Diffusion models are a class of probabilistic generative models that have been widely used as a prior for image processing tasks like text conditional generation and inpainting. We demonstrate that these models can be adapted to make predictions and provide uncertainty quantification for chaotic dynamical systems. In these applications, diffusion models can implicitly represent knowledge about outliers and extreme events; however, querying that knowledge through conditional sampling or measuring probabilities is surprisingly difficult. Existing methods for conditional sampling at inference time seek mainly to enforce the constraints, which is insufficient to match the statistics of the distribution or compute the probability of the chosen events. To achieve these ends, optimally one would use the conditional score function, but its computation is typically intractable. In this work, we develop a probabilistic approximation scheme for the conditional score function which provably converges to the true distribution as the noise level decreases. With this scheme we are able to sample conditionally on nonlinear userdefined events at inference time, and matches data statistics even when sampling from the tails of the distribution.

翻译：扩散模型是一类概率生成模型，已被广泛用作图像处理任务（如文本条件生成与图像修复）的先验。我们证明，这类模型可被调整用于混沌动力系统的预测与不确定性量化。在这些应用中，扩散模型能够隐式表征异常值与极端事件的知识；然而，通过条件采样或概率测量来查询这类知识却异常困难。现有的推理时条件采样方法主要致力于强制执行约束条件，这不足以匹配分布的统计特征或计算所选事件的概率。为实现这些目标，最理想的方法是使用条件得分函数，但其计算通常难以处理。在本研究中，我们开发了一种条件得分函数的概率近似方案，该方案被证明能够随着噪声水平的降低而收敛于真实分布。借助该方案，我们能够在推理时根据非线性用户定义事件进行条件采样，即便从分布尾部采样时，也能匹配数据统计特征。