For many important problems it is essential to be able to accurately quantify the statistics of extremes for specific quantities of interest, such as extreme atmospheric weather events or ocean-related quantities. While there are many classical approaches to perform such modeling tasks, recent interest has been increasing in the usage of generative models trained on available data. Despite the sporadic success of such methods, it is not clear for what systems or datasets a system-agnostic generative AI tool is capable of generating previously ‘unseen’ extreme events in a manner that accurately extrapolates the tails for the observable of interest. Here, we propose an apriori criterion, which based on the geometry of the training dataset, it can predict whether a generative AI tool will be able to extrapolate the tails, i.e. generate previously unseen extreme events. The idea is to quantify whether existing extreme events lie in the interior of the dataset or its boundary. In the former case it is shown that generative AI tools can work in an ‘interpolation’ mode and generate new extreme events. On the other hand, if the topology of the dataset is such that extremes live in the boundary of the domain then the generative AI algorithm needs to operate in an extrapolation mode which does not lead to accurate results. We illustrate our findings on a specific class of Diffusion Models (DMs) called Denoising Diffusion Probabilistic Models (DDPMs) and we test on three datasets, a simple on-hyperball dataset following a Weibull distribution for the radii of the data points of dimensionality 2⋅103, a dataset sampled from the so-called Majda–McLaughlin–Tabak Wave Model (MMT), of dimensionality 8.1⋅103 and a dataset consisting of Lagrangian turbulence trajectories, of dimensionality 2⋅103.

中文翻译：

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扩散模型可以捕获极端事件统计数据吗？


对于许多重要问题，必须能够准确量化特定感兴趣量的极端事件统计数据，例如极端大气天气事件或与海洋相关的量。虽然有许多经典方法可以执行此类建模任务，但最近人们对使用基于可用数据训练的生成模型的兴趣越来越大。尽管此类方法取得了零星的成功，但目前尚不清楚与系统无关的生成式 AI 工具能够以准确推断感兴趣的可观察对象的尾部的方式生成以前“看不见”的极端事件。在这里，我们提出了一个先验标准，该标准基于训练数据集的几何形状，它可以预测生成式 AI 工具是否能够推断尾部，即生成以前从未见过的极端事件。这个想法是量化现有的极端事件是位于数据集的内部还是其边界。在前一种情况下，表明生成式 AI 工具可以在 “插值” 模式下工作并生成新的极端事件。另一方面，如果数据集的拓扑结构是这样的，极端值存在于域的边界中，那么生成式 AI 算法需要在外推模式下运行，这不会产生准确的结果。我们在一类称为降噪扩散概率模型 （DDPM） 的特定扩散模型 （DM） 上展示了我们的发现，并在三个数据集上进行了测试，一个简单的 on-hyperball 数据集，遵循维数 2⋅103 数据点半径的 Weibull 分布，一个从维数为 8 的所谓 Majda-McLaughlin-Tabak 波模型 （MMT） 中采样的数据集。1⋅103 和一个由拉格朗日湍流轨迹组成的数据集，维度为 2⋅103。