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Energy Flow Networks: Deep Sets for Particle Jets

Patrick T. Komiske, Eric M. Metodiev, Jesse Thaler

arXiv:1810.05165v2hep-phhep-exstat.ML

TL;DR

The paper asks how to represent collider events that are naturally variable-length, unordered particle sets. It adapts Deep Sets into EFNs and PFNs with summed per-particle latent representations, then evaluates them on quark–gluon discrimination. The networks achieve favorable performance, unify existing event representations, and expose interpretable learned structure, with conclusions beyond parton-shower generators left for future testing.

  • Problem

    Collider events are variable-length, unordered particle sets, but standard particle-input architectures do not fully respect this structure.

  • Method

    The paper adapts Deep Sets by mapping particles into latent representations and summing them, defining IRC-safe EFNs and general PFNs that can use particle information such as charge and flavor.

  • Results

    The networks achieve favorable performance compared with many existing machine-learning techniques for particle physics in a quark–gluon discrimination case study.

  • Takeaways & Limitations

    The latent-space framework unifies image-based and moment-based event representations while enabling direct visualization and analytic study of learned collider information.

  • Takeaways & Limitations

    The paper’s conclusions beyond parton-shower generators remain to be tested, and particle-type studies were performed without detector simulation.

Abstract

from arXiv · show

A key question for machine learning approaches in particle physics is how to best represent and learn from collider events. As an event is intrinsically a variable-length unordered set of particles, we build upon recent machine learning efforts to learn directly from sets of features or "point clouds". Adapting and specializing the "Deep Sets" framework to particle physics, we introduce Energy Flow Networks, which respect infrared and collinear safety by construction. We also develop Particle Flow Networks, which allow for general energy dependence and the inclusion of additional particle-level information such as charge and flavor. These networks feature a per-particle internal (latent) representation, and summing over all particles yields an overall event-level latent representation. We show how this latent space decomposition unifies existing event representations based on detector images and radiation moments. To demonstrate the power and simplicity of this set-based approach, we apply these networks to the collider task of discriminating quark jets from gluon jets, finding similar or improved performance compared to existing methods. We also show how the learned event representation can be directly visualized, providing insight into the inner workings of the model. These architectures lend themselves to efficiently processing and analyzing events for a wide variety of tasks at the Large Hadron Collider. Implementations and examples of our architectures are available online in our EnergyFlow package.

1 Introduction

Collider events are variable-length, unordered particle sets, motivating architectures that process particles directly while respecting permutation symmetry. The paper develops latent-space decompositions and EFN/PFN networks, demonstrating competitive jet-classification performance and interpretable learned representations.

  • Motivation: Collider events contain variable-length, unordered particle lists, while common DNNs require fixed-size inputs and RNNs depend on particle ordering.Zero padding and ad hoc ordering provide workarounds, but an ideal architecture would respect set structure directly.
  • Observable decomposition: Deep Sets represents a symmetric particle observable by mapping each particle into a latent space, summing the representations, and applying a continuous function.The per-particle map can use kinematics or particle attributes such as charge and flavor.
  • Network architectures: Energy Flow Networks implement an IRC-safe decomposition using angular particle information and energy or transverse-momentum weighting.These energy weights and energy-independent angular inputs ensure IRC safety in the latent event representation.
  • Network architectures: Particle Flow Networks implement the more general decomposition, allowing learned functions of energy and additional particle-level information.The paper distinguishes PFNs from EFNs by permitting general energy dependence and inputs such as charge or flavor.
  • Contributions: The framework respects variable length and permutation invariance, unifies calorimeter-image and radiation-moment representations, and supports visualization of learned information.The paper also presents extraction of novel analytic observables from trained models.
  • Results: For quark–gluon discrimination, PFNs generally outperform EFNs, while particle-identification information further improves PFN classification.The authors interpret the PFN advantage as evidence that IRC-unsafe information helps this task.

2 A general framework for observables

The paper represents collider observables as permutation-invariant functions of variable-length particle sets, using per-particle mappings summed into an event-level latent space. Combining this framework with infrared and collinear safety yields an energy-weighted decomposition that encompasses common event representations and supports learned latent filters.

  • 2.1 Observables as functions of sets of particles: Collider events are modeled as unordered, variable-length sets of particles whose attributes can include momentum, charge, and flavor.This formulation treats observables as functions on particle sets rather than ordered lists.
  • 2.1 Observables as functions of sets of particles: The Deep Sets framework approximates permutation-invariant observables by summing per-particle mappings in a latent space and applying a continuous output map.Each component of the per-particle mapping is a latent filter, while the summed latent representation describes the full event.
  • 2.2 Enforcing infrared and collinear safety: Infrared and collinear safety constrain the per-particle mapping so it can be taken linear in energy and dependent only on particle geometry.Infrared safety removes the zero-energy constant term, while collinear safety removes higher-order energy terms.
  • 2.2 Enforcing infrared and collinear safety: At hadron colliders, the decomposition uses transverse momentum or normalized transverse-momentum weights with rapidity and azimuth as geometric inputs.The corresponding choices are z_i = pT,i and p̂_i = (y_i, φ_i), or dimensionless normalized weights in practice.
  • 2.3 Encompassing image and moment representations: The IRC-safe decomposition encompasses calorimeter images and radiation moments as different choices of latent filters.Pixel indicator functions produce image-like representations, while Φ(y, φ) = y^mφ^n produces global moment-like filters.
  • 2.3 Encompassing image and moment representations: Learned IRC-safe latent spaces can be visualized through their filters, revealing whether the learned representation is localized and image-like or global and moment-like.The paper presents latent-space visualization as a tool for understanding EFN behavior.

3 Network implementation

The implementation adapts Deep Sets to jets using preprocessing, dense neural networks, and permutation-invariant particle aggregation, with optional particle-identification inputs. The study also identifies scope boundaries involving detector simulation and future equivariant or iterative architectures.

  • Scope and limitations: The implementation focuses on individual jets for the quark–gluon case study, while emphasizing broader event-level applicability.The method is described as broadly applicable at the event level despite the jet-focused implementation.
  • Input preprocessing and particle information: Kinematic preprocessing divides each particle’s transverse momentum by the jet’s scalar pT sum and centers rapidity and azimuth around the jet axis.The preprocessing relies on the jet’s scalar-sum pT, rapidity, and azimuth.
  • Input preprocessing and particle information: The models process jets using kinematic inputs, with optional particle-identification information added through several PFN variants.PFN-ID, PFN-Ex, and PFN-Ch respectively use full particle IDs, experimentally realistic IDs, or electric charge.
  • Network architecture: EFNs use IRC-safe latent information, whereas PFNs allow general particle dependence and can incorporate particle-identification features.The latent observables sum per-particle representations, with EFNs weighting them by particle energy or pT.
  • Scope and limitations: Detector effects are not included in the particle-level comparison, and equivariant or iterative local-global extensions are left for future work.The paper notes that detector effects may alter the information available in different particle types.
  • Network architecture: The functions Φ and F are approximated with dense neural networks, using three layers for each and varying the latent dimension ℓ up to 256.Φ uses layer widths 100, 100, and ℓ; F uses three layers of 100 nodes.

4 Discriminating quark and gluon jets

PFNs outperform EFNs for quark/gluon discrimination, while latent-space visualizations reveal structured, complementary IRC-safe observables that can be converted into closed-form discriminants.

  • Classification performance: AUC performance generally increases with latent dimension, achieving good performance by ℓ=16 and saturating near ℓ=256.Models were trained ten times with different initializations; low-dimensional models showed higher variance because some filters failed to train to non-zero values.
  • Classification performance: PFN-ID performs best, followed by PFN-Ex, while PFNs outperform EFNs because IRC-unsafe information aids discrimination.Full particle identification gives the strongest performance, and experimentally realistic particle identification follows closely.
  • Comparison to other architectures: PFN-ID slightly outperforms RNN-ID, whereas PFN and RNN achieve comparable classification performance with similar particle information.The comparison is qualified because many possible RNN architectures could have been chosen.
  • Extracting new observables from the model: The learned function Φ can be directly visualized and quantitatively approximated to extract novel jet-substructure observables.For EFNs, Φ depends on angular information and can be displayed in the rapidity–azimuth plane before constructing closed-form observables.
  • Visualizing the singularity structure of QCD: Learned EFN filters show approximate radial symmetry, with complementary central and wide-angle responses resembling IRC-safe angularities.The filters’ sizes correlate with distance from the jet origin, and this pattern emerges across several strong models despite training-to-training profile variation.
  • Extracting new observables from the model: The closed-form estimate reproduces the trained EFN output and achieves similar performance in the ROC and SI comparisons.The model output and closed-form function are compared using the learned observables A_r0 and B_r1,β.
  • Extracting new observables from the model: The closed-form observables A_r0 and B_r1,β perform similarly to individual angularities, but their combination outperforms all single angularities and approaches the trained ℓ=2 EFN.Their combined advantage indicates that they probe complementary information, unlike the considered angularities.

5 Conclusions

The paper introduces EFNs and PFNs for learning from particle sets, combining additive latent representations with IRC safety or general particle information. Applications to quark/gluon discrimination and visualization demonstrate favorable performance and interpretable learned observables.

  • 5 Conclusions: Deep Sets expresses symmetric collider observables through per-particle mappings summed into an event-level latent representation.This decomposition encompasses image-based and moment-based event representations.
  • 5 Conclusions: EFNs impose IRC safety through energy-weighted geometric latent observables, while PFNs incorporate general particle information such as charge and flavor.Both architectures target variable-length unordered particle lists.
  • 5 Conclusions: In quark/gluon discrimination, the methods achieved favorable performance compared with many existing particle-physics machine-learning techniques.The paper also develops closed-form observables from learned latent structure.
  • 5 Conclusions: EFN visualization revealed dynamically sized rapidity-azimuth pixels, with finer resolution near the jet core and a pattern consistent with collinear radiation structure.The learned filters could be interpreted as probing collinear and wide-angle radiation.
  • 5 Conclusions: The learned two-observable classes were competitive with constituent multiplicity and outperformed other IRC-safe angularities on the studied quark/gluon samples.The authors note that the numerical parameter choices are specific to these samples.
  • 5 Conclusions: At event level, preprocessing may not fully solve new challenges, motivating equivariant or iterative Deep Sets extensions for hierarchical relationships.Possible extensions include pileup mitigation and adding high-level information to Φ or F.

A Details of models and observables

The appendix documents model implementations, training procedures, input representations, and computational comparisons for EFNs, PFNs, and baselines.

  • Details of models and observables: EFN and PFN implementations are included in the EnergyFlow Python package, with zero-padding used only as a technical input-format workaround.The padding is distinct from the conceptual fixed-size-input requirement of DNNs.
  • Details of models and observables: RNN baselines order particles by transverse momentum, whereas EFN and PFN models process particle sets without relying on that ordering.RNN variants use either kinematic information alone or kinematics plus particle identification.
  • Details of models and observables: The EFP baseline uses 996 Energy Flow Polynomials with degree d ≤7 and complexity χ ≤3 in a Fisher linear-discriminant model.The default hadronic measure uses β = 0.5; β = 1 performed slightly worse.
  • Details of models and observables: RNN implementations require substantially longer training times than EFN and PFN models.Table 4 reports median epoch times and training epochs for the considered models.
  • Details of models and observables: The DNN baseline uses the N-subjettiness basis, while the CNN baseline uses 33 × 33 grayscale jet images.The DNN has three 100-node fully connected layers before a two-node output layer.
  • Details of models and observables: Quark/gluon neural networks use one million training jets, 200k validation jets, and 200k test jets, with stated common optimization settings.The appendix specifies ReLU activations, Adam optimization, and a learning rate of 0.001.

B Tagging top jets

The top-tagging study evaluates EFNs and PFNs across latent dimensions and preprocessing choices on simulated top and QCD jets. PFNs perform best overall, while rotation and reflection preprocessing improves both architectures.

  • Tagging top jets: Top and QCD jets are Pythia-generated anti-kT, R = 0.8 jets at √s = 14 TeV without underlying event or pileup.Selected jets satisfy pT ∈ [550, 650] GeV and |η| < 2.
  • Tagging top jets: Models use jet centering and normalized particle transverse momenta, with additional rotation and reflection preprocessing tested separately.The additional transformations are designed to improve training efficiency for the different top-jet topology.
  • Tagging top jets: Performance competitive with Ref. is achieved, particularly for PFN models, across latent dimensions from 8 to 256.The study reports ROC and SI curves alongside comparisons to the linear EFP model.
  • Tagging top jets: Rotation and reflection preprocessing notably improves both EFN and PFN classification performance.Figure 16 shows AUC as latent dimension varies in powers of 2, with ten retrainings producing the plotted spread.
  • Tagging top jets: Top-tagging EFN filters are less strongly concentrated toward smaller central pixels than quark/gluon filters.This is consistent with top jets’ typical three-prong topology and lack of distinguished central-axis radiation.
  • Tagging top jets: PFNs achieve the best performance, while EFPs slightly outperform EFNs on AUC and background rejection at 0.5 signal efficiency.EFPs perform more poorly than EFNs in background rejection at 0.3 signal efficiency.

C Additional visualizations

Additional visualizations examine the spatial organization of EFN filters and compare their behavior in Cartesian and emission-plane coordinates.

  • Additional visualizations: Quark/gluon EFN filters showed a relationship between filter size and location that suggested uniform pixelization in the emission plane.The relationship was observed both qualitatively and quantitatively.
  • Additional visualizations: A coordinate change from Cartesian (y, φ) to polar (θ, ϕ) makes the learned filters appear more uniform in the emission plane.The visualizations include latent dimensions 16, 64, and 256.
  • Additional visualizations: The top-tagging EFN model did not exhibit as clear a filter-size relationship as the quark/gluon model.The comparison concerns both the visual and quantitative filter analyses.
  • Additional visualizations: A larger version of the quark/gluon ℓ = 256 EFN visualization is provided for framing.
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