Neural operators aim to approximate the solution operator of a system of differential equations purely from data. They have shown immense success in modeling complex dynamical systems across various domains. However, the occurrence of uncertainties inherent in both model and data has so far rarely been taken into account\textemdash{}a critical limitation in complex, chaotic systems such as weather forecasting. In this paper, we introduce the probabilistic neural operator (PNO), a framework for learning probability distributions over the output function space of neural operators. PNO extends neural operators with generative modeling based on strictly proper scoring rules, integrating uncertainty information directly into the training process. We provide a theoretical justification for the approach and demonstrate improved performance in quantifying uncertainty across different domains and with respect to different baselines. Furthermore, PNO requires minimal adjustment to existing architectures, shows improved performance for most probabilistic prediction tasks, and leads to well-calibrated predictive distributions and adequate uncertainty representations even for long dynamical trajectories. Implementing our approach into large-scale models for physical applications can lead to improvements in corresponding uncertainty quantification and extreme event identification, ultimately leading to a deeper understanding of the prediction of such surrogate models.

翻译：神经算子旨在仅从数据中逼近微分方程组的解算子。它们在建模跨多个领域的复杂动力系统方面已展现出巨大成功。然而，模型与数据中固有的不确定性迄今鲜少被纳入考量——这在天气预报等复杂混沌系统中是一个关键局限。本文提出概率神经算子（PNO），该框架用于学习神经算子输出函数空间上的概率分布。PNO基于严格恰当评分规则，通过生成式建模扩展了神经算子，将不确定性信息直接整合到训练过程中。我们为此方法提供了理论依据，并展示了其在跨领域及不同基线对比中量化不确定性的性能提升。此外，PNO仅需对现有架构进行最小调整，在多数概率预测任务中表现出更优性能，即使对于长时程动力学轨迹也能产生校准良好的预测分布与充分的不确定性表征。将本方法应用于物理领域的大规模模型，可提升相应不确定性量化与极端事件识别的能力，最终深化对此类代理模型预测机理的理解。