# Deep Generative Models for Detector Signature Simulation: An Analytical
Taxonomy

Hosein Hashemi
Faculty of Physics
Ludwig Maximilians University of Munich,
Munich, Germany
<EMAIL_ADDRESS>
& Claudius Krause
Institute for High-Energy Physics (HEPHY),
Austrian Academy of Sciences (OeAW),
Vienna, Austria
<EMAIL_ADDRESS>

###### Abstract

In modern collider experiments, the quest to explore fundamental interactions
between elementary particles has reached unparalleled levels of precision.
Signatures from particle physics detectors are low-level objects encoding the
physics of collisions. The complete simulation of them in a detector is a
memory and storage-intensive task. To address this computational bottleneck in
particle physics, “Fast Simulation” has been introduced and refined over the
years. The field has seen a surge in interest in surrogate modeling the
detector simulation, fueled by the advancements in deep generative models.
These models aim to generate responses that are statistically identical to the
observed data. In this paper, we conduct a comprehensive and exhaustive
taxonomic review of the existing literature on the simulation of detector
signatures from both methodological and application-wise perspectives.
Initially, we formulate the problem of detector signature simulation and
discuss its different variations that can be unified. Next, we classify the
state-of-the-art methods into four distinct categories based on their
underlying model architectures, summarizing their respective generation
strategies. We then identify and discuss three key application areas. Finally,
we shed light on the challenges and opportunities that lie ahead in detector
signature simulation, setting the stage for future research and development.

##  1 Introduction

The core concept of generative models is derived from the training of a
density estimator that produces samples approximating the distribution of the
training data. The initial generation of neural network-based generative
models, also known as energy-based models [1, 2], attempted to accomplish this
by establishing an energy function proportional to the likelihood for data
points. However, these models faced challenges in scaling up to high-
dimensional, complex data, like natural images. Thus, they necessitated the
use of Markov Chain Monte Carlo (MCMC) sampling [3], a method required during
both training and inference stages. The goal of energy-based models is to
adjust the model parameters so that the energy function assigns lower energy
(i.e., higher likelihood) to real data points and higher energy to improbable
ones. However, the challenge is that these energy landscapes can be very
complex, especially for high-dimensional data. Calculating likelihoods or
gradients directly becomes computationally infeasible or extremely difficult.
That’s where MCMC comes in. This method, characterized by its iterative
nature, often resulted in a very slow and inefficient process. The resurgence
of interest in generative models in the past few years can be attributed to
the availability of larger datasets and significant advancements in Deep
Generative Models (DGMs) [4] for natural images. These DGMs have pushed the
boundaries in terms of visual quality, sample diversity, and speed of
sampling.

In Particle Physics, the application of DGMs was first studied in the “Fast
and Efficient Simulation” campaign [5] that sparked the search for faster and
more storage-efficient surrogate models and simulation methods of collider
physics experiments [6, 7]. Surrogate models for fast detector simulation are
simplified, computationally efficient approximations that emulate the behavior
of more complex, detailed simulations of signatures from particle detectors.
These models are constructed using DGMs, trained on a dataset either generated
by the Geant4 simulation [8, 9, 10] or the real signatures from particle
detectors. Once trained, the model can generate samples that are statistically
similar to their training data and even can generalize to the data beyond the
training regime [11, 12, 13, 14] depending on the objective. There are some
key requirements that the model should meet to be effective and efficient:

  1. 1.

Low time-complexity: The surrogate model must be computationally efficient in
order to facilitate fast simulations. Computational speed is crucial when
performing large-scale simulations or when needing to iterate the model many
times for optimization or fine-tuning. Thus, the model should take advantage
of parallel processing capabilities and have to be optimized for the hardware
it is expected to run on, whether it’s a CPU or GPU.

  2. 2.

Low space-complexity: It should come with a minimal storage cost. Hence, the
underlying compression technique has to reduce the storage footprint without
sacrificing too much in terms of the downstream physics analysis.

  3. 3.

Realistic and Diverse: It has to generate samples as faithful and diverse as
possible from the downstream physics analysis point of view. Thus, the
sampling techniques should be capable of employing the nuanced behaviors and
symmetries of the detector as an inductive bias to ensure that the model
captures the diversity inherent in the real data.

  4. 4.

Generalization: For certain objectives, It has to be able to extrapolate to
parameter levels (such as energy levels, kinematic profile, beam parameters,
and luminosities) beyond the current experimental limits in order to analyze
the detector’s effect and do physics analysis in the extrapolation region.
Therefore, the model should be robust against overfitting and incorporate a
proper measure of control when extrapolating to give a range of plausible
outcomes.

The subsequent sections of this paper are organized in the following manner.
We begin by defining the challenge of simulating signatures from particle
detectors with deep generative models as surrogates. Subsequently, we present
a taxonomic breakdown of current methodologies, categorizing them into four
classes. We introduce a universal framework, delve into prevalent generation
strategies and tasks, and then provide an exhaustive review of research for
each category. In our concluding remarks, we address the challenges and
potential advancements in deep generative modeling for detector response
generation. Through this survey, we aim to offer a unified and consolidated
view of the present landscape of deep generative models for detector signature
simulation. We hope to bridge the existing gap in this sparse and occasionally
overlooked research, ensuring an inter-experimental perspective that remains
objective and free from bias.

##  2 Problem Definition

The detector signatures can be defined by a triplet
$\mathcal{D}_{e}=(\mathbf{L}_{e},\mathbf{C}_{e},\mathbf{H}_{e})$ for each
event $e$, where $\mathbf{L}\in\mathbb{Z}$ is the detector component such as
layers or sensors indicators, $\mathbf{C}\in\mathbb{R}^{n}$ is the global
attribute of the corresponding event such as incident energy or the beam
parameter, $\mathbf{H}\in\mathbb{R}^{d}$ is the hit points per sensor/layer
like tracks, clusters or jets which can be represented for instance by a grid-
like an image and sequence, or a multi-set like point clouds and graphs. Given
a set of $M$ observed detector responses
$\mathbf{D}=\\{\mathcal{D}^{i}_{e}\\}_{i=1}^{M}$, DGMs learn the distribution
of these signatures $p(\mathbf{D})$, from which new responses can be sampled
$\mathbf{H}_{\text{new}}\sim p(\mathbf{H})$.

This paper presents both the latent variable and non-latent variable
approaches as the mainstream detector response generation models. Latent
variable approaches follow an encoder–sampler–decoder pipeline. It firstly
maps the data into a hidden space through an encoding function, manipulates
the hidden variables to reflect the desired properties of the detector
response to be generated, and then generates new samples based on latent codes
through a decoding function. Unlike the latent variable models111 Here, the
term latent refers to a compressed space. In the literature, latent spaces
sometimes also refer to uncompressed spaces, such as for normalizing flows or
diffusion models., non-latent variable approaches directly map the input data
to the desired detector responses stochastically without the intermediary step
of encoding into a compressed hidden space.

##  3 Algorithm Taxonomy

Within the scope of latent variable techniques, the given data is mapped into
a stochastic latent space. An i.i.d sample from this distribution is then fed
into a decoder that reconstructs the original data structures. In non-latent
variable models, generation methods do not map the observation into the latent
space. In other words, they perform the generation in the raw space without
first compressing it into a latent form and directly generate the data
representation. These models, given initial conditions, apply transformations
or rules that map (encoder) the original input features straight to the
output. Prior to delving into an in-depth exploration of distinct models, we
initially established a bird-eye-view pipeline that encompasses encoding,
sampling, and decoding stages. This pipeline allows us to encapsulate the
majority of the existing generative models used for detector simulation within
a single unified framework. In accordance with this framework, we will
classify various methodologies based on their interaction with the following
three pivotal components:

The Encoder. The encoding function $f_{\theta}(z|\mathcal{D})$ maps the
detector response triplets to a dense, continuous (or quantized) topological
space. To ensure the learned latent space is meaningful for generation,
depending on the data type and the inductive bias, one incorporates various
morphisms (e.g., Convolutional Neural Networks [15], Graph Neural Networks
[16], DeepSets [17], etc) as the encoder. In this way, the encoder function
$f_{\theta}$ generates the parameters for a stochastic distribution that
adheres to a prior distribution denoted as $p(z)$.

The Sampler. Following the encoding process, the model that generates the
detector response draws samples from the latent distribution $z\sim p(z)$.
There are generally two prevalent strategies for this sampling process: random
sampling and controlled sampling [18]. Random sampling refers to randomly
selecting latent vectors from the learned or prior distribution. On the other
hand, guided or controlled sampling is designed to sample the stratified
latent vectors with the specific goal of generating samples that exhibit
certain preferred characteristics. These characteristics could be certain
kinematic profiles, beam parameters, physics processes, number of hits,
luminosities, incident angles, or incident energies. In most tasks, controlled
sampling is model-dependent and necessitates an additional optimization
component that goes beyond the scope of random generation.

The Decoder. Upon obtaining the latent features drawn from the learned
distribution, the decoding mechanism is responsible for reconstituting them
into a data manifold. Due to the multi-objective and fine-grained learning
nature of detector simulation, the decoding phase is inherently more complex
than the encoding stage. Typically, the decoders could be grouped into three
categories: sequential generation, one-shot generation, and zero-shot
generation. Sequential generation refers to generating the detector component
information in a set of consecutive and autoregressive steps, usually done
sensor-by-sensor, layer-by-layer, or hit-by-hit. One-shot generation, instead,
refers to generating the whole detector signatures in one single step. Zero-
shot generation is when the model generates an unseen set of detector
information where the control parameter goes beyond the training data. Thus,
the model has to generalize well to the Out-Of-Distribution (OOD) domain.
While zero-shot generation may employ one-shot or sequential methods for the
actual data construction, its main challenge lies in ensuring that these
generated responses are both accurate and plausible when dealing with
conditions that differ from the training data.

###  3.1 Deep Generative Models

At first, we discuss the following five representative generative models,
depicted in Figure 1, and their capabilities for simulating detector
signatures, summarized at Table 1.

Figure 1: Generic architectures of deep Generative models for simulating
detector signatures

Variational Auto Encoders (VAE). VAE estimates the distributions of data
$p(\mathcal{D})$ with the amortized variational posterior [19, 20], by
maximizing the Evidence Lower BOund (ELBO), i.e. the lower bound of the log-
likelihood function, as follows:

$\mathcal{L}_{\text{VAE}}=\mathbb{E}_{z\sim
f_{\theta}(\bm{z}\mid\mathcal{D})}\left[\log(p_{\phi}(\mathcal{D}\mid\bm{z}))\right]-D_{\text{KL}}\left(f_{\theta}(\bm{z}\mid\mathcal{D})\parallel
p_{\phi}(\bm{z})\right)$ (1)

where the former term is known as the negative reconstruction loss between the
input and the reconstructed data, while the latter is the disentanglement
enhancement regularizer that drives the amortized variational posterior
$f_{\theta}(z|\mathcal{D})$ to the prior (marginal) distribution
$p_{\theta}(z)$. For simulation-based inference where the tractability of the
likelihood is vital, one can use various sampling methods to estimate the
likelihood such as Importance Sampling [20, 21], Annealed Importance Sampling
[22, 23], and Nested Variational Inference [24] with a sufficiently large
sample space to estimate the log-likelihood. Generally, VAEs require fewer
computational resources than all the DGMs, converge faster, and one can have
more control over the conditional parameters.

However, VAEs face a few challenges. A potential issue is the “posterior
collapse” [25], which occurs when a powerful decoder treats the latent
variable $z$ as mere noise, leading to the regularization term being minimized
for priors like the standard Gaussian. Another problem, known as the “hole
problem” [26], arises from a mismatch between the aggregated posterior and the
prior. If there are regions where the prior assigns high probability, but the
aggregated posterior assigns low, sampling from these regions can result in
low-quality output.

Normalizing Flows. Normalizing flow (NF) estimates the density of data
$p(\mathcal{D})$ directly with an invertible and deterministic bijection
between the latent variables and the data manifold via the change of variables
formula. A typical instance of flow-based models takes the following form:

$p(\mathcal{D})=p(f(\mathcal{D}))\left|\det\left(\frac{\partial
f^{-1}(\mathcal{\mathcal{D}})}{\partial\mathcal{D}}\right)\right|,$ (2)

where $p(\bm{z})$ is the density of the latent variables $\bm{z}$. The term
$f^{-1}(\mathcal{D})$ is the inverse of the transformation $f$ that maps the
latent variables to the data. The determinant of the Jacobian matrix
$\left|\det\left(\frac{\partial
f^{-1}(\mathcal{D})}{\partial\mathcal{D}}\right)\right|$ accounts for the
change in volume induced by this transformation, allowing for efficient and
expressive density estimation. Flow-based models are proper candidates for
lossless compression [27, 28], and Approximate Bayesian Computation (ABC) [29]
in simulation-based inference since they allow the calculation of the exact
likelihood. However, the likelihood does not always correlate directly with
sample quality [30]. Conditional flow-based models could also be used to form
a flexible family of variational posteriors [31, 32] in VAE-based models with
which the lower bound to the log-likelihood function could be tighter.

The main problem with NF-based models is the computational overhead they
create with high-dimensional data. NF models can struggle with very high-
dimensional data spaces as the complexity of the necessary transformations
grows exponentially. This makes them less suitable for simulating complex and
high granular detector signatures like HGCAL [33] for HL-LHC [34] and PXD [35]
at Belle II [36].

Generative Adversarial Networks (GAN). A GAN [37, 38] is an implicit density
estimator [39], which learns to sample real data points in a zero-sum game.
GANs consist of two main components, namely, a generator $f_{G}$ for
estimating the density implicitly and generating realistic data and a
discriminator $f_{D}$ for distinguishing between the generated and real
detector responses. This adversarial mechanism encourages the generator to
produce synthetic data that the discriminator cannot distinguish from real
data, improving the model’s ability to capture the data distribution. Although
GANs do not learn the density directly, they do it implicitly, where one can
indeed incorporate it for simulation-based inference [40], for example, in
detector unfolding [41, 42]. Formally, its training objective is a min-max
game as follows:

$\min_{f_{\text{G}}}\max_{f_{\text{D}}}\,\mathcal{L}_{\text{GAN}}(f_{\text{G}},f_{\text{D}})=\mathbb{E}_{\mathcal{D}\sim
p(\mathcal{D})}[\log f_{\text{D}}(\mathcal{D})]+\mathbb{E}_{\bm{z}\sim
p(\bm{z})}[\log(1-f_{\text{D}}(f_{\text{G}}(\bm{z})))].$ (3)

In general, training GANs is a highly brittle and painful task. It requires a
significant amount of patience and hyperparameter tuning for domain-specific
tasks, but when it generalizes well, it can produce remarkably high-resolution
samples [43, 44] with high fidelity. As with almost all DGMs, GANs also suffer
from mode collapse and vanishing gradients. Nevertheless, GANs provide good
control over the conditioning and enable representation learning [45] for any
possible downstream task. However, a specific issue with GANs is their evasive
Nash equilibrium, a state where the discriminator and the generator cannot
unilaterally improve their position by deviating from their current strategy.
The models may continue to oscillate without settling into a stable state, or
they may converge to a non-optimal solution. This makes the model selection
non-trivial, since GANs have no straightforward way to select the best
training epoch.

Autoregressive Models Autoregressive Models (ARMs) [46] are based on the chain
rule of probability and decompose a joint distribution over $N$ random
variables. Specifically, this model factorizes the generation process as a
sequential step, which determines the next step action given an initial
detector layer, hit, jet constituent, or shower. The general formulation of
ARM models is as follows:

$p(\mathcal{D}^{e})=\prod_{i=1}^{N}p(\mathcal{D}^{e}_{i}\mid\mathcal{D}^{e}_{1},\mathcal{D}^{e}_{2},\cdots,\mathcal{D}^{e}_{i-1}),$
(4)

where it is possible to directly maximize the likelihood of the data by
training a recurrent neural network to model
$p(\mathcal{D}^{e}_{i}\mid\mathcal{D}^{e}_{1:i-1})$ by minimizing the negative
log-likelihood,

$-\ln p(\mathcal{D}^{e})=-\sum_{i}\ln
p(\mathcal{D}^{e}_{i}\mid\mathcal{D}^{e}_{1},\mathcal{D}^{e}_{2},\cdots,\mathcal{D}^{e}_{i-1}).$
(5)

Since ARMs work like sequential generation, applying AR models requires a pre-
specified ordering defined by hit orders $\mathbf{H}_{e}$ or sensor/layer
$\mathbf{L}_{e}$ orders in a detector signature triplet. While autoregressive
models are extremely powerful density estimators, sampling is inherently a
sequential process and can be exceedingly slow on high-dimensional data.
Additionally, data must be decomposed into a fixed ordering (e.g., temporal);
while the choice of ordering can be clear for detector layers in regular
detectors, it is not obvious for irregular and complex topologies, which can
affect performance.

Diffusion Models Diffusion models or deep score-based [47, 48] generative
models are a class of generative processes inspired by non-equilibrium
thermodynamics [49] and PDEs [50]. Score matching is based on the idea of
minimizing the difference between the derivatives of the data and the model’s
log density functions. Diffusion models contain two processes, the forward and
the reverse diffusion process. The forward diffusion process constantly adds
noise to the data sample using noise schedule $\beta_{t}\in(0,1)$ controlling
the step size, while the reverse diffusion process recreates the true data
sample from a Gaussian noise input. In the language of hierarchical VAEs [51,
52], the bottom-up path (i.e., the variational posteriors) can be a diffusion
process, and the top-down path is parameterized by a reversed diffusion. Ergo,
they provide only an approximation of the likelihood. Formally, the forward
diffusion process from step $(t-1)$ to $t$ is defined as a discrete Markov
chain

$\displaystyle q(\mathcal{D}_{t}\mid\mathcal{D}_{t-1})$
$\displaystyle=\mathcal{N}(\mathcal{D}_{t};\,\sqrt{1-\beta_{t}}\mathcal{D}_{t-1},\,\beta_{t}\bm{I}),$
(6) $\displaystyle q(\mathcal{D}_{1:T}\mid\mathcal{D}_{0})$
$\displaystyle=\prod_{t=1}^{T}q(\mathcal{D}_{t}\mid\mathcal{D}_{t-1}).$ (7)

Note that the reverse diffusion process $q(\mathcal{D}_{t-1}|\mathcal{D}_{t})$
will also be Gaussian if $\beta_{t}$ is small enough. However, as
$q(\mathcal{D}_{t-1}|\mathcal{D}_{t})$ is intractable,
$p_{\theta}(\mathcal{D}_{t-1}|\mathcal{D}_{t})$ is learned to approximate
$q(\mathcal{D}_{t-1}|\mathcal{D}_{t})$ as follows:

$p_{\theta}(\mathcal{D}_{t-1}\mid\bm{x}_{t})=\mathcal{N}(\mathcal{D}_{t-1};\,\bm{\mu}_{\theta}(\mathcal{D}_{t},t),\,\bm{\Sigma}_{\theta}(\mathcal{D}_{t},t)),$
(8)

By optimizing a re-weighted variant of the ELBO,

$-\log
p_{\theta}(\mathcal{D}_{0})\leq\mathbb{E}_{q(\mathcal{D}_{1:T}\mid\mathcal{D}_{0})}\left[\log\frac{q(\mathcal{D}_{1:T}\mid\mathcal{D}_{0})}{p_{\theta}(\mathcal{D}_{0:T})}\right],$
(9)

where
$p_{\theta}(\mathcal{D}_{0:T})=p(\mathcal{D}_{T})\prod_{t=1}^{T}p_{\theta}(\mathcal{D}_{t-1}\mid\mathcal{D}_{t})$.
The final objective takes expectation over $q(x_{0})$

$\mathcal{L}=\mathbb{E}_{q(\mathcal{D}_{0:T})}\left[\log\frac{q(\mathcal{D}_{1:T}\mid\mathcal{D}_{0})}{p_{\theta}(\mathcal{D}_{0:T})}\right].$
(10)

Despite the interoperability of Diffusion Models and their remarkable results
in natural images domain [53, 54, 55], they are unable to learn a
representation and like flow-based models, there is no bottleneck on the way
which makes them slow-samplers. Recently, Diffusion-based models have been
adapted to approximate priors in VAEs [56, 57]. A downside to consider when
applying diffusion models to high-granularity detector signatures is the
computational cost. The diffusion process involves multiple steps, each of
which typically requires its own round of computation, making them somewhat
slow samplers and computationally intensive.

Table 1: A generic comparison between Deep Generative Models (DGM) for simulating detector signatures DGMs |  Training |  Likelihood Estimation |  Granularity |  Sampling Time |  Compression |  Representation Learning
---|---|---|---|---|---|---
VAEs |  Stable |  Approximate |  High |  Fast |  Lossy |  ✓
Flows |  Stable |  Exact |  Low |  Slow/Fast |  Lossless |  ✓
GANs |  Unstable |  Implicit |  Ultra-High |  Fast |  Lossy |  ✓
ARMs |  Stable |  Exact |  Low/Mid |  Slow |  Lossless |  ✗
Diffusion |  Stable |  Approximate |  High |  Slow |  Lossy |  ✓

####  3.1.1 Sampling Strategies

After learning a latent space (or initial inputs in the case of ARMs) for
representing the input data, the generative model samples from the learned (or
latent) distribution during the inference (i.e. sampling). The sampling
strategies could be divided into two categories: _random sampling_ and
_controllable sampling_. Random sampling simply draws latent samples from a
simple prior distribution, in which the model learns to approximate the
distribution of the observed detector responses. The latter, on the contrary,
samples detector signatures with controls over the desired properties (e.g.,
detector geometries, kinematic parameters, energies, luminosities, and beam
parameters). Therefore, random sampling is relatively trivial, while
controllable sampling usually requires extra effort in algorithm design.

Controllable generation usually manipulates the randomly sampled $z\sim p(z)$
or the encoded vector $z\sim p(z,\mathbf{L},\mathbf{C})$ in the latent space
to obtain a final representation vector $z$, which is later decoded to the
detector response representation $\mathbf{H}$ with expected properties. There
are three types of commonly used approaches:

  * •

Latent Disentanglement Methods: Disentangled latent methods factorize the
latent vector $z$ to $n$ parts, each of which focuses on one property $p_{n}$,
following the disentanglement regularization methods that encourages the
learned latent variables to be disentangled from each other. Therefore,
varying one latent dimension $z_{n}$ of the latent vector $z$ will lead to
property change in the generated detector responses.

  * •

Control Code Methods: This method incorporates a conditional vector $c$
(discrete or continuous) that explicitly controls the property of generated
detector elements. In this case, the final latent representation is usually a
modulation of $z$, $\mathbf{L}$ and $\mathbf{C}$. This method is the most
common conditional sampling in detector signature simulation due to its
versatility.

  * •

Iterative Editing Methods: Edit-based methods focus on iterative refinements
and involve a multi-step process to modify the embedding of latent vector $z$
with conditional priors $c$ to achieve the desired characteristics in the
generated detector response. Unlike disentangled sampling or conditional
sampling, where the modification is done in a single pass, iterative editing
typically uses optimization-based strategies (e.g., Slot-Attention mechanism
[58, 59]) to iteratively update the embedding of the latent vector. This
enables fine-grained control of the generated data.

####  3.1.2 Generation Strategies

The decoder restores the latent vector back to the data manifold. Due to the
sparse, often high dimensional, and unordered nature of detector signatures,
the resulting outcome of the decoder faces challenges in accurately
reconstructing the data manifold or generating new signatures. This leads to
artifacts or inaccuracies in the generated detector responses. Thus, existing
works take three types of generation strategies for detector response
generation: one-shot generation, sequential generation, and zero-shot
generation.

One-shot generation. One-shot generation usually generates an event in one
single step. It is achieved by feeding the latent representations to neural
networks to obtain the desired representation. In practice, various neural
networks could be utilized, including 2D Locally Connected Networks [5], 2D
and 3D Convolutional Neural Networks (CNN), Graph Neural Networks (GNN),
Multi-layer Perceptron (MLP), or any combination of these modules according to
different types of detectors and representations to be generated. The
advantage of one-shot generation is that it generates the whole event data in
a single step.

If the detector topology has a non-sequential ordering (non-
cylindrical/cubical topology) or irregular geometry (where sensor sizes vary
with the detector depth as well as within a single layer), it is not always
feasible to treat the sensor information, $\mathbf{L}$ the same as the hit
manifold $\mathbf{H}$. Consequently, using a grid-based (homogenous)
representation to characterize the full detector signatures is not the most
efficient method. Instead, a variable multi-set (heterogeneous) representation
is preferable. Detector signatures in multi-set representation can be related
similarly to the Mesh representation of 3D objects, which consists of a
collection of vertices (like detector hits) and polygon faces (like layer
information). For example, the Pixel Vertex Detector (PXD) [35] detector at
Belle II carries such a non-trivial topology where the detector is a Toroid
with an octagon inner layer and a dodecagon outer layer. Or the ATLAS
experiment [60] electromagnetic calorimeter incorporates an irregular geometry
of calorimeter segmentation. As a result, in the one-shot generation of multi-
set representations of detector elements, it is still of paramount importance
to incorporate suitable inductive biases to capture the detector geometry and
topology.

Sequential generation. In contrast to one-shot generation, sequential
generation generates the detector responses consecutively in a few steps. As
there is ordering naturally defined for detector layers such as a calorimeter
detector, sequential generation has to follow a certain sequential inductive
bias for the generation. This is usually done by generating probabilistic
sensor or layer features while sampling and feeding step-by-step the
reconstructed detector response following a predefined ordering. Despite slow
sampling, sequential generative models enjoy the benefit of auto-regressive
and causal reasoning, which prompts a precise correlation modeling of the
data. Therefore, it could be easily combined with constraints in each of the
generation steps when the responses to be generated should obey certain
restrictions. Using sequential sampling, when generating detector signatures
with either high-granularity or long detector layers, the error will
accumulate at each step, possibly resulting in discrepancies in the final
generated and observed detector signatures. Also, in the case of non-
cylindrical detector topologies such as PXD at Belle II, pure autoregressive
reasoning could introduce a non-existing sequential bias into the generated
data. In this case, one has to change the perspective from the single “shower”
generation to the full “event” generation where correlations between different
directions (various angles) become important.

Zero-shot generation. Zero-shot detector response generation refers to the
process of generating detector responses without any prior exposure to the
specific representation or structure of the “new” target detector signatures.
The new target detector signatures could belong to beam parameters beyond the
training range, higher incident energies, higher luminosities, or detector
responses for uninstalled sensors, as will be discussed in Section 4.3. The
term Zero-Shot comes from the concept of zero-shot learning, which is when a
model can recognize or generate outputs for new, unseen categories or tasks
without any training examples. This is achieved by leveraging the model’s pre-
existing knowledge and generalization abilities, typically acquired during the
pre-training phase on large datasets by incorporating symmetries and
constraints directly into the generative model. This strategy leverages the
latent space’s ability to capture the essential characteristics of the data
manifold, enabling the generation of plausible detector responses for regions
and conditions beyond the current data at hand. The core idea that is
introduced in this study is to extrapolate to the Out-Of-Distribution (OOD)
data by designing a model to be more flexible and adaptive, allowing it to
accommodate detector geometries, representations, and conditions beyond the
training data.

####  3.1.3 Representative Work and Datasets

In this section, we discuss representative works for deep generative models in
detector signature simulation with an emphasis on how they handle generation
and sampling. These works deal with a wide variety of representations for
different detector signatures, including Shower Simulation, Jet Simulation,
and track (Hit) Simulation — that yield datasets with different granularity
and complexity.

Shower signatures correspond to datasets that encapsulate the complex dynamics
of particle showers in calorimeters. These datasets often include the spatial
distribution of energy deposits, the type of interacting particle, incident
angle, and other features that distinguish electromagnetic from hadronic
showers. Depending on the calorimeter design and the simulation framework, the
granularity can range from very fine to coarser bins that encapsulate multiple
layers of the calorimeter. The highest granularity studied by the surrogate
models for calorimeter showers was by Khattak et al. [61, 62] with 65k
information channels.

Jets are sprays of particles that are produced when high-energy quarks or
gluons (partons) propagate through the detector. These partons undergo a
parton shower and hadronization process, which results in a jet of hadrons.
Jet signatures contain not only the final state particles in a jet but also
their kinematic properties and relationships, enabling a multidimensional view
of jets. The granularity in jet signatures is usually more variable, often
depending on the event’s complexity. Kinematic properties like transverse
momentum, azimuthal angle, and pseudo rapidity can be considered at various
resolutions, depending on the level of detail required for the specific
analysis.

Track signatures come from the direct detector hits or fired pixels. They
contain the signals corresponding to charged particle interactions with the
detector elements. They often have high granularity, mimicking the detector’s
actual resolution, which can be on the scale $\mathcal{O}(10^{7})$ channels.
Each hit usually contains information like position, time, and the deposited
charge, offering a granular view suitable for the downstream track
reconstruction algorithms. The highest granularity studied by the surrogate
models for charged tracks was by Hashemi et al. [63, 45] with 7,680,000
information channels per event.

In general, based on granularity, one can categorize DGMs into $4$ classes:
_low granularity_ data with $\mathcal{O}(100)$ channels like jets, _mid
granularity_ data with $\mathcal{O}(10^{4})$, _high granularity_ data with
$\mathcal{O}(10^{5})$ channels, and _ultra-high granularity_ data with
$\mathcal{O}(10^{7})$ input channels such as High Granularity Calorimeter
(HGCAL) [33] at CMS [64] with roughly $6$ million channels, the PXD at Belle
II [36] with more than $7.5$ million-pixel channels, or the EPICAL-2 [65]
ultra-high granularity electromagnetic calorimeter with $12.5$ million-pixel
channels. In Table 2, we provide the top $15$ SOTA generative models with the
highest granularities ever studied.

Historically, each newly introduced architecture was focused on a specific
detector layout and hence worked with its own dataset. Such a situation is
unfavorable for a large-scale and inter-experiment deployment of DGMs for
detector simulation, as it is almost impossible to study improvements of new
architectures over older ones. To alleviate this problem, a public data
challenge called the Fast Calorimeter Simulation Challenge 2022 —
CaloChallenge [66] was created. It consists of simulated showers of increasing
dimensionality, ranging from $\mathcal{O}(10^{2})$ to $\mathcal{O}(10^{5})$
dimensions. These datasets can serve as a benchmark for the existing and
future DGMs for mid to high-granularity experiments and will help to improve
our understanding of common struggles, advantages, and disadvantages, as well
as the scaling behavior of different DGM approaches.

In addition, there are other public datasets with different granularities
coming from various experiments. The Electromagnetic Calorimeter Shower
dataset [67, 68] was the first public dataset on calorimeter showers
corresponding to energy deposits from positrons, photons, and charged Pions.
Diefenbacher et al. [69, 70] also provided a public dataset for high-
resolution photon showers conditioned on the incident energy and angle. JetNet
[71] is another important benchmarking contribution to jet datasets,
represented as multi-sets containing gluon, top quark, and light quark jets.
JetClass [72, 73], a more complex public jet dataset, contains 10 classes of
jets carrying three categories of features, kinematics, particle
identification, and trajectory displacement information. Hashemi et al. [74,
45] also recently introduced a public dataset for ultra-high granularity
detector signatures with a non-trivial detector topology of PXD at Belle II,
conditioned on the sensor position. (radius and angle). This dataset acts as a
benchmark for evaluating the DGMs geared towards real experiments with ultra-
high granularity and provides insights into the scalability of current DGM
approaches towards the HL-LHC era. To spur further research and development,
frameworks such as COCOA [75] and the Open Data Detector [76, 77] have been
introduced recently.

Table 2: Current top $15$ SOTA models based on the granularity of the signatures they can generate. This list does not provide a fair comparison for the surrogate models simulating jet signatures as they inherently carry rather low granularities. Model |  Algorithm |  Representation |  Conditioning |  Experiment |  Granularity $\uparrow$
---|---|---|---|---|---
IEA-GAN [45, 63] |  GAN |  grid/set |  sensor position (radius and angle) |  Belle II PXD (2023,2021) |  $40\times 250\times 768=\mathbf{7,680,000}$ ch
WGAN [78] |  GAN |  grid |  random |  Belle II PXD (2019) |  $40\times 250\times 768=\mathbf{7,680,000}$ ch
YonedaVAE [14] |  VAE/ARM |  multi-set |  sensor position and Luminosity |  Belle II PXD (2023) |  $\mathbf{110,000}$ points
3DGAN [61, 62] |  GAN |  grid |  incident energy and angle |  CLIC ECAL (2021, 2020) |  $25\times 51\times 51=\mathbf{65,025}$ ch
BIB-AE [70] |  VAE/GAN/NF |  grid |  incident energy and angle |  ILD ECAL (2023) |  $30\times 60\times 30=\mathbf{54,000}$ ch
CaloScore v2 [79] |  Diffusion |  grid |  incident energy and time information |  CaloChallenge D3 (2023) |  $45\times 50\times 18=\mathbf{40,500}$ ch
iCaloFlow [80] |  NF/ARM |  grid |  incident energy |  CaloChallenge D3 (2023) |  $45\times 50\times 18=\mathbf{40,500}$ ch
CaloDiffusion [81] |  Diffusion |  grid |  incident energy |  CaloChallenge D3 (2023) |  $45\times 50\times 18=\mathbf{40,500}$ ch
CaloScore [82] |  Diffusion |  grid |  incident energy and time information |  CaloChallenge D3 (2022) |  $45\times 50\times 18=\mathbf{40,500}$ ch
BIB-AE [83] |  VAE/GAN |  grid |  incident energy |  ILD AHCal (2021) |  $48\times 25\times 25=\mathbf{30,000}$ ch
BIB-AE [84] |  VAE/GAN |  grid |  incident energy |  ILD ECAL (2021) |  $30\times 30\times 30=\mathbf{27,000}$ ch
WGAN [85] |  GAN |  grid |  energy and the deposit coordinates |  EXO-200 (2023) |  $74\times 350=\mathbf{25,900}$ ch
CaloClouds [86, 87] |  VAE/Diffusion |  multi-set |  incident energy and cardinality |  ILD ECAL (2023) |  $\mathbf{6,000}$ points
NF-VAE [88] |  VAE/NF |  grid |  random |  Hyper-k IWCD (2019) |  $19\times 16\times 40=\mathbf{12,160}$ ch
SuperCalo [89] |  NF |  grid |  incident energy and geometry |  CaloChallenge D2 (2023) |  $45\times 16\times 9=\mathbf{6,480}$ ch

Variational Autoencoders. VAE is a simple yet flexible framework and could be
adopted for controllable sampling by either modifying the loss function to
enforce latent variables to be correlated with properties of interest or to
feed the conditional information to different parts of the model. VAEs can
also be incorporated to compress detector signatures in hybrid DGMs.

In [90], the ATLAS collaboration conditioned the encoder and decoder directly
on the energy of the incident particle to generate showers corresponding to a
specific energy. Abhishek et al. [88] utilized the VAE with the Normalizing
Flows [31] to recover the true posterior distribution and to improve sampling
for Water Cherenkov detector [91] simulation with grid-based data of size
$19\times 16\times 40$.

In [92], the authors leverage the Sinkhorn Autoencoder [93] to have a
trainable prior approximation, namely a noise generator to encode and generate
embeddings with the same distribution in the latent space. In order to
overcome the mode collapse issue and promote diversity, they include
additional regularisation on the autoencoder’s latent space. Following [94],
they compute a similarity matrix for the neural network’s weights according to
the cosine similarity between its different layers to assess the diversity.

DeepRICH [95], designs a conditional latent space as a combination of CVAE
[96] and infoVAE [97], where the latent variable $\sigma$ is determined using
a Bayesian Optimization [98]. The control variables in this conditional VAE
are the kinematic parameters of each particle learned by an auxiliary
classifier over the encoded latent manifold as a regularization. They only
consider the reconstruction of their dataset.

In [99, 84], the authors utilize the BIB-AE [100] model, conditioned on the
incident photon energy. Thus, the latent manifold is conditioned on the
Energy. They introduce a post-processing module that relaxes the trade-off
between the accuracy of the emulated hit energy spectrum and the reproduced
shower shape. This module is an MLP-based network that fixes the hit energy
spectrum resolution between the input and generated images. In the later
efforts [83] for hadronic showers, along with improving their model, inspired
by [101], the authors use a Kernel Density Estimator (KDE) [102] to fit the
learned latent manifold for the inference time and use rejection sampling for
the correct density estimation. In an updated version of their work, McKeown
et al. [70] add multi-parameter conditioning over the BIB-AE model, now
conditioning on both incident angles and energies. Due to the multi-parameter
space of their conditioning, they used Normalizing flows for latent space
sampling during inference.

GVAE [103] introduces a graph-based VAE architecture for learning the
representation of collision events without any controllable sampling for
emulation. Orzari et al. [104, 105] develop a VAE for generating constituents
of hadronic jets represented by sets. They incorporate a permutation invariant
loss, Chamfer distance [106], instead of the typical mean squared error (MSE)
as the reconstruction loss; however, their model does break the permutation
equivariance using 2d convolution layers in the encoder. They regularize the
model by constraining the $p_{T}$ and the invariant mass to follow the desired
jet characteristics. The authors later in [107], fix this issue by using DGCNN
[108] permutation equivariant layers.

Otten et al. [101] introduce a method called buffering density information
given the encoded events. They construct a prior by aggregating a subset of
the encoded training data by saving all the parameters of the Gaussian
distributions for all events in the training data to a file, which constitutes
the buffer. At inference time, to increase the variance and avoid overfitting
to the training data, they also sample from the buffered Gaussian
distributions with a variance control factor.

Collins et al. [109, 110], using a ParticleFlow [111] $\beta$-VAE [112], find
an interpretable and meaningful representation of the jets and their
information complexity by analyzing the VAE’s latent information. Moreover,
they leverage $\beta$ from a fixed hyperparameter to an input of both the
encoder and decoder networks.

Abhishek et al. [113] incorporated a Discrete Variational Autoencoder (DVAE)
based model [114, 115, 116] with hierarchical dependencies of latent variables
and a Restricted Boltzmann Machine (RBM) [117] latent prior using block Gibbs
sampling for generation of the calorimeter showers. They also tackle the
sparsity of the showers with a learnable masking like [118]. This solves the
assignment problem with the sparsity, but it will be difficult to extend this
to the simulation of high-granularity detector signatures.

Huang et al. [119], introduce an Autoencoder model for lossy compression of
the data from Time Projection Chambers (TPCs) in sPHENIX experiment [120].
Their model, BCAE++, is an improvement over the original BCAE (Bicephalous
Convolutional Autoencoder) [121] and is designed to handle the sparse data,
achieving a higher compression ratio and better reconstruction accuracy. It
utilizes a specialized loss function that combines a segmentation decoder for
voxel-wise bi-class classification and a regression decoder for
reconstruction. This dual-decoder approach allows the model to effectively
handle the sparse and irregular distribution of the TPC data. A novel aspect
of this work is the application of half-precision mode in the network without
too much loss in reconstruction accuracy.

Cresswell et al. [122], develops a manifold hypothesis-inspired model (density
estimation) [123, 124, 125] to make a dimensional reduction to the Calorimeter
data to speedup the inference-level sampling process. This method comprises a
two-step approach. Initially, the manifold of calorimeter showers is learned
using a generalized Autoencoder, which aids in constructing low-dimensional
latent encodings. Following this, density estimation is performed to capture a
probability density within the learned manifold, alleviating the
dimensionality mismatch typically encountered in maximum-likelihood
estimation. This structured procedure allows for a more efficient and
practical simulation of calorimeter showers. A possible drawback of this
approach might be the presumption that the manifold encompasses showers, which
could potentially limit the ability to generalize beyond the training data
[126].

Hashemi et al. [14], marks the first model that robustly addresses the
challenge of OOD simulation by introducing YonedaVAE, a novel multi-set
generative model inspired by Category Theory [127, 128, 129]. YonedaVAE
introduces a self-distilled set generator, capable of zero-shot creating sets
and estimating the number of hits per sensor with variable “inter-event” and
“intra-event” cardinality, facilitated by their Adaptive Top-q Sampling.
Adaptive Top-q dynamically determines the multiplicity of points to be sampled
based on the shape of the probability distribution for each event during
inference. This adaptability is crucial for handling variable intra-event
cardinality, especially when dealing with irregular detector geometries and
hit patterns. They showed that YonedaVAE could produce new detector signature
point clouds with cardinalities well beyond the training data and achieve
context extrapolation. Trained on low luminosity data of PXD at Belle II with
$\mathcal{O}(10^{2})$ hit multiplicity, YonedaVAE generates high luminosity
valid signatures with the correct intra-event correlation and ultra-high
granularity of $\mathcal{O}(10^{5})$ hits, without exposure to similar data
during training.

Generative Adversarial Networks. GAN-based models, by design, allow easy
implementation of controllable sampling due to introducing a sample
discriminator given the desired properties, though taming its convergence is
very difficult.

The application of GAN-based generative models as implicit density estimators
was embarked on by [5] where they simulated 2D jet images for high energy W
bosons and QCD jets as their conditional classes while introducing 2d locally
connected layers (LAGAN). CaloGAN [68, 130, 131] employs the LAGAN layers to
generate layer-wise two-dimensional images that were conditioned on the
primary particle energy ranging uniformly from 1–100 GeV. Vallecorsa et al.
[132, 133, 61] uses 3D ACGAN [134], to generate the calorimeter showers. In
[62], they add incident angle conditioning as well. Erdmann et al. [135], uses
WGAN [136] with continuous air shower energy conditioning using a constrainer
network (like ACGAN). Musella et al. [118], for the generation of sparse
hadronic jets using a U-net [137] module for the generator, introduce a
decision-making method by adding an additional channel to their output as a
mask probability to decide if a pixel should be zero or not. Srebre et al.
[78] used the WGAN-gp [138] with random sampling to provide a proof of concept
for the offline compression of the background hitmaps of ultra-high resolution
Pixel Vertex Detector.

In [139], the authors do controlled sampling by conditioning the WGAN-gp model
for the LHCb Calorimeter images. In [140], they introduce an emulator-
simulator setup that benefits from the Siamese Network [141, 142, 143]. They
show that using their parameter-to-image grid 2-stage training pipeline, they
can model complex functions. In the pre-training stage, the goal is to learn
an emulator distribution that matches the Monte-Carlo simulator distribution
using the Siamese network to learn the similarity of the simulated and
emulated images. Then, at the next stage, a generator will be trained to learn
to map the random noise to the parameter space. All these stages follow an
adversarial training regime.

Hashemi et al. [63], for the first time, generate ultra-high granularity PXD
detector track hits with more than $7.5$ million pixel channels per event [74]
— the highest spatial resolution detector simulation dataset ever analyzed
with deep generative models. Building upon their previous work, the authors
later introduce the Intra-Event Aware GAN (IEA-GAN) in [45]. As a fusion of
the Transformer [144] model and GANs with self-supervised learning and
conditional contrastive loss, they introduce the Relational Reasoning Module
to approximate the concept of an “event” in sampling. They do the conditioning
on the position of the PXD sensors (angles and radius) using methods from Deep
Metric Learning [145]. Introducing self-supervised learning objectives in deep
generative models, they show that they can generate fine-grained samples with
large intra-event and small inter-event variations. Their evaluation
encompasses a spectrum of results, extending from track-level (signature-
level) features to the downstream performance of Helix (impact) parameter
reconstruction.

Diefenbacher et al. [146] introduces a method for refining the precision of
GANs using a post-processing re-weighting and tuning function based on [147,
148]. Kansal et al. [149, 150] for the first time choose a more sparse
representation of the detector data and introduce a graph GAN based on Neural
Message Passing layers [151]. Shirobokov et al. [152] introduce a new approach
that synergizes deep generative models and non-differentiable simulators. They
show that one can both approximate the stochastic behavior of the simulator
and enable direct gradient-based optimization of an objective by
parameterizing the latent variable model with the relevant parameters of the
simulator. Jaruskova et al. [153], improves the calorimeter simulations over
the lower energy depositions with AdaGAN-based [154] ensemble of GANs. Li et
al. [85] use a WGAN model, conditioned on scintillation energy and the deposit
coordinates, for simulating photodetector signals in the EXO-200 experiment’s
time projection chamber with grid-shaped data of size $74\times 350$.

In [155], Winterhalder et al. introduce the Latent Space Refinement (LaSeR)
protocol to enhance the precision and the topological obstruction of sampling
by refining the predictions of a generator. In LaSeR, each generated sample is
assigned a weight, which is then mapped to the corresponding latent space
point. Rather than directly sampling from this weighted latent space, which
could lead to biased results, the authors propose training a second generative
model, the refiner, to transform the weighted latent space into an unweighted
one.

CaloShowerGAN [156] employs Dataset 1 from the Fast Calorimeter Simulation
Challenge 2022 [66] along with a well-structured training regime and data pre-
processing to demonstrate revival of the classic CaloGAN [68]. This study
defines a new GAN-based baseline and addresses the void in the baseline
comparisons for calorimeter shower simulation, superseding the CaloGAN.

Hashemi et al. [157], for the first time, uses GAN to directly emulate high-
level features computed from the reconstructed $Z\rightarrow\mu\mu$ events.
DijetGAN [158] a simple GAN with random sampling for the simulation of QCD
dijet events. Butter et al. [159], do top pair generation with a modified MMD-
GAN [160] where the MMD kernel helps to describe on-shell resonances as well
as tails of distributions, an improvement to [157]. They also study the
statistical uncertainties and do ablation studies of the GAN approach to the
event simulation using GANs. Carrazza et al. [161] employ CycleGAN [162] with
the cycle-consistency loss to create mappings between two domains of Lund
images [163], different categories of jets. Farrell et al. [164] apply GANs to
generate full particle physics events conditioned on physics theory
parameters.

Li et al. [165], introduce a style-based [166] conditional GAN [167] to
predict lepton decay angles in the rest frames of $W$ bosons in Vector Boson
Scattering (VBS) processes. Their approach addresses the challenge of missing
neutrino information in the final state, which traditionally hampers the full
determination of lepton angles. Alanazi et al. [168, 169] develop a GAN-based
model that does an importance sampling over the generated features that
improve the sensitivity of the discriminator. Prieto et al. [170] propose a
style-based quantum GAN to generate events with a 3-qubit model. Howard et al.
[171] incorporate the Sliced-Wasserstein VAE [172] and the theory-based
physics constraints in an unsupervised setting for event generation. Epic-GAN
[173], by Buhmann et al., unveils an equivariant point cloud module for
simulating particle jets as multi-sets, addressing the computational
challenges with existing models. The framework, grounded on deep sets,
showcases significant computational efficiency by avoiding pairwise
information sharing between jet constituents. A notable merit of EPiC-GAN lies
in its scalability to large particle multiplicities for jet data.

Käch et al. in [174] propose a new GAN-based model that leverages attention
mechanisms to generate particle cloud jets. The architecture is designed to
handle a variable inter-event number of particles. They utilize a Bert-like
[175] aggregation mechanism that scales linearly with the particle
multiplicity. As a result, they introduce a “mean-field” particle, which
serves as an aggregation point for information from individual particles in
the cloud. This mean-field particle is initialized by the sum of all particles
and is used in a cross-attention mechanism [144] to dynamically select which
particles are important for distinguishing real from generated jets. This
attention-based aggregation is particularly useful for handling large
variances in individual particle energies. They achieve better results in jet
simulation in comparison to the Epic-GAN [173] model.

Anderlini et al. [176] introduce a distilled GAN from an ensemble of models
[177] to reduce the variance for a maximally diverse set of models. In the
inference time, they test the performance of their model in the Out-Of-
Distribution (OOD) regions of the phase space.

Flow-based Models. Flow-based Models [31, 32] learn an explicit bijection
between the data distribution and a known distribution (usually a simple one,
like a multivariate normal). This allows for exact likelihood computation and
precise sampling but does not involve a compressed hidden latent space in the
same way as models like VAEs or GANs.

For the first time, the CaloFlow [178] applies the normalizing flow to the
simplified calorimeter geometry of CaloGAN [68] with $504$ cells. Their model
is a combination of Masked Autoencoder for Distribution Estimation (MADE)
blocks [179] and RQS transformations [180]. CaloFlow provides the additional
benefit of tractable likelihoods with application to parameter inference for
particle reconstruction. In CaloFlow II [181], they incorporate Knowledge
distillation to transfer the probability density of the stronger model
(teacher) to the student; an Inverse Autoregressive Flow (IAF) [182] model
which is much faster in sampling. CaloFlow was then subsequently applied to
CaloChallenge dataset 1 in [183].

JetFlow [184] employs Discrete Normalising Flows with multiple coupling
layers. Each coupling layer divides input features into two sets, with one
undergoing an identity transformation and the other a parameterized element-
wise transformation. This configuration, with the addition of mass
constraints, aids in modeling the generation of jets, showcasing a structured
approach toward capturing more complicated correlations between the jet
particles. Later, the authors [185] introduce a post-processing transformer
encoder module, adversarially trained, to refine the output of the normalizing
flow model. Xu et al. [186] develop a conditional Normalizing Flow based on
[187] with the emphasis on modeling the correlation between the kinematic
variables.

SuperCalo [89] embarks on a new perspective by employing a flow-based super-
resolution model. This model targets fast coarse-grained calorimeter shower
upscaling to high-dimensional fine-grained showers, providing a new point of
view to fast simulation. The coarse voxel geometry is obtained from the full
voxel geometry by grouping neighboring fine voxels to form coarse patches. The
underpinning idea here is the utilization of conditional normalizing flows to
recover the high resolution of voxel energies in calorimeters autoregressively
patch-by-patch. However, using this method, recovering the global layer-by-
layer correlation is still an issue.

Diffusion Models. Diffusion models or deep score-based generative models [47,
48, 53, 54, 55] employ the concept of a stochastic diffusion process to
gradually transform a known noise distribution into a target data
distribution. This is very much like simulating a random walk over a series of
time steps, each guided by a set of transition probabilities. By taking
numerous small steps to evolve from a simple distribution to a complex, high-
dimensional target, diffusion models can effectively model the subtle nuances
and details often required in high-fidelity simulations.

CaloScore [82] marks the first application of a score-based generative model
for detector simulation. They construct the score function using Conv3D U-nets
models [137], conditioned over the normalized incident energy. In CaloScore v2
[79], the authors introduce an upgraded version of their previous model, which
employs a single-shot diffusion model for detector simulation. The methodology
involves the use of attention layers and progressive distillation techniques
to improve model performance and reduce sampling times. Mikuni et al. [188]
apply a Transformer-based diffusion model to particle jets [150] conditioned
fully on the initial jet type, kinematics, and multiplicity. In order to
increase the sampling process, they use the progressive distillation mechanism
[189] to transfer the knowledge of the Transformer-based teacher to an MLP-
based student. PC-JeDi [190] introduces new methodologies for generating jets
as particle clouds. By leveraging transformers trained to reverse a diffusion
process, PC-JeDi generates jets through an initial noise sampling for particle
momenta, followed by denoising operations that capture complex underlying
correlations with conditional generation. This approach facilitates the
creation of jets with large transverse momentum from two distinct elementary
particles. Later in [191], the authors achieve a faster and more accurate jet
generation, using an EDM diffusion [192] approach to speed up the sampling
process and enhance the ability to perform the reverse diffusion process in
fewer steps, cross-attention encoder [144] as a faster and more memory
efficient permutation equivariant network, along with the consistency
distillation introduced by Song et al. [193].

Butter et al. [194] introduce two new diffusion models alongside an
autoregressive transformer architecture for jet simulation. The first model,
Denoising Diffusion Probabilistic Models (DDPM) [53], employs a time-dependent
process that transforms a physics distribution into a Gaussian noise
distribution. This is achieved by adding Gaussian noise in discrete steps,
with each step having a specific variance. The reverse process then denoises
this diffused data to recover the original distribution. The second model also
uses diffusion but with a continuous time evolution. The paper also introduces
JetGPT, an autoregressive transformer architecture [195], which aims to scale
better with the phase space dimensionality. Bayesian versions of all three
models are developed to control learning patterns and estimate uncertainties
in density estimation. For the DDPM, the sampling process makes the sampling
process slower compared to classic generative networks. They illustrate the
results first using two toy datasets, a two-dimensional linear ramp, and a
Gaussian ring. Then, they applied all three networks to generate jet events,
which allowed for comparative analysis, highlighting the advantages and
disadvantages of the proposed architectures and the trade-offs between
precision and computational efficiency.

CaloClouds by Buhmann et al. [86] as a novel combination of a VAE and
diffusion model consists of four main components: the PointWise Net, EPiC
Encoder [173], Latent Flow, and Shower Flow. The PointWise Net serves as a
permutation-invariant, diffusion-based point cloud generator. It is trained in
parallel with the EPiC Encoder. The Latent Flow model is conditioned on energy
and particle number and is trained simultaneously with the EPiC Encoder and
the diffusion model. The Shower Flow is a separately trained normalizing flow
model that estimates the number of points for a given incident energy and also
serves for post-diffusion calibration. During inference, the CaloClouds
architecture employs a three-step approach for sampling. Initially, a Shower
Flow model is used to generate an appropriate number of points based on the
incident energy. This Shower Flow also produces the total visible energy of
the calorimeter point cloud and the number of points per layer for post-
diffusion calibration. Once these variables are generated, the encoded latent
space is produced using a conditional Latent Flow model. This Latent Flow is
conditioned on the incident energy and the number of points generated by the
Shower Flow. The final step involves using the PointWise Net to generate the
point cloud through reverse diffusion. Currently, they could study point cloud
signatures up to $6000$ cardinality. The computational cost associated with
the PointWise Net and the reverse diffusion process could be a bottleneck for
high-granularity applications.

The authors later in [87], with CaloClouds II introduce several advancements
over its predecessor, CaloClouds [86]. It adopts an EDM diffusion [192]
approach to speed up the sampling process by reducing diffusion iterations.
For calibration, CaloClouds II shifts focus to per-layer energy calibration
and X- and Y-direction center of gravity adjustments, moving away from the
total energy calibration of CaloClouds, thereby improving longitudinal energy
distribution fidelity. It also uses the consistency distillation introduced by
Song et al. [193], enabling single-step data generation, which dramatically
accelerates sampling speed.

Imani et al. [196] focuses on the implementation and evaluation of a score-
matching diffusion model for generating images from Liquid Argon Time
Projection Chambers (LArTPCs) signatures. The sampling involves the use of
stochastic differential equations (SDEs) [197] to describe the diffusion
process, with specific choices for drift and diffusion functions. In [198],
the authors introduce a novel proof of concept called Schrödinger bridge
Quality Improvement via Refinement of Existing Lightweight Simulations
(SQuIRELS) to augment the quality of existing calorimeter simulation methods.
The paper incorporates diffusion-based Schrödinger Bridge matching [199, 200]
to map between samples where the probability density is not explicitly known.
The paper proposes a two-step process for refinement. The first step focuses
on mapping the total energy sum of a given fast simulation to that of a high-
fidelity simulation (like Geant4). This model is a fully connected network
consisting of three encoding networks. These networks are conditioned on the
current timestep and the incident particle energy. The second step involves a
high-dimensional Schrödinger Bridge [201] that refines the spatial
distribution of the energy in the calorimeter. The paper employs forward and
backward Gaussian transition kernels, utilizing neural networks to approximate
forward and backward drift functions.

Amram et al. [81] presents a denoising diffusion approach to simulating
calorimeter showers, CaloDiffusion. Unlike CaloScore [82], a score-matching
network, they focus on the optimal denoising network [202]. In order to solve
the translation invariance problem of typical CNN models for calorimeter
showers represented in a voxelized cylindrical geometry, they use 3D
cylindrical convolutions [203]. Moreover, they introduce an interesting
geometry latent mapping (GLaM) module, which is able to map irregular detector
geometries into a regular structure suitable for grid-based operations.

Autoregressive Models (ARM). Autoregressive Models [204, 205, 206], or ARMs,
are designed to generate data in a sequential manner. Each new data point or
feature generated by these models is conditioned on the preceding elements.
Although these models might utilize internal hidden states during the
generation process, it’s worth noting that these states are not considered
latent variables in the traditional probabilistic sense. In the context of
detector simulation, ARMs offer the advantage of generating highly correlated
data, thanks to their sequential conditioning on previously generated
elements. This enables intricate probabilistic relationships to be captured,
offering the potential for more realistic simulations. However, this strength
comes with notable weaknesses, including computational inefficiency, slow
sampling, and potential error accumulation.

Lu et al. [207] introduce a pioneering work, SARM (Sparse Autoregressive
Models for Scalable Generation of Sparse Images in Particle Physics), to model
the joint distribution of the sparse jet images directly by breaking it down
into a product of conditional distributions. Thus, each data point as a pixel
is modeled autoregressively, conditionally dependent on the previous pixel in
a certain reading pattern. This model takes sparseness into account by
explicitly learning it with a tractable likelihood, providing a more stable
and interpretable solution than the previous candidates for jet generation.

Liu et al. [208, 209], generate calorimeter responses autoregressively while
taking into account variable detector sizes as a geometrical (detector size)
conditioning for OOD detector geometries. The authors employ a geometry-aware
ARM that adapts its energy deposition based on the size and position of the
calorimeter cells. The model is trained on a range of calorimeter geometries,
including challenging transition regions where cell sizes abruptly change. The
ARM framework consists of three Discrete modules, each trained separately for
different layers of the calorimeter. They are based on MADE [179] and are
designed to generate all desired parameters in a single pass, thus enabling
faster training on GPU. The model takes into account not only the energy
deposits but also the sizes of the cells, allowing it to generate energy
distributions for different geometries adaptively. During training and
inference, the model starts generating energy deposits from the central cell
of the calorimeter, which typically receives the most energy deposition. The
model then proceeds to generate energy deposits in the surrounding cells based
on learned multinomial probability distributions in a spiral path. This is
particularly useful for handling non-uniform cell sizes and complex
geometries.

Di Bello et al. [59] introduce a modified Transformer model that is a
combination of GNN as an encoding module and Slot-Attention [58] layers as a
context-injecting layer to the probabilistic decoder which is a GRU-based
[210] model [210]. Finke et al. [211], develops a masked Transformer Encoder-
based model [144], based on TraDE (Transformers for Density Estimation) [212]
for discretized and ordered jet constituents.

Layer-to-Layer-Flows [213] introduce an autoregressive flow-based model for
fast calorimeter emulation. The main idea of L2LFlows is to use CaloFlow with
a causal inductive bias. Conditioned on the incident energy and a subset of
preceding calorimeter layers for layer-to-layer associations, each layer of
the calorimeter is learned by a separate flow. Inductive CaloFlow [80]
enhances L2LFlows, implementing a singular flow for generating calorimeter
layer $i$ based on the shower shape of layer $i-1$, permitting an “inductive”
learning of calorimeter showers while training not on entire events over all
layers, but rather on pairs of layers. iCaloFlow employs in total three
normalizing flows, one for the energy depositions per layer, one for the
shower in the first layer, and one for the inductive step. Their teacher-
student knowledge distillation framework also facilitates faster sampling
while retaining high fidelity. However, the sampling time and scalability are
still an issue with this approach.

##  4 Applications

In this section, we give an overview of the applications of DGMs in detector
signature simulation. Specifically, we categorize them into three concrete
branches: _statistics amplification_ , _Amortised generation_ , and _OOD
simulation_. Then, we illustrate their formulations in detector response
generation and how different approaches and incorporation of inductive biases
lead to success in various applications. “Inductive biases” in DGMs refer to
the assumptions the model makes to predict outputs for both in- and out-of-
distribution inputs. These biases are inherent in the model’s architecture and
learning algorithm such as symmetries and adversarial robustness that guide
the model’s learning process [214, 215, 216, 217, 218, 219, 220, 221, 45,
222]. In general, for DGMs in particle physics, the inductive biases could
include distributional assumptions such as smoothness inductive bias,
structural assumptions such as relational inductive bias or geometry
awareness, and Physics-informed Assumptions such as Energy-Momentum
conservation bias.

###  4.1 Statistics Amplification

As surrogate models are being used for detector signature emulation to do
sample amplification [223], it is very important to quantify the statistical
power of the generated dataset. Sample amplification is a procedure where
there exists a map that takes a finite, initial subset of data and generates
an extended one. In this regime, DGMs work as classical parametric fits [224,
225] to the training that also works as data augmentation methods. The
question then would be to which extent they can interpolate. In other words,
_how can one quantify the diversity and uncertainty of DGMs?_ This involves
understanding the limitations and potential biases of the DGMs and possibly
introducing methods to quantify the diversity or amplification factors [226,
227]. For amplification of statistics, the smoothness assumption where the
physics probability densities are smooth is very natural. In DGMs, the
smoothness inductive bias is an assumption that similar inputs will produce
similar outputs up to model interpolation. Regularization techniques in
machine learning, which are used to encourage the model to learn smoother
functions, are a common manifestation of the smoothness inductive bias.

The smoothness inductive bias is studied in [228, 11, 12] for a simplified
VAE-GAN [84]. Their study revealed that individual-generated shower samples
contain less information compared to a single real data point. When the number
of generated shower samples increases significantly, the information contained
within the generated sample set eventually levels off. As a result, they
demonstrate the ability of DGMs to not only sample from implicitly defined
distributions but also to leverage enhanced interpolation or fitting
capabilities. Similar results are being achieved in [14] from an Information
Theoretic perspective while introducing a versatile diversity measure for
detector signature simulation.

To improve interpolation, it’s crucial to consider how to diversify samples.
In their work, Hashemi et al. [45] enhance the diversity and complexity [222,
229] of generated samples by introducing Self-Supervised Learning [230]
techniques that can be incorporated in any DGMs. In particular, they introduce
a Uniformity loss that helps the model to avoid mode collapse and to generate
more diverse samples by maintaining the “uniformity of information” inductive
bias for the discriminator. Kansal et al. [231] study in detail the
quantifying of the diversity of the generated samples by analyzing various
metrics and measures for the jet point cloud generation task. They show that
the message-passing GAN [150] performs better than set-transformer [232].

###  4.2 Amortised Generation

For detector simulation, amortized generation generally refers to a strategy
where a computationally expensive simulation process is replaced or
approximated by a faster and more efficient surrogate model and goes beyond
just pure enhancement of statistics. Detector effect simulations and
parametrization can be computationally intensive due to a high space- and
time-computation complexity. An “amortized” approach would involve training
DGMs on a large set of data. Once trained, this model can generate new
simulations much faster than the original simulation process. The term
amortized in this context refers to the fact that the upfront cost of training
the model is spread out, or “amortized”, over the many simulations that the
model generates.

In an amortized generation, the sub-tasks are either “Fast Simulation” to
overcome the high time complexity of detector simulation or “Data Compression”
[78, 121, 110, 233, 119] to overcome the high space complexity of detector
simulation. Based on the different types of data manifolds, various inductive
biases are being incorporated. One can categorize different representations
into two main classes, “geometry-dependent” and “geometry-independent”
representations. Geometry-dependent approaches look at the detector signatures
as grid-like structures. For example, 1d fixed representations come with
sequential local and translation invariance [234, 235]. 2D representation with
local and translation-invariant inductive bias, which was studied for showers,
jets, and tracks in depth for different detectors. Then, in order to capture
the detector’s layer-by-layer association and correspondence, 3d grid-based
models [132, 133, 62, 61, 82] are studied. However, as a consequence of
translation invariance, it has the drawback of stationary assumption over the
temporal/spatial features.

Geometry-independent approaches, on the other hand, are more suitable for
simulating detector signatures due to the inherent variable length of data and
their heterogeneity and sparsity. For example, graph-based models [103, 149,
107, 150, 59], besides the variable-length assumption and relational inductive
bias, it assumes graph isomorphism inductive bias as well. This bias ensures
that the model focuses on the structural information contained in the graph
rather than the specific labeling of nodes. Another geometry-independent
approach is considering the set representation of detector signatures. The
most important property with set-based models are the permutation equivariant
encoders for jet, shower or track constituents [211, 173, 190, 188, 86, 14] or
sensors [45], and possibly permutation invariant loss functions [107, 59,
231]. For normalizing flows, equivariance under permutation group action is
more non-trivial. Proved by Köhler et al. [236], given a Flow-based model $F$
such that the set creation yields an exchangeable distribution, the update is
permutation equivariant and invertible, and $p_{\theta}$ denotes the model
likelihood, then $(F,-\log p_{\theta})$ is permutation equivariant.

While maintaining the variable-length assumption, dropping the permutation
equivariance would correspond to a sequential inductive bias of detector
responses either layer-by-layer [14, 213], or hit-by-hit [207, 208, 209].
While autoregressive models show a better predictive capability [194], the
sequential nature of these models can be a disadvantage when it comes to high-
dimensional computation, as the sampling is rather slow.

###  4.3 Out-Of-Distribution (OOD) Generation

DGMs for OOD and zero-shot learning is an exciting area of research that holds
significant potential across various fields [237, 238, 239, 240, 241, 242,
243, 244] including, but not limited to, Drug Discovery, material design, and
weather forecasting. Traditional methods for the synthetic generation of
objects with enhanced or specific properties are often iterative and costly,
requiring extensive manual work or heavy computational resources.

In contrast, DGMs with zero-shot capability can deal with new scenarios that
are not explicitly present in the training data, making them highly desirable
in a wide range of problems. OOD generation of detector signatures is an
emerging field in High Energy Physics (HEP) as well [68, 158, 245, 176, 209,
14]. For example, for real PXD data at Belle II [36], an important challenge
in working is its reliance on actual experiments for gathering the necessary
random triggers. Consequently, real luminosity and beam-parameter-dependent
PXD background data that go beyond current experimental limits are
unavailable, leaving us reliant on computationally demanding simulations. This
underscores the critical need for a surrogate model that can effectively
generalize to OOD luminosity regions. The current main challenge remains to be
the optimization of DGMs, based on available information while avoiding
overfitting, and the generalization to cases in which information is scarce or
altogether absent, such as extrapolation to beam parameters, energies,
luminosities, and geometries where there are no data.

The first example of such OOD detector response generation is the CaloGAN
paper [68], where they very briefly show they can generate showers beyond the
training incident energy conditions. DijetGAN [158], also discusses
extrapolation to new Beyond the Standard Model (BSM)-dependent OOD regions of
dijet invariant mass. Anderlini et al. [176] evaluates the uncertainty of GANs
in new momentum regions through Background efficiency comparison. Their
analysis shows that for Kaons, where the background efficiency does not
decrease linearly in the OOD, high-momentum phase-space regions, their
ensemble model fails to capture this behavior. Whereas, for Muons, due to the
monotonic behavior of the background efficiency, the extrapolation shows
better results.

A very important inductive bias for OOD generation is the variable-length
assumption, especially in the context of detector signature simulation, due to
the inherently variable nature of the data, characterized by a fluctuating
number of detector responses in individual events. DGMs with a variable-length
inductive bias have a promising capability to generate sets/sequences of
differing lengths, contingent on the inherent complexity or dynamism of the
event being generated. This equips the model with better adaptability when
faced with novel and unseen scenarios. Liu et al. [208, 209], by using an ARM
approach (length-independent), tries to extrapolate to unseen detector
geometries where the extrapolation domain is calorimeter cell sizes. However,
they show that the high granularity is a bottleneck to their approach. Hashemi
et al. [14] address the challenge of OOD simulation for high granular and
sparse events (variable intra-event cardinality) by introducing YonedaVAE that
can achieve both length and context extrapolation. Their model is trained on
random trigger PXD background data from an early experiment of Belle II with a
peak recorded luminosity of $1.42\times 10^{34}\text{cm}^{-2}\text{s}^{-1}$
and a mean occupancy of 0.06% ($\mathcal{O}(100)$ hit multiplicity). It was
then tested on the previously unseen data from a later experiment of Belle II,
which had nearly double the peak luminosity — $2.68\times
10^{34}\text{cm}^{-2}\text{s}^{-1}$ — and a mean occupancy of 0.32%
($\mathcal{O}(10^{5})$ hit multiplicity). The length extrapolation here was
when the model had to infer how an event with a higher (than training)
cardinality looks like. Context extrapolation means that the model has to
produce the correct distribution of sparse cardinalities over all sensors,
given only the maximum cardinality per event.

##  5 Outlook: Challenges and Opportunities

Despite the remarkable advancements in deep generative models for efficient
simulation of detector signatures, the field is far from reaching its full
potential. In this section, we underscore various challenges and opportunities
that have emerged from existing research and explore prospective directions
for future investigations.

Physics-Informed Generative Models. One of the most pressing challenges in the
field of deep generative models for simulating detector signatures in particle
physics is the incorporation of physics-based constraints or rules into the
model architecture. While current methodologies often focus on data-driven
approaches (permutation equivariance is a special case), they sometimes
overlook the underlying physics and symmetries that govern the data. This
could lead to models that are statistically accurate but physically
implausible. Future work could focus on models that not only learn from data
but also respect the governing symmetries [221], thereby ensuring both
statistical and physical fidelity in the generated signatures.

Precision and Uncertainty Quantification. One of the less explored but
critically important areas in the application of deep generative models for
simulating high-granular detector signatures is the quantification of model
uncertainty and the evaluation of precision [13]. While these models can
generate data that statistically resemble the target data, the question of how
accurately they capture the underlying physical processes remains. Traditional
metrics [178, 231, 45] like log-likelihood or NN-based metrics provide some
measure of the model’s performance but often fall short in capturing the
nuances of physical validity and uncertainty. Moreover, the stochastic nature
of these models introduces an inherent level of uncertainty in the
predictions, which needs to be rigorously quantified to make the models truly
useful. Future work could focus on the development of new evaluation metrics
and techniques that not only assess the model’s ability to replicate known
phenomena but also its reliability in predicting new high-granular detector
signatures. This could involve Bayesian approaches [246, 247, 248], ensemble
methods [176], or even hybrid models [249] to provide error bars along with
predictions.

Computational Efficiency and Real-Time Optimization. The increasing complexity
of particle physics experiments, characterized by high event rates and pile-up
conditions, poses significant challenges for real-time data analysis and
decision-making for ultra-high granular detectors. Surrogate models, as
discussed in this paper, provide very affordable solutions; however,
optimizing their real-time performance is still a bottleneck. One promising
avenue is the use of Federated Learning (FL) [250], which allows for
decentralized optimization across multiple nodes, thereby leveraging the
computational resources of the entire experimental setup. FL enables models to
train on large real-time data sets and reduce biases associated with locally
trained models. Meta-learning techniques [251] are also strong candidates to
enable models to quickly adapt to new experimental data or detector
conditions, reducing the time required for retraining. Recently, the MetaHEP
[252] project embarked on this path. Additionally, another promising avenue
for future research lies in the direction of differentiable programming (DP)
[6, 253, 254, 255]. Through DP, software becomes differentiable through
automatic differentiation (AD) [256], enabling efficient gradient computation
to understand the influence of input variations on output predictions. Such
gradient-based insights could be invaluable for various downstream tasks.
Utilizing DP frameworks would allow particle physics simulation tools to be
integrated into machine learning pipelines in an end-to-end manner, thereby
facilitating joint optimization for enhanced computational efficiency.

Quantum Generative Models. The advent of quantum computing offers a new avenue
for tackling the computational challenges in particle physics simulations
[257]. Quantum Generative Models (QGM) could potentially revolutionize the
field by providing exponential speed-ups for certain types of problems.
However, the practical implementation of QGMs for particle physics is still in
its infancy [170, 258, 259, 260], and significant challenges related to error
correction, qubit stability, and algorithmic design remain. Future research
could focus on the development and validation of QGM architectures
specifically tailored for detector signature simulations.

Extrapolation Beyond Training Data. One of the limitations of current
generative models is their ability to generalize beyond the scope of the
training data. Given the high-dimensional nature of HEP data, the
inevitability of sparse datasets arises due to limited sample sizes, leading
us into zones with inadequate training data. If the model demonstrates strong
extrapolation capabilities, it doesn’t just fill gaps in areas completely
lacking data; it also provides a more reliable interpretation of regions where
data is minimal. This is particularly crucial in particle physics, where there
is a need to explore uncharted territories of parameter space [245], kinematic
regions [7, 176], luminosities [14] or detector geometries [209]. Developing
models capable of reliable extrapolation is an open challenge and involves
techniques like uncertainty quantification and the incorporation of prior
physical knowledge and inductive bias into the model.

Scalability and Ultra-High Granularity Challenges. The next generation of
particle detectors will feature ultra-high granularities, as we defined in
Section 3.1.3, leading to an explosion in the dimensionality of the data
[261], for instance, the high Granularity Calorimeter (HGCAL) [33] at CMS [64]
with roughly $6$ million channels, the ultra-high resolution PXD at Belle II
[36] with more than $7.5$ million-pixel channels, or the EPICAL-2 [65]
electromagnetic calorimeter with $12.5$ million-pixel channels. Except for
some efforts [74, 14], current generative models struggle to handle such
ultra-high-dimensional data efficiently. Even these efforts have not delved
deeply into the issue of uncertainty quantification. This gap underscores the
urgent need for future work that not only addresses the development of
scalable algorithms and architectures that can handle ultra-high granularity
detectors with irregular geometries but also rigorously quantifies the
uncertainties to ensure the physical validity and reliability of the generated
detector signatures.

##  6 Conclusion

In this paper, we offered an exhaustive and unified review of deep generative
models for the efficient simulation of detector signatures in diverse
experimental settings. As a result, we introduced a cohesive framework to
articulate the task and provide an up-to-date and comprehensive taxonomy of
algorithms. Subsequently, we conducted an in-depth survey of existing
techniques and models, elaborating on their key characteristics. We then
turned our attention to three primary application domains where deep
generative models are making significant contributions to the simulation of
detector signatures. In the end, we spotlighted the prevailing challenges in
current research and offered a panoramic perspective on the future avenues for
advancing deep generative models in the realm of efficient detector signature
simulation.

## Acknowledgments

This research was part of Hosein Hashemi’s PhD thesis, supported by the
collaborative project IDT-UM (Innovative Digitale Technologien zur Erforschung
von Universum und Materie) funded by the German Federal Ministry of Education
and Research (BMBF) and the Deutsche Forschungsgemeinschaft under Germany’s
Excellence Strategy – EXC 2094 “ORIGINS” – 390783311. The authors would like
to express their gratitude to the HEP ML Living Review [262], which provided
invaluable assistance in compiling and locating an extensive body of research.

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