scales”
(20220063DR). The study has been approved for unlimited release with LA-
UR-23-26562. The authors acknowledge significant support from the Darwin test
bed at Los Alamos National Laboratory (LANL), which is funded by the
Computational Systems and Software Environments subprogram of LANL’s Advanced
Simulation and Computing program (NNSA/DOE), and Institutional High
Performance Computing.

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