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Getting High: High Fidelity Simulation of High Granularity Calorimeters with High Speed

Erik Buhmann, Sascha Diefenbacher, Engin Eren, Frank Gaede, Gregor Kasieczka, Anatolii Korol, Katja Krüger

arXiv:2005.05334v3physics.ins-dethep-exhep-phphysics.data-an

TL;DR

Calorimeter shower simulation is costly, and high-granularity detectors require accurate modeling of shower properties. This paper combines a modified BIB-AE with an MMD-based post-processor and reports high-fidelity simulation of challenging energy distributions, including the per-cell MIP peak.

  • Problem

    Calorimeter simulation is costly, while high-granularity detectors require accurate reproduction of physically relevant shower properties.

    Source: §1 · pp. 1–2

  • Method

    The paper modifies the BIB-AE with an MMD loss and a separate post-processor network to refine generated shower hit energies.

    Source: §1 · p. 2, §3.3 · p. 5, §3.4 · p. 6

  • Results

    The study reports the first high-fidelity simulation of overall energy response and per-cell energy distribution around the MIP peak for a realistic high-granularity calorimeter.

    Source: §5 · p. 10

  • Takeaways & Limitations

    The MMD and post-processor improvements can potentially also be applied to other generative architectures and models.

    Source: §5 · p. 11

    The post-processed BIB-AE still deposits additional energy at shower edges, which is problematic for reconstruction methods relying on precise shower start and end positions.

    Source: §4.2 · p. 9

Abstract

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Accurate simulation of physical processes is crucial for the success of modern particle physics. However, simulating the development and interaction of particle showers with calorimeter detectors is a time consuming process and drives the computing needs of large experiments at the LHC and future colliders. Recently, generative machine learning models based on deep neural networks have shown promise in speeding up this task by several orders of magnitude. We investigate the use of a new architecture -- the Bounded Information Bottleneck Autoencoder -- for modelling electromagnetic showers in the central region of the Silicon-Tungsten calorimeter of the proposed International Large Detector. Combined with a novel second post-processing network, this approach achieves an accurate simulation of differential distributions including for the first time the shape of the minimum-ionizing-particle peak compared to a full GEANT4 simulation for a high-granularity calorimeter with 27k simulated channels. The results are validated by comparing to established architectures. Our results further strengthen the case of using generative networks for fast simulation and demonstrate that physically relevant differential distributions can be described with high accuracy.

1 Introduction

The introduction frames calorimeter simulation as an increasingly costly bottleneck and emphasizes that speedups must preserve shower physics, especially in high-granularity detectors. It presents generative models, including a modified BIB-AE, as a promising route to faster, more faithful simulation.

1 Introduction

  • The paper focuses on improving the fidelity of physically relevant distributions for a high-resolution calorimeter prototype in the ILD.The introduction situates this work amid broader HEP uses of generative models, including muon interactions, parton showers, event generation, and unfolding [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49].Source: §1 · pp. 1–2
  • Generative models can sample encoded particle-interaction densities efficiently, avoiding the individual tracking of many secondary shower particles through detector material.The introduction describes generative models as a promising way to accelerate simulation, pioneered in Ref. [1].Source: §1 · pp. 1–2
  • High-granularity calorimeters demand accurate reproduction of shower properties, while simulation costs are already a potential bottleneck and are expected to grow at future colliders.The introduction notes that ATLAS uses more than half of its total CPU time on the LHC Computing Grid for Monte Carlo simulation, dominated by calorimeter simulation.Source: §1 · pp. 1–2
  • The study applies a modified BIB-AE to model tested physics distributions more accurately than traditional GANs.The BIB-AE architecture unifies several generative models, including GANs and VAEs [24].Source: §1 · p. 2
  • Earlier calorimeter GAN simulations reported speed-ups of up to five orders of magnitude over Geant4, alongside WGAN and other generative approaches for shower simulation [1] [25] [26].Related work also includes WGANs for air-shower detectors and high-granularity calorimeters [27] [28], and GAN-inspired or VAE/WGAN approaches for future colliders and ATLAS [29].Source: §1 · p. 2

2 Data Set

The data set consists of Geant4 photon showers in the ILD Si-W ECal, represented on a regular grid despite the detector’s staggered geometry. Photons enter centrally at a perpendicular angle, and the study uses a large training set plus independent evaluation samples.

2 Data Set

  • For this study, the ILD Si-W ECal consists of 30 active silicon layers in a tungsten absorber stack, with 20 layers of 2.1 mm followed by 10 additional layers.Source: §2 · p. 2
  • The study maps the sensors to a 30×30×30 grid, with one sensor per cell, although the actual calorimeter geometry is staggered.This mismatch makes learning more challenging; more geometry-aware architectures are outside the study’s focus [51].Source: §2 · p. 3
  • 950k Geant4 photon showers with energies from 10–100 GeV form the training set; evaluations use independent samples of 40k events and 4k events at each of seven discrete energies.Photons enter perpendicular to the ECal at the grid center; the figures show a 60 GeV example and an overlay of 2000 50 GeV showers projected along y.Source: §2 · p. 3, Figure 1 · p. 3, Figure 2 · p. 3

3 Generative Models

The section surveys GAN and WGAN benchmarks, then presents BIB-AE and MMD and post-processing refinements for calorimeter-shower generation.

3 Generative Models

  • GAN and WGAN serve as calorimeter-generation benchmarks, followed by BIB-AE and refinements intended to improve generated-data quality.The GAN and WGAN architectures and where true energy enters are shown in Fig. 3.Source: §3 · p. 3, Figure 3 · p. 4

3.2 Wasserstein-GAN

  • WGAN uses Wasserstein-1 loss with a 1-Lipschitz critic enforced by gradient penalty, while conditioning generation on photon energy.The generator and critic are parameterized by energy, and a constrainer network supplies an additional conditioning loss.Source: §3.2 · p. 4

3.3 Bounded Information Bottleneck-Autoencoder

  • BIB-AE combines adversarial reconstruction, reconstruction-norm, and latent-space regularization losses; this implementation replaces the reconstruction norm with a third critic after norm-based comparisons smeared images.The architecture and added MMD and post-processor are shown in Fig. 4.Source: §3.3 · p. 5, Figure 4 · p. 5

3.4 Maximum Mean Discrepancy

  • Sorted-Kernel-MMD sorts real and generated hit energies, compares top-n windows shifted by m, and reduces computation to linear in N while matching similar energies.The reported settings are m=25, n=100, and N=2000.Source: §3.4 · p. 6
  • With α = 200, MMD makes the per-cell hit-energy spectrum near-identical to training data, but often places added pixels at unphysical cube edges.This illustrates the trade-off between reproducing the hit-energy spectrum and preserving shower shape.Source: §3.4 · p. 6, §3.5 · p. 6

3.5 Post Processing

  • Post Processor Network follows shape-correct but spectrum-inaccurate GAN or BIB-AE showers, using 1×1 convolutions and MMD plus MSE to minimally adjust existing pixels.Its convolutions can modify existing pixel values but do not easily add or remove pixels.Source: §3.5 · p. 6

4 Results

The evaluated generative models reproduce several calorimeter-shower distributions, with BIB-AE plus post-processing capturing the MIP peak that GAN and WGAN miss. GPU generation is up to nearly 3,000 times faster than Geant4 on CPU, while post-processing and some model-specific mismatches remain important qualifications.

4 Results

  • The study evaluates GAN, WGAN, and BIB-AE with energy-MMD and post-processing on the same sample of 950k Geant4 showers.Tests use either the full momentum range or specified incident photon energies.Source: §4 · pp. 6–7

4.1 Physics Performance

  • The BIB-AE with post-processing reproduces the per-cell MIP bump missed by GAN and WGAN, and models hit-count mean and width accurately after the 0.1 MeV cutoff.The cutoff makes hit counts sensitive to the modeled spectrum near that threshold.Source: §4.1 · p. 7, Figure 6 · p. 8
  • GAN matches Geant4 correlations exceptionally well; the reported maximum absolute correlation differences are 0.2 for GAN, 0.36 for BIB-AE, and 0.57 for WGAN.The scatter plots also show slight correlation mismatches for WGAN and BIB-AE.Source: §4.1 · p. 8
  • All models reproduce visible-energy peak shape, center, and width, while BIB-AE shows slight z-axis center-of-gravity deviations and WGAN deviates near spatial-profile edges.The BIB-AE z-axis mismatch is traced to a latent variable sampled from a normal distribution rather than the observed latent distribution.Source: §4.1 · pp. 7–8, Figure 7 · p. 9

4.2 The importance of post processing

  • Post-processing improves the BIB-AE MIP spectrum, but edge energy deposits impair longitudinal and radial profiles and may hinder reconstruction of shower start and end positions.The authors implement the post-processor as a separate network because applying MMD loss across the decoder led to similar edge deposits.Source: §4.2 · p. 9

4.3 Computational Performance

  • GPU evaluation provides up to nearly a factor of three thousand speed-up over Geant4 on CPU, and generative-model evaluation time is independent of incident photon energy.Geant4 evaluation time is not independent of incident photon energy.Source: §4.3 · p. 10

5 Conclusion

The study applies a modified BIB-AE to high-granularity calorimeter simulation, targeting overall energy response and per-cell deposits around the MIP peak. It also compares correlation fidelity with GAN and WGAN models and identifies extensions for future work.

5 Conclusion

  • Earlier calorimeter GAN work used an idealized 504-cell detector, while prior WGAN work covered 1,260 cells but mismodeled cell energies below 10 MIPs and hit multiplicity.The proposed BIB-AE with MMD loss and post-processing is reported to describe low-energy deposits reliably.Source: §5 · p. 10
  • A prior GAN study considered a 65k-cell calorimeter, but did not report per-cell energy distributions; its hit-multiplicity disagreement implied MIP-peak mismodeling.The present work targets the per-cell energy distribution around the MIP peak.Source: §5 · p. 10
  • The modified BIB-AE with an MMD loss and Post Processor Network reliably describes low-energy deposits and supports high-fidelity overall response and per-cell MIP-peak distributions in a realistic high-granularity calorimeter.The contribution is described as the first high-fidelity simulation of these quantities for such a calorimeter.Source: §5 · pp. 10–11
  • Mean absolute correlation differences from Geant4 are 0.058 for GAN, 0.187 for WGAN, and 0.132 for BIB-AE.Fig. 9 compares the correlation coefficients for the three models with Geant4.Source: Figure 9 · p. 11
  • The authors identify possible application of the modifications to other generative models and plan to extend the approach across particle types, incident positions, and angles.The cited future work is aimed at a complete, fast, physically reliable synthetic calorimeter simulation.Source: §5 · p. 11

A Network architectures and training procedure

This appendix describes implementation and training procedures for GAN, WGAN, and BIB-AE calorimeter simulators. The BIB-AE combines encoder-decoder and critic components with a post-processing network; systematic hyperparameter tuning was not performed.

Training procedure

  • No systematic hyperparameter tuning was performed because of computational cost; good modeling of Geant4 training distributions served as the stopping criterion.The BIB-AE training ran for four days in parallel on four NVIDIA Tesla V100 GPUs.Source: A Network architectures and training… · p. 12

GAN implementation and training

  • The GAN baseline uses a noise vector and true energy as generator inputs, alternates discriminator and generator updates, and stops after 125k weight updates.The implementation has 1.5M generator weights and 2.0M discriminator weights and does not use a gradient penalty.Source: §A.1 · p. 13

WGAN implementation and training

  • The WGAN uses a generator, critic, and energy constrainer; it trains for 131k updates over 20 epochs, with learning rates reduced during training.The constrainer weights are frozen after 30k iterations; training took one week on three NVIDIA Tesla V100 GPUs.Source: §A.2 · pp. 13–14

BIB-AE implementation

  • The BIB-AE implementation combines an encoder-decoder, latent-space critic, critic and difference critic, and post-processing network, with 71M weights in total.The encoder accounts for 35M weights; the authors attribute much of the larger model size relative to GANs to its fully connected layers.Source: §A.3 · p. 14
  • The BIB-AE encoder samples 24 Gaussian latent variables, and its decoder combines them with 488 Gaussian-noise values and the true energy label to generate a shower.The encoder and decoder architecture is described in detail, including convolutional and dense layers.Source: §A.3 · p. 14
  • The Post Processor Network takes the shower image, true energy, and image sum as inputs and applies shared convolutions with normalization to produce position- and shower-tailored outputs.The network’s kernel-size 1 convolutions apply the same function across pixels, while LayerNorm makes the functions vary by shower and pixel.Source: §A.3 · p. 14

BIB-AE training

  • BIB-AE training runs 35 epochs without post-processing, then one MSE-only post-processor epoch, followed by three epochs using MSE and MMD.Critics are updated five times per encoder-decoder update; checkpoint selection uses a composite figure of merit based on 1D distributions.Source: §A.3 · pp. 14–15
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