Multi-Resolution Laplacian Diffusion Models for Kilometer-Scale Climate Emulation

Generative models have shown promise in emulating kilometer-scale climate simulations at a small fraction of their cost. A common design pairs a coarse-resolution global generator with a diffusion-based super-resolution stage that generates fine-scale fields conditioned on the coarse output. The two stages are trained independently across a single resolution gap and share no multiresolution structure, so they inhabit different representations and nothing constrains them to compose exactly. Weather data make this harder. Each channel distributes its variance across scales differently, with heavy tails spanning orders of magnitude, and any multiresolution construction must be realized on a spherical grid. We introduce SPyD (Spherical Pyramid Diffusion), which formulates coarse generation and super-resolution as a single diffusion over a Laplacian pyramid. The state splits into a coarse band and orthogonal finer bands, each with its own noise schedule, and the bands evolve under a system of probability-flow ODEs coupled through a shared denoiser, which we instantiate with separately trained conditional band models. In this formulation, the conditioning is an exact projection of the diffusing state, so the super-resolution stage trains directly on this projected state and the coarse and fine bands compose exactly under the pyramid operators. We extend the EDM framework to channel-dependent noise schedules, calibrated per band from measured statistics, and we realize the pyramid on the sphere with global wavelet operators on HEALPix. On ICON simulation output at 5 km, a single SPyD model spanning the full $16\times$ gap in one stage (HPX64 to HPX1024) reproduces the spherical power spectra and pixel distributions of temperature and precipitation and extreme-precipitation percentiles through the $99.999$th, improves on the deployed state of the art for global kilometer-scale emulation on every channel in CRPS at equal conditioning and equal cost, and produces well-calibrated conditional ensembles as measured by rank histograms. SPyD composes directly with pretrained coarse-resolution generators, supporting end-to-end sampling and reanalysis downscaling.