Inductive Simulation of Calorimeter Showers with Normalizing Flows

TL;DR

This work tackles the heavy computational cost of high-granularity calorimeter shower simulation by introducing iCaloFlow, an inductive framework that generates showers layer-by-layer using three normalizing flows conditioned on incident energy and prior layer information. It leverages a teacher-student distillation (PDD) to convert slow, high-capacity MAFs into fast IAFs, enabling rapid generation while preserving fidelity. The approach achieves high fidelity across CaloChallenge2022 Dataset 2 and 3, with substantial speedups over Geant4, particularly when using the IAF student on GPUs (e.g., ~1.3–5.9 ms per shower). These results suggest a practical path toward fast, high-resolution detector simulations, albeit with trade-offs in expressivity and geometric generalization; future work may explore even faster architectures and coupling-layer variants to broaden applicability.

Abstract

Simulating particle detector response is the single most expensive step in the Large Hadron Collider computational pipeline. Recently it was shown that normalizing flows can accelerate this process while achieving unprecedented levels of accuracy, but scaling this approach up to higher resolutions relevant for future detector upgrades leads to prohibitive memory constraints. To overcome this problem, we introduce Inductive CaloFlow (iCaloFlow), a framework for fast detector simulation based on an inductive series of normalizing flows trained on the pattern of energy depositions in pairs of consecutive calorimeter layers. We further use a teacher-student distillation to increase sampling speed without loss of expressivity. As we demonstrate with Datasets 2 and 3 of the CaloChallenge2022, iCaloFlow can realize the potential of normalizing flows in performing fast, high-fidelity simulation on detector geometries that are ~ 10 - 100 times higher granularity than previously considered.

Figures (15)

• Figure 1: Geometry of the detector voxels in each layer for Dataset 2 (left) and Dataset 3 (center) and the three-dimensional geometry of Dataset 2 (right). Dataset 2 has 9 concentric rings, each divided into 16 voxels in $\alpha$, while Dataset 3 has 18 rings, each divided into 50 segments. Both Dataset 2 and 3 contain 45 layers in depth (as shown for Dataset 2).
• Figure 2: Schematic of the three iCaloFlow flows. Solid lines are bidirectional --- the direction into each flow denotes the density estimation step and the direction out of the flow denotes the sample generation step. Note that there are postprocessing steps (see main text) after each generation step, which are omitted in the schematic. Dashed lines indicate the conditional input to the respective flows. Flow-3 is used iteratively on subsequent layers.
• Figure 3: Illustration of OneCycle LR schedule with annihilation phase Krause:2022jna.
• Figure 4: Pattern of energy deposition from two example events generated by Geant4 in Dataset 2 (top row), iCaloFlow teacher (middle row), and iCaloFlow student (bottom row). Events have $E_{\rm inc} = 693$ GeV (left column) and $E_{\rm inc} = 86$ GeV (right column). For visual clarity, voxels with less than 1 MeV of energy have been suppressed. The beam axis is shown with a black line. For display purposes, the separation between layers and voxels within a layer have both been artificially increased from the real detector geometry.
• Figure 5: Averaged energy deposition pattern of events in layers 1, 10, 20, and 45 (from left to right) for the Geant4 data in Dataset 3 (top row), events sampled from iCaloFlow teacher trained on Dataset 3 (middle row), and events sampled from iCaloFlow student trained on Dataset 3 (bottom row)