Gauge-Equivariant Spherical U-Nets on HEALPix for Global SST Forecasting: Encoding local orientation on the sphere

Many climate variables are naturally defined on the sphere and exhibit strong anisotropy and directionality (e.g., fronts, jets, boundary currents). Yet most deep-learning forecasting models still rely on planar projections and Euclidean convolutions, which introduce geometric distortions and artificial discontinuities. Graph-based spherical models alleviate some of these issues, but typically remain isotropic and do not explicitly represent local orientation, a key ingredient to model directional transport-like patterns.

Here we introduce and evaluate a gauge-equivariant spherical U-Net implemented directly on the HEALPix grid, designed to encode local orientation consistently across the sphere. Our approach leverages gauge-equivariant convolutions that transform predictably under changes of local reference frame, allowing the network to learn directional filters while preserving spherical geometry. This provides a principled alternative to both planar U-Nets (with longitude-periodic padding) and graph U-Nets, and addresses a core limitation of most spherical models: the lack of explicit orientation handling. This work benchmarks this model against two strong baselines: a planar U-Net with longitude-periodic padding and a spherical graph U-Net defined on the same HEALPix discretization.

We apply this architecture to multi-horizon forecasting of global sea-surface temperature (SST) anomalies at NSIDE=32, using a controlled experimental design with matched training protocols and comparable parameter budgets, with emphasis on low-capacity regimes relevant to data-limited climate settings (≈30–40 years of monthly observations). We report quantitative metrics across horizons and analyze qualitative error modes, showing how gauge-equivariant spherical convolutions mitigate projection artefacts while enabling orientation-aware feature extraction on the sphere. Our results highlight when and why encoding orientation through gauge equivariance provides added value beyond “spherical-but-isotropic” baselines, and offer practical guidance for deploying spherical equivariant models in climate forecasting pipelines.