LCG-SFNO: Low-Rank Clebsch–Gordan Corrections for Global Atmospheric Forecasting on the Sphere

Spherical-harmonic neural operators provide an efficient representation for global forecasting, but their discretized pointwise nonlinearities can break rotational equiv- ariance through aliasing. Clebsch–Gordan (CG) tensor products provide exactly equivariant interactions between spherical-harmonic representations, but their com- putational cost grows rapidly with spectral bandwidth and has limited their use at forecasting scale. We introduce LCG-SFNO, which combines low-rank CG interactions within a controlled low-degree band with coefficient-space residual propagation and an exactly projected quadratic nonlinearity, preserving SO(3) equivariance throughout the learned operator. To make the CG interaction practi- cal, we further develop SSHCG, a sparse GPU implementation with a factorized alternative for wider coupled bands. On spherical shallow-water dynamics and the ERA5-based WeatherBench benchmark, LCG-SFNO reduces rotational error to numerical precision—more than three orders of magnitude below SFNO on shallow water and five orders of magnitude below it on WeatherBench—while matching SFNO on shallow-water forecasting and substantially improving fore- cast skill relative to SFNO on WeatherBench. Matched ablations show that CG coupling accelerates convergence, while SSHCG keeps the coupling to about 10% of a training step at the operating bandwidth, making exact equivariant coupling practical at forecasting scale