Modern weather forecasting has increasingly transitioned from numerical weather prediction (NWP) to data-driven machine learning forecasting techniques. While these new models produce probabilistic forecasts to quantify uncertainty, their training and evaluation may remain hindered by conventional scoring rules, primarily MSE, which are designed for single time point predictions and ignore the highly correlated data structures present in weather behaviour. This work introduces the signature kernel scoring rule to the domain of weather forecasting, which reframes weather variables as continuous paths to encode temporal and spatial dependencies through iterated integrals. Validated as strictly proper through the use of path augmentations to guarantee uniqueness, the signature kernel provides a theoretically robust metric for forecast verification and model training. Empirical evaluations through weather scorecards on WeatherBench 2 models demonstrate the signature kernel scoring rule's high discriminative power and unique capacity to capture path-dependent interactions. Following previous demonstration of successful adversarial-free probabilistic training, we train sliding window generative neural networks using a predictive-sequential scoring rule on ERA5 reanalysis weather data. Using a lightweight model, we demonstrate that signature kernel based training outperforms climatology for forecast paths of up to fifteen timesteps.

翻译：现代天气预报正日益从数值天气预报（NWP）转向数据驱动的机器学习预测技术。尽管这些新模型通过生成概率预测来量化不确定性，但其训练与评估仍受限于传统评分规则（主要是均方误差MSE）——这些规则专为单时间点预测设计，忽略了天气行为中高度相关的数据结构。本研究将签名核评分规则引入天气预报领域，通过迭代积分将天气变量重新表述为连续路径，从而编码时空依赖性。通过路径增广保证唯一性并验证其严格适当性后，签名核为预报验证和模型训练提供了理论稳健的度量标准。基于WeatherBench 2模型的气象评分卡实证评估表明，签名核评分规则具有高区分能力及捕获路径依赖相互作用的独特优势。继先前成功展示无对抗概率训练后，我们利用ERA5再分析气象数据，采用预测序贯评分规则训练滑动窗口生成神经网络。通过轻量级模型验证，基于签名核的训练在长达十五个时间步的预报路径上优于气候学方法。