Low-rank clebsch-gordan coupling improves spherical weather forecasts
Low-Rank Clebsch--Gordan Coupling for Equivariant Spherical Forecasting
Machine Learning
Summary
Forecasting global weather patterns often involves working with data on spherical surfaces, but existing methods can break symmetry when predicting changes, leading to errors. The paper introduces a new technique that keeps the math perfectly aligned with the Earth's rotation, reducing rotational errors dramatically and improving forecast accuracy. The authors also developed efficient software to make this method practical for large-scale weather prediction.
What this means in practice
- •For weather forecasters: Produce more precise global weather predictions by using a new method that preserves Earth's rotational symmetry in forecast models.
- •For climate modelers: Integrate exact equivariant coupling to reduce errors in spherical data simulations, enhancing large-scale climate model fidelity.
Authors
Yaojun Li, Zhixuan Zhao, Dongzhe Zheng, Christine Allen-Blanchette
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.