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

arXiv: https://arxiv.org/abs/2501.19374 · HTML: https://mlsanity.com/paper/2501.19374/

## Key facts

- Cited by 28 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 38 references that are in the map.
- First posted on arXiv 31 Jan 2025.

## Builds on (most cited first, 11 of 11)

- [WeatherBench 2: A benchmark for the next generation of data-driven global weather models](https://mlsanity.com/paper/2308.15560/) (2023)
- [Spherical Fourier Neural Operators: Learning Stable Dynamics on the Sphere](https://mlsanity.com/paper/2306.03838/) (2023)
- [PDE-Refiner: Achieving Accurate Long Rollouts with Neural PDE Solvers](https://mlsanity.com/paper/2308.05732/) (2023)
- [FourCastNet: Accelerating Global High-Resolution Weather Forecasting using Adaptive Fourier Neural Operators](https://mlsanity.com/paper/2208.05419/) (2022)
- [Forecasting Global Weather with Graph Neural Networks](https://mlsanity.com/paper/2202.07575/) (2022)
- [Improving data-driven global weather prediction using deep convolutional neural networks on a cubed sphere](https://mlsanity.com/paper/2003.11927/) (2020)
- [ArchesWeather & ArchesWeatherGen: a deterministic and generative model for efficient ML weather forecasting](https://mlsanity.com/paper/2412.12971/) (2024)
- [A Practical Probabilistic Benchmark for AI Weather Models](https://mlsanity.com/paper/2401.15305/) (2024)
- [Leveraging data-driven weather models for improving numerical weather prediction skill through large-scale spectral nudging](https://mlsanity.com/paper/2407.06100/) (2024)
- [Efficient fine-tuning of 37-level GraphCast with the Canadian global deterministic analysis](https://mlsanity.com/paper/2408.14587/) (2024)
- [Can we integrate spatial verification methods into neural-network loss functions for atmospheric science?](https://mlsanity.com/paper/2203.11141/) (2022)

## Built on by (most cited first, 28 of 28)

- [FourCastNet 3: A geometric approach to probabilistic machine-learning weather forecasting at scale](https://mlsanity.com/paper/2507.12144/) (2025)
- [High-Resolution Probabilistic Data-Driven Weather Modeling with a Stretched-Grid](https://mlsanity.com/paper/2511.23043/) (2025)
- [HRRRCast: a data-driven emulator for regional weather forecasting at convection allowing scales](https://mlsanity.com/paper/2507.05658/) (2025)
- [STCast: Adaptive Boundary Alignment for Global and Regional Weather Forecasting](https://mlsanity.com/paper/2509.25210/) (2025)
- [FastNet: Improving the physical consistency of machine-learning weather prediction models through loss function design](https://mlsanity.com/paper/2509.17601/) (2025)
- [(Sparse) Attention to the Details: Preserving Spectral Fidelity in ML-based Weather Forecasting Models](https://mlsanity.com/paper/2604.16429/) (2026)
- [PhysMetrics.Weather: An Evaluation Framework for Physical Consistency in ML Weather Models](https://mlsanity.com/paper/2606.10642/) (2026)
- [A comparison of stretched-grid and limited-area modelling for data-driven regional weather forecasting](https://mlsanity.com/paper/2507.18378/) (2025)
- [Learning to Advect: A Neural Semi-Lagrangian Architecture for Weather Forecasting](https://mlsanity.com/paper/2601.21151/) (2026)
- [WIND: Weather Inverse Diffusion for Zero-Shot Atmospheric Modeling](https://mlsanity.com/paper/2602.03924/) (2026)
- [On Neural Scaling Laws for Weather Emulation through Continual Training](https://mlsanity.com/paper/2603.25687/) (2026)
- [Multi-Scale Wavelet Transformers for Operator Learning of Dynamical Systems](https://mlsanity.com/paper/2602.01486/) (2026)
- [Toward generative machine learning for boosting ensembles of climate simulations](https://mlsanity.com/paper/2602.06287/) (2026)
- [Revealing the Potential of Learnable Perturbation Ensemble Forecast Model for Tropical Cyclone Prediction](https://mlsanity.com/paper/2510.23794/) (2025)
- [MeltwaterBench: Deep learning for spatiotemporal downscaling of surface meltwater](https://mlsanity.com/paper/2512.12142/) (2025)
- [WADEPre: A Wavelet-based Decomposition Model for Extreme Precipitation Nowcasting with Multi-Scale Learning](https://mlsanity.com/paper/2602.02096/) (2026)
- [The Rise of AI in Weather and Climate Information and its Impact on Global Inequality](https://mlsanity.com/paper/2603.05710/) (2026)
- [3DTCR: Vortex-following generative reconstruction improves tropical cyclone intensity prediction from global AI weather forecasts](https://mlsanity.com/paper/2603.13049/) (2026)
- [FuXiWeather2: Learning accurate atmospheric state estimation for operational global weather forecasting](https://mlsanity.com/paper/2603.15358/) (2026)
- [Improving Ensemble Forecasts of Abnormally Deflecting Tropical Cyclones with Fused Atmosphere-Ocean-Terrain Data](https://mlsanity.com/paper/2603.29200/) (2026)
- [ThousandWorlds: A benchmark for climate emulation of potentially habitable exoplanets](https://mlsanity.com/paper/2606.18338/) (2026)
- [Do AI weather models miss extremes?](https://mlsanity.com/paper/2608.09972/) (2026)
- [Diffusion-Based Refinement for Kilometer-Scale Probabilistic Precipitation Nowcasting](https://mlsanity.com/paper/2608.30205/) (2026)
- [Neptune: An AI model for Global Ocean Subseasonal Prediction](https://mlsanity.com/paper/2609.08606/) (2026)
- [Aries: A Proprietary Medium-Range Weather Prediction Model for the Energy Industry](https://mlsanity.com/paper/2609.13292/) (2026)
- [Lightweight Probabilistic Downscaling from a Deterministic Base Model](https://mlsanity.com/paper/2609.29383/) (2026)
- [Varda-single-1.0: deterministic data-driven weather forecasting at 1 km resolution over Switzerland's complex topography](https://mlsanity.com/paper/2610.01835/) (2026)
- [Xaurora: Generative Weather Forecasting with Denoising Stochastic Interpolants from a Foundation Model Prior](https://mlsanity.com/paper/2610.06509/) (2026)

## Abstract (arXiv)

Recent advancements in data-driven weather forecasting models have delivered deterministic models that outperform the leading operational forecast systems based on traditional, physics-based models. However, these data-driven models are typically trained with a mean squared error loss function, which causes smoothing of fine scales through a "double penalty" effect. We develop a simple, parameter-free modification to this loss function that avoids this problem by separating the loss attributable to decorrelation from the loss attributable to spectral amplitude errors. Fine-tuning the GraphCast model with this new loss function results in sharp deterministic weather forecasts, an increase of the model's effective resolution from 1,250km to 160km, improvements to ensemble spread, and improvements to predictions of tropical cyclone strength and surface wind extremes.