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 ignore the highly correlated data structures present in weather and atmospheric systems. This work introduces the signature kernel scoring rule, grounded in rough path theory, 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：现代天气预报已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），这些规则忽略了天气与大气系统中存在的高度相关数据结构。本研究基于粗糙路径理论提出签名核评分规则，将天气变量重构为连续路径，通过迭代积分编码时空依赖性。该核评分规则通过路径增广技术保证唯一性，被验证为严格恰当，为预报验证与模型训练提供了理论稳健的度量标准。在WeatherBench 2模型上进行的天气评分卡实证评估表明，签名核评分规则具有高判别力及捕捉路径依赖相互作用的独特能力。继先前成功实现无对抗概率训练的演示后，我们利用ERA5再分析天气数据，采用预测-序列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，基于签名核的训练方法在长达十五个时间步的预测路径上优于气候学基准。