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：科学机器学习领域现有研究表明，数据驱动的解算子学习可为经典数值偏微分方程求解器提供快速的近似替代方案。其中，神经算子展现出特别广阔的前景。我们观察到，即使模型在内域任务中能较好逼近解，当前多种针对神经算子的不确定性量化方法在面对中度外域测试输入时仍会失效。为突破此局限，我们证明集成多个神经算子可识别高误差区域，并提供与预测误差高度相关的优质不确定性估计。基于此，我们提出一种经济高效的替代方案——DiverseNO，该模型通过激励其最后前馈层中多头结构的多样性预测来模拟集成特性。随后，我们在ProbConserv框架内引入Operator-ProbConserv方法，利用这些经良好校准的不确定性量化估计来更新模型。实证结果表明，Operator-ProbConserv能提升模型在多种挑战性偏微分方程问题中的外域性能，并满足守恒律等物理约束。