aainstitutetext: CERN,
1211 Geneva 23, Switzerlandbbinstitutetext: University of Engineering and
Technology Peshawar,
Jamrud Road Peshawar Khyber Pakhtunkhwa, Pakistan

# Fast Simulation of a High Granularity Calorimeter by Generative Adversarial
Networks

Gul Rukh Khattak,111Corresponding author. a Sofia Vallecorsa a Federico
Carminati b Gul Muhammad Khan<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We present the 3DGAN for the simulation of a future high granularity
calorimeter output as three-dimensional images. We prove the efficacy of
Generative Adversarial Networks (GANs) for generating scientific data while
retaining a high level of accuracy for diverse metrics across a large range of
input variables. We demonstrate a successful application of the transfer
learning concept: we train the network to simulate showers for electrons from
a reduced range of primary energies, we then train further for a five times
larger range (the model could not train for the larger range directly). The
same concept is extended to generate showers for other particles (photons and
neutral pions) depositing most of their energies in electromagnetic
interactions. In addition, the generation of charged pion showers is also
explored, a more accurate effort would require additional data from other
detectors not included in the scope of the current work. Our further
contribution is a demonstration of using GAN-generated data for a practical
application. We train a third-party network using GAN-generated data and prove
that the response is similar to a network trained with data from the Monte
Carlo simulation.

The showers generated by GAN present accuracy within $10\%$ of Monte Carlo for
a diverse range of physics features, with three orders of magnitude speedup.
The speedup for both the training and inference can be further enhanced by
distributed training.

##  1 Introduction

Particles undergo complex stochastic interactions upon contact with materials.
The modeling of these interactions is further complicated by the large number
of secondary particles involved. The Monte Carlo simulation depends on
repeated random sampling to produce a snapshot of the particle interactions
with a detector. Simulation is crucial in most High Energy Physics (HEP)
experiments and is extremely resource-intensive. More than $50\%$ of the
current computing resources of the HEP community are utilized in simulation
alone hsf2019 . In the future that need will increase further due to the
higher luminosity and granularity of future experiments, and it will not be
possible to create a corresponding increase in computing resources. The main
motivation for the fast simulation is to incorporate other faster alternatives
to decrease the cost of future experiments. Current fast simulation approaches
are mostly based on parametrization Altfast ; gflash ; nomad  or lookup table
frozenShower  approaches, providing between 10 and 100 times speedup while
achieving different levels of accuracy.

Generative Adversarial Network (GAN) 2014Goodfellow  is a training paradigm
for deep generational neural networks. Other approaches for generative
networks include Variational autoencoders (2014autoencoding, ) and
autoregressive models (2016pixelrnn, ), etc. All of these approaches have
their own strengths and weaknesses. The autoencoder-based methods often
produce blurry images while the pixel-based methods are not only slow to
evaluate but also suffer from limited capacity. The GAN approach has been able
to demonstrate highly realistic and sharp images as compared to other
approaches 2014Goodfellow . There have been many recent variants of the GAN
methodology, such as WGAN wgan , StackGAN stackgan , and Progressive GAN
progressiveGAN  further enhancing the quality and resolution of generated
images. The generative problem often involves intractable probability
densities and thus methods involving likelihood estimates are not practical
for most purposes. GAN can learn a distribution implicitly since it does not
rely on the explicit computation of probability densities. Therefore the GAN
approach is suitable for the generation of a wider range of data such as
musical notes (musicGAN, ), natural language (langGAN, ), medical data
(medicalGAN, ; medicalGAN2, ), natural scenes (2014Goodfellow, ), faces
(progressiveGAN, ) and image denoising(denoiseGAN, ). Since simulation is
essentially a generative problem thus a successful image generation model can
be exploited in this domain. A potential advantage of deep generative models
is that the generated distribution does not have to be explicitly defined, and
thus even the real data from a detector can be simulated directly.

We leverage the GAN methodology to generate HEP calorimeter output.
Calorimeters are special HEP detectors that record particles through the
measurement of the energies deposited by them. These detectors can be regarded
as huge cameras taking pictures of particle interactions. The Monte Carlo
simulation for the detector output is extremely precise but highly expensive
both in regards to the simulation time and resources. For most HEP experiments
the calorimeter is a simulation bottleneck, consuming, for example, more than
$80\%$ of the simulation time for the ATLAS experiment Altfast . We generate
the calorimeter cells as monochromatic pixelated images with the cell energy
depositions as our pixel intensities.

The 3DGAN icip18_3dgan ; 3dgan_icmla ; Belayneh2019  is the first effort where
the detector output is generated retaining correlations in all three spatial
dimensions. We employ three-dimensional convolutional layers in our network,
while previously detector output was generated either as a two-dimensional
image or a concatenated set of two-dimensional images. We demonstrate our
approach for a high granularity detector with higher spatial resolution and
thus consequently much larger image dimensions than previous such efforts. We
pre-process the cell depositions by taking a power less than one, thus
decreasing the dynamic range of corresponding pixel intensities and improving
the convergence. We employ a multi-step training process to generate images,
from a complex multivariate distribution, for a large range of input
conditions. We also perform extensive validations from diverse viewpoints
including vision and deep learning, as well as, physics-based evaluation. The
network scores highly on all the platforms both for the pertinence to the
training data and for maintaining sufficient diversity. The details of the
network development and the validation from different perspectives have been
presented previously 3dgan_icmla . The current work is geared more towards the
physics community and thus is limited to only physics-based validation.
Previously we simulated electrons coming with energies from a wide spectrum by
employing our multi-step training. Now we successfully extend the same
approach to simulate additional particle types such as photon and neutral pion
where most of the energy is lost in electromagnetic interactions. We perform
an additional investigation to prove that the GAN could accurately reproduce
the signature features of a particular particle type. We also undertake some
preliminary exploration of the charged pion simulation and generation of rare
events. We finally present a successful practical example of using the GAN-
generated data in a typical reconstruction tool, demonstrating that the GAN-
generated images could provide similar performance as Monte Carlo images.

The current paper is organized as follows. Section 2 presents the overview of
past efforts for fast simulation of HEP calorimeters exploiting deep neural
networks. The next section (Section 3) describes the Monte Carlo training
dataset. The basic structure of the calorimeter and the important features of
our data are discussed. Section 4 describes how the GAN approach is adapted to
the problem of HEP detector simulation. The approach has been exploited for
the generation of particles with predominantly electromagnetic showers. The
results for comparison to Monte Carlo simulation are presented in Section 5.
Some preliminary work is also carried out for the charged pion simulation as
presented in Section 6. Another study exploring the simulation of rare modes
is presented in Section 7. Section 8 presents a practical application for the
use of GAN-generated data. Finally, Section 9 summarises the main
contributions and presents some future suggestions.

##  2 Previous Work

Fast simulation is already incorporated in existing experiments through
approaches like parametrization nomad ; gflash ; Altfast ; fastcalosim  and
Lookup tables frozenShower , etc. Usually a part of the simulation is replaced
by fastsim where some tradeoff between speed and accuracy can be feasible.
Following the same concept deep learning has also been explored to generate
simulation data. Fast simulation using neural networks can be regarded as a
special type of parametrization, with the weights of the neural network as
parameters, optimized through a training process.

Table 1: Calorimeter simulation with deep learning No. |  Model |  Algorithm |  Architecture |  Condition |  Output
---|---|---|---|---|---
1 |  LAGAN de2017learning  (2017) |  GAN |  2D Locally connected |  Particle Type as discrete labels |  $25\times 25$
2 |  CALOGAN paganini2017calogan  (2017) |  GAN |  2D Locally connected |  $E_{P}\sim{}U(1,100)$ GeV |  layer1: $3\times 96$ layer2: $12\times 12$ layer3: $12\times 6$
3 |  3DGAN initial prototype icip18_3dgan  (2018) |  ACGAN |  Conv3D |  $E_{P}\sim{}U(2,500)$ GeV |  $25\times 25\times 25$
4 |  ATLAS Salamani2018  (2018) |  WGAN and VAE |  Dense |  $E_{P}\sim{}U(1,260)$ GeV |  vector of 266 cells.
5 |  LHCb gan_lhcb  (2019) |  WGAN |  Conv2D |  Five variables related to position and momentum |  $30\times 30$
6 |  HGCAL precise  (2019) |  WGAN |  Conv2D and Locally connected |  $E_{P}$ and initial impact position (x,y) |  Concatenation of 7 ($12\times 15$) layers
7 |  3DGAN Belayneh2019  (2019) |  ACGAN |  Conv3D |  $E_{P}\sim{}U(2,500)$ GeV and $\theta\sim{}U(60^{\circ},120^{\circ})$ |  $51\times 51\times 25$
8 |  DijetGAN Dijet  (2020) |  WGAN |  Conv2D |  |  vector of 7 jet variables.
9 |  ILD high2020  (2021) |  GAN, WGAN and BIB-AE |  Conv3D |  $E_{P}\sim{}U(10,100)$ GeV) |  $30\times 30\times 30$.

The GAN technique is an unsupervised training methodology. The power of GAN
lies in the fact that the target distribution does not have to be tractable
and instead the training relies on a Minimax game between a discriminator (D)
network and a generator (G) network. The discriminator is trained to
differentiate between the target and the generated distributions, while the
generator is trained to confuse the discriminator. Both the networks compete
with each other till the generator manages to completely confuse the
discriminator, given enough capacity for both models. At this point, the GAN
is said to have converged.

Calorimeter data have been simulated through deep generative networks in a
number of recent approaches as presented in Table 1. LAGAN de2017learning  was
one of the first fast simulation approaches based on deep learning. A
simplified calorimeter was simulated as 2D jet images for high energy W bosons
(signal) and generic quark/gluon jets (background). CALOGAN
paganini2017calogan  employed the LAGAN architecture to generate sets of three
two-dimensional images that were then concatenated to obtain the output for a
three-layered simplified calorimeter conditioned on the primary particle
energy ($E_{P}$) ranging from $1-100$ GeV. Since then, there have been other
demonstrations employing deep learning for HEP calorimeter simulation. Deep
learning has been used for fast simulation of the ATLAS calorimeter
Salamani2018 . Showers with energies $1-260$ GeV and pseudorapidity $|\eta|$
in the range of $0.2-0.25$ were generated as flattened arrays of pixels, by a
dense network employing both VAE and GAN methodologies. The images were also
conditioned on the primary particle energy and constrained on the total energy
deposition. The GAN-generated showers were reported to have better performance
as compared to the VAE generated showers. WGAN has also been used to simulate
the LHC detector output collapsed to a two-dimensional array of cells gan_lhcb
. A simplified version of HGCAL was simulated as seven 2D images concatenated
together, conditioned on the primary particle energy and impact position
precise . The DijetGAN Dijet  employed GAN for the simulation of diject
events: a background process for important physics studies at LHC. A more
recent approach high2020  experimented with several GAN architectures for the
simulation of high granularity calorimeter for the ILD ild  detector as
$30\times 30\times 30$ three-dimensional images. The simulation was limited to
the orthogonally incident photons coming with $10-100$ GeV primary energy.

The 3DGAN initial prototype icip18_3dgan  exploited 3D convolutional networks
to simulate the response of a high granularity calorimeter as $25\times
25\times 25$ image. The GAN setup was used to train the network for a
simplified scenario involving only orthogonally incident electrons. The
approach was then extended to condition $51\times 51\times 25$ images on both
the particle energy and incident angle Belayneh2019 ; 3dgan_icmla . The more
complex distribution could be generated through multi-step training,
architecture, and loss function modifications (details in access2021 ). We now
simulate the detector output for all the particle types available in the
dataset and further validate the results. The 3DGAN greatly surpasses existing
efforts in the granularity and dimensions of the generated images, conditioned
on both the incident particle angle and energy from a wide range, and
validated in great detail from diverse viewpoints. Finally, a culmination of
the effort is to test the GAN-generated data for a practical use case.

##  3 Calorimeter Dataset

We present a solution for the needs of future experiments with higher demands
for computing resources due to increased luminosity and granularity. We,
therefore, select the proposed Linear Collider Detector (LCD), designed in the
context of the future Compact Linear Collider (CLIC) CLIC  accelerator for our
study. The dataset employs the GEANT4 toolkit (geant4, ) for the generation of
the simulation data for several particle types (i.e., electrons $e$, photons
$\gamma$, neutral pions $\pi^{0}$ and charged pions $\pi$) and is publicly
available on Zenodo at https://zenodo.org/communities/mpp-hep.

###  3.1 Detector Geometry

Figure 1 shows the proposed detector design, highlighting the main detector
concepts. The basic design consideration for improving the jet energy
resolution is to resolve the energy depositions of the individual particles in
a jet, through a high cell granularity and precise time information. The
electron and positron will collide in the central region. The trackers are
shown in blue. The surrounding grey region will comprise the calorimeters. The
calorimeter will be highly segmented with an electromagnetic (ECAL) and a
hadronic (HCAL) calorimeter.

The data used for the current work is that of the ECAL central barrel region.
This region has a cylindrical shape with an inner radius of $1.5~{}m$ with
$25$ concentric layers. The proposed granularity for the ECAL cells is
$5.1\times 5.1~{}mm^{2}$. The cells are arranged in the form of $25$
cylindrical layers with silicon sensor planes (active), alternating with
tungsten absorber planes (passive). The simulation is carried out considering
the entire detector geometry, including the material in front of the
calorimeter, and the effect of the solenoid magnetic field.

Figure 1: Schematic diagram for the CLIC calorimeter (clic_report, )

###  3.2 Data Features

Figure 2: The calorimeter barycenter computation was done in global
coordinates (left) and the cells were saved based on the local coordinates
(right).

The energy deposits in the calorimeter cells result from the interaction of an
incoming primary particle with the calorimeter material. These deposits form a
characteristic shape that can be termed as an "event" or "shower". A slice
around the barycenter of each shower is saved as a 3D array of energy
depositions. The slicing is carried out by taking a projection of all the
deposited energy on the ECAL inner surface. The barycenter of this 2D image
and the point of origin of the incoming particle are then used to compute the
polar angles ($\theta$ and $\phi$) corresponding to each shower. Due to the
different $\phi$ granularity for each depth layer in the ECAL, a
multiplicative transformation is also applied to scale every layer to look
like the innermost ECAL layer. Finally, the data is saved in the HDF5 format.
Each entry in the dataset comprises the 3D array with cell energy deposits,
the incoming particle energy $E_{P}$, and the incidence angles $\theta$ and
$\phi$. The energies for the incoming particles are uniformly distributed from
$2$ to $500$ GeV and an incident angle ($\theta$) uniformly distributed from
$60^{\circ}$ ($1.047$ radians) to $120^{\circ}$ ($2.094$ radians).

Figure 2 shows the particle gun position with respect to the calorimeter
surface. In the global coordinate system, the $Z$ axis lies along the axis of
the calorimeter cylinder. While in the local coordinates of the shower it is
perpendicular to the calorimeter surface. Similarly, other axes are also
transformed to the local coordinates of each sample. The $Z$ axis of our 3D
images lies along with the detector depth and $X$, $Y$ are the transverse
axes. The calorimeter is isotropic in the $\phi$ direction, therefore only the
$\theta$ direction is incorporated for the current work. The $\theta$ value is
recomputed as a weighted mean of the angles computed using the barycenter of
the event and the barycenters of the $XY$ planes for each position along $Z$
(weighted by the position along $Z$).

##  4 3DGAN

The HEP simulation depends on a set of variables that impact the underlying
physics processes described by the simulation. Therefore, 3DGAN uses the
$E_{P}$ and $\theta$ of a particle striking the calorimeter surface as inputs,
to generate the appropriate detector response. In order to provide feedback on
the correspondence between generated showers and input conditions, we exploit
the concept of auxiliary tasks acgan2  together with domain-related
constraints. The current works mainly describe the final optimized version of
the 3DGAN model while additional details about the development process can be
obtained from access2021 . The 3DGAN is implemented using Keras 2.2.4 keras
deep learning python library with Tensorflow 1.14.0 tensorflow2015-whitepaper
as a backend. The code is available at https://github.com/svalleco/3Dgan.

###  4.1 Pre-processing

Figure 3: The cell energy distribution for GEANT4 MC events (red) and GAN
generated events after a pre-processing by taking the power of pixel
intensities: power =$0.85$ (blue); power =$0.75$ (green); power =$0.5$ (cyan);
power =$0.25$ (magenta).

One of the main challenges for generating scientific data through techniques
developed for computer vision lies in the inherent difference between the
dynamic ranges of the pixel intensities. The pixel intensities in a typical
RGB image have a range from $0$ to $255$ while the energy deposited in
detector cells covers more than $10$ orders of magnitude. We have implemented
a pre-processing procedure aimed at reducing this dynamic range. Initial tests
conducted taking the logarithms of the pixel intensities, resulted in the
generation of highly distorted images. Taking a less drastic approach we
calculate the power function of pixels intensities using an exponent smaller
than one. A smaller exponent results in faster convergence but greater
distortion in generated images, while a larger exponent slows down convergence
yet retaining image quality. Figure 3 shows how the value of the exponent
($p$) affected the distribution of the generated pixel intensities for the
individual cells. The value of $p$ is adjusted to an optimum value of $0.85$,
where a faster convergence is achieved while retaining an acceptable level of
accuracy at both ends of the spectrum. The generated images are then post-
processed by simply taking the inverse of the power function. The data is also
subjected to a threshold of $0.2$ GeV for the total energy deposited in the
event. This rejection is aimed at removing some spurious events with little or
no energy deposition.

###  4.2 Architecture

Figure 4: The 3DGAN architecture, see the text for details.

The 3DGAN architecture is presented in Figure 4. The generator network
implements stochasticity through a latent vector of $254$ random numbers drawn
from a Gaussian distribution. The generator input includes $E_{P}$ and
$\theta$ concatenated to the latent vector. The generator network then maps
the input to a layer of linear neurons followed by seven 3D convolutional
layers. The discriminator input is an image while the network has only four 3D
convolutional layers. Batch normalization BN  is performed after all except
the first convolutional layer in the discriminator and the last two layers in
the generator. The leakyRelu leakyrelu  activation function is used for the
discriminator hidden layers while the Relu relu  activation function is used
for the generator layers to induce sparsity. The discriminator uses dropout
dropout  of $20\%$ for regularization and a single average pooling layer after
the last convolutional layer since additional pooling layers result in
substantial loss of performance.

The discriminator network has two trainable outputs: a sigmoid neuron predicts
the $O_{GAN}$ and a linear neuron $O_{P}$ predicts $E_{P}$. The other two
additional outputs are simple analytical measurements: $O_{sum}$ is the total
deposited energy and $O_{\theta}$ is the measured incident angle (geometrical
angle of the shower energy depositions). These non-trainable outputs represent
physics-based constraints.

###  4.3 Loss Function

The 3DGAN loss function is the weighted sum of individual losses pertaining to
the discriminator outputs and constraints. The domain-related constraints are
essential to achieve a high level of agreement over the very large dynamic
range of the image pixel intensity distribution. Equation 1 presents the
discriminator loss related to the output $O_{GAN}$ as $L_{G}$, the loss
related to the output $O_{sum}$ as $L_{E}$, the output $O_{\theta}$ as $L_{A}$
and the predicted $E_{P}$ as $L_{P}$ balanced by corresponding weights $W$.

$L_{3DGAN}=W_{G}L_{G}+W_{P}L_{P}+W_{A}L_{A}+W_{E}L_{E}$ (1)

The $L_{P}$ and $L_{\theta}$ both provide feedback on how well the generated
images correspond to the input conditions. The loss $L_{E}$ ensures energy
conservation. $L_{G}$ is evaluated as binary cross-entropy. $L_{P}$ and
$L_{E}$ are implemented on mean percentage errors, while $L_{A}$ as mean
absolute error. The generator loss is implemented as the inverse of $L_{G}$
together with the auxiliary losses and constraints. The weights (presented in
Appendix A) are considered as hyperparameters and chosen to balance the loss
ranges and their relative importance (in this case the loss $L_{G}$ is given
higher priority as compared to the auxiliary losses).

###  4.4 Training

The 3DGAN training is inspired by the concept of transfer learning. The GAN
could not converge for the highly complex multivariate distribution directly
thus a two-step training is applied. In order to successfully train the
network, we reduce the complexity by training the GAN first for electron
events having $E_{P}\sim~{}U(100,200)$ GeV. After the GAN converges, the same
trained model is further trained with the data from the whole $E_{P}$ range of
$E_{P}\sim~{}U(2,500)$ GeV. The first training step exploits $137,342$
electron events. The GAN is then trained for the larger $E_{P}$ range,
utilizing a much larger size of training data ($400$ k events) from each
particle type (electrons, photons, and neutral pions). The train and test
losses are evaluated on the data divided in a ratio of nine to one. The first
training step is run for $130$ epochs (2 hours per epoch on GTX 1080) while
the second step is run for $30$ epochs (4 hours per epoch on GTX 1080).
Finally, the best network is selected according to the minimum relative error
for the $SF$ on additional validation data ($20k$ events are filtered around
specific $E_{P}$ bins). This last step is aimed to further improve the
accuracy for the $SF$.

initialize

for  _number of epochs_ do

#Training

for  _all batches in the training data_ do

get real $E_{P}$, $\theta$, $E_{sum}$ and image batches from data

latent batch $\sim\mathcal{N}(0\,,1)$

generator input =
$E_{P}\mathbin{{+}\mspace{-8.0mu}{+}}\theta\mathbin{{+}\mspace{-8.0mu}{+}}$
latent

generate fake events for the same $E_{P}$ and $\theta$

train discriminator on real batch

train discriminator on fake batch

for  _2 times_ do

latent batch $\sim\mathcal{N}(0\,,1)$

generator input =
$E_{P}\mathbin{{+}\mspace{-8.0mu}{+}}\theta\mathbin{{+}\mspace{-8.0mu}{+}}$
latent

use real $E_{P}$ and $\theta$ for fake events

train the generator (minimizing the discriminator loss on the generated
images)

end for

$\\#$ Testing

Evaluate discriminator on real and fake data

Evaluate generator using the inverse of discriminator loss on generated data

end for

end for

Algorithm 1 Training the 3DGAN model

Figure 5: The $L_{G}$ losses for the generator (blue) and the discriminator
(orange) converge for the first training step with restricted $E_{P}$ range.
(Left) training losses. (Middle) test losses. (Right) the discriminator output
$O_{GAN}$ for MC (red) and GAN (blue) images is similar and the discriminator
is confused.

The training process for each epoch is presented by the Algorithm 1. For each
training iteration the discriminator is trained twice: once on a batch of real
data, and next on a batch of generated data. For a balanced approach, the
generator is also trained twice while freezing the weights of the
discriminator. The RMSProp rmsProp  optimizer is utilized to train the network
through Stochastic Gradient Descent. Figure 5 shows the $L_{G}$ losses
associated to the discriminator (blue) and the generator (orange). It can be
seen that the loss for the discriminator increases, while the loss for the
generator increases till both losses are converged at around $0.6$ (log(4)).
At this point, the $O_{GAN}$ output for both the data (red) and the GAN images
(blue), shown in Figure 5 right panel has similar distributions centered
around $0.5$. The discriminator is indeed confused and the GAN converges.

###  4.5 Generation Time

The 3DGAN greatly reduces the simulation time. Table 2 compares the time taken
to generate a shower using Monte Carlo and GAN. The inference time is around
$4$ ms/particle on GeForce GTX 1080. A similar shower can be simulated using
the GEANT4 in about $17$ second per particle on an Intel Xeon 8180 (currently
it is not possible to run a full Geant4-based simulation on GPUs). The speedup
of many orders of magnitude is achieved.

Table 2: Inference Timing for 3DGAN Method |  Platform |  Time/shower (msec) |  Speedup
---|---|---|---
Monte Carlo |  2S Intel Xeon Platinum 8180 |  $17000$ |  $1.0$
3DGAN CPU |  $16$ |  $2500$
3DGAN GPU |  GTX 1080 |  $4$ |  $4250$

##  5 Results and Discussion

The performance assessment for GAN models is a subject of much debate and
diverse viewpoints empirical . The GAN evaluation is nontrivial due to the
intractable probability densities and thus is mainly sample-based and
application-specific. We have validated the realism and diversity of our
generated data from several independent viewpoints, such as the output of a
third-party neural network and image quality assessment as presented
previously access2021  but the current work focuses mainly on the physics-
based comparison to a Monte Carlo simulation.

The particle showers have specific characteristics due to the underlying
physics processes, depending on the detector material and the type, energy,
and direction of the particle initiating the shower. We validate these
characteristics as a function of our inputs by dividing the data in $5$ GeV
$E_{P}$ and $0.1$ radian ($5.73^{\circ}$) $\theta$ bins. To ensure an unbiased
comparison, GAN events are generated with the same $E_{P}$ and $\theta$ values
as the GEANT4 events. The bin-wise comparison of each physics-based feature,
results in hundreds of histograms, for each particle type. We present here a
selected subset of the detailed and exhaustive validation that we consider to
be the most essential and representative of performance. The results presented
in this section are also alternated among different bins and particle types,
in order to convey the overall level of accuracy.

###  5.1 Visual Inspection

Figure 6: GEANT4 vs. GAN electrons showers with $E_{p}=202.78$ GeV and
$\theta=91.12^{\circ}$.

Figure 7: Projections of Monte Carlo vs. GAN events on the YZ, XZ and XY
planes. Top) neutral pions with $E_{P}=97.72$ GeV and $\theta=62.16^{\circ}$.
Bottom) photons with $E_{P}=403.62$ GeV and $\theta=116.33^{\circ}$.

An initial qualitative assessment can be performed by comparing the events
visually. Figure 6 shows an example of a 3D electron event. The event on the
left has been generated by GEANT4, while the event on the right has been
generated by the GAN for the same input values. It can be observed that both
events have very similar visual characteristics while retaining uniqueness for
individual cell deposits. The graphical projections on different planes
further illustrates the shower correlation to the incident angle ($\theta$)
and energy ($E_{P}$). Figure 7 compares the projections of the GEANT4 showers
to the corresponding GAN showers. The top panel presents neutral pion events,
while the bottom panel display photons events, with the $E_{P}$ and $\theta$
from both ends of the spectrum. The GAN images appear similar to the
respective GEANT4 images with the deposited cell energies correlated to the
input conditions while retaining stochasticity, for all particle types and
input conditions.

###  5.2 Particle Shower Features

Shower shapes define the structure of the deposited energy distribution, as a
shower develops through the detector material. The profiles of the energy
deposition along the detector axes are important observables related to the
shower geometry and crucial for most particle identification techniques. We
would like to point out that these geometrical features are not included in
the 3DGAN loss function as presented in Equation 1 and are learned by the GAN
implicitly. Figure 8 presents the shower shapes for the $X$, $Y$ and $Z$ axes
as a function of $\theta$ and $E_{P}$. In order to summarize the performance
for all particle types, we present a different particle in each column: the
shapes for electrons in the first column from the left, the photons in the
middle column, and the neutral pions in the rightmost column. The top row
presents the transverse shape distribution for the $Y$ axis corresponding to
the different $\theta$ bins since the $Y$ axis profile is most relevant for
$\theta$. The plots are displayed in the log scale to enhance the sparse
distributions along the tails. The second row presents the shapes along the
$Z$ axis (longitudinal direction) in linear scale for the different $E_{P}$
bins. The network is able to reproduce a similar shape distribution as the
GEANT4 showers, furthermore, the network can correctly relate it to the
inputs. In the transverse profiles, some discrepancies are observed in the log
scale. These discrepancies occur at the volume edges, where smaller energy
depositions occur. This region is also highly sparse and outside the main body
of the shower with expected energies well below $0.1$ MeV, which is comparable
to the pedestal values.

Moments are another aspect of the shower geometry. The GEANT4 showers are all
centered on the barycenter of the energy deposition by design thus the first
moment ($M1$) defining the shower center is easily replicated by GAN.
Therefore, we present here the performance related to the second moment ($M2$)
having a more complex distribution depending on both $E_{P}$ and $\theta$.
Figure 9 left panel presents the distribution of the second moment or the
width of the shower for electrons. Here it can be appreciated that the GAN has
learned the non-Gaussian width distribution. The mid plot shows the difference
between internal correlation present between the shower inputs ($E_{P}$ and
$\theta$) and the shower features (shapes, moments, total deposition, hits,
and ratios of energy deposition in different parts of the shower) for GEANT4
and GAN photons. The GAN showers are able to reproduce the internal
correlations present between the different shower observables. The right panel
displays the close agreement between the $\theta$ measured from GEANT4 neutral
pion events and that measured from the GAN events generated for the same
$\theta$ values.

Figure 8: Shower Shapes for the GEANT4 vs. GAN events as a function of inputs.
Top row) longitudinal shower shapes along the $Y$ axis: left) electrons with
$\theta$ in the $62^{\circ}$ bin; mid) photons with $\theta$ in the
$90^{\circ}$ bin; right) neutral pions with $\theta$ in the $118^{\circ}$ bin.
Bottom row) transverse shower shapes: left) electrons with $E_{P}$ in the
$100$ GeV bin; mid) photons with $E_{P}$ in the $200$ GeV bin; right) neutral
pions with $E_{P}$ with the $400$ GeV bin. Figure 9: The GAN vs. GEANT4 shower
features. Right) the shower width (M2) along the $Y$ axis for electrons. Mid)
difference between internal correlations present between physics features and
the inputs for photons. Left) the correlation between the measured angle from
GEANT4 events and the GAN events generated for the same $\theta$ values for
the neutral pions. Figure 10: Shower features related to the pixel intensities
(cell depositions) for the GEANT4 (red) vs. GAN (blue) events. Left) sparsity
level as a function of cutoff threshold for electrons. Mid) distribution of
cell energy deposits for photons. Right) sampling fraction for neutral pion
showers.  Figure 11: The GAN could successfully learn the photon features
after further training a network generating electrons with reduced $E_{P}$
range. Left) the $SF$ for photons is lower than electrons both in GEANT4
events and GAN events. The bottom panel shows the relative error
($\delta_{SF}$) for the sampling fraction. Right) the distribution of the
energy fraction deposited in the first part of the shower ($R1$) for the
electron and photon showers. The distribution for $R1<0.1$ (zoomed in the
inset) has very different distributions for electrons and photons. The GAN
also demonstrates a similar behavior.

The energies deposited in detector cells are the pixel intensities of our
images. The images are mostly empty, centered around a shower. The energies
are deposited only in around $20\%$ of the cells. Figure 10 left panel shows
sparsity (S) as the fraction of cells with some deposition against the
threshold used for cutoff. The GAN images have similar sparsity distributions
as the GEANT4 events, without specifically constraining the image. The
distributions for cell energy depositions (photon) shown in the mid panel have
a similar shape for GEANT4 and GAN events. We had reported that a sharp,
vertical drop around $0.2$ MeV was present in the GEANT4 cell energy
distribution that the GAN could not learn, yet tried to smooth out in the best
manner 3dgan_icmla . Since then a recent work high2020  also recognizes this
feature and tries to replicate the effect by additional post-processing on the
generated images. They note that their network improves the performance on the
simulation of the correct pixel distribution at the cost of reduced
performance for other features. We also report a reduction in performance when
constraining the pixel intensity distribution through our loss function. We
believe in the future the concept of ensembling ensambling  can be used to
employ two networks to generate the pixels above and below this region. The
sampling fraction ($SF$) is an important characteristic of the detector
response. Figure 10 left panel presents $SF$ for neutral pions. The $SF$ is
presented as a function of $E_{P}$. There is a close agreement for most of the
input range with some discrepancies at low energies, where events are highly
sparse with low cell energy deposits.

The characteristic features for the electromagnetic showers are faithfully
reproduced in the GAN-generated showers for different particle types. We test
if the generated showers for different particle types are mutually
distinguishable through their corresponding features. This is crucial as the
final networks for all particle types, use the same initial weights trained to
generate electron-induced showers for a reduced energy range. The photon-
initiated showers have some minor differences from electron-initiated showers.
The photons penetrate more distance into the detector material before starting
to interact Fabjan2020 ; calo_presentation . This effect can be evaluated by
studying the $SF$ and the fraction of energy deposited in the first ($8$ cells
along the $Z$ axis) part of the shower. Figure 11 left panel shows the
profiles of $SF$ as a function of $E_{P}$ for the central region of the
spectrum. The $SF$ for photons is lower than electrons for a similar value of
$E_{P}$. The right panel compares the distribution of energy deposited in the
first part of the shower for electrons and photons. The photons present more
entries for the region where the fraction of energy deposited in the first
part of the shower $R1$ is less than $10\%$. The GAN-generated photons clearly
demonstrate these identifiable features.

##  6 Simulating Charged Pions

The charged pions deposit a much smaller part of their energies in the ECAL
while most of the energy is deposited in the HCAL. The current project is only
limited to the ECAL data due to the limitation of the computing resources,
thus the work for charged pion is only a preliminary study. A more accurate
approach will also need to incorporate data from HCAL. We will present the
results of our study to lay the foundation for any future work.

---
Figure 12: Shower Shapes for GEANT4 vs. GAN charged pion events along $X$
(left), $Y$ (mid) and $Z$ (right) axes with random $E_{P}$ and $\theta$.
Figure 13: Shower moments along $Y$ axis for GEANT4 vs. GAN charged pions.
Left) $M1Y$ (shower center). Mid) $M2Y$ (shower width). Right) $M3Y$ (shower
skewness).

The transfer learning approach could not be extended to charged pions as the
showers have a very different distributions. The GAN is trained for the full
$E_{P}$ range of the charged pions, directly from random weights, for about
$200$ epochs using ($300k$) events. There is a great diversity in the charged
pion events, and most of the events have low energy deposition in the ECAL.
Since visual inspection will not be helpful to understand the performance,
thus we present only the distributions of physics-based features. Figure 12
presents the overall shower shapes in log scale. The shape distributions along
the $X$ and $Z$ axis show slightly better performance as compared to the $Y$
axis, probably due to the higher variance present in this dimension. Figure 13
compares the distribution of the first three moments defining the shower
center, width, and skewness along the $Y$ axis, display similar distributions.

Figure 14: Shower features for GAN vs. GEANT4 charged pion events. Left) $SF$
as a function of $E_{P}$ with shaded area representing standard deviation.
Mid) pixel intensities with log yscale. Right) difference between internal
correlations among shower features.

Figure 14 left panel displays the sampling fraction for GEANT4 and GAN charged
pion events. It can be seen that there is a difference between the means of
the two distributions, while the shaded area representing the standard
deviation shows some overlap. The mid panel presents the distribution of the
pixel intensities that are energies deposited in the calorimeter cells. There
is a deterioration in performance as compared to other particles, particularly
the effect of the presence of a cut around $0.2$ MeV (see Section 5) is more
pronounced. The left panel presents the difference in internal correlations
between physics-based features like shapes, moments, hits, $E_{P}$ and
$\theta$ for the GEANT4 and GAN events. There is a less than $10\%$ error for
the correlations, even given the highly diverse and incomplete showers. In
order to convey an idea of the diversity present in these showers, we
investigate the barycenters of the shower energy depositions along the
longitudinal axis (Z) for electrons and charged pions, as a function of
$E_{P}$. Figure 15 compares the first moment for the GEANT4 and GAN events.
The charged pion showers show greater diversity and depth as compared to
electrons, also apparent in the GAN events.

Figure 15: The distribution for the first moment along the $Z$ axis (Mz1) for
electrons and charged pions. The barycenter of deposition along the $Z$ axis
shows very different distributions for both particles with the charged pion
showers starting much later.

##  7 Rare modes

Figure 16: Graphical projections for the $XY$ and $XZ$ planes for MC events
with high discriminator probability of being real images; top) GEANT4 event;
bottom) GAN event.

The discriminator assigns a higher probability of being realistic to images
exhibiting features that the GAN cannot reproduce correctly . We visually
investigate such events having $O_{GAN}$ value greater than $0.6$, with the
help of graphical projections. It is observed that most of these events
manifest rare modes in data like pre-showering, late showering, and
incorrectly centered events. As other modes are found to be even rarer, thus
only the early showering events are further investigated. Figure 16 top row
presents an example of a GEANT4 pre-showering event. The particles that start
depositing their energies before entering the calorimeter volume have multiple
particles striking the detector surface, thus resulting in multiple branches.
Only a few percent of the Monte Carlo samples present such behavior and the
percentage decreases with increasing $E_{P}$.

Figure 16 bottom row presents an example of a pre-showering event generated by
GAN. It must be mentioned here that in the first training step more such
events are generated. The performance for these rare modes deteriorates with
further training for the full $E_{P}$ range due to a decrease in the
percentage of such events for higher energies. In order to further improve the
performance, methods like the ensembling ensambling  can be explored.

##  8 Training with GAN data

We present a practical use case for the GAN-generated events. Triforce
Belayneh2019  is a deep learning model developed by a third-party study for
the identification of particle type and primary energy for particle showers
from the calorimeter dataset used for 3DGAN. We had previously employed the
pre-trained Triforce DNN model for classification and regression of the GAN
events 3dgan_icmla  and proved that the type and the primary particle energy
for the GAN-generated events was correctly predicted.

Triforce can also be considered as an example of a typical reconstruction tool
used in HEP simulation. We test the performance of our 3DGAN generated images
for training this tool. The Triforce requires two types of particles
(electrons and charged pions ) for training. We train the Triforce GoogleNet
model from scratch on GEANT4 electron events and then on GAN electron events.
The charged pion events in both trainings are those generated by the GEANT4.
Figure 17 presents a comparison for the primary particle energy regression for
the network trained on GEANT4 against the network trained on GAN electrons
showing a similar performance. The particle type classification accuracy
presented in Table 3 also manifest similar values. Thus we prove that the GAN
simulation can be used to replace the GEANT4 simulation without any loss of
accuracy.

In the context of GAN evaluation, the GAN-train and GAN-test are two very
interesting concepts 2018gan_test . The accuracy of a classifier network
trained on GAN generated events and tested on data events is termed as GAN-
train. When GAN images are high quality and as diverse as the training set,
the score on the validation set should be similar to training accuracy. A
lower accuracy would indicate that GAN images are not covering the entire
distribution of the training data. GAN-test is the accuracy of a network
trained on true data and validated on GAN images. A lower accuracy would
indicate that the GAN images are not sufficiently realistic while a higher
accuracy could be related to mode dropping. The 3DGAN shows similar
performance as the GEANT4 training data and thus the results of this test can
also be regarded as proof of high accuracy and diversity of the GAN generated
events.

Figure 17: Performance of the network trained on GEANT4 electrons and charged pions (left) vs. that of the network trained GAN electrons and GEANT4 charged pions (right). The profile plot for true and predicted $E_{P}$ for the GAN and Monte Carlo samples are very similar. There were $9834$ events for each type. Table 3: Triforce classification results Trained on |  Accuracy for MC |  Accuracy for GAN
---|---|---
Monte Carlo electrons |  $0.9928\pm 0.014$ |  $0.9992\pm 0.014$
GAN electrons |  $0.9884\pm 0.014$ |  $0.9998\pm 0.014$

##  9 Conclusions and Future Suggestions

Simulation is crucial for most HEP experiments. Monte Carlo methodology can
successfully simulate particle interactions at the cost of time and resources.
Fast simulation is a set of faster alternatives that have been successfully
used to replace detailed simulation where some loss in accuracy can be
acceptable. Recent advances in deep learning have had a tremendous impact on
the HEP community and are especially interesting for simulation due to the
possibility of training directly from the detector data, not possible using
other methodologies. The 3DGAN is an effort aimed to simulate the detector
output, as images generated by a neural network. These images are conditioned
on a number of variables, having large dynamic range of values. We exploit a
multi-step training process, resulting in accurate simulation for electrons,
photons, and neutral pions. The accuracy for individual data features varies
but is within $10\%$ of the GEANT4 simulation for all quantities. Preliminary
work on charged pion simulation is also promising where the ECAL contains only
partial showers, manifesting higher variance. The GAN is able to reproduce the
essential features of a charged pion shower. Another exploratory study
demonstrate the possibility of generating rare modes present in the data. We
further provide an example of a practical use of the GAN-generated events. The
GAN simulated events are able to train a third-party particle classification
and regression tool for the correct classification of the GEANT4 events. The
response of the tool trained on GAN data is similar to that trained on GEANT4
data. Finally, we would like to state that the GAN-generated showers are
simulated with a speedup of three orders of magnitude.

We would like to point out some insights from our endeavour that maybe helpful
in the context of any future work. Certain domain-related features like the
total deposited energy need to be hardcoded in the loss function while other
features can be learned implicitly. These features include geometrical
properties such as shapes and moments, level of sparsity, pixel intensity
distribution, and correlation among the different features as well as the
inputs. The model can learn complex distributions for individual features and
the only part where the GAN struggles is reproducing the sharp drop in pixel
intensities as discussed in Section 5. Apart from that, the sparse peripheral
regions of the images are more difficult to be correctly generated, and there
is some loss in performance for very low $E_{P}$ particles. The current
training allows learning of an average response for the total deposited energy
since the training relies on direct comparison for small data batches. We
believe that a better formulation of the loss might result in better agreement
in future efforts. The preliminary work on charged pion and rare mode
simulation shows great promise. The charged pion simulation can be improved by
including the HCAL data. The generation of rare modes can benefit by exploring
methods like the ensembling approach, where multiple networks can be trained
simultaneously for different modes present in the data. The speedup can also
be further increased by exploiting parallel hardware cray2018  that cannot yet
be done for the sequential logic employed by the standard Monte Carlo tools. A
distributed training aisis2020  will be most essential for future
generalization of the approach through hyperparameter scan.

###### Acknowledgements.

This work has been conducted with the support of Intel in the framework of the
CERN openlab-Intel collaboration agreement. Part of this work was conducted at
"iBanks", the AI GPU cluster at Caltech. We acknowledge NVIDIA, SuperMicro and
the Kavli Foundation for their support of "iBanks". We thank Matt Zhang from
the University of Illinois at Urbana-Champaign for help regarding the Triforce
Belayneh2019  model.

##  Appendix A Hyper-parameters for 3DGAN

Table 4 presents the values for the different hyperparameters selected for
3DGAN.

Table 4: Hyperparameters for 3DGAN fully optimized version No. |  param |  value |  description
---|---|---|---
1 |  epochs |  $130$ for step $1$ and $30$ epochs for step $2$ |  Number of iterations through entire training data
2 |  batch size |  $64$ |  Number of samples in a minibatch
3 |  latent size |  $254$ |  Size of latent vector sampled from Gaussian with mean=0 and std =1
4 |  discriminator layers |  $4$ |  Convolutional layers in the discriminator
5 |  generator layers |  $7$ |  Convolutional layers in the generator
6 |  optimizer |  RMSprop rmsProp  |  type of optimizer
7 |  lr |  $0.01$ |  learning rate
8 |  pre-processing cell intensities by power |  $0.85$ |  taking a power of cell intensities
9 |  pre-processing target $E_{p}$ |  scaling by $1/100$ |  Dividing $E_{p}$ by a factor of $100$
10 |  $W_{G}$ |  $3$ |  Weight for loss associated to real/fake probability
11 |  $W_{P}$ |  $0.1$ |  Weight for loss associated to auxilliary energy regression task
12 |  $W_{E}$ |  $0.1$ |  Weight for loss associated to sum of intensities
13 |  $W_{A}$ |  $25$ |  Weight for loss associated to measured angle

## References

  * (1) The HEP Software Foundation, J. Albrecht et al., _A Roadmap for HEP Software and Computing R &D for the 2020s_, _Computing and Software for Big Science_ 3 (2019) 7.
  * (2) W. Lucas, _Fast simulation for atlas: Atlfast-ii and isf_ , in _International Conference on Computing in High Energy and Nuclear Physics_ , vol. 396, 2012.
  * (3) D. Orbaker, _Fast simulation of the cms detector_ , in _International Conference on Computing in High Energy and Nuclear Physics_ , vol. 219, 2010.
  * (4) NOMAD Collaboration collaboration, _Parameterization of $e$ and $\gamma$ initiated showers in the NOMAD lead-glass calorimeter_, _Nucl. Instrum. Methods Phys. Res., A_ 425 (1998) 188. 28 p.
  * (5) E. Barberio et al., _Fast simulation of electromagnetic showers in the ATLAS calorimeter: Frozen showers_ , _J. Phys. Conf. Ser._ 160 (2009) 012082.
  * (6) I.J. Goodfellow et al., _Generative adversarial nets_ , in _Proceedings of the 27th International Conference on Neural Information Processing Systems - Volume 2_ , NIPS’14, (Cambridge, MA, USA), pp. 2672–2680, MIT Press, 2014, http://dl.acm.org/citation.cfm?id=2969033.2969125.
  * (7) D.P. Kingma and M. Welling, _Auto-encoding variational bayes._ , in _ICLR_ , Y. Bengio and Y. LeCun, eds., 2014, http://dblp.uni-trier.de/db/conf/iclr/iclr2014.html#KingmaW13.
  * (8) A. Van Den Oord, N. Kalchbrenner and K. Kavukcuoglu, _Pixel recurrent neural networks_ , in _Proceedings of the 33rd International Conference on International Conference on Machine Learning - Volume 48_ , ICML’16, p. 1747–1756, JMLR.org, 2016.
  * (9) M. Arjovsky, S. Chintala and L. Bottou, _Wasserstein gan_ , 2017.
  * (10) H. Zhang, T. Xu, H. Li, S. Zhang, X. Wang, X. Huang et al., _Stackgan++: Realistic image synthesis with stacked generative adversarial networks_ , _IEEE Transactions on Pattern Analysis and Machine Intelligence_ 41 (2019) 1947.
  * (11) T. Karras, T. Aila, S. Laine and J. Lehtinen, _Progressive growing of gans for improved quality, stability, and variation_ , _CoRR_ abs/1710.10196 (2017) [1710.10196].
  * (12) S. Li, S. Jang and Y. Sung, _Automatic melody composition using enhanced gan_ , _Mathematics_ 7 (2019) .
  * (13) S. Subramanian, S. Rajeswar, F. Dutil, C. Pal and A. Courville, _Adversarial generation of natural language_ , in _Proceedings of the 2nd Workshop on Representation Learning for NLP_ , (Vancouver, Canada), pp. 241–251, Association for Computational Linguistics, Aug., 2017, DOI.
  * (14) X. Yi, E. Walia and P. Babyn, _Generative adversarial network in medical imaging: A review_ , _Medical Image Analysis_ 58 (2019) 101552.
  * (15) Z. Chen, Z. Zeng, H. Shen, X. Zheng, P. Dai and P. Ouyang, _Dn-gan: Denoising generative adversarial networks for speckle noise reduction in optical coherence tomography images_ , _Biomedical Signal Processing and Control_ 55 (2020) 101632.
  * (16) A. Alsaiari, R. Rustagi, A. Alhakamy, M.M. Thomas and A.G. Forbes, _Image denoising using a generative adversarial network_ , in _2019 IEEE 2nd International Conference on Information and Computer Technologies (ICICT)_ , pp. 126–132, 2019.
  * (17) G. Khattak, S. Vallecorsa and F. Carminati, _Three dimensional energy parametrized generative adversarial networks for electromagnetic shower simulation_ , in _2018 25th IEEE International Conference on Image Processing (ICIP)_ , pp. 3913–3917, Oct, 2018, DOI.
  * (18) G. Khattak, S. Vallecorsa, F. Carminati and G.M. Khan, _Particle detector simulation using generative adversarial networks with domain related constraints_ , in _2019 18th IEEE International Conference On Machine Learning And Applications (ICMLA)_ , pp. 28–33, Dec, 2019, DOI.
  * (19) D. Belayneh et al., _Calorimetry with deep learning: Particle simulation and reconstruction for collider physics_ , _Eur. Phys. J. C_ 80 (2019) 688. 31 p [1912.06794].
  * (20) ATLAS Collaboration, _The atlas calorimeter simulation fastcalosim_ , in _IEEE Nuclear Science Symposuim Medical Imaging Conference_ , pp. 1–5, 2010\.
  * (21) L. de Oliveira, M. Paganini and B. Nachman, _Learning particle physics by example: Location-aware generative adversarial networks for physics synthesis_ , _arXiv preprint arXiv:1701.05927_ (2017) .
  * (22) M. Paganini, L. de Oliveira and B. Nachman, _Calogan: Simulating 3d high energy particle showers in multilayer electromagnetic calorimeters with generative adversarial networks_ , _Phys. Rev. D_ 97 (2018) 014021.
  * (23) D. Salamani et al., _Deep Generative Models for Fast Shower Simulation in ATLAS_ , in _Proceedings, 14th International Conference on e-Science: Amsterdam, Netherlands, October 29-November 1, 2018_ , p. 348, 2018, DOI.
  * (24) V. Chekalina, E. Orlova, F. Ratnikov, D. Ulyanov, A. Ustyuzhanin and E. Zakharov, _Generative Models for Fast Calorimeter Simulation: the LHCb case_ , _EPJ Web Conf._ 214 (2019) 02034 [1812.01319].
  * (25) M. Erdmann, J. Glombitza and T. Quast, _Precise simulation of electromagnetic calorimeter showers using a Wasserstein Generative Adversarial Network_ , _Comput. Softw. Big Sci._ 3 (2019) 4 [1807.01954].
  * (26) R. Di Sipio, M.F. Giannelli, S.K. Haghighat and S. Palazzo, _DijetGAN: A Generative-Adversarial Network Approach for the Simulation of QCD Dijet Events at the LHC_ , _JHEP_ 08 (2020) 110 [1903.02433].
  * (27) E. Buhmann, S. Diefenbacher, E. Eren, F. Gaede, G. Kasieczka, A. Korol et al., _Getting High: High Fidelity Simulation of High Granularity Calorimeters with High Speed_ , 2005.05334.
  * (28) T.I. Collaboration, _International large detector: Interim design report_ , 2020\.
  * (29) G.R. Khattak, S. Vallecorsa, F. Carminati and G.M. Khan, _High energy physics calorimeter detector simulation using generative adversarial networks with domain related constraints_ , _IEEE Access_ 9 (2021) 108899.
  * (30) CERN, “Welcome to the Compact Linear Collider Website | clic-study.web.cern.ch.”
  * (31) GEANT4 collaboration, _GEANT4: A Simulation toolkit_ , _Nucl. Instrum. Meth._ A506 (2003) 250.
  * (32) P. Lebrun, L. Linssen, A. Lucaci-Timoce, D. Schulte, F. Simon, S. Stapnes et al., _The CLIC Programme: Towards a Staged e+e- Linear Collider Exploring the Terascale : CLIC Conceptual Design Report_ , 1209.2543.
  * (33) A. Odena, C. Olah and J. Shlens, _Conditional image synthesis with auxiliary classifier GANs_ , in _Proceedings of the 34th International Conference on Machine Learning_ , D. Precup and Y.W. Teh, eds., vol. 70 of _Proceedings of Machine Learning Research_ , (International Convention Centre, Sydney, Australia), pp. 2642–2651, PMLR, 06–11 Aug, 2017, http://proceedings.mlr.press/v70/odena17a.html.
  * (34) F. Chollet et al., “Keras.” https://github.com/fchollet/keras, 2015.
  * (35) M. Abadi et al., _TensorFlow: Large-scale machine learning on heterogeneous systems_ , 2015.
  * (36) S. Ioffe and C. Szegedy, _Batch normalization: Accelerating deep network training by reducing internal covariate shift_ , in _Proceedings of the 32nd International Conference on International Conference on Machine Learning \- Volume 37_ , ICML’15, p. 448–456, JMLR.org, 2015.
  * (37) A.L. Maas, A.Y. Hannun and A.Y. Ng, _Rectifier nonlinearities improve neural network acoustic models_ , in _in ICML Workshop on Deep Learning for Audio, Speech and Language Processing_ , 2013.
  * (38) V. Nair and G.E. Hinton, _Rectified linear units improve restricted boltzmann machines_ , in _Proceedings of the 27th International Conference on International Conference on Machine Learning_ , ICML’10, (USA), pp. 807–814, Omnipress, 2010, http://dl.acm.org/citation.cfm?id=3104322.3104425.
  * (39) N. Srivastava, G.E. Hinton, A. Krizhevsky, I. Sutskever and R. Salakhutdinov, _Dropout: a simple way to prevent neural networks from overfitting._ , _Journal of Machine Learning Research_ 15 (2014) 1929.
  * (40) G. Hinton, N. Srivastava and K. Swersky, “Lecture 6a overview of mini–batch gradi-ent descent..” 2012.
  * (41) Q. Xu, G. Huang, Y. Yuan, C. Guo, Y. Sun, F. Wu et al., _An empirical study on evaluation metrics of generative adversarial networks_ , 2018.
  * (42) J. Xie, B. Xu and C. Zhang, _Horizontal and vertical ensemble with deep representation for classification_ , _CoRR_ abs/1306.2759 (2013) [1306.2759].
  * (43) C.W. Fabjan and D. Fournier, _Calorimetry_ , in _Particle Physics Reference Library: Volume 2: Detectors for Particles and Radiation_ , C.W. Fabjan and H. Schopper, eds., (Cham), pp. 201–280, Springer International Publishing (2020), DOI.
  * (44) D. Bortoletto, “Detectors for particle physics.” url: https://indico.cern.ch/event/318531/attachments/612850/843143/daniela_l5.pdf.
  * (45) K. Shmelkov, C. Schmid and K. Alahari, _How good is my GAN?_ , _arXiv e-prints_ (2018) arXiv:1807.09499 [1807.09499].
  * (46) S. Vallecorsa, D. Moise, F. Carminati and G.R. Khattak, _Data-parallel training of generative adversarial networks on hpc systems for hep simulations_ , in _2018 IEEE 25th International Conference on High Performance Computing (HiPC)_ , pp. 162–171, 2018, DOI.
  * (47) F. Carminati et al., _Generative Adversarial Networks for Fast Simulation: distributed training and generalisation_ , in _Proceedings of Artificial Intelligence for Science, Industry and Society — PoS(AISIS2019)_ , vol. 372, p. 012, 2020, DOI.