With the increased prevalence of neural operators being used to provide rapid solutions to partial differential equations (PDEs), understanding the accuracy of model predictions and the associated error levels is necessary for deploying reliable surrogate models in scientific applications. Existing uncertainty quantification (UQ) frameworks employ ensembles or Bayesian methods, which can incur substantial computational costs during both training and inference. We propose a lightweight predictive UQ method tailored for Deep operator networks (DeepONets) that also generalizes to other operator networks. Numerical experiments on linear and nonlinear PDEs demonstrate that the framework's uncertainty estimates are unbiased and provide accurate out-of-distribution uncertainty predictions with a sufficiently large training dataset. Our framework provides fast inference and uncertainty estimates that can efficiently drive outer-loop analyses that would be prohibitively expensive with conventional solvers. We demonstrate how predictive uncertainties can be used in the context of Bayesian optimization and active learning problems to yield improvements in accuracy and data-efficiency for outer-loop optimization procedures. In the active learning setup, we extend the framework to Fourier Neural Operators (FNO) and describe a generalized method for other operator networks. To enable real-time deployment, we introduce an inference strategy based on precomputed trunk outputs and a sparse placement matrix, reducing evaluation time by more than a factor of five. Our method provides a practical route to uncertainty-aware operator learning in time-sensitive settings.

翻译：随着神经算子被日益广泛地用于为偏微分方程提供快速解，理解模型预测的准确性及相关误差水平对于在科学应用中部署可靠的代理模型至关重要。现有的不确定性量化框架采用集成方法或贝叶斯方法，这些方法在训练和推理阶段都可能产生巨大的计算成本。我们提出了一种专为深度算子网络量身定制的轻量级预测不确定性量化方法，该方法也能推广到其他算子网络。在线性和非线性偏微分方程上的数值实验表明，该框架的不确定性估计是无偏的，并且在训练数据集足够大的情况下，能提供准确的分布外不确定性预测。我们的框架提供了快速的推理和不确定性估计，能够高效地驱动外循环分析，而使用传统求解器进行此类分析的成本将极其高昂。我们展示了如何将预测不确定性用于贝叶斯优化和主动学习问题中，以提高外循环优化过程的准确性和数据效率。在主动学习设置中，我们将该框架扩展到傅里叶神经算子，并描述了一种适用于其他算子网络的通用方法。为实现实时部署，我们引入了一种基于预计算主干输出和稀疏放置矩阵的推理策略，将评估时间减少了五倍以上。我们的方法为在时间敏感场景下实现具有不确定性感知的算子学习提供了一条实用途径。