A multi-scale loss formulation for learning a probabilistic model with proper score optimisation

Over the last few years, probabilistic machine-learned weather prediction models have begun to rival physics-based numerical weather prediction (NWP) systems in skill (Kochkov et al., 2024; Price et al., 2023; Lang et al., 2024c,b). AIFS-CRPS (Lang et al., 2024b) is based on the machined-learned weather forecasting model AIFS (Lang et al., 2024a), developed at the European Centre for Medium-Range Weather Forecasts (ECMWF). AIFS-CRPS produces skilful predictions by directly optimising a score based on a proper scoring rule, the almost fair continuous ranked probability score (afCRPS). The model learns to shape Gaussian noise to represent uncertainty in the atmospheric state and achieves ensemble forecast skill that is competitive with, or superior to, the physics-based IFS ensemble (Molteni et al., 1996; Leutbecher and Palmer, 2008; Lang et al., 2021, 2023) at ECMWF. The afCRPS loss function used in AIFS-CRPS is computed point-wise on the full output field. However, atmospheric processes are inherently multi-scale, and different scales contribute to a different degree to the loss function.