PELICAN: Permutation Equivariant and Lorentz
Invariant or Covariant Aggregator Network for
Particle Physics

Alexander Bogatskiy
Center for Computational Mathematics
Flatiron Institute, New York, NY, U.S.A.
abogatskiy@flatironinstitute.org

Timothy Hoffman
Department of Physics, University of Chicago
Chicago, IL, U.S.A.
hoffmant@uchicago.edu

David W. Miller
Department of Physics, University of Chicago
Enrico Fermi Institute
Chicago, IL, U.S.A.
David.W.Miller@uchicago.edu

Jan T. Offermann
Department of Physics, University of Chicago
Enrico Fermi Institute
Chicago, IL, U.S.A.
jano@uchicago.edu

Abstract
Many current approaches to machine learning in particle physics use genericarchitectures that require large numbers of parameters, often adapted from unrelated
data science or industry applications, and disregard underlying physics principles,
thereby limiting their applicability as scientific modeling tools. In this work,
we present a machine learning architecture that uses a set of inputs maximallyreduced with respect to the full 6-dimensional Lorentz symmetry, and is fully
permutation-equivariant throughout. We study the application of this networkarchitecture to the standard task of classifying the origin of jets produced byeither hadronically-decaying massive top quarks or light quarks, and show that theresulting network outperforms all existing competitors despite significantly lower
model complexity. In addition, we present a Lorentz-covariant variant of the same
network applied to a 4-momentum regression task in which we predict the full
4-vector of the W boson from a top quark decay process.

1

Introduction

Neural networks have played a significant role in data analysis for particle physics experiments,
perhaps most significantly in the study of jets, the collimated streams of hadronic particles producedby the decay, showering and hadronization of quarks and gluons. As demonstrated in Ref. [6],convolutional neural networks can be leveraged for identifying the type of particle that initiated a jet.
However, such network architectures do not explicitly respect (or leverage) the symmetries inherent
in particle physics, in particular those of the Lorentz group. Designing a network architecture that
respects these symmetries may benefit model interpretability while reducing model complexity via
physically-meaningful constraints, without sacrificing performance.

2

Equivariance and jet physics

Lorentz Invariance In what follows, “Lorentz group” refers to the proper orthochronous Lorentz
group SO+ (1, 3), i.e. the identity component of the full Lorentz group O(1, 3) of linear transformations
on R4 that preserve the Minkowski metric 𝜂 = diag(1, −1, −1, −1). The classification task considered
Machine Learning and the Physical Sciences workshop, NeurIPS 2022.

2

EQUIVARIANCE AND JET PHYSICS

2

in this work is Lorentz invariant, that is, the output of the network is invariant under the application of
any Lorentz transformation Λ ∈ SO+ (1, 3) to all of the 4-vector inputs (energy-momentum vectors inour case, for which 𝑝 = ( 𝑝 0 , 𝑝)
® = (𝐸, 𝑝 𝑥 , 𝑝 𝑦 , 𝑝 𝑧 ) and 𝑝 2 = 𝐸 2 − 𝑝®2 ). The simplest way to enforce
invariance is to hand-pick a set of invariant observables (e.g. particle masses, identification labels) as
inputs to a generic neural network architecture, as summarized in Ref. [2]. Another approach inspiredby convolutional neural networks (CNN’s) is to preserve group-equivariant latent representations in
the hidden layers, see e.g. Refs. [1, 9]. As in CNN’s, equivariant latent representations, as opposed toinvariant ones, can regularize the network via efficient weight-sharing [25].
Here, we take a slightly different approach. Given a set of 4-vector inputs 𝑝 1 , . . . , 𝑝 𝑁 , we computethe complete set of Lorentz invariants on that set. Weyl’s work [23] characterized the set of all Lorentz
invariant functions of a collection of 4-vector inputs. Namely, all totally symmetric Lorentz invariants𝐼 ( 𝑝 1 , . . . , 𝑝 𝑁 ) depend only on the invariant dot products (see related discussion in Ref. [21]):𝐼 ( 𝑝 1 , . . . , 𝑝 𝑁 ) = 𝐼 {𝑝 𝑖 · 𝑝 𝑗 }𝑖, 𝑗 .
(1)
The array of all 𝑁 × 𝑁 pairwise dot products will be the network input. We note that the idea toinclude dot products as inputs was recently used in Ref. [15]. As we will show, these invariant dot
products alone can provide state-of-the-art performance in a significantly simpler architecture.
Permutation Equivariance Particle data is often naturally represented by a point cloud, or a set.
For such problems the ordering of the particles in the set is not physically meaningful, and thus itmakes sense to use one of the permutation-equivariant architectures. One approach is that of Deep
Sets [24], applied to jet tagging e.g. in Ref. [14]. ItÍis based on the fact that any symmetric functionof inputs 𝑥1 , . . . , 𝑥 𝑁 can be written in the form 𝜓 ( 𝑖 𝜑(𝑥 𝑖 )), where 𝜓 and 𝜑 can be approximated byneural networks. However, since aggregation happens only once, the network can struggle at modelingcomplex higher-order interactions between the particles. The sub-network representing 𝜓 is forced to
be a relatively complex (wide) fully-connected network, which makes it difficult to train [22, 25].
The alternative to permutation-invariant architectures is provided by permutation-equivariant ones.
Equivariance is a key property of all convolutional networks – for example, in CNN’s convolutions
are manifestly equivariant with respect to translations (up to edge effects). Similarly, Graph Neural
Networks (GNN’s) use permutation equivariance, usually in the form of message passing (MP), toforce architectures to respect the underlying graph structure.
Despite the benefits of MP (previously used in jet tagging [1, 9]), attempts to combine MP withLorentz invariance run into an obstacle: the key inputs to the network are nothing but edge data
𝑑𝑖 𝑗 = 𝑝 𝑖 · 𝑝 𝑗 . Since traditional MP architectures use only single-label vertices, we employ thegeneral permutation-equivariant layers proposed in Refs. [7, 19]. In the general setting, permutationequivariance is a constraint on mappings 𝐹 between arrays 𝑇𝑖1 𝑖2 ···𝑖𝑟 of any rank 𝑟, where every index
𝑖 𝑘 ∈ {1, . . . , 𝑁 } refers to a particle label, whereby permutations 𝜋 ∈ 𝑆 𝑁 of the particles “commute”
with the map:𝐹 𝜋 ◦ 𝑇𝑖1 𝑖2 ···𝑖𝑟 = 𝜋 ◦ 𝐹 𝑇𝑖1 𝑖2 ···𝑖𝑠 , 𝜋 ∈ 𝑆 𝑁 .
(2)Here, the action of permutations is diagonal: 𝜋 ◦ 𝑇𝑖1 𝑖2 ···𝑖 𝑝 = 𝑇 𝜋 (𝑖1 )... 𝜋 (𝑖 𝑝 ) . Thus a higher-ordergeneralization of the MP layer can be defined as 𝑇 (ℓ+1) = Agg ◦ Msg 𝑇 (ℓ ) . Here, Msg is a
node-wise nonlinear map (“message forming”) shared between all nodes, and Agg is a general
permutation-equivariant linear mapping (“aggregation”) acting on the particle indices of 𝑇.
Elementary Equivariant Aggregators It remains to describe the exact structure of the equivariant
aggregation layers introduced above. Since the general case is presented in [7, 19], here we will only
present the layers needed for jet physics. Since the input is an array of rank 2 (of dot products), themain equivariant layer in this case is one that maps between arrays of rank 2: 𝑇𝑖 𝑗 ↦→ 𝑇𝑖′𝑗 . The spaceof all linear maps of this type is 15-dimensional and its basis elements can be defined using binary
arrays of rank 4. There are 15 such arrays 𝐵𝑖𝑎𝑗 𝑘𝑙 , 𝑎 = 1, . . . , 15, (see Ref. [19] for exact expressions)Í𝑁and the action of the equivariant layer can be written as 𝑇𝑖′𝑎𝑗 = 𝑘,𝑙=1
𝐵𝑖𝑎𝑗 𝑘𝑙 𝑇𝑘𝑙 . Five of the basis
elements in fact contain only one non-zero component for each 𝑖 𝑗 pair, which includes the identity
and the transposition maps, so they can be thought of as sorts of “skip connections”. The rest involve
aggregations over 𝑁 or 𝑁 2 elements of the input.
More generally, instead of a simple summation, aggregators can involve arbitrary (nonlinear) symmetricfunctions, e.g. maximum. In practice, we define aggregation as the mean of its inputs followed by an

3

PELICAN ARCHITECTURE

3𝛼additional scaling by a factor of 𝑁/ 𝑁¯ with learnable exponents 𝛼, where 𝑁¯ is a constant describing
the typical number of particles expected in the input.
Equivariance and Jet Physics There are several reasons for enforcing the full Lorentz symmetry in
our ML models. First and foremost, it is a fundamental symmetry of the space to which the inputs
belong. If an analyzer in the lab frame establishes that a given collection of particles resulted from a
top quark decay, then the same is true for all other reference frames. The breaking of the Lorentz
symmetry implicit in the running of the QCD couplings notwithstanding, there is no question that boththe original protons and the final (asymptotic) decay products are accurately represented by a collectionof 4-vectors subject to the global Lorentz symmetry. Another reason for symmetry-restricted modelingis that, from the geometric perspective, only some mathematical operations are permissible whenworking with objects that transform in a certain way under a symmetry group. A non-equivariant
neural network effectively neglects the vector nature of the inputs by treating individual componentsof the input vectors as scalars. Despite a priori improving network expressivity, non-equivariance fails
to deliver physically interpretable models. Ultimately, a statement about equivariance is a statement
about what quantities are not truly self-contained features of the inputs – e.g. a single 𝑥-component of
a 2D vector is not a feature of that vector unless the input also contains the vector (1, 0).

3

PELICAN architecture

Permutation Equivariant Blocks The main equivariant block, Eq2→2 ,consists of a simple dense layer Msg and an aggregation block Agg. The
aggregation block applies 15 linear aggregation functions (LinEq2→2 ) as
outlined in Section 2. Note that this is a non-parametric transformation𝑙 × 15 resulting
performed on each channel separately. Each of the 𝐶eq
aggregation values is then independently multiplied by 𝑁 𝛼 / 𝑁¯ 𝛼 with a
trainable exponent 𝛼 (initialized as a random float in [0, 1], allowed to
become negative), where 𝑁 is the number of particles in the corresponding
event. This allows for some flexibility in the aggregation process, forexample 𝛼 = 1 returns the sum aggregation function, and combining
multiple aggregators is known to boost accuracy [7]. PELICAN stacks
five such blocks for optimal results according to a loss-minimizing hyperparameter search.
PELICAN Classifier To build a classifier, aside from the Eq2→2 equivariant layer one needs a
Eq2→0 layer that reduces the rank 2 array to permutation-invariant scalars. This layer involves just
2 aggregation functions instead of 15 – the trace and the total sum of the input square matrix, but
is otherwise identical to the equivariant block described above. The inputs 𝑑𝑖 𝑗 are positive with avery steeply decaying distribution at large values, therefore after forming the matrix of pairwise dot
products the input layer applies a set of encoding functions of the form ((1 + 𝑥) 𝛿 − 1)/𝛿, with 𝛿 = 𝛽2and learnable 𝛽’s. From the input block, the tensor is passed through several equivariant Eq2→2
blocks, and a Eq2→0 block, all with dropout. One final dense layer mixes the channels down to 2classification weights per event. A cross-entropy loss function is then used for optimization.
PELICAN Regressor The same architecture can also be easily adapted for 4-momentum regression
tasks, such as momentum reconstruction. Any Lorentz-equivariant map from a collection of
4-momenta 𝑝 1 , . . . , 𝑝 𝑁 to one (or several) 4-momentum has the form𝑁∑︁𝐹 ( 𝑝1, . . . , 𝑝 𝑁 ) =𝑓𝑖 ( 𝑝 1 , . . . , 𝑝 𝑁 ) · 𝑝 𝑖 ,(3)𝑖=1

where 𝑓𝑖 ’s are Lorentz-invariant functions [21]. Combining this with permutation-invariance, we
conclude that the multi-valued map ( 𝑝 1 , . . . , 𝑝 𝑁 ) ↦→ ( 𝑓1 , . . . , 𝑓 𝑁 ) must also be equivariant withrespect to the permutations of the inputs. The only change required to the architecture we’ve introduced
for classification is that Eq2→0 must be replaced with Eq2→1 and the final output layer must have
only one output channel. The Eq2→1 layer is again identical to Eq2→2 except that it uses only 4 linear
aggregation functions. For the loss function we use 5 𝑚 𝑝 − 𝑚 𝑡 + 𝑝® 𝑝 − 𝑝®𝑡 where subscripts 𝑝, 𝑡
stand for predicted and true values, respectively. We avoid squares due to their sensitivity to outliers,
and the coefficients of the two terms are chosen to roughly balance their magnitudes on our dataset,
forcing the network to simultaneously predict the mass and the spatial momentum.

4

EXPERIMENTS

4

Experiments

4

Top tagging We perform top-tagging on the reference dataset [13] explored in Ref. [2]. This dataset
consists of 2M entries, each entry corresponding to a single hadronic top jet or the leading jet froma QCD dĳet event. The events were generated with PYTHIA 8.2 [20], with DELPHES [8] used for
detector interactions. For each jet, the four-momenta of up to 200 constituents are listed. The model
was trained in batches of 100 events on A100 80GB GPU’s using the AdamW optimizer [17] witha linear warmup of the learning rate up to 2.5 · 10−3 for the first 4 epochs, followed by 28 epochsof CosineAnnealingWarmRestarts with 𝑇0 = 4, 𝑇mult = 2, and 3 more epochs of exponential
schedule with 𝛾 = 0.5. A dropout rate of 1% was used. The Msg blocks output 35 channels and theAgg blocks output 60. Hyperparameters were tuned by manual optimization.

Architecture Accuracy AUC

1/𝜖 𝐵# Params

LGN
PFN
ResNeXt
ParticleNet
LorentzNet

0.929(1)
0.932
0.936
0.938
0.942

0.964(14) 424 ± 82
0.982
891 ± 180.984
1122 ± 47
0.9851298 ± 46
0.98682195 ± 173

4.5k
82k
1.46M
498k
220k

PELICAN

0.9425(1) 0.9869(1) 2289 ± 204

45k

PFN
ParticleNet
ResNeXt50
LorentzNet
LGN
PELICAN104
Background rejection 1B

The metrics of several top taggers are compared inTable 1, quoted from Refs. [2, 9]. 1/𝜖 𝐵 stands forthe background rejection rate at efficiency rate of
0.3. Among these PFN, LGN and LorentzNet arephysics-motivated architectures, with LorentzNet using a Lorentz-invariant version of Message Passing.
Table 1: Comparison of top-taggers.

103

102

101
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Signal efficiency S

Fig. 1: Top-tagger ROC curves.

Our results are averaged over 5 random initialization seeds and the uncertainties are given by thestandard deviation. As can be seen from these results, equivariant architectures provide good models
for top-tagging despite much lower model complexity as measured by the number of trainable
parameters. PELICAN in particular provides state-of-the-art accuracy with almost 5 times fewer
parameters than the next best model. Model complexity is of great importance for many real-world
applications, such as online detector triggering, which requires extremely low microsecond latencies[4, 5, 10, 11, 16]. Normally reduction in model size comes at the cost of accuracy, but equivariant
architectures can avoid such compromises. Aside from accuracy, physics applications also demandhigh background rejection rates, where PELICAN also provides state-of-the-art performance.

𝜎𝑝𝑇 (%)𝜎𝑚 (%)

Without
DELPHES

Table 2: Momentum reconstruction results for the Johns
Hopkins (JH) tagger and PELICAN. We report the relative
𝑝 𝑇 and mass resolutions, and the interquantile range forthe angle 𝜓 ∈ (0, 𝜋) between predicted and true momenta.
PELICAN uncertainties are within the last significant digit.

JH
PELICANPELICAN|JH
PELICAN|FC

0.70%
0.83%
0.28%
0.32%

1.29%
1.21%
0.60%
0.76%

0.162
0.388
0.089
0.111

With
DELPHES

4-Momentum reconstruction For
regression, we use a dataset [18] containing 1M entries, each corresponding to a single hadronic top jet from an
event generated with PYTHIA 8, withsettings as in Ref. [13]. We prepare
two versions of this dataset: One withjets clustered from truth-level, finalstate particles, the other clustered fromthe output of detector simulation with
DELPHES, each using the same events.
Each entry contains the 200 leading
jet constituents, and the parton-level4-momenta of the top quark, and of
the b-quark and 𝑊-boson to which it
decays.

Method𝜎𝜓 (centirad)

JH
PELICANPELICAN|JH
PELICAN|FC

10.8 %
5.6 %
3.8 %
4.4 %

8.3 %
3.2 %
2.9 %
3.1 %

8.9
4.2
2.7
3.0

Our regression task consists of predicting the 4-momentum of the parton-level 𝑊-boson in the lab frame. The results are summarized inTable 2 by the resulting 𝑝 𝑇 and mass resolutions – given by half of the central 68th interquantile
range of (𝑥predict − 𝑥 true )/𝑥true , where 𝑥 is 𝑚 or 𝑝 𝑇 – and the lower 68th interquantile range for 𝜓,
the angle between predicted and true momenta. To serve as a baseline regression method, we use

5

ACKNOWLEDGEMENTS

5

the 𝑊-boson identification of the Johns Hopkins top tagger [12] implemented in FASTJET [3]. The
tagger has a 36% efficiency on the dataset and can only identify 𝑊-boson candidates for jets it tags,
so we report PELICAN results both on the tagged jets (PELICAN|JH) and on the full dataset. In
addition, we report PELICAN results on the subset of jets that are fully-contained (PELICAN|FC),
which we define as jets where both quarks from 𝑊-boson decay are within the jet radius. This is
in fact a strict subset of the tagged jets, and we highlight this subpopulation of jets as we do not
expect accurate 𝑊-boson momentum reconstruction in the case of jets that fail to capture a significant
fraction of the 𝑊-boson decay products. Notably, fully-contained events comprise about 75% of thedataset, which is still significantly higher than JH tagger’s efficiency. The regression network uses
36k parameters and was trained in the same way as the classifier. The code for PELICAN can befound at github.com/abogatskiy/PELICAN.
Conclusion We have introduced a new neural network architecture designed to respect some basic
symmetry constraints in particle physics. PELICAN delivers state-of-the-art results in a top-tagging
benchmark despite its relatively low complexity. It also shows potential for more complex tasks suchas momentum reconstruction, significantly outperforming an established non-ML approach from
Ref. [12].

5Acknowledgements

We would like to acknowledge Brian Nord and the Deep Skies Lab as a community of multi-domain
experts and collaborators who’ve facilitated an environment of open discussion, idea-generation, andcollaboration.

6

Broader Impacts Statement

This work will potentially have a positive impact on basic physics research: Specifically throughthe use of PELICAN as a method in particle physics measurements, and perhaps more broadly
in providing further inspiration for the use of symmetry-respecting/physically-constrained neuralnetwork architectures in scientific research. PELICAN’s ability to not only tag particles, but also
accurately reconstruct their 4-momenta, opens up possibilities of improving precision measurementsof the Standard Model and searches for new physics. This basic research, in turn, will likely continue
to yield positive impacts through the innovation and development of new technologies that it drives.
Beyond its potential influence in furthering basic high-energy physics research, this work is unlikely
to have other societal impacts (good or bad), as PELICAN does not have clear direct applications
outside of physics research, and in fact the use of symmetry-preserving architectures is already present
to some extent in industry and government applications – specifically, the use of convolutional neuralnetworks for image recognition.

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7

1. For all authors...
(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s
contributions and scope? [Yes](b) Did you describe the limitations of your work? [No] In this context, it’s not exactly
clear what the "limitations" would be – the network design is quite general for particle
physics measurements, and we do not make any claims of using it outside of thatdomain.
(c) Did you discuss any potential negative societal impacts of your work? [N/A] This work
does not have any clear direct applications that would result in negative societal impact.
(d) Have you read the ethics review guidelines and ensured that your paper conforms tothem? [Yes]2. If you are including theoretical results...
(a) Did you state the full set of assumptions of all theoretical results? [Yes](b) Did you include complete proofs of all theoretical results? [Yes]3. If you ran experiments...
(a) Did you include the code, data, and instructions needed to reproduce the main
experimental results (either in the supplemental material or as a URL)? [Yes] A link toa GitHub repository with the network code is provided, and the dataset is cited with aZenodo link – this Zenodo listing, in turn, provides a link to the GitHub repository forthe code used to generate that dataset.
(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they
were chosen)? [Yes](c) Did you report error bars (e.g., with respect to the random seed after running experiments
multiple times)? [Yes] For the classification benchmark, we provide explicit errors.
For the (new) momentum reconstruction benchmark we do not provide these explicitly,
but comment on their magnitude.
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(a) If your work uses existing assets, did you cite the creators? [Yes] We cite the reference
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