Lantern: Conflict-Aware Gradient Blending for Physics-Guided Diffusion Models in Calorimeter Simulation

Monte Carlo simulation of calorimeter showers is a principal bottleneck for the High-Luminosity LHC, and diffusion models have emerged as fast, high-fidelity surrogates. Their denoising objective is purely statistical, however: a model can minimize it while placing the physics wrong. Existing physics-informed generative methods cannot close this gap, because they assume a closed-form law, a governing PDE residual or a hard per-sample constraint, that a shower does not supply: no per-sample PDE governs a stochastic cascade, and energy conservation fixes only one scalar per shower. Standard metrics ignore the correlation structure across calorimeter layers and voxels, comparing showers only in a physics feature space. We address both gaps. We introduce the Correlation Frobenius Distance (CFD), a single normalized score for correlation fidelity at layer-wise and voxel-wise scales. We then encode the soft per-sample structure available in a shower as two physics-aware auxiliary losses: a variance-stabilized voxel residual loss grounded in counting statistics, and a graph Laplacian loss over the detector geometry. We combine both with denoising through GradBlend, which anchors the step magnitude to the denoising gradient while letting the auxiliary steer its direction, yielding Lantern, a physics-guided diffusion surrogate. On CaloChallenge Dataset 2, injecting the physics losses through task-symmetric rules such as PCGrad, GradNorm, IMTL-G, and ConFIG inflates FPD by 2-100x relative to denoising alone, whereas GradBlend admits the same signal without regression and, with the Laplacian loss, Lantern improves both FPD and CFD. Our ablation on the auxiliary loss scheduler shows that the voxel residual loss, whose gradient conflicts with denoising, requires a terminal denoising-only phase to preserve shower fidelity, whereas the non-conflicting Laplacian loss is insensitive to the schedule.

SPADE: Split-and-Delay Embeddings for Autoregressive High-Granularity Calorimeter Simulation

We introduce SPADE (SPlit And Delay Embeddings), an autoregressive transformer for sequences whose tokens carry multiple features. Rather than embedding these features jointly, SPADE embeds them independently. Delaying each feature stream relative to the previous one allows intra-token correlations to be learned by the standard self-attention mechanism. Applied to point-cloud calorimeter shower generation in the highly granular ILD detector, SPADE is competitive with the state of the art AllShowers model on photon showers, and substantially outperforms its VQ-VAE-based predecessor OmniJet-$α_C$. The mechanism is applicable to any generative task with multi-feature tokens, enabling LLM-style pretraining workflows for higher-dimensional data.

CaloTrilogy: Toward a Breakthrough in One-Step, End-to-End, Physics-Guided Shower Generation for Modern Calorimeters

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 ${\cal O}(100)$ 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.