[Paper Review] Score-based Generative Models for Calorimeter Shower Simulation

V. M. Mikuni, Benjamin Nachman|arXiv (Cornell University)|Jun 17, 2022

Computational Physics and Python Applications4 citations

TL;DR

This paper introduces CaloScore, the first application of score-based generative models to calorimeter shower simulation in collider physics. It uses diffusion models to generate high-fidelity, high-dimensional calorimeter showers across three datasets from the Fast Calorimeter Simulation Challenge 2022, achieving state-of-the-art realism with stable, scalable training and exact likelihood estimation, though generation time remains a bottleneck compared to GANs.

ABSTRACT

Score-based generative models are a new class of generative algorithms that have been shown to produce realistic images even in high dimensional spaces, currently surpassing other state-of-the-art models for different benchmark categories and applications. In this work we introduce CaloScore, a score-based generative model for collider physics applied to calorimeter shower generation. Three different diffusion models are investigated using the Fast Calorimeter Simulation Challenge 2022 dataset. CaloScore is the first application of a score-based generative model in collider physics and is able to produce high-fidelity calorimeter images for all datasets, providing an alternative paradigm for calorimeter shower simulation.

Motivation & Objective

• To explore score-based generative models as a novel alternative to existing fast simulation methods in collider physics.
• To evaluate the performance of diffusion-based score models on high-dimensional calorimeter shower datasets from the Fast Calorimeter Simulation Challenge 2022.
• To address limitations of prior models such as mode collapse in GANs and scalability issues in normalizing flows.
• To enable scalable, stable training with exact likelihood estimation while maintaining high-fidelity generation.
• To assess the impact of inductive biases and voxelization strategies on model efficiency and performance.

Proposed method

• The method employs score-based generative models that learn the gradient of the data log-density (the 'score') via score matching on a diffusion process.
• Three different diffusion processes are evaluated: the sub-VP, VP, and VE stochastic differential equations (SDEs), each with distinct drift and diffusion coefficients.
• The model uses a U-Net architecture with residual blocks and sinusoidal positional encoding to process voxelized calorimeter showers.
• Training is performed via convex score-matching loss, enabling stable optimization without adversarial competition.
• Reverse SDE sampling is used at inference time to generate realistic showers from noise, with multiple steps required per sample.
• Inductive biases such as geometric voxelization and coordinate transformations are applied to improve sample quality and reduce model parameters.

Experimental results

Research questions

• RQ1 Can score-based generative models produce high-fidelity calorimeter showers across diverse, high-dimensional datasets?
• RQ2 How do different SDE formulations (sub-VP, VP, VE) affect the realism and fidelity of generated showers?
• RQ3 Can score-based models achieve exact likelihood estimation while maintaining scalability and stability?
• RQ4 How does voxelization strategy and geometric inductive bias impact model efficiency and performance?
• RQ5 What is the trade-off between generation quality and inference speed compared to GANs and normalizing flows?

Key findings

• CaloScore successfully generates realistic calorimeter showers across all three datasets from the Fast Calorimeter Simulation Challenge 2022, including the highest-dimensional dataset with tens of thousands of voxels.
• The sub-VP SDE variant outperformed VP and VE SDEs on the highest-dimensional dataset, showing better agreement with Geant4 simulations across multiple observables.
• CaloScore achieved better fidelity than the WGAN-GP baseline in all distributions tested, except for generation time, where WGAN-GP was 3–16 times faster.
• The model demonstrated scalability, with parameter count dependent on architecture rather than input dimensionality, especially when using geometric inductive biases in datasets 2 and 3.
• Regular voxelization enabled convolutional operations only after a coordinate transformation, which introduced nonphysical empty voxels and could be improved with better preprocessing.
• Energy conservation was challenging to enforce; a normalization head was added to dataset 1, but a two-step conditional approach (as in CaloFlow) would be more effective for future improvements.