Spherical Fourier Neural Operators: Learning Stable Dynamics on the Sphere

TL;DR

This work extends Fourier Neural Operators to spherical geometry by deriving an $SO(3)$-equivariant spherical convolution via the Spherical Harmonic Transform. The resulting Spherical Fourier Neural Operator (SFNO) combines geometry-respecting spectral processing with grid-invariant learning, enabling stable, year-long autoregressive rollouts for geophysical dynamics and efficient climate forecasting. Empirical results on the rotating shallow-water equations and ERA5 data show SFNO achieves competitive short-term forecast skill while dramatically improving long-range stability and speed relative to FFT-based methods. The approach promises impactful applications for ML-driven climate modeling and digital twins by delivering physically plausible, scalable, and fast sphere-based simulations.

Abstract

Fourier Neural Operators (FNOs) have proven to be an efficient and effective method for resolution-independent operator learning in a broad variety of application areas across scientific machine learning. A key reason for their success is their ability to accurately model long-range dependencies in spatio-temporal data by learning global convolutions in a computationally efficient manner. To this end, FNOs rely on the discrete Fourier transform (DFT), however, DFTs cause visual and spectral artifacts as well as pronounced dissipation when learning operators in spherical coordinates since they incorrectly assume a flat geometry. To overcome this limitation, we generalize FNOs on the sphere, introducing Spherical FNOs (SFNOs) for learning operators on spherical geometries. We apply SFNOs to forecasting atmospheric dynamics, and demonstrate stable auto\-regressive rollouts for a year of simulated time (1,460 steps), while retaining physically plausible dynamics. The SFNO has important implications for machine learning-based simulation of climate dynamics that could eventually help accelerate our response to climate change.

Figures (6)

• Figure 1: Qualitative comparison of temperature predictions (t850) over Antarctica at 4380h (730 autoregressive steps). The SFNO shows no visible artifacts even after six-month-long rollouts. Models which use FFT and do not incorporate spherical geometry are not stable for long rollouts compared to SFNO. The AFNO model breaks down early and shows large visible artifacts everywhere. In the non-linear FNO model, artifacts are less pronounced but increase in magnitude with time and towards the poles.
• Figure 2: The structure of a single SFNO block. Multi-layer perceptrons (MLPs) act point-wise in the spatial domain and allow for channel mixing. The generalized Fourier transform $\mathcal{F}$ and its inverse $\mathcal{F}^{-1}$ allow for the learning of long-range spatial dependencies. $\kappa$ is a learned filter, which is applied linearly to the frequency components.
• Figure 3: Diagram of the overall SFNO architecure. Encoder and decoder MLPs inflate the channel dimension. A learned position embedding is added in cases where position-dependent information should be learned by the network. At the core lie $N$ SFNO blocks, where the first and last blocks perform up- or down-scaling. A skip connection is added for autoregressive maps close to the identity.
• Figure 4: Solutions to the Shallow Water Equations on the rotating Sphere predicted by SFNO and FNO architectures in comparison to the ground truth solution computed using a classical spectral solver. Plots depict the geopotential height at 5 and 10 hours, corresponding to 5 and 10 autoregressive steps respectively. The view is centered on the south pole to highlight artifacts arising due to the non-geometrical treatment of the sphere.
• Figure 5: Year-long rollout (1,450 autoregressive steps) of absolute wind speed 10m above the surface depicting stable behavior over exceptionally long timescales for an ML model, which has important implications for ML-based climate modeling. In contrast, the FFT-based architecture has spurious waves and artifacts and excessive diffusion. SFNO faithfully captures the dynamics of weather within the predictability horizon of two weeks and shows physically- and statistically-consistent behaviour over longer timescales up to a year.