Quantifying and predicting rare and extreme events persists as a crucial yet challenging task in understanding complex dynamical systems. Many practical challenges arise from the infrequency and severity of these events, including the considerable variance of simple sampling methods and the substantial computational cost of high-fidelity numerical simulations. Numerous data-driven methods have recently been developed to tackle these challenges. However, a typical assumption for the success of these methods is the occurrence of multiple extreme events, either within the training dataset or during the sampling process. This leads to accurate models in regions of quiescent events but with high epistemic uncertainty in regions associated with extremes. To overcome this limitation, we introduce Extreme Event Aware (e2a or eta) or $\eta$-learning which does not assume the existence of extreme events in the available data. $\eta$-learning reduces the uncertainty even in `uncharted' extreme event regions, by enforcing the extreme event statistics of an observable indicative of extremeness during training, which can be available through qualitative arguments or estimated with unlabeled data. This type of statistical regularization results in models that fit the observed data, while enforcing consistency with the prescribed observable statistics, enabling the generation of unprecedented extreme events even when the training data lack extremes therein. Theoretical results based on optimal transport offer a rigorous justification and highlight the optimality of the introduced method. Additionally, extensive numerical experiments illustrate the favorable properties of the $\eta$-learning framework on several prototype problems and real-world precipitation downscaling problems.

翻译：量化与预测罕见极端事件始终是理解复杂动力系统的关键而艰巨的任务。这些事件的低频性与严重性带来了诸多实际挑战，包括简单采样方法的高方差问题以及高保真数值模拟的巨大计算成本。近年来已开发出众多数据驱动方法以应对这些挑战。然而，这些方法成功的一个典型前提是需要在训练数据集或采样过程中出现多次极端事件。这导致模型在平稳事件区域表现准确，但在与极端事件相关的区域存在较高的认知不确定性。为突破这一局限，本文提出极端事件感知学习（e2a或eta），即η-学习，该方法不假设现有数据中存在极端事件。η-学习通过在训练过程中强化表征极端性的可观测量的统计特性（可通过定性分析获取或利用未标注数据估计），有效降低即使在“未知”极端事件区域的不确定性。此类统计正则化使模型既能拟合观测数据，又能保持与预设可观测统计量的一致性，从而即使在训练数据缺乏极端事件的情况下，也能生成前所未有的极端事件。基于最优传输的理论结果为该方法提供了严格证明，并凸显了其最优性。此外，大量数值实验展示了η-学习框架在多个原型问题及实际降水降尺度问题上的优越特性。