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Event Generation and Statistical Sampling for Physics with Deep Generative Models and a Density Information Buffer

Sydney Otten, Sascha Caron, Wieske de Swart, Melissa van Beekveld, Luc Hendriks, Caspar van Leeuwen, Damian Podareanu, Roberto Ruiz de Austri, Rob Verheyen

arXiv:1901.00875v5hep-phhep-exphysics.data-an

TL;DR

Monte Carlo event generation is computationally costly, and learning both event production and correct occurrence frequencies remains a challenge. The paper evaluates GANs and VAEs on three physical processes and introduces a VAE-based density information buffer. The B-VAE yields event distributions in very good agreement with Monte Carlo data, while the tested GANs and standard VAE do not reliably produce the correct distributions.

  • Problem

    Monte Carlo event generation can be computationally expensive, and the study asks whether generative models can learn event distributions while producing samples quickly and with correct frequencies.

    Source: §I · pp. 1–2

  • Method

    The authors compare GANs and VAEs across three event-generation processes and introduce a B-VAE that estimates a decoder prior from encoded ground-truth events.

    Source: §I · p. 2, B. e+e−→Z →l+l− · p. 8

  • Results

    The B-VAE produces event distributions in very good agreement with Monte Carlo data, whereas the standard VAE and tested GAN architectures do not reliably produce events with the right frequencies.

    Source: §IV · p. 7, §VI · p. 11

  • Takeaways & Limitations

    The results indicate the method may be useful for computationally expensive Monte Carlo simulations and sampling from high-dimensional, complex distributions.

    Source: §VI · p. 11

    The degree to which the learned event distribution generalizes beyond observed ground truth remains to be established, and processes with up to twenty or more final-state objects remain an open modeling challenge.

    Source: §VI · p. 11

Abstract

from arXiv · show

We present a study for the generation of events from a physical process with deep generative models. The simulation of physical processes requires not only the production of physical events, but also to ensure these events occur with the correct frequencies. We investigate the feasibility of learning the event generation and the frequency of occurrence with Generative Adversarial Networks (GANs) and Variational Autoencoders (VAEs) to produce events like Monte Carlo generators. We study three processes: a simple two-body decay, the processes $e^+e^-\to Z \to l^+l^-$ and $p p \to t\bar{t} $ including the decay of the top quarks and a simulation of the detector response. We find that the tested GAN architectures and the standard VAE are not able to learn the distributions precisely. By buffering density information of encoded Monte Carlo events given the encoder of a VAE we are able to construct a prior for the sampling of new events from the decoder that yields distributions that are in very good agreement with real Monte Carlo events and are generated several orders of magnitude faster. Applications of this work include generic density estimation and sampling, targeted event generation via a principal component analysis of encoded ground truth data, anomaly detection and more efficient importance sampling, e.g. for the phase space integration of matrix elements in quantum field theories.

I. INTRODUCTION

The paper addresses the computational cost and inefficiency of Monte Carlo event generation by investigating deep generative models that can learn event distributions and generate samples quickly. It compares GANs and VAEs across increasingly complex processes and reports that the B-VAE performs well where standard VAEs and tested GANs do not reliably capture event frequencies.

I. INTRODUCTION

  • Monte Carlo event generation can take up to 10 minutes per event, while targeted analyses often generate and discard many events to find kinematically similar ones.The authors identify simulation speed and inefficient selection as constraints on scientific progress and signal studies.Source: §I · p. 1
  • The study tests whether a model can transform random latent samples into events following the data distribution and generate additional samples quickly.It compares GANs and VAEs, including a B-VAE with a heavily weighted reconstruction loss and density estimation from encoded ground-truth data.Source: §I · p. 2
  • The evaluation spans a two-body decay toy model, leptonic Z production, and 26-dimensional top-pair production with five or six final-state objects.The top-pair process includes top decays and detector-response simulation.Source: §I · p. 2
  • The Z study finds that standard VAEs cannot reliably model the process, while the B-VAE achieves good agreement; for top-pair events, the B-VAE produces realistic collections following Monte Carlo distributions.The introduction also reports GAN architectures with recommended hyperparameters, without specifying their results here.Source: §I · p. 2
  • PCA of encoded top-pair events offers a way to steer generation, and proposed applications include anomaly detection and phase-space integration of matrix elements.The paper also notes prior GAN work on two-object processes and azimuthal-density difficulties as context for its studies.Source: §I · p. 2

II. MONTE CARLO DATA

The paper constructs three generated-event datasets of increasing dimensionality and complexity. For the toy model, sampled masses and angles determine two-body decay kinematics, while the training variables include a deliberate degeneracy for testing whether models learn relativistic dispersion relations.

II. MONTE CARLO DATA

  • The study uses generated events for a two-body toy model, leptonic Z production, and top-pair production and decay.These datasets provide the study's increasing levels of dimensionality and complexity.Source: §II · p. 2

A. 10-dimensional toy model

  • The toy model starts with a stationary particle of mass M decaying into particles of masses m1 and m2, with their momentum four-vectors calculated from sampled quantities.The passage begins to specify the sampling procedure but does not provide its full details.Source: A. 10-dimensional toy model · p. 2
  • The daughter masses m1 and m2 and direction angles θ and φ are sampled from uniform distributions 10^6 times.The angles describe the direction of particle 1; momentum conservation fixes particle 2's direction.Source: A. 10-dimensional toy model · p. 3
  • Training data comprise both daughter energies, their three momentum components, and their masses, creating a degeneracy to test whether models learn relativistic dispersion relations.The passage states that the angles and momentum conservation determine particle 2's direction.Source: A. 10-dimensional toy model · p. 3

B. 16-dimensional e+e−→Z →l+l−

The paper describes a 16-dimensional leptonic Z-decay dataset and the generative-model methods used in the study. Its VAE variant weights reconstruction strongly, while GANs are evaluated using recommended hyperparameters as baselines.

B. 16-dimensional e+e−→Z →l+l−

  • Leptonic Z-decay events use the four-momenta of produced leptons as 16-dimensional model inputs and outputs, with eight entries zero because each event contains either an electron or muon pair.The data are generated at matrix-element level for a 91 GeV center-of-mass energy.Source: B. 16-dimensional e+e−→Z →l+l− · p. 3

III. METHODS

  • The B-VAE combines a density information buffer with a VAE variant to generate realistic event collections.The methods section also covers GANs, traditional latent-density methods, and figures of merit.Source: §III · p. 3

A. Generative Models

  • GANs are trained as generator-discriminator systems, and the reported GAN models use recommended hyperparameters as baselines rather than results from an extensive hyperparameter scan.The authors note that a broader scan could yield better-performing GANs.Source: A. Generative Models · p. 3
  • The VAE loss includes a factor B on the KL-divergence term, and the study chooses B ≪ 1 to emphasize reconstruction quality.This reduces the penalty for latent codes deviating from a standard Gaussian; at B = 0, the loss matches that of a standard autoencoder.Source: A. Generative Models · p. 4

B. Latent Code Density Estimation

The section addresses the mismatch between a VAE's standard-normal sampling prior and the latent codes produced by encoding true events. It describes estimating a data-driven prior from those codes, including a density information buffer and alternative density estimators.

B. Latent Code Density Estimation

  • A standard-normal prior may not match the encoded true-event latent distribution, so decoding samples from it may not reproduce the reference event distribution.The proposed response estimates the decoder prior from encoded ground-truth events.Source: B. Latent Code Density Estimation · p. 5
  • The density information buffer aggregates encoded training events and can construct a prior directly or support KDE, Gaussian mixture models, and a staged VAE.The buffer stores encoder means and variances, explicitly conserving correlations, but its density can be biased toward training data.Source: B. Latent Code Density Estimation · pp. 5–6
  • The study also considers KDE, Gaussian mixture models, and a staged VAE as ways to model latent-code density.KDE uses a bandwidth h to trade off bias and variance; the staged VAE learns a decoder for latent codes.Source: B. Latent Code Density Estimation · p. 5
  • The buffer's training-data bias is addressed with a smudge factor α that increases sampled latent variances when σ < 0.05.The tested α values are 1, 5, and 10.Source: B. Latent Code Density Estimation · p. 6

C. Figures of Merit

The section defines density-comparison and overfitting measures for ranking generative models against Monte Carlo data. It sets expected reference values by comparing Monte Carlo samples with Monte Carlo samples and extrapolating to the target sample size.

C. Figures of Merit

  • The evaluation uses 105 of 1.2 · 10^6 Monte Carlo samples for training, then generates 1.2 · 10^6 events with each model for comparison.The comparison is against Monte Carlo data.Source: C. Figures of Merit · p. 6
  • The correlated-performance measure averages a test statistic over 528 two-dimensional histograms built from 33 observables, using 252 bins per histogram.The observables include particle kinematics and multi-object masses; the section reports that Kullback-Leibler divergence and Wasserstein distance give identical conclusions.Source: C. Figures of Merit · p. 6
  • Overfitting is probed by measuring empty-bin fractions in η_j1 versus η_j2 histograms as bin counts increase, relative to the Monte Carlo equivalent.Increasing histogram granularity probes progressively smaller phase-space regions; the empty-bin fraction depends on the underlying distribution.Source: C. Figures of Merit · pp. 6–7
  • For 1.2 · 10^6 events, the expected references are δ ≈ 0.0003 and δ_OF ≈ 0.5; the selected model keeps δ_OF ⪅ 1 while minimizing δ.The δ reference is extrapolated from Monte Carlo evaluations up to 600,000 events using a fitted curve; δ_OF is assumed roughly flat around 0.5.Source: C. Figures of Merit · p. 7

IV. RESULTS

Across three datasets, the B-VAE produces events in very good agreement with Monte Carlo distributions, while many default-parameter GAN architectures and the standard VAE do not perform well. The study also examines alternative density-buffer improvements, sanity checks, and traditional-method benchmarks.

IV. RESULTS

  • The B-VAE captures the underlying distribution and yields events in very good agreement with Monte Carlo data at 12 times the training-data size, while many GAN architectures with default parameters and the standard VAE do not perform well.The best GAN results come from DijetGAN with the authors’ implementation and LSGAN with recommended hyperparameters; the standard VAE’s failure is attributed to encoded events not following a standard-normal latent distribution, which the density information buffer circumvents.Source: §IV · p. 7
  • The study tests KDE, Gaussian mixture models, and another VAE for density-buffer improvements, and also checks traditional benchmarks and whether the method merely learns the identity function.A brief parameter scan varies the latent dimension, B, smudge factors α, and offsets γ.Source: §IV · p. 7

A. Two-body decay toy model

In the two-body toy model, the B-VAE with an adjusted prior best reproduces the momentum-component distributions among the tested methods. The models also learn the relativistic dispersion relation, while GANs struggle with the uniform azimuthal distribution.

A. Two-body decay toy model

  • The B-VAE with an adjusted prior most reliably models p_x, p_y, and p_z compared with regular GANs and VAEs using a standard-normal prior, though those models remain good approximations.The tested GANs have small capacities in this dataset.Source: A. Two-body decay toy model · p. 7
  • All models learn the relativistic dispersion relation, consistent with findings in [49] [50].Source: A. Two-body decay toy model · p. 7
  • GANs have difficulty learning the uniform φ distribution; manually sampling φ_j1 in [20] addresses it for two jets, but its generalization to higher multiplicities is questioned.Source: A. Two-body decay toy model · p. 7

B. e+e−→Z →l+l−

For the leptonic Z decay, B-VAE follows the physical-event distributions more accurately than the standard VAE. In the more complex t¯t study, B-VAE generally agrees with Monte Carlo distributions, with some remaining artifacts and tail deviations.

B. e+e−→Z →l+l−

  • For Z decays, B-VAE agrees very well with pT, θ and φ distributions and well with the lepton-pair invariant mass near 91 GeV; standard VAE distributions are smeared and can violate momentum conservation.Standard VAE can also generate four-lepton events; B-VAE better reproduces the lepton-pair kinematics.Source: B. e+e−→Z →l+l− · p. 7
  • The t¯t setup has 26 input/output dimensions, four jets and up to two leptons, without distinguishing jet flavors or lepton types.Jets are clustered objects with Monte Carlo pT of at least 20 GeV.Source: B. e+e−→Z →l+l− · p. 8
  • The best δ1D settings use the highest latent dimension, lowest B and α = 1, γ = 0, but perform poorly in δOF; the selected model balances these metrics.The selected configuration is dim z = 20, B = 10−6, α = 1, γ = 0.05; simple 5% Gaussian smearing offers no advantage over the model described as reproducing training data.Source: B. e+e−→Z →l+l− · p. 8
  • 177.5 seconds were needed to generate 107 t¯t events with the VAE, several orders of magnitude faster than traditional Monte Carlo methods.The reported hardware was an Intel i7-4790K.Source: B. e+e−→Z →l+l− · p. 9
  • In t¯t, B-VAE smears visible φ detector granularity and produces artifacts near (±π, 0) and (0, ±π), while E, pT and η show larger tail deviations.The granularity smearing is attributed most likely to γ; the φ artifacts were expected from one-dimensional azimuthal deviations around ±π.Source: B. e+e−→Z →l+l− · p. 9
  • The tested GANs generally perform unsatisfactorily on δ and worse than traditional KDE and GMM methods, although LSGAN surpasses traditional and several B-VAE models on δOF.The GANs used recommended hyperparameters; their large hyperparameter spaces mean the poor results do not rule out GANs as event generators.Source: B. e+e−→Z →l+l− · pp. 9–10

D. Sanity Checks

The sanity checks compare density estimators and Gaussian smearing on t¯t events, and test whether a VAE simply learns the identity function. The tested density estimators perform poorly, while smearing’s limited success does not extend to correlation-sensitive performance.

D. Sanity Checks

  • KDE bandwidth is grid-searched on 10^4 samples then fit on 10^5; GMMs with 50, 100, or 1000 components (500-iteration limit) are evaluated on 1.2·10^6 generated samples using δ and δOF.Separately, 10^5 events are Gaussian-smeared after the same preprocessing.Source: D. Sanity Checks · p. 10
  • Gaussian smearing performs well for δ but poorly for δOF because it does not respect correlations in the data.Table V reports the KDE and GMM comparisons on δ and δOF; the prose gives the smearing result.Source: D. Sanity Checks · p. 10, Table V · p. 10
  • The VAE does not simply learn the identity function: reconstructions of t¯t events differ from reconstructions of uniform-noise inputs.This check uses a VAE with a 20-dimensional latent space and B = 10^-6.Source: D. Sanity Checks · p. 10

V. APPLICATIONS

The applications section explores the B-VAE for targeted event generation, anomaly detection, and phase-space integration. Its latent-space representation can be inspected to steer decoded events toward selected characteristics.

V. APPLICATIONS

  • A two-dimensional PCA map of latent-space representations can steer generation toward events with selected characteristics.The authors demonstrate sampling around r = 3.5 and φ = 225° in PCA space to target events resembling event 62.Source: §V · pp. 10–11
  • Repeated exploration of latent-space regions may allow event characteristics to be narrowed further, while a classifier could define the desired region.The authors describe these as options for targeted generation.Source: §V · p. 11
  • B-VAE event weights may support efficient phase-space integration of multi-leg matrix elements.The paper presents this as a possible application and relates it to prior machine-learning approaches to Monte Carlo and matrix-element integration.Source: §V · p. 11
  • Relative reconstruction losses may help detect anomalies as departures from training data in individual events or their frequency of occurrence.The authors suggest training a B-VAE on mixed standard-model events to look for new-physics anomalies in experimental data.Source: §V · p. 11

VI. DISCUSSION

The discussion concludes that the B-VAE’s density information buffer can produce event distributions in good agreement with ground truth, while identifying tuning choices and open limits on generalization and event multiplicity.

VI. DISCUSSION

  • The density information buffer yields generated events whose probabilistic characteristics agree very well with ground truth, whereas tested GANs and a standard-normal-prior VAE fail to reproduce the right frequencies.The authors report that latent dimensionality should be smaller than, but close to, the input dimension.Source: §VI · p. 11
  • Accurate generation requires heavily weighting reconstruction loss and tuning the underestimated latent-code variance; the tested GAN results may improve with stricter selection and broader hyperparameter searches.The authors also mention alternative GAN objectives and training approaches as possible improvements.Source: §VI · p. 11
  • The degree to which implicit interpolation generalizes beyond observed ground truth remains to be identified, and modeling events with up to twenty or more final-state objects remains an open question.Auxiliary features, regression and classification loss terms, and flow-based models are proposed as directions for improvement or comparison.Source: §VI · p. 11
  • The authors see potential usefulness beyond particle physics in sciences using computationally expensive Monte Carlo simulations or needing to sample complex, high-dimensional distributions.Source: §VI · p. 11

VIII. AUTHOR CONTRIBUTIONS

The paper records author contributions and states that the custom code for its main results is publicly available.

VIII. AUTHOR CONTRIBUTIONS

  • S. O. contributed to the idea, wrote most of the paper, invented the B-VAE, and performed most training and analysis.The passage also credits S. C. with the idea and B-VAE invention, W. d. S. with toy-model data and an initial GAN, and M. v. B. with t¯t data and figures.Source: §VIII · p. 12

XI. CODE AVAILABILITY STATEMENT

  • The custom code created for the paper’s main results is published in a public GitHub repository.Source: §XI · p. 12
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