DeepMuon: generating cosmic-ray Muons based on optimal transport

Deep learning can learn distributions directly from data, enabling fast event simulation without hand-crafted designs or costly sampling. Yet many mainstream generators, such as diffusion models, require sophisticated architectures and multi-step denoising, which reduces throughput. Neural generators also struggle with sharp, high-kurtosis distributions common in physics, such as the cosmic-ray muon energy spectrum. Cosmic-ray muon simulation underpins non-destructive imaging, positioning and navigation, timing, and cryptography, and is traditionally performed with Monte Carlo or parametric models. Here we propose DeepMuon, a deep learning-based cosmic-ray muon generator that targets both efficiency and accuracy while addressing high-kurtosis fitting. We first apply an inverse Box--Cox transformation to reduce energy-spectrum kurtosis and simplify the learning task. We then optimize the generator with the Sliced Wasserstein Distance loss, achieving statistically consistent, one-step cosmic-ray muon generation with only a 2-layer Transformer encoder. DeepMuon also learns muon distribution patterns from limited data, enabling simulation of real detector-measured distributions. At sea level, DeepMuon substantially accelerates muon generation relative to the Cosmic-ray Shower Library (CRY). We further develop a DeepMuon-based pipeline to directly simulate underwater muon distributions and compare against CRY, demonstrating suitability for simulation and imaging tasks. For more details about our open-source project, please visit: https://github.com/wangab0/deepmuon.