Machine Learning Visualization Tool for Exploring Parameterized Hydrodynamics

Abstract

We are interested in the computational study of shock hydrodynamics, i.e. problems involving compressible solids, liquids, and gases that undergo large deformation. These problems are dynamic and nonlinear and can exhibit complex instabilities. Due to advances in high performance computing it is possible to parameterize a hydrodynamic problem and perform a computational study yielding $\mathcal{O}\left({\rm TB}\right)$ of simulation state data. We present an interactive machine learning tool that can be used to compress, browse, and interpolate these large simulation datasets. This tool allows computational scientists and researchers to quickly visualize "what-if" situations, perform sensitivity analyses, and optimize complex hydrodynamic experiments.

Full text

MACHINE LEARNING VISUALIZATION TOOL FOR EXPLORING
PARAMETERIZED HYDRODYNAMICS
C. F. Jekel*D. M. Sterbentz T. M. Stitt P. Mocz R. N. Rieben D. A. White J. L. Belof
Lawrence Livermore National Laboratory, PO Box 808, Livermore, CA,94551, USA†
June 25, 2024
ABS TR ACT
We are interested in the computational study of shock hydrodynamics, i.e. problems involving
compressible solids, liquids, and gases that undergo large deformation. These problems are dynamic
and nonlinear and can exhibit complex instabilities. Due to advances in high performance computing
it is possible to parameterize a hydrodynamic problem and perform a computational study yielding
O(TB)
of simulation state data. We present an interactive machine learning tool that can be used to
compress, browse, and interpolate these large simulation datasets. This tool allows computational
scientists and researchers to quickly visualize “what-if” situations, perform sensitivity analyses, and
optimize complex hydrodynamic experiments.
1 Introduction
For hydrodynamics, it can be very difficult to understand the sensitivity of physical instabilities to small perturbations
in initial conditions. It is possible for a human to understand the impact of one or two inputs on the temporal evolution
of an instability. However, as the number of system parameters grow, so does the complexity of the system. Consider
the Rayleigh-Taylor instability (RTI) and Richtmyer-Meshkov instabilities (RMI) which can have several inputs that
influence the transient state. In these cases, ensembles of simulations are required to understand the sensitivity of
these instabilities with respect to their initial states. Often these simulation results are computationally expensive, and
it is difficult for researchers to look at every simulation result. The Cinema project [
1
] set out to aid researchers in
understanding ensemble calculations by providing tools to quickly look through simulation results. The tools provided
an intuitive interface for researchers to explore pictures of simulation results. While this tool is quite useful, the results
are limited to only the performed calculations. Our work builds upon this concept of allowing researchers to quickly
explore ensemble calculations, with the main advantage being the ability to quickly visualize results that were not
previously calculated. This interpolation is accomplished with a machine learning (ML) model that allows a user to
view the temporal evolution of instabilities by seamlessly changing initial conditions.
The RTI and RMI are closely related. A RTI occurs at the interface of two fluids mixing with different densities. A RMI
occurs when a shock wave amplifies perturbations at a material interface, causing large jet-like growths [
2
,
3
,
4
,
5
]. The
use and understanding of the transient behavior of these instabilities is important in many applications. For example,
experimentally measuring RMI formations is useful for calibrating high strain rate material models [
6
,
7
]. Additionally,
in inertial confinement fusion (ICF) experiments, where lasers are used to heat and compress a fuel capsule to the point
of starting a self-sustaining fusion reaction [
8
]. Unfortunately, RMI have been known to form within ICF capsules. The
∗jekel1@llnl.gov
†
This manuscript has been authored by Lawrence Livermore National Security, LLC under Contract No. DE-AC52-07NA2
7344 with the US. Department of Energy. The United States Government retains, and the publisher, by accepting the article for
publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license
to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes.
LLNL-JRNL-865692
arXiv:2406.15509v1 [physics.comp-ph] 20 Jun 2024

Machine Learning Visualization Tool for Exploring Parameterized Hydrodynamics
RMI often destroys the fuel target before fusion ignition is achieved [
9
]. Increasing our ability to design and control
RMI would have profound impacts in fusion research.
Our ability to perform simulations with the onset of exascale computing[
10
] may exceed our ability to comprehend
those results. Several tools, for instance Merlin[
11
], currently exist to orchestrate millions of simulations with high
performance computing (HPC). Typically, a researcher processes simulation results one at a time. The tool Cinema [
1
]
can be used to quickly swap between images of the ensemble of simulation results, but is limited to only those results.
Our work proposed to use ML to interpolate and quickly infer from datasets of several simulations, while still allowing
researchers to quickly build their own intuition for these complex systems.
There happen to be several applications of ML work applied specifically to the flow around airfoils. These simulations
are typically 2D and are utilized to understand the flow properties (e.g. lift, drag) of the airfoil. Wu et al.
[12]
and Wu
et al.
[13]
trained General Adversarial Networks (GAN) to airfoil flow fields. The airfoil shape was represented as a
vector of 14 values, which the generator model then learned the mapping from the parameterized geometry to the CFD
solution. For more detail on GANs we refer the reader to the review by Chakraborty et al.
[14]
. Li et al.
[15]
used the
same methodology to learn force fields of a hypersonic vehicle from parameterized flight conditions. Wang et al. [16]
then goes on to demonstrate how transfer learning can be used on subsequent datasets in a related domain. Hariansyah
and Shimoyama
[17]
used Deep Convolutional Generative Adversarial Networks (DCGAN) model to produce fake
airfoils in an optimization. Nandal et al.
[18]
trained a GAN model to pressure fields around parameterized airfoils.
Hou et al.
[19]
predicted hydrodynamic solutions around a submarine using a time series of previous fluid flow states.
A prediction at an arbitrary time requires iteratively feeding past ML predictions into the model. Kashefi and Mukerji
[20]
used neural networks to predict the permeability from 2D or 3D point clouds of the boundary surface of porous
media. These studies all had very specific applications in mind, and our work will build upon these methods to show
how to approach a general hydrodynamics problem.
It is also important to mention that ML has not been limited to just simulation results, as there has been recent work
applying to experiments. Chen et al.
[21]
trained a neural network to predict how a flame would develop within a
scramjet from pictures of the previous states of the flame. Li et al.
[22]
trained a system of neural networks to learn
the mapping of pressure sensors to images of the shockwave structure. The structure of the network also follows the
DCGAN model. Fernandez-Grande et al.
[23]
used GANs for the reconstruction of acoustic fields. The work shows that
the potentially machine learning can be used to enhance bandwidth limited acoustic sensors by being able to recover
some lost sound energy at high frequencies.
A number of the previous literature utilized the Deep Convolutional Generative Adversarial Networks (DCGAN)[
24
]
architecture on physics based datasets. Akkari et al.
[25]
trained DCGAN models to CFD solutions of flow fields
around a parameterized square obstacle. Instead of using the parameterized position of the obstacle, the DCGAN model
learned an unsupervised latent representations of the obstacle placement. Cheng et al.
[26]
used a DCGAN to predict
fluid flow solutions. Drygala et al.
[27]
trained a DCGAN model to fluid flow around a cylinder and a low pressure
turbine stator, with the goal focusing on the model being able to quickly generate several synthetic fluid flows.
Our work will also utilize the DCGAN model architecture as it is fast in inference, easy to train, and simple to construct.
It also works well for predicting multiple 2D physical fields. The focus of this work is largely limited to 2D simulations
to demonstrate how the real-time tool can be used to intuitively explore the complex transient space of these instabilities.
Belinchon and Gallucci
[28]
used GANs to investigate 1D stochastic fields for multiscale physics of turbulence, and
expressed a desire to extend their methods to 2D. 3D ML methods are currently a bit more expensive in inference time.
For 3D applications, we refer the reader to the work of Wiewel et al.
[29]
, who used an autoencoder to compress 3D
Eulerian solutions of dropping liquid and buoyant smoke. Then a LSTM network was used to learn concurrent time
sequences in the compressed latent space. The two models working together allowed for the generation of quick 3D
fluid predictions.
The generic multi-material hydrodynamic problem of interest consists of materials (in the solid, liquid, or gas state)
that undergo large deformations. These problems are driven by high velocity impacts or rapid energy deposition e.g.
chemical explosives or incident laser pulses. These problems are dynamic and nonlinear and exhibit complex behavior
such as RTI and RMI. A key aspect of this work is studying the control, e.g. enhancement or suppression, of these
hydrodynamic instabilities, while most of the previous literature applied ML to steady-state fluid dynamics.
The outline is as follows: Section
??
reviews the hydrodynamic simulation methodology, Section 3 reviews specific
applications (high-velocity impacts, RTI mixing, linear shaped charges), and Section 4 details the machine learning
approach. The key results are discussed in Section 5, and “screenshots” of the real-time interactive visualization tool
are shown in Section 6. Finally, we will discuss some of the limitations of the approach.
2

Machine Learning Visualization Tool for Exploring Parameterized Hydrodynamics
Copper target
and impactor
Distributed data
parallelism
across many
GPUs
Parametric initial
velocity and geometry
Train a Generative model to learn the mapping
from parametric designs to full-field solutions
Real-time visualization and exploration of full-field solutions
ML Model is machine portable, can be
run on a laptop, and easily shared
Quickly explore and optimize for any
parametric configuration within the domain and
see the results in real-time by performing
inference on the ML model
P1
13
5
24
6
P1
P2
P2
P3
P3
P4
P4
P5
P5
P1
-+X
P2
P3
P4
P5
t
P1
P2
P3
P4
P5
t
Randomly sample
multiphysics simulation
initial conditions
O(10,000) configurations
HPC
Cloud
Asynchronously run simulations
accross many resources
Multiphsyics simulations run
using either CPUs or GPUs
Dataset of high fidelity
solutions O(10) TiB
Deep Generative
Machine Learning
(ML) Model
-+X
P1
P2
P3
P4
P5
t
Figure 1: This is an overview of the proposed method. 1) A hydrodynamic simulation is parameterized to study the
effect of these parameters on the resulting instabilities. 2) These parameters are randomly sampled. 3) Simulations are
performed on HPC using an asynchronous queue. 4) This yields a large dataset of full-field hydrodynamic solutions. 5)
A generative machine learning model is then trained to learn the temporal mapping from the parameters to the full-field
solutions. 6) This machine learning model can be used for real-time visualization, dissemination of results, optimization,
and more.
2 Hydrodynamic simulations
A general overview of the methodology used to create the machine learning model is shown in Figure 1. The key
steps are: (1) parameterize initial conditions of a hydrodynamic problem, (2) define sample points in parameter space
for which to perform high-fidelity hydrodynamic simulations, (3) asynchronously run the independent hydrodynamic
simulations on large computational resources, (4) create a dataset of the full-field solutions, which is typically O(T B)
of data, (5) train a generative deep convolutional neural network to “learn” the hydrodynamic response. The final
step (6) is to use the ML model to quickly predict time dependent full-field results for arbitrary parameters, including
parameters not included the training space.
The hydrodynamic model consists of the Euler Equations (conservation of mass, momentum, and energy) combined
with equation of state formula and strength models such as Steinberg-Guinan[
30
]. The initial condition consists of
the geometry of the materials and the initial density, velocity, and internal energy. Boundary conditions such as wall,
periodic, or outflow are enforced. Source terms such as energy deposition may be applied. The Euler Equations
are solved using an Arbitrary Lagrangian-Eulerian (ALE) Finite Element Method (FEM). The Euler Equations are
integrated in time using explicit conditionally stable Runge-Kutta with adaptive time stepping. At fixed intervals, e.g. 1
µ
s, the fields are projected from the high order ALE-FEM mesh onto a uniform Cartesian mesh, i.e. an image, and
are recorded to disk. A single hydrodynamics simulation thus results in a sequence of images of
l
physical fields (e.g.
density, velocity, energy) at fixed snapshots of time. The images are of a constant
n×m
size, and we have
k
temporal
instances of these images. Each value in the image arrays is a single precision float.
A hydrodynamic study consists of a hydrodynamic problem with an initial condition that is parameterized via
d
real-
valued parameters. Clearly these parameters must have bounds and it is assumed these parameters can be normalized to
the interval [0,1]. The overall goal is to construct an accurate and compact model of the form
fML :Rd+1
□→Rl×n×m(1)
where
R
is the set of real numbers in a specified dimension. The input space consists of the
d
initial condition parameters
plus time (where time has also been normalized to
[0,1]
. The output consists of images of hydrodynamic fields e.g.
3

Machine Learning Visualization Tool for Exploring Parameterized Hydrodynamics
Table 1: An overview of the datasets created.
Name # of sims. Time Steps Fields Pixels # of pixels (millions)
PCHIP impact 2,985 51 6 1024 x 1024 957,779
Double sine wave 1,626 51 3 1024 x 1024 260,862
Linear shaped-charge 2,299 41 7 512 x 1664 566,153
Rayleigh-Taylor 2,000 51 6 768 x 256 120,324
density, velocity, energy. The model function
fMl
has millions of free parameters, the optimal value of these parameters
are “learned” from simulation data without knowledge of the Euler Equations, material properties, etc.
The machine learning tool that we have developed has three major advantages that have the potential to revolutionize
the way analysis and postprocessing of large sets of hydrodynamic simulations. One key point is that the size of this
model (in bytes) is many orders of magnitude smaller than the raw image data, this enables efficient archiving and
sharing of large ensembles of hydrodynamic studies with colleagues. A second key point is that the model can be
evaluated at some new point in parameter space many orders of magnitude faster than performing a new hydrodynamic
simulation, enabling a researcher to quickly visualize a hydrodynamics simulation that was not actually performed. A
third point is that since by construction the model
fMl
is continuous, the derivative can be computed (exactly, using the
automatic differentiation capability of ML libraries) enabling sensitivity analysis and inverse design optimization. The
architecture of the model function fMl , and the training process, is described in more detail in Section ??.
The hydrodynamic simulations were performed using an Arbitrary Lagrangian-Eulerian (ALE) Finite Element Method
(FEM) simulation code called MARBL. Generally speaking, Lagrangian formulations allow for precise tracking of
material interfaces, but the computational mesh can become excessively distorted. On the other hand, Eulerian methods
do not attempt to maintain sharp interfaces between material interfaces, and materials are allowed to mix. In the ALE
method the simulation begins as pure Lagrangian but at later time allows mesh relaxation and remap as the material
deformation becomes large, the flow becomes turbulent, or material mixing occurs. A particular feature of MARBL is
higher order elements [
31
],[
32
],[
33
], this allows for greater accuracy for a given mesh, and results in high efficiency on
Graphical Processing Units (GPU) due to high ratio of flops per mesh element. Other considerations include the need for
high order artificial viscosity [
34
] and of course the important ALE remap step [
35
]. The Livermore Equation of State
Library (LEOS) [
36
] is used for the equation of state for all materials, and Steinberg-Cochran-Guinan strength model
[
37
] is used for solid materials. The MARBL simulation code is not in the public domain, but a more limited high-order
Lagrangian-FEM code restricted to ideal gases and single materials, named Laghos [38], is publicly available.
3 Parameterized hydrodynamic examples
Four parameterized hydrodynamic problems are presented here. An ensemble of simulations is performed to understand
the influence of the parameters on the complex time-dependent instabilities. Latin Hypercube Sampling [
39
] (LHS)
was used to generate samples from the bounded parameters. The workflow utility Merlin[
11
] was used to manage
the ensemble of simulations. The physics simulations were either run in a single large allocation, or multiple smaller
allocations. Merlin executes the simulations asynchronously as soon as either resources became available, or a
simulation completed. The studies were performed on the LLNL Lassen3HPC.
An overview of the datasets generated is shown in Table 2. The full-field solutions were stored as float 32 data using
hdf5 [
40
]. All datasets were on order of a hundred billion pixels, with the Rayleigh-Taylor dataset having the fewest
number of pixels at 120 billion. Lossless compression with gzip was used to reduce storage requirements of the data.
The Rayleigh-Taylor dataset is 328 GB on disk.
3.1 High velocity impact study
The high velocity impact studies consist of an initially stationary copper target with a perturbation machined into the
right-hand side (the copper-air interface), and a copper impactor with velocity of
2km/s
. As the shock wave reaches
the interface perturbations, vorticity deposition occurs along the interface due to misalignments between pressure and
density gradients at the perturbations. This creates the RMI that generally results in the jetting of the copper target
material. The impactor is
1×9
cm and the target is nominally
0.5×9
cm. These dimensions and velocities are chosen
to be compatible with the two-stage gas gun at LLNL’s High Explosive Applications Facility (HEAF)[
41
,
42
]. Whereas
in actual HEAF experiments the impactor/target are circular with 9cm diameter, the simulations are 2D. The simulations
3
Lassen is a HPC with 795 number of nodes. Each node contains 2 power 9 ppc64 CPU sockets and 4 NVidia 16 GB V100
GPUs. https://hpc.llnl.gov/hardware/compute-platforms/lassen
4

Machine Learning Visualization Tool for Exploring Parameterized Hydrodynamics
begin at time
t=0
with the impactor and target in contact with a discontinuous velocity. An example figure of one of
these impacts is shown in Figure 2.
Figure 2: Illustration of the high velocity impact. The left figure is at
t=0
, the right figure is at some later time i.e.
t=10µs.
Two different parameterizations of these high velocity impacts were studied to investigate the formation of Richtmyer-
Meshkov Instabilities (RMI) that occur at the target-air interface, as small perturbations evolve into a sharp jet. The first
study is the PCHIP impact which looks at how changes in the perturbation in the target-air interface influence the RMI.
The second study is the double sine wave study, which looks at how a sine wave can be placed on the impactor side of
the target to influence RMI, when there is a fixed sine wave along the target-air interface.
3.1.1 PCHIP impact
An example of the experimental setup of the PCHIP impact is shown in Figure 3. The target-air interface is parameterized
with four parameters defining a Piecewise Cubic Hermite Interpolating Polynomial (PCHIP) [
43
]. The PCHIP parameter
range was [−0.25,0.25]cm.
The hydrodynamic solutions were computed using a nominal mesh of
144 ×144
quadratic (
Q2Q1
) elements, the mesh
was morphed to have conformal interfaces between air and copper. During the simulation the fields density, velocity
x
,
Figure 3: An illustrative depiction of the parameterized PCHIP impact.
5

Machine Learning Visualization Tool for Exploring Parameterized Hydrodynamics
Figure 4: This shows the geometry and materials of the double sine wave impact.
velocity
y
, energy, pressure, and the material indicator are projected onto a
1024 ×1024
Cartesian image and exported
at 51 uniform timesteps from 0 to 15 µs.
3.1.2 Double sine wave
The other high velocity impact involved sinusoidal waves on both sides of the impactor. The impactor side of the target
was parameterized with the sinusoidal wave
Bcos2πQx −sπ
9.0(2)
to seed initial RMI growths. The free side of the target (copper-air interface) utilized a fixed wave of
0.5+0.1cos2π10x
9.0(3)
which also seed RMI growth. The purpose of this parameterization was to study how parameterized RMI growths
interact with a known RMI seed (on the free side), the notion is that an optimized sinusoidal perturbation can initiate
vorticity that “cancels out” the primary RMI. The following bounds were placed on the three parameters of the impactor
side wave: Bfrom [0.1,0.25],Qfrom [5.0,25.], and sfrom [0.0,3.14].
The copper impactor was
1×9
cm and traveling at 2 km/s. Lucite was used to fill in the material between the target
and the impactor, creating a flush interface for an initial impact. The simulations were ran out for 7
µ
s after the initial
impact. An overview of the simulation setup is seen in Figure 4.
The solutions were computed using quadratic (
Q2Q1
) elements on a nominal grid of
144 ×144
. The mesh was morphed
to have conformal interfaces. The simulation results were saved on a
1024 ×1024
uniform grid for the following fields:
density, velocity x, velocity y. These fields were exported along 51 ideally uniform timesteps from 0 to 7 µs.
3.2 Linear shaped charge study
Linear shaped charges have a variety of industrial use cases such as structural demolition [
44
], geo-engineering [
45
],
[
46
], and aerospace [
47
]. They utilize an explosive which propels a liner (typically copper) into a high velocity jet
6

Machine Learning Visualization Tool for Exploring Parameterized Hydrodynamics
that will penetrate or cut into materials. Like the previous high velocity impactor study, the shaped charge jet is
initiated by RMI. Design exploration of shaped charge jet formation can be non-intuitive and require many thousands of
hydrodynamic simulations to explore the parameter space [
48
]. As a case study, this work proposes a parameterized
linear shaped charge design involving liner shape and detonator locations.
An overview of the parameterized linear shaped charge is shown in Figure 5. The shape of the copper liner facing the
explosive is parameterized with four spline parameters ranging from
[0.05,0.3]
cm, while jet side of the liner is fixed at
a
60◦
angle. One additional parameter controls the location of two detonation points, which are kept symmetric about
the center of the liner. The detonator location was parameterized along the steel case, where 0 represents a placement
along the center of the liner, and 1 represents a placement against the liner.
Figure 5: The planar shaped charge in this study consists of a steel case, explosive, and a copper liner. One face of the
copper liner is fixed, the other face is parameterized with a polynomial; the polynomial coefficients can be optimized to
enhance performance.
The linear shaped charge hydrodynamic simulations used a mesh of 11,000 quadratic (
Q2Q1
) elements. The full field
solutions were saved on a
512 ×1664
uniform grid. The results contain the following fields: density of the liner, density,
velocity
x
, velocity
y
, energy, pressure, and the volume fraction of the liner. The simulations were run from initial
detonation to t=20
µ
s, and results were saved every 0.5
µ
s. The high explosive was modeled using the Cochran-Tarver
ignition and growth reactive flow model.
3.3 Rayleigh-Taylor study
The final study is a single-mode Rayleigh-Taylor instability (RTI). The setup for the initial RTI was based off of the
example in Athena
++
[
49
] and the work of Liska and Wendroff
[50]
. The problem has a
x
domain of
[−1/6,1/6]
cm
and a
y
domain of
[−0.5,0.5]
cm. A heavier ideal gas is placed on-top of a lighter ideal gas. An initial velocity was
applied in ydirection to seed the instability growth as
vy=vinit(u0∗(1+cos(6πx))(1+cos(2πy))/4)(4)
with u0=0.01 cm/s. There is a constant gravitational acceleration of 1.0 cm/s2.
The simulations were parameterized for three physical parameters: density ratio of the two gases, the heat capacity ratio
of
γ
for both gases, and the initial velocity of
vinit
. The parameters were randomly sampled in the following ranges
respectively: [1.1,6.7],[1.1,1.6], and [0.6,10.0].
It is well known that this RTI problem becomes turbulent with decreasing feature size as time progresses. Thus, the
simulation based on the Euler equations is never fully resolved. Our solutions were computed on a
192 ×64
grid using
cubic (Q3Q2) elements.
7

Machine Learning Visualization Tool for Exploring Parameterized Hydrodynamics
The full field solutions were saved on a
768 ×256
uniform grid. The results contain the following fields: density,
velocity
x
, velocity
y
, energy, pressure, and the materials. The simulations were run to 10 s, and results were saved
every 0.2 s (with 51 total snapshots per simulation).
4 Machine learning
The purpose of the proposed Machine Learning (ML) model is to learn the mapping Eq 1 which maps the parameterized
simulation to 2D image arrays of physics fields. The ML architecture originates from a generative model based on
deep, but sparse, convolutional layers. The typical application of these deep convolutional networks is in unsupervised
learning of pictures e.g. human faces[
24
]. In these applications, the faces have no labels and no known a-prior
parameterization. In our work, neural network layers are added to the generative model to connect the simulation
parameters to the layers that have the smallest image representation. This allows us to take just about any generative
model architecture and perform regression to learn the parameterized simulation solutions.
The DCGAN [
24
] generator architecture is perhaps one of the simplest generative models to learn the mapping of
parameters to full-field solutions. The models require only a couple milliseconds for inference which is approximately
a million times faster than a full hydrodynamic solution. The model uses transposed convolution layers (sometimes also
called inverse convolutional, or deconvolutional [
51
]). The first transposed convolutional layer creates an initial kernel
representation (e.g.
4×4
pixels) of the entire field. Then, each subsequent transposed convolutional layer doubles the
previous layer’s full field representation (e.g.
4×4→8×8
). These layers can be stacked until the output is the correct
size of the final images.
Our use of the DCGAN model differs in a couple key characteristics from [
52
]. We discovered that we could achieve
superior accuracy by using the image channels (normally used for e.g. red, green, blue) to learn multiple physical
fields (e.g. density, velocity, pressure) rather than using an entirely separate generator models for each physical field.
The model takes advantage of the many correlations between fields. Each physical field has different units that may
be orders of magnitude different, which would cause issues with the transposed convolution layers. To address the
imbalance of units within the physical fields, we propose the model learn the following linear mapping as the final layer,
αX+β(5)
where
α
and
β
are vectors that are each the size of the number of
l
physical fields. This allows to output the physical
fields of different orders of magnitude into the correct units, which would be important when applying physics informed
constraints[53].
A visual overview of the model architecture for the PCHIP impact is shown in Figure 6, and ML models for the other
datasets was similar. The initial kernel was
4×4
, and the number of channels ranged from 1024 down to the final 6
fields. Each transposed convolution layer is followed by batch norm [
54
] and ReLU activations [
55
]. The layer by
layer
4
breakdown with learnable parameters for each ML model is shown in Appendix A. All models generally have
around 80 million learnable parameters.
The number of features in the model were limited to between 128 through 1024 (in the dimension where RGB is
typically used in images). Reducing to 64 through 512 reduces model quality while also reducing the total number of
learnable parameters. Increasing these features to 256 through 2048 greatly increases the total number of learnable
parameters while making a negligible impact on accuracy, however this remains an area of active investigation. It is still
not well understood how the sizes in this dimension affect the accuracy of the model, as well as which datasets would
benefit from the additional features.
The models were implemented using PyTorch [
56
]. Training was performed using Adam[
57
] to minimize the mean
absolute error. All models were trained using distributed data-parallel training and a zero redundancy optimizer [
58
].
There is a copy of the model on each GPU which processes some unique
n
mini-batch fraction of the data. The
mini-batch size was chosen as the largest value that maximizes the available GPU memory. The effective batch size for
k
GPUs is given by
nk
. An epoch represents one complete training iteration through the entire dataset. The learning rate
ηwas selected to be 1e−5. The learning rate scaling is inspired by Goyal et al. [59] and follows
ηeffective =ηnk (6)
which should result in similar models when trained on different resources. The models were trained on NVIDIA 16GB
V100 GPUs on the Lassen HPC using 12 hour allocations.
4
The PyTorch layers, shapes, and learnable parameters are outputs of a packaged called torchinfo available online at
https:
//github.com/TylerYep/torchinfo
8

Machine Learning Visualization Tool for Exploring Parameterized Hydrodynamics
Figure 6: The convolution neural network architecture of the generator model starts with a
4×4
kernel (right). Each
subsequent layer doubles pixels until the final
1024 ×1024
fields are created (left). This is actual ML output for the
PCHIP impact, showing fields of density, energy, pressure, materials, velocity x, and velocity y.
5 Results
Machine learning (ML) models were trained on the four datasets. The training details including batch size, number of
GPUs, learning rates, and effective batch size are reported in Table 2. A single image array from the Rayleigh-Taylor
dataset was roughly a third of the size of the images from the other datasets, and thus could support approximately three
times the batch size on a single GPU compared to the others.
The training errors for each epoch are shown in Figure 7. All models were trained on mean absolute error, and there is
strong correlation with mean absolute error, mean squared error, and L-infinity error. Diminishing training returns with
respect to additional epochs is observed with all models. This is most pronounced in the Rayleigh-Taylor case which
shows negligible training improvements for the last 100 epochs.
Figure 7: The mean absolute error, mean squared error, and L-infinity training errors for each dataset.
One necessary aspect that enables the ML models to be machine portable is having a reasonable file size. Table 3
shows the file sizes of the ML model weights compared to the dataset size. The ML models were generally around 0.9
Table 2: An overview of the datasets created.
Name Batch size # of GPUs η ηeffective effective batch size
PCHIP impact 14 160 1e-5 2.24e-2 2240
Double sine wave 14 40 1e-5 5.6e-3 560
Linear shaped-charge 14 60 1e-5 8.4e-3 840
Rayleigh-Taylor 48 32 1e-5 1.536e-2 1536
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Machine Learning Visualization Tool for Exploring Parameterized Hydrodynamics
Table 3: Dataset vs machine learning model size in gigibytes.
Name Dataset size (GiB) ML model size (GiB)
PCHIP impact 2098 0.900
Double sine wave 729 0.896
Linear shaped-charge 1151 0.709
Rayleigh-Taylor 328 0.899
gigibytes, while the dataset were on the order of a tebibyte
5
. In this aspect, the ML model can be thought of as a lossy
compression of the data. In these cases, the achieved compression factor was around 1,000 times. The dataset sizes
discussed here is the size on disk of the hdf5 files which utilize gzip lossless compression.
For all models, a simulation not in the training set is compared to the ML model’s density predictions in Figures 8-11.
Density is just one of the multi-field outputs from these models. The density field provides a good illustration on
how the materials are mixing in the hydrodynamic simulations. All of the ML predictions appear to be an excellent
representation of the actual field. It is nearly impossible to notice the errors in the ML predictions of the PCHIP impact.
The double sine wave impact has much finer Richtmyer-Meshkov instabilities than the PCHIP impact, and the ML
model blurs a high-low density interface of the finer instabilities (e.g. at the copper-lexan-copper interface). The ML
predictions of the linear shaped charge look excellent, with only some minor fine details missing on the jet as it forms.
The Rayleigh-Taylor results (shown in Figure 11) are perhaps the most interesting. For all times the ML prediction
is tracking the overall interface between the high-low density fluids. However, as the simulation time progresses the
overall interface becomes significantly more complicated with many very fine perturbations along the mixing interface.
The ML model appears incapable of modeling these finer features and appears to blur the many very fine high-low
density features with a smooth "middle" density prediction. This spatial averaging behavior of the ML predictions is
similar to the expected behavior of a RANS solver [60].
Additional results showing predictions and errors for each field are shown in Appendix B. Predictions are compared
to the simulation results for a simulation not in the training set. All model fields are shown at different times within
the simulation domain. The mean absolute error for each field is roughly two orders of magnitude better than the ML
model initiated with random weights. Additional results showing predictions and errors for each field are shown in
Appendix B. Predictions are compared to the simulation results for a simulation not in the training set. All model fields
are shown at different times within the simulation domain. The mean absolute error for each field is roughly two orders
of magnitude better than the ML model initiated with random weights.
6 Real time visualization
A desired application that the ML models enable is real-time visualization and exploration. The ML model can be
queried at any place within the parameterized space. These full-field solutions out of the ML model can be thought of
as an interpolation in the parameterized space. While it would be possible to quickly view results from an ensemble
dataset in a similar fashion, the ML model offers the ability to quickly explore anywhere in the high dimensional space.
Additionally, inference from the ML model can be performed on a portable laptop computer, while both the dataset and
simulations are solidly in the realm of HPC. The full-field inference for the PCHIP impact model takes 0.2 s using an
Apple M2 Max on battery power.
A real-time visualization tool was created using napari[
61
]. Static images taken from the tool are shown in Figure 12.
The user is presented with 4 inputs to the Rayleigh-Taylor ML model (density ratio, heat capacity ratio
γ
, initial velocity,
and time) as a slider interface. The user can either click or slide through the input parameters and view the resulting
full-field solutions in real-time. There is a dropdown menu to switch between which output field is displayed (e.g.
density, velocity, pressure). Inference was done using an AMD MI250X, which takes an average time of 0.002 ms using
float16 precision. Having fast ML inference is key to visualize the full-field solutions in real-time.
Videos of the slider interface while doing live ML inference are included in the supplemental material for all four ML
models. The interface allows for the interpolation of the physical fields at any point within the ensemble domain. The
predictions from the tool look excellent. With the linear shaped charge model, users can interactively explore how
the liner shape influences the long-term jet development, to build intuition on how these parameters work together to
enhance or mitigate the jet. With the double sine wave and PCHIP models, users can gain intuition on how the initial
shape of the copper target influences the development of the RMI. This interactive ability allows users to explore and
interpolate an ensemble of hydrodynamic simulations to further their understanding of complex physical instabilities.
5One tebibyte is 1.099511627776 terabytes.
10

Machine Learning Visualization Tool for Exploring Parameterized Hydrodynamics
The real-time visualization tool also makes it easy to explore how well ML models behave when extrapolating outside
of the ensemble domain. While it would be incredibly desirable for a fast-running hydrodynamics tool to extrapolate,
unfortunately these ML models possess no ability to extrapolate the hydrodynamic instabilities. As all input parameters
are extrapolated, the ML model’s predictions go to zero everywhere. Even with subtle extrapolations like 10%, users
will see predictions quickly begin to look unrealistic as interfaces begin to blur with cloud-like pockets of material
disappearing. The material interfaces do not move when extrapolating in simulation time, but rather begin to disappear.
In summary as the model begins to extrapolate then the predictions quickly look both nonphysical and unbelievable.
7 Conclusion
Performing ensembles of simulations is one way to understand the complex sensitivity of physical instabilities to initial
conditions. We present results from four ensemble datasets involving the Richtmyer-Meshkov Instability (RMI) and
Rayleigh-Taylor instabilities. A machine leaning (ML) model framework was proposed to learn the full-field solutions
as functions of initial physical and geometric conditions. The ML model weights were a thousand times smaller than
the datasets (as a form of lossy compression). Additionally inference from the ML model can be millions of times faster
than a full hydrodynamic solution, while also not requiring HPC resources demanded by the ensemble. The ML model
enables real-time visualization of the instabilities by sliding through both initial conditions and time. This enables users
to explore and gain intuition about complicated hydrodynamic relationships. Such tools also enable the dissemination
of hydrodynamic simulations and results to additional audiences (e.g. experimentalists) with reduced barriers of entry
(e.g. high performance computing). Lastly, the output of the ML model is fully differentiable with respect to the input,
which can be useful for applications in design optimization or uncertainty quantification.
11

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Figure 8: Figure of density predictions and truth for the PCHIP impact. Rows show machine learning model (ML) next
to simulation (Sim) results.
12

Machine Learning Visualization Tool for Exploring Parameterized Hydrodynamics
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1.0
1.1
1.2
1.3
1.4
1.5
1.6
0 200
0
100
200
300
400
500
600
700
Marbl Sim. at 1.54 1.41 5.33 7.00
1.0
1.1
1.2
1.3
1.4
1.5
1.6
0 200
0
100
200
300
400
500
600
700
Marbl Sim. at 1.54 1.41 5.33 8.00
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
0 200
0
100
200
300
400
500
600
700
Marbl Sim. at 1.54 1.41 5.33 10.00
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
Figure 11: Figure of density predictions and truth for the Rayleigh-Taylor simulation. Rows show machine learning
model (ML) next to simulation (Sim) results.
15

Machine Learning Visualization Tool for Exploring Parameterized Hydrodynamics
Figure 12: Real-time visualization of the Rayleigh-Taylor model showing the full-field density solution. The first two
rows show a user sliding through time for a particular parameterization that the model was not trained on. The last row
shows the user sliding through the input velocity at a fixed time. These are still frames from a video of a user interacting
with the tool.
16

Machine Learning Visualization Tool for Exploring Parameterized Hydrodynamics
Acknowledgments
This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National
Laboratory under Contract DE-AC52-07NA27344 and was supported by the LLNL-LDRD Program under Project No.
21-SI-006.
We would like to acknowledge Kareem Hegazy for pointing out that placing the multiple physics fields into the image
channel would enable the model to learn the correlations between physical fields, which thus resulted in more accurate
ML models.
Data Availability Statement
The Rayleigh-Taylor and PCHIP datasets and ML model weights will be available upon request.
A Model architectures
Table 4: PCHIP impact ML model architecture.
Layer name Output shape # of learnable parameters
Input parameters [6, 1, 1] -
ConvTranspose2d →BatchNorm2d →ReLU [1024, 4, 4] 100,352
ConvTranspose2d →BatchNorm2d →ReLU [1024, 8, 8] 16,779,264
ConvTranspose2d →BatchNorm2d →ReLU [1024, 16, 16] 16,779,264
ConvTranspose2d →BatchNorm2d →ReLU [1024, 32, 32] 16,779,264
ConvTranspose2d →BatchNorm2d →ReLU [1024, 64, 64] 16,779,264
ConvTranspose2d →BatchNorm2d →ReLU [512, 128, 128] 8,389,632
ConvTranspose2d →BatchNorm2d →ReLU [256, 256, 256] 2,097,664
ConvTranspose2d →BatchNorm2d →ReLU [128, 512, 512] 524,544
ConvTranspose2d [6, 1024, 1024] 12,288
Linear map [6, 1024, 1024] 12
Total number of learnable parameters 78,241,548
17

Machine Learning Visualization Tool for Exploring Parameterized Hydrodynamics
Table 5: Double sine wave ML model architecture.
Layer name Output shape # of learnable parameters
Input parameters [4, 1, 1] -
ConvTranspose2d →BatchNorm2d →ReLU [1024, 4, 4] 67,584
ConvTranspose2d →BatchNorm2d →ReLU [1024, 8, 8] 16,779,264
ConvTranspose2d →BatchNorm2d →ReLU [1024, 16, 16] 16,779,264
ConvTranspose2d →BatchNorm2d →ReLU [1024, 32, 32] 16,779,264
ConvTranspose2d →BatchNorm2d →ReLU [1024, 64, 64] 16,779,264
ConvTranspose2d →BatchNorm2d →ReLU [512, 128, 128] 8,389,632
ConvTranspose2d →BatchNorm2d →ReLU [256, 256, 256] 2,097,664
ConvTranspose2d →BatchNorm2d →ReLU [128, 512, 512] 524,544
ConvTranspose2d [3, 1024, 1024] 6,144
Linear map [3, 1024, 1024] 6
Total number of learnable parameters 78,202,630
Table 6: Linear shaped charge ML model architecture.
Layer name Output shape # of learnable parameters
Input parameters [6, 1, 1] -
ConvTranspose2d →BatchNorm2d →ReLU [1024, 4, 13] 321,536
ConvTranspose2d →BatchNorm2d →ReLU [1024, 8, 26] 16,779,264
ConvTranspose2d →BatchNorm2d →ReLU [1024, 16, 52] 16,779,264
ConvTranspose2d →BatchNorm2d →ReLU [1024, 32, 104] 16,779,264
ConvTranspose2d →BatchNorm2d →ReLU [512, 64, 208] 8,389,632
ConvTranspose2d →BatchNorm2d →ReLU [256, 128, 416] 2,097,664
ConvTranspose2d →BatchNorm2d →ReLU [128, 256, 832] 524,544
ConvTranspose2d [128, 512, 1664] 14,343
Linear map [6, 1024, 1024] 14
Total number of learnable parameters 61,685,518
Table 7: Rayleigh Taylor ML model architecture.
Layer name Output shape # of learnable parameters
Input parameters [4, 1, 1] -
ConvTranspose2d →BatchNorm2d →ReLU [1024, 3, 1] 14,336
ConvTranspose2d →BatchNorm2d →ReLU [1024, 6, 2] 16,779,264
ConvTranspose2d →BatchNorm2d →ReLU [1024, 12, 4] 16,779,264
ConvTranspose2d →BatchNorm2d →ReLU [1024, 24, 8] 16,779,264
ConvTranspose2d →BatchNorm2d →ReLU [1024, 48, 16] 16,779,264
ConvTranspose2d →BatchNorm2d →ReLU [512, 96, 32] 8,389,632
ConvTranspose2d →BatchNorm2d →ReLU [256, 192, 64] 2,097,664
ConvTranspose2d →BatchNorm2d →ReLU [128, 384, 128] 524,544
ConvTranspose2d [6, 768, 256] 12,288
Linear map [6, 768, 256] 12
Total number of learnable parameters 78,155,532
18

Machine Learning Visualization Tool for Exploring Parameterized Hydrodynamics
B Model results
B.1 PCHIP impact results
Figure 13: The epoch vs mean absolute error for each field while training the PCHIP impact model.
19

Machine Learning Visualization Tool for Exploring Parameterized Hydrodynamics
Field Ml prediction Simulation Abs. Error
density
0 200 400 600 800 1000
0
200
400
600
800
1000
ML Prediction at -0.18 0.01 0.06 -0.07
0
2
4
6
8
0 200 400 600 800 1000
0
200
400
600
800
1000
Marbl Sim. at -0.18 0.01 0.06 -0.07
0
2
4
6
8
0 200 400 600 800 1000
0
200
400
600
800
1000
Abs. Error at -0.18 0.01 0.06 -0.07
1
2
3
4
5
6
7
8
materials
0 200 400 600 800 1000
0
200
400
600
800
1000
ML Prediction at -0.18 0.01 0.06 -0.07
0
2
4
6
8
10
0 200 400 600 800 1000
0
200
400
600
800
1000
Marbl Sim. at -0.18 0.01 0.06 -0.07
0
2
4
6
8
10
0 200 400 600 800 1000
0
200
400
600
800
1000
Abs. Error at -0.18 0.01 0.06 -0.07
0.2
0.4
0.6
0.8
1.0
1.2
1.4
energy
0 200 400 600 800 1000
0
200
400
600
800
1000
ML Prediction at -0.18 0.01 0.06 -0.07
0.002
0.000
0.002
0.004
0.006
0.008
0.010
0.012
0 200 400 600 800 1000
0
200
400
600
800
1000
Marbl Sim. at -0.18 0.01 0.06 -0.07
2.0
1.5
1.0
0.5
0.0
0.5
1.0
1e 13+1e 6
0 200 400 600 800 1000
0
200
400
600
800
1000
Abs. Error at -0.18 0.01 0.06 -0.07
0.002
0.004
0.006
0.008
0.010
0.012
pressure
0 200 400 600 800 1000
0
200
400
600
800
1000
ML Prediction at -0.18 0.01 0.06 -0.07
0.0035
0.0030
0.0025
0.0020
0.0015
0 200 400 600 800 1000
0
200
400
600
800
1000
Marbl Sim. at -0.18 0.01 0.06 -0.07
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1e 5
0 200 400 600 800 1000
0
200
400
600
800
1000
Abs. Error at -0.18 0.01 0.06 -0.07
0.0015
0.0020
0.0025
0.0030
0.0035
velocity x
0 200 400 600 800 1000
0
200
400
600
800
1000
ML Prediction at -0.18 0.01 0.06 -0.07
0
2
4
6
8
10
0 200 400 600 800 1000
0
200
400
600
800
1000
Marbl Sim. at -0.18 0.01 0.06 -0.07
0
2
4
6
8
10
0 200 400 600 800 1000
0
200
400
600
800
1000
Abs. Error at -0.18 0.01 0.06 -0.07
0.025
0.050
0.075
0.100
0.125
0.150
0.175
0.200
velocity y
0 200 400 600 800 1000
0
200
400
600
800
1000
ML Prediction at -0.18 0.01 0.06 -0.07
0
2
4
6
8
10
0 200 400 600 800 1000
0
200
400
600
800
1000
Marbl Sim. at -0.18 0.01 0.06 -0.07
0
2
4
6
8
10
0 200 400 600 800 1000
0
200
400
600
800
1000
Abs. Error at -0.18 0.01 0.06 -0.07
0.00280
0.00285
0.00290
0.00295
0.00300
0.00305
0.00310
Figure 14: Figure of predictions, truth, and absolute error for linear shaped charge predictions at t=0.
20

Machine Learning Visualization Tool for Exploring Parameterized Hydrodynamics
Field Ml prediction Simulation Abs. Error
density
0 200 400 600 800 1000
0
200
400
600
800
1000
ML Prediction at -0.18 0.01 0.06 -0.07
0
2
4
6
8
0 200 400 600 800 1000
0
200
400
600
800
1000
Marbl Sim. at -0.18 0.01 0.06 -0.07
0
2
4
6
8
0 200 400 600 800 1000
0
200
400
600
800
1000
Abs. Error at -0.18 0.01 0.06 -0.07
1
2
3
4
materials
0 200 400 600 800 1000
0
200
400
600
800
1000
ML Prediction at -0.18 0.01 0.06 -0.07
0
2
4
6
8
10
0 200 400 600 800 1000
0
200
400
600
800
1000
Marbl Sim. at -0.18 0.01 0.06 -0.07
0
2
4
6
8
10
0 200 400 600 800 1000
0
200
400
600
800
1000
Abs. Error at -0.18 0.01 0.06 -0.07
0.0
0.2
0.4
0.6
0.8
1.0
1.2
energy
0 200 400 600 800 1000
0
200
400
600
800
1000
ML Prediction at -0.18 0.01 0.06 -0.07
0.000
0.002
0.004
0.006
0.008
0.010
0.012
0.014
0.016
0 200 400 600 800 1000
0
200
400
600
800
1000
Marbl Sim. at -0.18 0.01 0.06 -0.07
0.01
0.00
0.01
0.02
0.03
0 200 400 600 800 1000
0
200
400
600
800
1000
Abs. Error at -0.18 0.01 0.06 -0.07
0.005
0.010
0.015
0.020
0.025
0.030
pressure
0 200 400 600 800 1000
0
200
400
600
800
1000
ML Prediction at -0.18 0.01 0.06 -0.07
0.00325
0.00300
0.00275
0.00250
0.00225
0.00200
0.00175
0.00150
0 200 400 600 800 1000
0
200
400
600
800
1000
Marbl Sim. at -0.18 0.01 0.06 -0.07
0.001
0.000
0.001
0.002
0.003
0 200 400 600 800 1000
0
200
400
600
800
1000
Abs. Error at -0.18 0.01 0.06 -0.07
0.001
0.002
0.003
0.004
0.005
0.006
velocity x
0 200 400 600 800 1000
0
200
400
600
800
1000
ML Prediction at -0.18 0.01 0.06 -0.07
0
2
4
6
8
10
0 200 400 600 800 1000
0
200
400
600
800
1000
Marbl Sim. at -0.18 0.01 0.06 -0.07
0
2
4
6
8
10
0 200 400 600 800 1000
0
200
400
600
800
1000
Abs. Error at -0.18 0.01 0.06 -0.07
0.05
0.10
0.15
0.20
0.25
velocity y
0 200 400 600 800 1000
0
200
400
600
800
1000
ML Prediction at -0.18 0.01 0.06 -0.07
0
2
4
6
8
10
0 200 400 600 800 1000
0
200
400
600
800
1000
Marbl Sim. at -0.18 0.01 0.06 -0.07
0
2
4
6
8
10
0 200 400 600 800 1000
0
200
400
600
800
1000
Abs. Error at -0.18 0.01 0.06 -0.07
0.02
0.04
0.06
0.08
Figure 15: Figure of predictions, truth, and absolute error for linear shaped charge predictions at t=7.
21

Machine Learning Visualization Tool for Exploring Parameterized Hydrodynamics
Field Ml prediction Simulation Abs. Error
density
0 200 400 600 800 1000
0
200
400
600
800
1000
ML Prediction at -0.18 0.01 0.06 -0.07
0
2
4
6
8
0 200 400 600 800 1000
0
200
400
600
800
1000
Marbl Sim. at -0.18 0.01 0.06 -0.07
0
2
4
6
8
0 200 400 600 800 1000
0
200
400
600
800
1000
Abs. Error at -0.18 0.01 0.06 -0.07
1
2
3
4
5
6
7
materials
0 200 400 600 800 1000
0
200
400
600
800
1000
ML Prediction at -0.18 0.01 0.06 -0.07
0
2
4
6
8
10
0 200 400 600 800 1000
0
200
400
600
800
1000
Marbl Sim. at -0.18 0.01 0.06 -0.07
0
2
4
6
8
10
0 200 400 600 800 1000
0
200
400
600
800
1000
Abs. Error at -0.18 0.01 0.06 -0.07
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
energy
0 200 400 600 800 1000
0
200
400
600
800
1000
ML Prediction at -0.18 0.01 0.06 -0.07
0.0000
0.0025
0.0050
0.0075
0.0100
0.0125
0.0150
0 200 400 600 800 1000
0
200
400
600
800
1000
Marbl Sim. at -0.18 0.01 0.06 -0.07
0.005
0.000
0.005
0.010
0.015
0.020
0.025
0.030
0.035
0 200 400 600 800 1000
0
200
400
600
800
1000
Abs. Error at -0.18 0.01 0.06 -0.07
0.005
0.010
0.015
0.020
0.025
0.030
pressure
0 200 400 600 800 1000
0
200
400
600
800
1000
ML Prediction at -0.18 0.01 0.06 -0.07
0.0035
0.0030
0.0025
0.0020
0.0015
0 200 400 600 800 1000
0
200
400
600
800
1000
Marbl Sim. at -0.18 0.01 0.06 -0.07
0.002
0.000
0.002
0.004
0.006
0 200 400 600 800 1000
0
200
400
600
800
1000
Abs. Error at -0.18 0.01 0.06 -0.07
0.002
0.004
0.006
0.008
velocity x
0 200 400 600 800 1000
0
200
400
600
800
1000
ML Prediction at -0.18 0.01 0.06 -0.07
0
2
4
6
8
10
0 200 400 600 800 1000
0
200
400
600
800
1000
Marbl Sim. at -0.18 0.01 0.06 -0.07
0
2
4
6
8
10
0 200 400 600 800 1000
0
200
400
600
800
1000
Abs. Error at -0.18 0.01 0.06 -0.07
0.05
0.10
0.15
0.20
0.25
velocity y
0 200 400 600 800 1000
0
200
400
600
800
1000
ML Prediction at -0.18 0.01 0.06 -0.07
0
2
4
6
8
10
0 200 400 600 800 1000
0
200
400
600
800
1000
Marbl Sim. at -0.18 0.01 0.06 -0.07
0
2
4
6
8
10
0 200 400 600 800 1000
0
200
400
600
800
1000
Abs. Error at -0.18 0.01 0.06 -0.07
0.02
0.04
0.06
0.08
0.10
Figure 16: Figure of predictions, truth, and absolute error for linear shaped charge predictions at t=15.
22

Machine Learning Visualization Tool for Exploring Parameterized Hydrodynamics
B.2 Double sine wave results
Figure 17: The epoch vs mean absolute error for each field while training the double sine wave model.
23

Machine Learning Visualization Tool for Exploring Parameterized Hydrodynamics
Field Ml prediction Simulation Abs. Error
density
0 200 400 600 800 1000
0
200
400
600
800
1000
ML Prediction at 0.15 18.12 1.94 0.00
0
2
4
6
8
0 200 400 600 800 1000
0
200
400
600
800
1000
Marbl Sim. at 0.15 18.12 1.94 0.00
0
2
4
6
8
0 200 400 600 800 1000
0
200
400
600
800
1000
Abs. Error at 0.15 18.12 1.94 0.00
1
2
3
4
5
6
7
velocity x
0 200 400 600 800 1000
0
200
400
600
800
1000
ML Prediction at 0.15 18.12 1.94 0.00
0
2
4
6
8
10
0 200 400 600 800 1000
0
200
400
600
800
1000
Marbl Sim. at 0.15 18.12 1.94 0.00
0
2
4
6
8
10
0 200 400 600 800 1000
0
200
400
600
800
1000
Abs. Error at 0.15 18.12 1.94 0.00
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
velocity y
0 200 400 600 800 1000
0
200
400
600
800
1000
ML Prediction at 0.15 18.12 1.94 0.00
0
2
4
6
8
10
0 200 400 600 800 1000
0
200
400
600
800
1000
Marbl Sim. at 0.15 18.12 1.94 0.00
0
2
4
6
8
10
0 200 400 600 800 1000
0
200
400
600
800
1000
Abs. Error at 0.15 18.12 1.94 0.00
0.00
0.02
0.04
0.06
0.08
Figure 18: Figure of predictions, truth, and absolute error for double sine wave predictions at t=0.
24

Machine Learning Visualization Tool for Exploring Parameterized Hydrodynamics
Field Ml prediction Simulation Abs. Error
density
0 200 400 600 800 1000
0
200
400
600
800
1000
ML Prediction at 0.15 18.12 1.94 3.50
0
2
4
6
8
0 200 400 600 800 1000
0
200
400
600
800
1000
Marbl Sim. at 0.15 18.12 1.94 3.50
0
2
4
6
8
0 200 400 600 800 1000
0
200
400
600
800
1000
Abs. Error at 0.15 18.12 1.94 3.50
1
2
3
4
5
6
7
velocity x
0 200 400 600 800 1000
0
200
400
600
800
1000
ML Prediction at 0.15 18.12 1.94 3.50
0
2
4
6
8
10
0 200 400 600 800 1000
0
200
400
600
800
1000
Marbl Sim. at 0.15 18.12 1.94 3.50
0
2
4
6
8
10
0 200 400 600 800 1000
0
200
400
600
800
1000
Abs. Error at 0.15 18.12 1.94 3.50
0.025
0.050
0.075
0.100
0.125
0.150
0.175
0.200
velocity y
0 200 400 600 800 1000
0
200
400
600
800
1000
ML Prediction at 0.15 18.12 1.94 3.50
0
2
4
6
8
10
0 200 400 600 800 1000
0
200
400
600
800
1000
Marbl Sim. at 0.15 18.12 1.94 3.50
0
2
4
6
8
10
0 200 400 600 800 1000
0
200
400
600
800
1000
Abs. Error at 0.15 18.12 1.94 3.50
0.00
0.02
0.04
0.06
0.08
0.10
0.12
Figure 19: Figure of predictions, truth, and absolute error for double sine wave predictions at t=3.5.
25

Machine Learning Visualization Tool for Exploring Parameterized Hydrodynamics
Field Ml prediction Simulation Abs. Error
density
0 200 400 600 800 1000
0
200
400
600
800
1000
ML Prediction at 0.15 18.12 1.94 7.00
0
2
4
6
8
0 200 400 600 800 1000
0
200
400
600
800
1000
Marbl Sim. at 0.15 18.12 1.94 7.00
0
2
4
6
8
0 200 400 600 800 1000
0
200
400
600
800
1000
Abs. Error at 0.15 18.12 1.94 7.00
1
2
3
4
5
6
7
8
velocity x
0 200 400 600 800 1000
0
200
400
600
800
1000
ML Prediction at 0.15 18.12 1.94 7.00
0
2
4
6
8
10
0 200 400 600 800 1000
0
200
400
600
800
1000
Marbl Sim. at 0.15 18.12 1.94 7.00
0
2
4
6
8
10
0 200 400 600 800 1000
0
200
400
600
800
1000
Abs. Error at 0.15 18.12 1.94 7.00
0.0
0.2
0.4
0.6
0.8
1.0
velocity y
0 200 400 600 800 1000
0
200
400
600
800
1000
ML Prediction at 0.15 18.12 1.94 7.00
0
2
4
6
8
10
0 200 400 600 800 1000
0
200
400
600
800
1000
Marbl Sim. at 0.15 18.12 1.94 7.00
0
2
4
6
8
10
0 200 400 600 800 1000
0
200
400
600
800
1000
Abs. Error at 0.15 18.12 1.94 7.00
0.0
0.2
0.4
0.6
0.8
Figure 20: Figure of predictions, truth, and absolute error for double sine wave predictions at t=7.
26

Machine Learning Visualization Tool for Exploring Parameterized Hydrodynamics
B.3 Linear shaped charge results
Figure 21: The epoch vs mean absolute error for each field while training the linear shaped charge model.
27

Machine Learning Visualization Tool for Exploring Parameterized Hydrodynamics
Field Ml prediction Simulation Abs. Error
density copper (Cu)
0 200 400 600 800 1000 1200 1400 1600
0
200
400
ML Prediction at 0.28 0.09 0.27 0.23
0.0
2.5
5.0
7.5
10.0
0 200 400 600 800 1000 1200 1400 1600
0
200
400
Marbl Sim. at 0.28 0.09 0.27 0.23
0.0
2.5
5.0
7.5
10.0
0 200 400 600 800 1000 1200 1400 1600
0
200
400
Abs. Error at 0.28 0.09 0.27 0.23
0
2
4
6
8
density
0 200 400 600 800 1000 1200 1400 1600
0
200
400
ML Prediction at 0.28 0.09 0.27 0.23
0.0
2.5
5.0
7.5
10.0
0 200 400 600 800 1000 1200 1400 1600
0
200
400
Marbl Sim. at 0.28 0.09 0.27 0.23
0.0
2.5
5.0
7.5
10.0
0 200 400 600 800 1000 1200 1400 1600
0
200
400
Abs. Error at 0.28 0.09 0.27 0.23
0
2
4
6
energy
0 200 400 600 800 1000 1200 1400 1600
0
200
400
ML Prediction at 0.28 0.09 0.27 0.23
0.05
0.00
0.05
0.10
0.15
0 200 400 600 800 1000 1200 1400 1600
0
200
400
Marbl Sim. at 0.28 0.09 0.27 0.23
0.05
0.00
0.05
0.10
0 200 400 600 800 1000 1200 1400 1600
0
200
400
Abs. Error at 0.28 0.09 0.27 0.23
0.05
0.10
0.15
pressure
0 200 400 600 800 1000 1200 1400 1600
0
200
400
ML Prediction at 0.28 0.09 0.27 0.23
0.00
0.05
0.10
0.15
0 200 400 600 800 1000 1200 1400 1600
0
200
400
Marbl Sim. at 0.28 0.09 0.27 0.23
0.1
0.0
0.1
0.2
0 200 400 600 800 1000 1200 1400 1600
0
200
400
Abs. Error at 0.28 0.09 0.27 0.23
0.05
0.10
0.15
0.20
velocity x
0 200 400 600 800 1000 1200 1400 1600
0
200
400
ML Prediction at 0.28 0.09 0.27 0.23
0
2
4
6
8
0 200 400 600 800 1000 1200 1400 1600
0
200
400
Marbl Sim. at 0.28 0.09 0.27 0.23
0
2
4
6
8
0 200 400 600 800 1000 1200 1400 1600
0
200
400
Abs. Error at 0.28 0.09 0.27 0.23
0.05
0.10
0.15
velocity y
0 200 400 600 800 1000 1200 1400 1600
0
200
400
ML Prediction at 0.28 0.09 0.27 0.23
0
2
4
6
8
0 200 400 600 800 1000 1200 1400 1600
0
200
400
Marbl Sim. at 0.28 0.09 0.27 0.23
0
2
4
6
8
0 200 400 600 800 1000 1200 1400 1600
0
200
400
Abs. Error at 0.28 0.09 0.27 0.23
0.05
0.10
0.15
volume fraction Cu
0 200 400 600 800 1000 1200 1400 1600
0
200
400
ML Prediction at 0.28 0.09 0.27 0.23
0.0
0.2
0.4
0.6
0.8
1.0
0 200 400 600 800 1000 1200 1400 1600
0
200
400
Marbl Sim. at 0.28 0.09 0.27 0.23
0.0
0.2
0.4
0.6
0.8
1.0
0 200 400 600 800 1000 1200 1400 1600
0
200
400
Abs. Error at 0.28 0.09 0.27 0.23
0.2
0.4
0.6
0.8
Figure 22: Figure of predictions, truth, and absolute error for linear shaped charge predictions at t=0.
28

Machine Learning Visualization Tool for Exploring Parameterized Hydrodynamics
Field Ml prediction Simulation Abs. Error
density copper (Cu)
0 200 400 600 800 1000 1200 1400 1600
0
200
400
ML Prediction at 0.28 0.09 0.27 0.23
0
2
4
6
8
0 200 400 600 800 1000 1200 1400 1600
0
200
400
Marbl Sim. at 0.28 0.09 0.27 0.23
0
2
4
6
8
0 200 400 600 800 1000 1200 1400 1600
0
200
400
Abs. Error at 0.28 0.09 0.27 0.23
2
4
6
8
density
0 200 400 600 800 1000 1200 1400 1600
0
200
400
ML Prediction at 0.28 0.09 0.27 0.23
0
2
4
6
8
0 200 400 600 800 1000 1200 1400 1600
0
200
400
Marbl Sim. at 0.28 0.09 0.27 0.23
0
2
4
6
8
0 200 400 600 800 1000 1200 1400 1600
0
200
400
Abs. Error at 0.28 0.09 0.27 0.23
2
4
6
energy
0 200 400 600 800 1000 1200 1400 1600
0
200
400
ML Prediction at 0.28 0.09 0.27 0.23
0.00
0.05
0.10
0.15
0 200 400 600 800 1000 1200 1400 1600
0
200
400
Marbl Sim. at 0.28 0.09 0.27 0.23
0.00
0.05
0.10
0.15
0.20
0 200 400 600 800 1000 1200 1400 1600
0
200
400
Abs. Error at 0.28 0.09 0.27 0.23
0.05
0.10
0.15
pressure
0 200 400 600 800 1000 1200 1400 1600
0
200
400
ML Prediction at 0.28 0.09 0.27 0.23
0.00
0.05
0.10
0 200 400 600 800 1000 1200 1400 1600
0
200
400
Marbl Sim. at 0.28 0.09 0.27 0.23
0.00
0.01
0.02
0 200 400 600 800 1000 1200 1400 1600
0
200
400
Abs. Error at 0.28 0.09 0.27 0.23
0.05
0.10
velocity x
0 200 400 600 800 1000 1200 1400 1600
0
200
400
ML Prediction at 0.28 0.09 0.27 0.23
0
2
4
6
8
0 200 400 600 800 1000 1200 1400 1600
0
200
400
Marbl Sim. at 0.28 0.09 0.27 0.23
0
2
4
6
8
0 200 400 600 800 1000 1200 1400 1600
0
200
400
Abs. Error at 0.28 0.09 0.27 0.23
0.1
0.2
0.3
0.4
velocity y
0 200 400 600 800 1000 1200 1400 1600
0
200
400
ML Prediction at 0.28 0.09 0.27 0.23
0
2
4
6
8
0 200 400 600 800 1000 1200 1400 1600
0
200
400
Marbl Sim. at 0.28 0.09 0.27 0.23
0
2
4
6
8
0 200 400 600 800 1000 1200 1400 1600
0
200
400
Abs. Error at 0.28 0.09 0.27 0.23
0.05
0.10
0.15
0.20
volume fraction Cu
0 200 400 600 800 1000 1200 1400 1600
0
200
400
ML Prediction at 0.28 0.09 0.27 0.23
0.0
0.2
0.4
0.6
0.8
1.0
0 200 400 600 800 1000 1200 1400 1600
0
200
400
Marbl Sim. at 0.28 0.09 0.27 0.23
0.0
0.2
0.4
0.6
0.8
1.0
0 200 400 600 800 1000 1200 1400 1600
0
200
400
Abs. Error at 0.28 0.09 0.27 0.23
0.2
0.4
0.6
0.8
Figure 23: Figure of predictions, truth, and absolute error for linear shaped charge predictions at t=0.
29

Machine Learning Visualization Tool for Exploring Parameterized Hydrodynamics
Field Ml prediction Simulation Abs. Error
density copper (Cu)
0 200 400 600 800 1000 1200 1400 1600
0
200
400
ML Prediction at 0.28 0.09 0.27 0.23
0
2
4
6
8
0 200 400 600 800 1000 1200 1400 1600
0
200
400
Marbl Sim. at 0.28 0.09 0.27 0.23
0
2
4
6
8
0 200 400 600 800 1000 1200 1400 1600
0
200
400
Abs. Error at 0.28 0.09 0.27 0.23
2
4
6
8
density
0 200 400 600 800 1000 1200 1400 1600
0
200
400
ML Prediction at 0.28 0.09 0.27 0.23
0
2
4
6
8
0 200 400 600 800 1000 1200 1400 1600
0
200
400
Marbl Sim. at 0.28 0.09 0.27 0.23
0
2
4
6
8
0 200 400 600 800 1000 1200 1400 1600
0
200
400
Abs. Error at 0.28 0.09 0.27 0.23
2
4
6
energy
0 200 400 600 800 1000 1200 1400 1600
0
200
400
ML Prediction at 0.28 0.09 0.27 0.23
0.0
0.1
0.2
0.3
0 200 400 600 800 1000 1200 1400 1600
0
200
400
Marbl Sim. at 0.28 0.09 0.27 0.23
0.0
0.1
0.2
0.3
0 200 400 600 800 1000 1200 1400 1600
0
200
400
Abs. Error at 0.28 0.09 0.27 0.23
0.1
0.2
0.3
pressure
0 200 400 600 800 1000 1200 1400 1600
0
200
400
ML Prediction at 0.28 0.09 0.27 0.23
0.00
0.05
0.10
0 200 400 600 800 1000 1200 1400 1600
0
200
400
Marbl Sim. at 0.28 0.09 0.27 0.23
0.00
0.02
0.04
0 200 400 600 800 1000 1200 1400 1600
0
200
400
Abs. Error at 0.28 0.09 0.27 0.23
0.05
0.10
velocity x
0 200 400 600 800 1000 1200 1400 1600
0
200
400
ML Prediction at 0.28 0.09 0.27 0.23
0
2
4
6
8
0 200 400 600 800 1000 1200 1400 1600
0
200
400
Marbl Sim. at 0.28 0.09 0.27 0.23
0
2
4
6
8
0 200 400 600 800 1000 1200 1400 1600
0
200
400
Abs. Error at 0.28 0.09 0.27 0.23
0.1
0.2
0.3
0.4
velocity y
0 200 400 600 800 1000 1200 1400 1600
0
200
400
ML Prediction at 0.28 0.09 0.27 0.23
0
2
4
6
8
0 200 400 600 800 1000 1200 1400 1600
0
200
400
Marbl Sim. at 0.28 0.09 0.27 0.23
0
2
4
6
8
0 200 400 600 800 1000 1200 1400 1600
0
200
400
Abs. Error at 0.28 0.09 0.27 0.23
0.025
0.050
0.075
0.100
volume fraction Cu
0 200 400 600 800 1000 1200 1400 1600
0
200
400
ML Prediction at 0.28 0.09 0.27 0.23
0.0
0.2
0.4
0.6
0.8
1.0
0 200 400 600 800 1000 1200 1400 1600
0
200
400
Marbl Sim. at 0.28 0.09 0.27 0.23
0.0
0.2
0.4
0.6
0.8
1.0
0 200 400 600 800 1000 1200 1400 1600
0
200
400
Abs. Error at 0.28 0.09 0.27 0.23
0.2
0.4
0.6
0.8
Figure 24: Figure of predictions, truth, and absolute error for linear shaped charge predictions at t=0.
30

Machine Learning Visualization Tool for Exploring Parameterized Hydrodynamics
B.4 Rayleigh-Taylor results
The mean absolute error for each field separately in training the Rayleigh Taylor simulation is shown in Figure 25. The
final fields errors (from highest to lowest) are density, energy, materials, velocity
y
, velocity
x
, and pressure. The error
in density was roughly an order of magnitude larger than pressure and velocity x.
Figure 25: The epoch vs mean absolute error for each field while training the Rayleigh-Taylor model.
Samples of ML predictions for multiple fields are plotted in Figures 26-28 with the simulation results and the absolute
error (between simulation and prediction). This was a simulation within the training set, that is representative of the
general trends within the Rayleigh-Taylor results. Essentially at early to middle times (Figure 26 and Figure 27) we see
the ML predictions are excellent at tracking the Rayleigh-Taylor instability. The errors are largest at the interface, which
is generally tracked well. However, at late times as shown in Figure 28, a large number of eddies are formed along
the interface of the two gases where the ML predictions are less accurate at maintaining a crisp interface. For these
high detail and complex predictions, it appears as if the ML model is locally averaging or smoothing the complicated
interface.
31

Machine Learning Visualization Tool for Exploring Parameterized Hydrodynamics
density velocity xvelocity ypressure energy materials
ML prediction
0 200
0
100
200
300
400
500
600
700
ML Prediction at 1.54 1.41 5.33 0.00
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
0 200
0
100
200
300
400
500
600
700
ML Prediction at 1.54 1.41 5.33 0.00
0.002
0.001
0.000
0.001
0.002
0.003
0 200
0
100
200
300
400
500
600
700
ML Prediction at 1.54 1.41 5.33 0.00
0.00
0.01
0.02
0.03
0.04
0.05
0 200
0
100
200
300
400
500
600
700
ML Prediction at 1.54 1.41 5.33 0.00
0.62
0.64
0.66
0.68
0.70
0.72
0.74
0.76
0 200
0
100
200
300
400
500
600
700
ML Prediction at 1.54 1.41 5.33 0.00
1.0
1.2
1.4
1.6
1.8
0 200
0
100
200
300
400
500
600
700
ML Prediction at 1.54 1.41 5.33 0.00
1.0
1.2
1.4
1.6
1.8
2.0
Simulation
0 200
0
100
200
300
400
500
600
700
Marbl Sim. at 1.54 1.41 5.33 0.00
0.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
0 200
0
100
200
300
400
500
600
700
Marbl Sim. at 1.54 1.41 5.33 0.00
0.002
0.001
0.000
0.001
0.002
0.003
0 200
0
100
200
300
400
500
600
700
Marbl Sim. at 1.54 1.41 5.33 0.00
0.00
0.01
0.02
0.03
0.04
0.05
0 200
0
100
200
300
400
500
600
700
Marbl Sim. at 1.54 1.41 5.33 0.00
0.64
0.66
0.68
0.70
0.72
0.74
0 200
0
100
200
300
400
500
600
700
Marbl Sim. at 1.54 1.41 5.33 0.00
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
0 200
0
100
200
300
400
500
600
700
Marbl Sim. at 1.54 1.41 5.33 0.00
1.0
1.2
1.4
1.6
1.8
2.0
Abs. Error
0 200
0
100
200
300
400
500
600
700
Abs. Error at 1.54 1.41 5.33 0.00
0.00
0.05
0.10
0.15
0.20
0 200
0
100
200
300
400
500
600
700
Abs. Error at 1.54 1.41 5.33 0.00
0.0005
0.0010
0.0015
0.0020
0.0025
0.0030
0 200
0
100
200
300
400
500
600
700
Abs. Error at 1.54 1.41 5.33 0.00
0.005
0.010
0.015
0.020
0 200
0
100
200
300
400
500
600
700
Abs. Error at 1.54 1.41 5.33 0.00
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0 200
0
100
200
300
400
500
600
700
Abs. Error at 1.54 1.41 5.33 0.00
0.05
0.10
0.15
0.20
0.25
0.30
0 200
0
100
200
300
400
500
600
700
Abs. Error at 1.54 1.41 5.33 0.00
0.0
0.1
0.2
0.3
0.4
Figure 26: Figure of predictions, truth, and absolute error for Rayleigh-Taylor predictions at t=0.
32

Machine Learning Visualization Tool for Exploring Parameterized Hydrodynamics
density velocity xvelocity ypressure energy materials
ML prediction
0 200
0
100
200
300
400
500
600
700
ML Prediction at 1.54 1.41 5.33 5.00
1.0
1.1
1.2
1.3
1.4
1.5
1.6
0 200
0
100
200
300
400
500
600
700
ML Prediction at 1.54 1.41 5.33 5.00
0.06
0.04
0.02
0.00
0.02
0.04
0.06
0 200
0
100
200
300
400
500
600
700
ML Prediction at 1.54 1.41 5.33 5.00
0.075
0.050
0.025
0.000
0.025
0.050
0.075
0.100
0 200
0
100
200
300
400
500
600
700
ML Prediction at 1.54 1.41 5.33 5.00
0.62
0.64
0.66
0.68
0.70
0.72
0.74
0.76
0 200
0
100
200
300
400
500
600
700
ML Prediction at 1.54 1.41 5.33 5.00
1.0
1.2
1.4
1.6
1.8
0 200
0
100
200
300
400
500
600
700
ML Prediction at 1.54 1.41 5.33 5.00
1.0
1.2
1.4
1.6
1.8
2.0
Simulation
0 200
0
100
200
300
400
500
600
700
Marbl Sim. at 1.54 1.41 5.33 5.00
1.0
1.1
1.2
1.3
1.4
1.5
1.6
0 200
0
100
200
300
400
500
600
700
Marbl Sim. at 1.54 1.41 5.33 5.00
0.06
0.04
0.02
0.00
0.02
0.04
0.06
0 200
0
100
200
300
400
500
600
700
Marbl Sim. at 1.54 1.41 5.33 5.00
0.075
0.050
0.025
0.000
0.025
0.050
0.075
0.100
0 200
0
100
200
300
400
500
600
700
Marbl Sim. at 1.54 1.41 5.33 5.00
0.67
0.68
0.69
0.70
0.71
0.72
0.73
0 200
0
100
200
300
400
500
600
700
Marbl Sim. at 1.54 1.41 5.33 5.00
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
0 200
0
100
200
300
400
500
600
700
Marbl Sim. at 1.54 1.41 5.33 5.00
1.0
1.2
1.4
1.6
1.8
2.0
Abs. Error
0 200
0
100
200
300
400
500
600
700
Abs. Error at 1.54 1.41 5.33 5.00
0.0
0.1
0.2
0.3
0.4
0.5
0 200
0
100
200
300
400
500
600
700
Abs. Error at 1.54 1.41 5.33 5.00
0.01
0.02
0.03
0.04
0 200
0
100
200
300
400
500
600
700
Abs. Error at 1.54 1.41 5.33 5.00
0.01
0.02
0.03
0.04
0.05
0.06
0 200
0
100
200
300
400
500
600
700
Abs. Error at 1.54 1.41 5.33 5.00
0.01
0.02
0.03
0.04
0.05
0 200
0
100
200
300
400
500
600
700
Abs. Error at 1.54 1.41 5.33 5.00
0.1
0.2
0.3
0.4
0.5
0.6
0 200
0
100
200
300
400
500
600
700
Abs. Error at 1.54 1.41 5.33 5.00
0.0
0.2
0.4
0.6
0.8
Figure 27: Figure of predictions, truth, and absolute error for Rayleigh-Taylor predictions at t=5.
density velocity xvelocity ypressure energy materials
ML prediction
0 200
0
100
200
300
400
500
600
700
ML Prediction at 1.54 1.41 5.33 10.00
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
0 200
0
100
200
300
400
500
600
700
ML Prediction at 1.54 1.41 5.33 10.00
0.10
0.05
0.00
0.05
0.10
0 200
0
100
200
300
400
500
600
700
ML Prediction at 1.54 1.41 5.33 10.00
0.2
0.1
0.0
0.1
0.2
0 200
0
100
200
300
400
500
600
700
ML Prediction at 1.54 1.41 5.33 10.00
0.62
0.64
0.66
0.68
0.70
0.72
0.74
0.76
0 200
0
100
200
300
400
500
600
700
ML Prediction at 1.54 1.41 5.33 10.00
1.0
1.2
1.4
1.6
1.8
0 200
0
100
200
300
400
500
600
700
ML Prediction at 1.54 1.41 5.33 10.00
1.0
1.2
1.4
1.6
1.8
2.0
Simulation
0 200
0
100
200
300
400
500
600
700
Marbl Sim. at 1.54 1.41 5.33 10.00
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
0 200
0
100
200
300
400
500
600
700
Marbl Sim. at 1.54 1.41 5.33 10.00
0.10
0.05
0.00
0.05
0.10
0 200
0
100
200
300
400
500
600
700
Marbl Sim. at 1.54 1.41 5.33 10.00
0.2
0.1
0.0
0.1
0.2
0 200
0
100
200
300
400
500
600
700
Marbl Sim. at 1.54 1.41 5.33 10.00
0.64
0.66
0.68
0.70
0.72
0.74
0.76
0.78
0 200
0
100
200
300
400
500
600
700
Marbl Sim. at 1.54 1.41 5.33 10.00
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
0 200
0
100
200
300
400
500
600
700
Marbl Sim. at 1.54 1.41 5.33 10.00
1.0
1.2
1.4
1.6
1.8
2.0
Abs. Error
0 200
0
100
200
300
400
500
600
700
Abs. Error at 1.54 1.41 5.33 10.00
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0 200
0
100
200
300
400
500
600
700
Abs. Error at 1.54 1.41 5.33 10.00
0.02
0.04
0.06
0.08
0 200
0
100
200
300
400
500
600
700
Abs. Error at 1.54 1.41 5.33 10.00
0.02
0.04
0.06
0.08
0 200
0
100
200
300
400
500
600
700
Abs. Error at 1.54 1.41 5.33 10.00
0.01
0.02
0.03
0.04
0.05
0 200
0
100
200
300
400
500
600
700
Abs. Error at 1.54 1.41 5.33 10.00
0.1
0.2
0.3
0.4
0.5
0.6
0 200
0
100
200
300
400
500
600
700
Abs. Error at 1.54 1.41 5.33 10.00
0.0
0.2
0.4
0.6
0.8
Figure 28: Figure of predictions, truth, and absolute error for Rayleigh-Taylor predictions at t=10.
33

Machine Learning Visualization Tool for Exploring Parameterized Hydrodynamics
References
[1]
James Ahrens, Sebastien Jourdain, Patrick O’Leary, John Patchett, David H. Rogers, and Mark Petersen. An
image-based approach to extreme scale in situ visualization and analysis. In SC ’14: Proceedings of the
International Conference for High Performance Computing, Networking, Storage and Analysis, pages 424–434,
2014. doi:10.1109/SC.2014.40.
[2]
Qian Chen, Li Li, Yousheng Zhang, and Baolin Tian. Effects of the atwood number on the richtmyer-meshkov
instability in elastic-plastic media. Phys. Rev. E, 99:053102, May 2019.
[3]
Hye-Sook Park, K. T. Lorenz, R. M. Cavallo, S. M. Pollaine, S. T. Prisbrey, R. E. Rudd, R. C. Becker, J. V. Bernier,
and B. A. Remington. Viscous rayleigh-taylor instability experiments at high pressure and strain rate. Phys. Rev.
Lett., 104:135504, Apr 2010. doi:10.1103/PhysRevLett.104.135504. URL
https://link.aps.org/doi/10.
1103/PhysRevLett.104.135504.
[4]
Z. Sternberger, B. R. Maddox, Y. P. Opachich, C. E. Wehrenberg, R. G. Kraus, B. A. Remington, G. C. Randall,
M. Farrell, and G. Ravichandran. A comparative study of rayleigh-taylor and richtmyer-meshkov instabilities
in 2d and 3d in tantalum. AIP Conference Proceedings, 1793(1):110006, 2017. doi:10.1063/1.4971669. URL
https://aip.scitation.org/doi/abs/10.1063/1.4971669.
[5]
W. T. Buttler, D. M. Oró, D. L. Preston, K. O. Mikaelian, F. J. Cherne, R. S. Hixson, F. G. Mariam, C. Morris,
J. B. Stone, G. Terrones, and et al. Unstable richtmyer–meshkov growth of solid and liquid metals in vacuum.
Journal of Fluid Mechanics, 703:60–84, 2012. doi:10.1017/jfm.2012.190.
[6]
Guy Dimonte, G. Terrones, F. J. Cherne, T. C. Germann, V. Dupont, K. Kadau, W. T. Buttler, D. M. Oro, C. Morris,
and D. L. Preston. Use of the richtmyer-meshkov instability to infer yield stress at high-energy densities. Phys.
Rev. Lett., 107:264502, Dec 2011. doi:10.1103/PhysRevLett.107.264502. URL
https://link.aps.org/doi/
10.1103/PhysRevLett.107.264502.
[7]
Jonathan L Belof, R. M. Cavallo, Russel T. Olson, Robert S. King, George Thompson Gray III, David Holtkamp,
Shuh-Rong Chen, R. E. Rudd, Nathan Barton, Athanasios Arsenlis, B. Remington, Hye-Sook Park, S. T. Prisbrey,
Peter Vitello, Grant Bazan, Karnig O. Mikaelian, A. J. Comley, Brian Maddox, and M. J. May. Rayleigh-taylor
strength experiments of the pressure-induced phase transition in iron. AIP Conference Proceedings, 1426(1):1521–
1524, 2012. doi:10.1063/1.3686572. URL https://aip.scitation.org/doi/abs/10.1063/1.3686572.
[8] A. B. Zylstra, O. A. Hurricane, D. A. Callahan, A. L. Kritcher, J. E. Ralph, H. F. Robey, J. S. Ross, C. V. Young,
K. L. Baker, D. T. Casey, T. Döppner, L. Divol, M. Hohenberger, S. Le Pape, A. Pak, P. K. Patel, R. Tommasini,
S. J. Ali, P. A. Amendt, L. J. Atherton, B. Bachmann, D. Bailey, L. R. Benedetti, L. Berzak Hopkins, R. Betti,
S. D. Bhandarkar, J. Biener, R. M. Bionta, N. W. Birge, E. J. Bond, D. K. Bradley, T. Braun, T. M. Briggs,
M. W. Bruhn, P. M. Celliers, B. Chang, T. Chapman, H. Chen, C. Choate, A. R. Christopherson, D. S. Clark,
J. W. Crippen, E. L. Dewald, T. R. Dittrich, M. J. Edwards, W. A. Farmer, J. E. Field, D. Fittinghoff, J. Frenje,
J. Gaffney, M. Gatu Johnson, S. H. Glenzer, G. P. Grim, S. Haan, K. D. Hahn, G. N. Hall, B. A. Hammel, J. Harte,
E. Hartouni, J. E. Heebner, V. J. Hernandez, H. Herrmann, M. C. Herrmann, D. E. Hinkel, D. D. Ho, J. P. Holder,
W. W. Hsing, H. Huang, K. D. Humbird, N. Izumi, L. C. Jarrott, J. Jeet, O. Jones, G. D. Kerbel, S. M. Kerr,
S. F. Khan, J. Kilkenny, Y. Kim, H. Geppert Kleinrath, V. Geppert Kleinrath, C. Kong, J. M. Koning, J. J. Kroll,
M. K. G. Kruse, B. Kustowski, O. L. Landen, S. Langer, D. Larson, N. C. Lemos, J. D. Lindl, T. Ma, M. J.
MacDonald, B. J. MacGowan, A. J. Mackinnon, S. A. MacLaren, A. G. MacPhee, M. M. Marinak, D. A. Mariscal,
E. V. Marley, L. Masse, K. Meaney, N. B. Meezan, P. A. Michel, M. Millot, J. L. Milovich, J. D. Moody, A. S.
Moore, J. W. Morton, T. Murphy, K. Newman, J. M. G. Di Nicola, A. Nikroo, R. Nora, M. V. Patel, L. J. Pelz,
J. L. Peterson, Y. Ping, B. B. Pollock, M. Ratledge, N. G. Rice, H. Rinderknecht, M. Rosen, M. S. Rubery, J. D.
Salmonson, J. Sater, S. Schiaffino, D. J. Schlossberg, M. B. Schneider, C. R. Schroeder, H. A. Scott, S. M. Sepke,
K. Sequoia, M. W. Sherlock, S. Shin, V. A. Smalyuk, B. K. Spears, P. T. Springer, M. Stadermann, S. Stoupin,
D. J. Strozzi, L. J. Suter, C. A. Thomas, R. P. J. Town, E. R. Tubman, C. Trosseille, P. L. Volegov, C. R. Weber,
K. Widmann, C. Wild, C. H. Wilde, B. M. Van Wonterghem, D. T. Woods, B. N. Woodworth, M. Yamaguchi,
S. T. Yang, and G. B. Zimmerman. Burning plasma achieved in inertial fusion. Nature, 601(7894):542–548, 2022.
doi:10.1038/s41586-021-04281-w. URL https://doi.org/10.1038/s41586-021-04281-w.
[9]
T.R. Desjardins, C.A. Di Stefano, T. Day, D. Schmidt, E.C. Merritt, F.W. Doss, K.A. Flippo, T. Carde-
nas, B. DeVolder, P. Donovan, S. Edwards, F. Fierro, R. Gonzales, L. Goodwin, C. Hamilton, T. Quin-
tana, R. Randolph, A.M. Rasmus, T. Sedillo, C. Wilson, and L. Welser-Sherrill. A platform for thin-layer
richtmyer-meshkov at omega and the nif. High Energy Density Physics, 33:100705, 2019. ISSN 1574-
1818. doi:https://doi.org/10.1016/j.hedp.2019.100705. URL
https://www.sciencedirect.com/science/
article/pii/S1574181819300461.
[10]
Reuben D. Budiardja, Mark Berrill, Markus Eisenbach, Gustav R. Jansen, Wayne Joubert, Stephen Nichols,
David M. Rogers, Arnold Tharrington, and O. E. Bronson Messer. Ready for the frontier: Preparing applications
34

Machine Learning Visualization Tool for Exploring Parameterized Hydrodynamics
for the world’s first exascale system. In Abhinav Bhatele, Jeff Hammond, Marc Baboulin, and Carola Kruse,
editors, High Performance Computing, pages 182–201, Cham, 2023. Springer Nature Switzerland. ISBN 978-3-
031-32041-5.
[11]
J Luc Peterson, Ben Bay, Joe Koning, Peter Robinson, Jessica Semler, Jeremy White, Rushil Anirudh, Kevin
Athey, Peer-Timo Bremer, Francesco Di Natale, et al. Enabling machine learning-ready hpc ensembles with
merlin. Future Generation Computer Systems, 131:255–268, 2022.
[12]
Haizhou Wu, Xuejun Liu, Wei An, Songcan Chen, and Hongqiang Lyu. A deep learning approach for efficiently
and accurately evaluating the flow field of supercritical airfoils. Computers & Fluids, 198:104393, 2020. ISSN
0045-7930. doi:https://doi.org/10.1016/j.compfluid.2019.104393. URL
https://www.sciencedirect.com/
science/article/pii/S0045793019303512.
[13]
Haizhou Wu, Xuejun Liu, Wei An, and Hongqiang Lyu. A generative deep learning framework for airfoil flow
field prediction with sparse data. Chinese Journal of Aeronautics, 35(1):470–484, 2022. ISSN 1000-9361.
doi:https://doi.org/10.1016/j.cja.2021.02.012. URL
https://www.sciencedirect.com/science/article/
pii/S1000936121000728.
[14]
Tanujit Chakraborty, Ujjwal Reddy K S, Shraddha M Naik, Madhurima Panja, and Bayapureddy Manvitha. Ten
years of generative adversarial nets (gans): a survey of the state-of-the-art. Machine Learning: Science and
Technology, 5(1):011001, jan 2024. doi:10.1088/2632-2153/ad1f77. URL
https://dx.doi.org/10.1088/
2632-2153/ad1f77.
[15]
Ni Li, Yunqin Liu, Guanghong Gong, Luming Zhao, and Haitao Yuan. A generative deep learning approach
for real-time prediction of hypersonic vehicles in fluid-thermo-structural coupling fields. Aerospace Science
and Technology, 139:108398, 2023. ISSN 1270-9638. doi:https://doi.org/10.1016/j.ast.2023.108398. URL
https://www.sciencedirect.com/science/article/pii/S127096382300295X.
[16]
Zhipeng Wang, Xuejun Liu, Jian Yu, Haizhou Wu, and Hongqiang Lyu. A general deep transfer learning
framework for predicting the flow field of airfoils with small data. Computers & Fluids, 251:105738, 2023. ISSN
0045-7930. doi:https://doi.org/10.1016/j.compfluid.2022.105738. URL
https://www.sciencedirect.com/
science/article/pii/S0045793022003309.
[17]
Muhammad Alfiyandy Hariansyah and Koji Shimoyama. Deep learning techniques for high-dimensional surrogate-
based aerodynamic design. In 33rd Congress of the International Council of the Aeronautical Sciences, ICAS
2022, pages 164–182. International Council of the Aeronautical Sciences, 2022.
[18]
Tanishk Nandal, Vaibhav Fulara, and Raj Kumar Singh. A synergistic framework leveraging autoencoders and
generative adversarial networks for the synthesis of computational fluid dynamics results in aerofoil aerodynamics.
arXiv preprint arXiv:2305.18386, 2023.
[19]
Yuqing Hou, Hui Li, Hong Chen, Wei Wei, Jiayue Wang, and Yicang Huang. A novel deep u-net-lstm framework
for time-sequenced hydrodynamics prediction of the suboff aff-8. Engineering Applications of Computational
Fluid Mechanics, 16(1):630–645, 2022. doi:10.1080/19942060.2022.2030802. URL
https://doi.org/10.
1080/19942060.2022.2030802.
[20]
Ali Kashefi and Tapan Mukerji. Point-cloud deep learning of porous media for permeability prediction. Physics of
Fluids, 33(9):097109, 09 2021. ISSN 1070-6631. doi:10.1063/5.0063904. URL
https://doi.org/10.1063/
5.0063904.
[21]
Erda Chen, Mingming Guo, Ye Tian, Yi Zhang, Hao Chen, Jialing Le, Fuyu Zhong, and Hua Zhang. Flame
development prediction of supersonic combustion flow based on lightweight cascaded convolutional neural
network. Physics of Fluids, 35(2):025134, 02 2023. ISSN 1070-6631. doi:10.1063/5.0140624. URL
https:
//doi.org/10.1063/5.0140624.
[22]
Yunfei Li, Juntao Chang, Ziao Wang, and Cheng Kong. Inversion and reconstruction of supersonic cascade passage
flow field based on a model comprising transposed network and residual network. Physics of Fluids, 31(12):
126102, 12 2019. ISSN 1070-6631. doi:10.1063/1.5124133. URL https://doi.org/10.1063/1.5124133.
[23]
Efren Fernandez-Grande, Xenofon Karakonstantis, Diego Caviedes-Nozal, and Peter Gerstoft. Generative models
for sound field reconstruction. The Journal of the Acoustical Society of America, 153(2):1179–1190, 02 2023.
ISSN 0001-4966. doi:10.1121/10.0016896. URL https://doi.org/10.1121/10.0016896.
[24]
Alec Radford, Luke Metz, and Soumith Chintala. Unsupervised representation learning with deep convolutional
generative adversarial networks. In Yoshua Bengio and Yann LeCun, editors, 4th International Conference on
Learning Representations, ICLR 2016, San Juan, Puerto Rico, May 2-4, 2016, Conference Track Proceedings,
2016. URL http://arxiv.org/abs/1511.06434.
35

Machine Learning Visualization Tool for Exploring Parameterized Hydrodynamics
[25]
Nissrine Akkari, Fabien Casenave, Marc-Eric Perrin, and David Ryckelynck. Deep convolutional generative
adversarial networks applied to 2d incompressible and unsteady fluid flows. In Science and Information Conference,
pages 264–276. Springer, 2020.
[26]
M. Cheng, F. Fang, C.C. Pain, and I.M. Navon. Data-driven modelling of nonlinear spatio-temporal fluid
flows using a deep convolutional generative adversarial network. Computer Methods in Applied Mechanics
and Engineering, 365:113000, 2020. ISSN 0045-7825. doi:https://doi.org/10.1016/j.cma.2020.113000. URL
https://www.sciencedirect.com/science/article/pii/S0045782520301845.
[27]
C. Drygala, B. Winhart, F. di Mare, and H. Gottschalk. Generative modeling of turbulence. Physics of Fluids, 34(3):
035114, 03 2022. ISSN 1070-6631. doi:10.1063/5.0082562. URL https://doi.org/10.1063/5.0082562.
[28]
Carlos Granero Belinchon and Manuel Cabeza Gallucci. A multiscale and multicriteria generative adversarial
network to synthesize 1-dimensional turbulent fields. Machine Learning: Science and Technology, 5(2):025032,
may 2024. doi:10.1088/2632-2153/ad43b3. URL https://dx.doi.org/10.1088/2632-2153/ad43b3.
[29]
S. Wiewel, M. Becher, and N. Thuerey. Latent space physics: Towards learning the temporal evolution of
fluid flow. Computer Graphics Forum, 38(2):71–82, 2019. doi:https://doi.org/10.1111/cgf.13620. URL
https:
//onlinelibrary.wiley.com/doi/abs/10.1111/cgf.13620.
[30]
JIANXIANG PENG, CHANGMING HU, YINGLEI LI, LIN ZHANG, and FUQIAN JING. Determination of
parameters of steinberg-guinan constitutive model with shock wave experiments. International Journal of Modern
Physics B, 22(09n11):1111–1116, 2008. doi:10.1142/S0217979208046396. URL
https://doi.org/10.1142/
S0217979208046396.
[31]
Veselin A. Dobrev, T. E. Ellis, and Robert N. Rieben. Curvilinear finite elements for lagrangian hydrodynamics.
Int. J. Numer. Meth. Fluid, 65:1295–1310, 2011.
[32]
Veselin A. Dobrev, Tzanio V. Kolev, and Robert N. Rieben. High order curvilinear finite element methods for
lagrangian hydrodynamics. SIAM J. Sci. Comp, 34(5):606–641, 2012.
[33]
Robert W. Anderson, Veselin A. Dobrev, Tzanio V. Kolev, Robert N. Rieben, and Vladimir Z. Tomov. High-order
multi-material ale hydrodynamics. SIAM J. Sci. Comp., 40(1):B32–B58, 2018.
[34]
Tzanio V. Kolev and Robert N. Rieben. A tensor artificial viscosity using a finite element approach. J. Comp.
Phys., 228:8336–8366, 2009.
[35]
R. W. Anderson, V. A. Dobrev, Tzanio V. Kolev, and Robert N. Rieben. Monotonicity in high-order curvilinear
finite element arbitrary lagrangian–eulerian remap. Int. J. Numer. Meth. Fluids, 77:249–273, 2015.
[36]
F N. Fritsch. The leos interpolation package. LLNL Techical Report, 2003. doi:doi:10.2172/15005830. URL
https://www.osti.gov/biblio/15005830.
[37]
D. J. Steinberg, S. G. Cochran, and M. W. Guinan. A constitutive model for metals applicable at high-strain rate.
J. Appl. Phys, 41(3):1498–1504, 1980. doi:https://doi.org/10.1063/1.327799.
[38] URL https://github.com/CEED/Laghos.
[39] F. A. C. Viana. A tutorial on latin hypercube design of experiments. Qual. Reliab. Engng., 32:1975–1985, 2016.
[40]
Mike Folk, Gerd Heber, Quincey Koziol, Elena Pourmal, and Dana Robinson. An overview of the hdf5 technology
suite and its applications. In Proceedings of the EDBT/ICDT 2011 workshop on array databases, pages 36–47,
2011.
[41]
Michael Armstrong, Jeffrey Nguyen, Sylvie Aubry, William Schill, Jonathan Belof, and Hector Lorenzana. Use of
shock wavefront curvature to modulate rmi jet growth. Bulletin of the American Physical Society, 67, 2022.
[42]
Jeffrey Nguyen, Sylvie Aubry, Michael Armstrong, Andrew Hoff, Jonathan Belof, Hector Lorenzana, Matthew
Staska, and Brandon LaLone. Modulation of richtmyer-meshkov instability in gas gun experiments. Bulletin of
the American Physical Society, 67, 2022.
[43]
Dane M. Sterbentz, Charles F. Jekel, Daniel A. White, Sylvie Aubry, Hector E. Lorenzana, and Jonathan L.
Belof. Design optimization for Richtmyer–Meshkov instability suppression at shock-compressed material
interfaces. Physics of Fluids, 34(8):082109, 08 2022. ISSN 1070-6631. doi:10.1063/5.0100100. URL
https:
//doi.org/10.1063/5.0100100.
[44]
B. Duan, Y. Zhou, S. Zheng, M. Bao, and J. Dong. Blasting demolition of steel structure using linear cumulative
cutting technology. Adv. Mech. Engng., 9(11):1–11, 2017.
[45]
E. N. Sher. Development of a system of plane radial cracks during explosion of linear blasthole and borehole
charges. J. Appl. Mech. Tech. Phys., 58(5):933–938, 2017.
36

Machine Learning Visualization Tool for Exploring Parameterized Hydrodynamics
[46]
B. Wu, G. Xu, and et. al. Meng. Study on dynamic evolution law of blasting cracks in elliptical bipolar linear
shaped charge blasting. Shock and Vibration, pages 1–10, 2021. doi:https://doi.org/10.1155/2021/5272878.
[47]
D. Novotny and M. Mallery. Historical development of linear shaped charge. In Joint Propioulsion Conference.
AIAA/ASME, 2007. URL https://arc.aiaa.org/doi/pdf/10.2514/6.2007-5141.
[48]
Dane M. Sterbentz, Charles F. Jekel, Daniel A. White, Robert N. Rieben, and Jonathan L. Belof. Linear shaped-
charge jet optimization using machine learning methods. Journal of Applied Physics, 134(4):045102, 07 2023.
ISSN 0021-8979. doi:10.1063/5.0156373. URL https://doi.org/10.1063/5.0156373.
[49]
James M. Stone, Kengo Tomida, Christopher J. White, and Kyle G. Felker. The athena
++
adaptive mesh
refinement framework: Design and magnetohydrodynamic solvers. The Astrophysical Journal Supplement Series,
249(1):4, June 2020. doi:10.3847/1538-4365/ab929b. URL
https://doi.org/10.3847%2F1538-4365%
2Fab929b.
[50]
Richard Liska and Burton Wendroff. Comparison of several difference schemes on 1d and 2d test problems for the
euler equations. SIAM Journal on Scientific Computing, 25(3):995–1017, 2003. doi:10.1137/S1064827502402120.
URL https://doi.org/10.1137/S1064827502402120.
[51]
Augustus Odena, Vincent Dumoulin, and Chris Olah. Deconvolution and checkerboard artifacts. Distill, 1(10):e3,
2016.
[52]
Charles F Jekel, Dane M Sterbentz, Sylvie Aubry, Youngsoo Choi, Daniel A White, and Jonathan L Belof. Using
conservation laws to infer deep learning model accuracy of richtmyer-meshkov instabilities. arXiv preprint
arXiv:2208.11477, 2022.
[53]
Derek Hansen, Danielle C. Maddix, Shima Alizadeh, Gaurav Gupta, and Michael W. Mahoney. Learning physical
models that can respect conservation laws. Physica D: Nonlinear Phenomena, 457:133952, 2024. ISSN 0167-
2789. doi:https://doi.org/10.1016/j.physd.2023.133952. URL
https://www.sciencedirect.com/science/
article/pii/S0167278923003068.
[54]
Sergey Ioffe and Christian Szegedy. Batch normalization: Accelerating deep network training by reducing internal
covariate shift. In International conference on machine learning, pages 448–456. PMLR, 2015.
[55]
Vinod Nair and Geoffrey E Hinton. Rectified linear units improve restricted boltzmann machines. In Icml, 2010.
[56]
Adam Paszke, Sam Gross, Francisco Massa, Adam Lerer, James Bradbury, Gregory Chanan, Trevor Killeen,
Zeming Lin, Natalia Gimelshein, Luca Antiga, et al. Pytorch: An imperative style, high-performance deep
learning library. Advances in neural information processing systems, 32, 2019.
[57]
Diederik P Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980,
2014.
[58]
Samyam Rajbhandari, Jeff Rasley, Olatunji Ruwase, and Yuxiong He. Zero: Memory optimizations toward training
trillion parameter models. In SC20: International Conference for High Performance Computing, Networking,
Storage and Analysis, pages 1–16. IEEE, 2020.
[59]
Priya Goyal, Piotr Dollár, Ross Girshick, Pieter Noordhuis, Lukasz Wesolowski, Aapo Kyrola, Andrew Tulloch,
Yangqing Jia, and Kaiming He. Accurate, large minibatch sgd: Training imagenet in 1 hour. arXiv preprint
arXiv:1706.02677, 2017.
[60]
Brandon E. Morgan. Large-eddy simulation and reynolds-averaged navier-stokes modeling of three
rayleigh-taylor mixing configurations with gravity reversal. Phys. Rev. E, 106:025101, Aug 2022.
doi:10.1103/PhysRevE.106.025101. URL https://link.aps.org/doi/10.1103/PhysRevE.106.025101.
[61]
Napari Contributors. napari: a multi-dimensional image viewer for python. Zenodo https://doi. org/10.5281/zenodo,
3555620, 2019.
37