Existing work in scientific machine learning (SciML) has shown that data-driven learning of solution operators can provide a fast approximate alternative to classical numerical partial differential equation (PDE) solvers. Of these, Neural Operators (NOs) have emerged as particularly promising. We observe that several uncertainty quantification (UQ) methods for NOs fail for test inputs that are even moderately out-of-domain (OOD), even when the model approximates the solution well for in-domain tasks. To address this limitation, we show that ensembling several NOs can identify high-error regions and provide good uncertainty estimates that are well-correlated with prediction errors. Based on this, we propose a cost-effective alternative, DiverseNO, that mimics the properties of the ensemble by encouraging diverse predictions from its multiple heads in the last feed-forward layer. We then introduce Operator-ProbConserv, a method that uses these well-calibrated UQ estimates within the ProbConserv framework to update the model. Our empirical results show that Operator-ProbConserv enhances OOD model performance for a variety of challenging PDE problems and satisfies physical constraints such as conservation laws.

翻译：现有科学机器学习（SciML）研究表明，基于数据驱动的解算子学习可作为经典数值偏微分方程求解器的快速近似替代方案。其中，神经算子（NOs）已展现出显著前景。我们观察到，即使模型在域内任务中能良好逼近解，多种针对NOs的不确定性量化（UQ）方法仍会在轻度域外测试输入时失效。为解决此局限，我们证明集成多个NOs可识别高误差区域，并提供与预测误差高度相关的优质不确定性估计。基于此，我们提出经济高效的替代方案DiverseNO——通过鼓励最后前馈层多个头部产生多样化预测来模拟集成特性。随后引入Operator-ProbConserv方法，在ProbConserv框架内利用这些校准良好的UQ估计更新模型。实验结果表明，Operator-ProbConserv能提升多种挑战性PDE问题的域外模型性能，并满足守恒律等物理约束。