Low-Rank Clebsch--Gordan Coupling for Equivariant Spherical Forecasting

Abstract

Spherical-harmonic neural operators provide an efficient representation for global forecasting, but their discretized pointwise nonlinearities can break rotational equivariance through aliasing. Clebsch-Gordan (CG) tensor products provide exactly equivariant interactions between spherical-harmonic representations, but their computational 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 practical, 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 forecast skill relative to SFNO on WeatherBench. Matched ablations show that CG coupling accelerates convergence, while SSHCG keeps the coupling to about 10 percent of a training step at the operating bandwidth, making exact equivariant coupling practical at forecasting scale.