Fixing the Double Penalty in Data-Driven Weather Forecasting Through a Modified Spherical Harmonic Loss Function

Beginning in 2023, the release of data-driven atmospheric forecasting models powered by deep neural network architectures began a revolution in medium-range weather forecasting, with some commenters [Bauer, 2024] anticipating that data-driven forecasting will soon supplant traditional numerical weather prediction (NWP) systems in all operational contexts. GraphCast [Lam et al., 2023], FourCastNet [Kurth et al., 2023], and Pangu-Weather [Bi et al., 2023] demonstrated forecast skill superior to that of the high-resolution forecast system (IFS) of the European Centre for Medium Range Weather Forecasts (ECMWF) at lead times (forecast lengths) up to 10 days. Since the publication of these models, the field has been joined by many others, including the Artificial Intelligence Forecasting System (AIFS) developed by ECMWF itself [Lang et al., 2024a]. From the standpoint of machine learning, atmospheric forecasting is a large-scale generative problem comparable to predicting the next frame of a video. As a typical example, the version of the GraphCast model deployed experimentally by the National Oceanic and Atmospheric Administration (NOAA) [NOAA, 2024] predicts the 6-hour forecast for six atmospheric variables at each of 13 vertical levels plus five surface variables, on a latitude/longitude grid, for about 86 million output degrees of freedom in aggregate. GraphCast takes two time-levels as input, so the input for this model has about 170 million degrees of freedom. These first-generation data-driven weather models generally act as deterministic forecast systems, where each unique initial condition is mapped to a single forecast and verified against a "ground truth" from a data analysis system. The ERA5 atmospheric reanalysis [Hersbach et al., 2020] of ECWMF is most often used as the source of initial and verifying data for these forecast systems owing to its high quality and consistent behaviour from 1979 to present.