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Convolutional L2LFlows: Generating Accurate Showers in Highly Granular Calorimeters Using Convolutional Normalizing Flows

Thorsten Buss, Frank Gaede, Gregor Kasieczka, Claudius Krause, David Shih

TL;DR

This work extends L2LFlows to simulate showers with a 9-times larger profile in the lateral direction, introduces convolutional layers and U-Net-type connections, moves from masked autoregressive flows to coupling layers, and demonstrates the successful modelling of showers in the ILD Electromagnetic Calorimeter as well as Dataset 3 from the public CaloChallenge dataset.

Abstract

In the quest to build generative surrogate models as computationally efficient alternatives to rule-based simulations, the quality of the generated samples remains a crucial frontier. So far, normalizing flows have been among the models with the best fidelity. However, as the latent space in such models is required to have the same dimensionality as the data space, scaling up normalizing flows to high dimensional datasets is not straightforward. The prior L2LFlows approach successfully used a series of separate normalizing flows and sequence of conditioning steps to circumvent this problem. In this work, we extend L2LFlows to simulate showers with a 9-times larger profile in the lateral direction. To achieve this, we introduce convolutional layers and U-Net-type connections, move from masked autoregressive flows to coupling layers, and demonstrate the successful modelling of showers in the ILD Electromagnetic Calorimeter as well as Dataset 3 from the public CaloChallenge dataset.

Convolutional L2LFlows: Generating Accurate Showers in Highly Granular Calorimeters Using Convolutional Normalizing Flows

TL;DR

This work extends L2LFlows to simulate showers with a 9-times larger profile in the lateral direction, introduces convolutional layers and U-Net-type connections, moves from masked autoregressive flows to coupling layers, and demonstrates the successful modelling of showers in the ILD Electromagnetic Calorimeter as well as Dataset 3 from the public CaloChallenge dataset.

Abstract

In the quest to build generative surrogate models as computationally efficient alternatives to rule-based simulations, the quality of the generated samples remains a crucial frontier. So far, normalizing flows have been among the models with the best fidelity. However, as the latent space in such models is required to have the same dimensionality as the data space, scaling up normalizing flows to high dimensional datasets is not straightforward. The prior L2LFlows approach successfully used a series of separate normalizing flows and sequence of conditioning steps to circumvent this problem. In this work, we extend L2LFlows to simulate showers with a 9-times larger profile in the lateral direction. To achieve this, we introduce convolutional layers and U-Net-type connections, move from masked autoregressive flows to coupling layers, and demonstrate the successful modelling of showers in the ILD Electromagnetic Calorimeter as well as Dataset 3 from the public CaloChallenge dataset.
Paper Structure (22 sections, 10 equations, 12 figures, 9 tables)

This paper contains 22 sections, 10 equations, 12 figures, 9 tables.

Table of Contents

  1. Introduction
  2. Datasets
  3. GettingHigh
  4. Dataset 3 of the CaloChallenge
  5. Model
  6. Energy Distribution Flow
  7. Causal Flows
  8. Training
  9. Postprocessing
  10. Metrics
  11. Observables
  12. Classifier
  13. Results
  14. Observables
  15. Classifier
  16. ...and 7 more sections

Figures (12)

  • Figure 1: Visualization of single showers from the training data. The upper row shows examples from the GettingHigh dataset, the lower row from CaloChallenge dataset 3. The left examples are for low incident particle energies, the right ones for high energies.
  • Figure 2: Diagram illustrating the overall architecture of L2LFlows. Arrows going into flows illustrate the conditional input of the flow. Arrows going out of flows illustrate what the flow has generated. The post processing is only applied to generated GettingHigh showers.
  • Figure 3: Diagrams illustrating the structure of a causal flow. Left: The overall structure of one single causal flow. Right: The U-Net used in the coupling blocks.
  • Figure 4: Average deposited energy for all cells in layer 1, 10, 20 and 30 overall flow generated (upper row) and Geant4 simulated (lower row) sample. Shown are the results for GettingHigh.
  • Figure 6: Top row: longitudinal (left) and $X$ profile for GettingHigh. Bottom row: longitudinal (left) and radial (right) profile for CaloChallenge dataset 3.
  • ...and 7 more figures