Generative AI for HGCal

In CaloDiffusion we showed that diffusion models can generate calorimeter showers that are nearly indistinguishable from Geant. But that was demonstrated on the CaloChallenge datasets, which are either idealized cylindrical detectors or a simplified slice of ATLAS.

The goal throughout has been to get these models running inside a real experiment. This project is that step: building generative fast simulation for the CMS High-Granularity Calorimeter (HGCal), with deployment in the CMS simulation workflow as the target.

Why HGCal is hard

HGCal is the upgrade that will replace the CMS endcap calorimeters for the High-Luminosity LHC. It is a very complex detector — 47 layers, roughly six million silicon channels, and sensors laid out on an irregular hexagonal grid whose cell sizes and materials change as you move through the different layers.

Cross section of the CMS High-Granularity Calorimeter

Hexagonal sensor cell layout on an HGCal silicon wafer

Left: a cross section of HGCal. The electromagnetic section (CE-E) is 26 layers of silicon interleaved with lead and copper-tungsten; the hadronic section (CE-H) adds 21 more layers of silicon and scintillator in steel and copper. Right: the sensor layout on an 8-inch silicon wafer, with the larger 1.18 cm² cells (left) and the smaller 0.52 cm² cells (right) used in different regions of the detector colours mark the cells. Between the hexagonal tiling and the change in cell size across the detector, the data has none of the regular structure that traditional ML architectures rely on. Figures from the HGCal Technical Design Report.

This complexity also makes simulation expensive. Geant4’s cost scales with the detail of the geometry, and HGCal is far more detailed than any calorimeter before it. Meanwhile the simulation share of the CMS computing budget is shrinking, not growing, as reconstruction gets more demanding. So fast simulation will be a requirement in the High Luminosity era. Generative AI can ensure we don’t lose physics performance because of it.

For a sense of the jump in scale: the showers here have roughly 500,000 active cells, about ten times the highest-granularity dataset in the CaloChallenge. That is after restricting to a per-layer region of interest around the shower axis: 25 cm for photon showers, growing from 42 to 127 cm with depth for pion showers.

Raw cell count is not the only challenge; the irregular geometry is also quite difficult to handle. The cells are hexagonal, their sizes vary across the detector, and the layout has none of the regularity that convolutional models are built to exploit. Irregular geometries were tested in the CaloChallenge, but only at much smaller scales of around 500 cells.

A common CMS dataset

Rather than have everyone build their own setup and report incomparable numbers, we put together a common CMS dataset and a shared evaluation framework. The dataset is single particles fired into HGCal in CMSSW: photons and charged pions, one million showers each, split 700k for training and 300k for evaluation, at incident energies from 1 to 1000 GeV sampled log-uniformly.

The evaluation follows the CaloChallenge template. We compute a few hundred physically meaningful features per shower — per-layer energy fractions, centroids, widths, cell occupancy, the transverse profile in rings of hexagonal neighbours — and then compare the generated and Geant distributions using 1D separation power, unbinned Kolmogorov-Smirnov tests, the multivariate FPD and KPD scores, and the AUC of a classifier trained to tell real showers from generated ones.

We plan to release this dataset and evaluation code publicly in the future!

Five models

We trained and evaluated five approaches, deliberately spanning the different generative paradigms in play right now:

HGCaloDiffusion (Amram & Pedro, 2023) — an improved version of our CaloDiffusion model. A diffusion U-Net on a voxelized representation, carrying over the cylindrical convolutions and GLaM geometry embedding from CaloDiffusion but with a new model for the layer energies and better training/sampling methods.
HGCaloTrilogy (Jiang et al., 2026) — a next-generation model on top of HGCaloDiffusion, sharing its preprocessing and geometry handling but replacing the generative core with mean-velocity flow transport, a learned Gaussian mixture prior, and a physics-guided loss on high-level observables. This is the line of work that upgrades CaloDiffusion to handle the complexities of HGCal.
HGCaloDream — flow matching with a vision-transformer backbone, in a two-stage factorization.
AllShowers — a point cloud representation with conditional flow matching, and a single model handling both particle types.
GraphCNF — continuous normalizing flows on graphs, chaining a total energy flow, a per-layer graph flow, and a Riemannian flow-matching step for the individual cell energies. Photons only so far.

So how did they do?

Pretty good! Most of the important features of this challenging dataset are well described by multiple models.

The marginal distributions are in good shape. Total energy response, shower centroids and widths, and the shape as a function of depth are all reproduced closely by most of the models.

Average fraction of the shower energy landing in each of the 47 layers, for photons (left) and pions (right), with the models overlaid on Geant4 in black and the ratio to Geant4 underneath. The pion profile shows the jumps where the detector changes character between the electromagnetic and hadronic sections. Most models sit within a few percent of Geant across the bulk of the shower, with GraphCNF being an outlier on the photons.

Zooming in on a single layer tells the same story, and also shows where it starts to break down. Layer 30 sits well inside the hadronic section, where pion showers are at their broadest and messiest.

$Energy fraction, pions, layer 30$

Pion showers in layer 30: the fraction of the shower energy deposited in the layer (left), the transverse position of the shower centroid (middle), and the cell occupancy (right). The first two hold up across the bulk of the distribution, HGCaloDream being the exception on the centroid. Occupancy was found to be the most challenging feature to capture and the agreement with Geant is visibly worse; no model captures the full tail behaviour.

The multivariate metrics, which are sensitive to correlations between features that the one-dimensional distributions miss, show good performance but with room for improvement. No model yet reaches Geant-Geant indistinguishability. HGCaloDream, the best model on the photon dataset, reports a classifier AUC of $\sim$ 0.56. On the more challenging pion dataset HGCaloTrilogy and AllShowers are the best, at AUCs of $\sim$ 0.65. These values are not far from full indistinguishability (AUC = 0.5).

There is also no single winner: the best model for pions is not the best for photons, and within one particle type one model will lead on bulk shape while another leads on width and occupancy. The architectures have complementary strengths, and it is too early to consolidate around one.

What’s next

This is a first demonstration that generative models can actually handle the complexities of CMS HGCal. Further evaluation will be needed to assess the computational performance of the models and to validate them on downstream physics observables. However, these promising early results set us on track to deploy these models in the HL-LHC era, with a significant benefit to the physics program of CMS.

I am leading this effort within CMS, coordinating the teams behind the different models along with the dataset and evaluation framework that lets us compare them on equal terms.

Results so far are public as a CMS Detector Performance note, CMS-DP-2026-049, with more detail on the public TWiki.

A full paper describing the dataset, the models, and the evaluation in detail is in preparation! CMS members can follow the analysis here.

High-precision calorimeter simulation at current and future colliders imposes rapidly growing computational demands, motivating the development of machine-learning surrogates for traditional Monte Carlo tools such as Geant4. Flow matching and diffusion-based generative models have become leading approaches for high-dimensional fast simulation because of their sample quality, but typically require O(100) function evaluations at inference and often rely on auxiliary networks to constrain global observables, compromising streamlined end-to-end generation. We introduce a unified framework that improves the balance between speed, shower quality, and physics fidelity. The method combines: (i) an average velocity field integrator that enables sampling in one or a few evaluations; (ii) a learned generative prior in shower space, constructed from data rather than random noise; and (iii) physics-guided loss terms that impose inductive biases on key observables during training. These elements are training time regularizers, preserving end-to-end inference with no additional cost. With only one or a few evaluation steps, the model achieves shower quality competitive with state-of-the-art flow and diffusion approaches, tested on several public high granularity calorimeter datasets. The results demonstrate inter-layer shower structure consistent with the underlying physics, providing a strong candidate for future fast simulation workflows.

Simulation is crucial for all aspects of collider data analysis, but the available computing budget in the High Luminosity LHC era will be severely constrained. Generative machine learning models may act as surrogates to replace physics-based full simulation of particle detectors, and diffusion models have recently emerged as the state of the art for other generative tasks. We introduce CaloDiffusion, a denoising diffusion model trained on the public CaloChallenge datasets to generate calorimeter showers. Our algorithm employs 3D cylindrical convolutions, which take advantage of symmetries of the underlying data representation. To handle irregular detector geometries, we augment the diffusion model with a new geometry latent mapping (GLaM) layer to learn forward and reverse transformations to a regular geometry that is suitable for cylindrical convolutions. The showers generated by our approach are nearly indistinguishable from the full simulation, as measured by several different metrics.

Why HGCal is hard

A common CMS dataset

Five models

So how did they do?

What’s next

References

2026

2023