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.

在科学机器学习中，研究人员发现利用数据驱动的解算器学习可以提供快速的近似解决方案，作为传统数值偏微分方程求解器的替代方法。本研究通过聚合多个神经操作器，识别高误差区域并提供与预测误差相关的良好不确定性估计，从而解决了现有神经操作器方法在域外测试输入上的不确定性量化问题。基于这一结果，提出了一种经济高效的方法DiverseNO，通过鼓励多个神经操作器在最后的前馈层中产生不同预测结果来模拟集成的特性。同时，引入了Operator-ProbConserv方法，将这些经过良好校准的不确定性估计嵌入ProbConserv框架以更新模型。实验结果表明，Operator-ProbConserv提高了挑战性偏微分方程问题的域外模型性能，并满足了物理约束条件如守恒定律。

利用不确定性量化来描述和改善偏微分方程的领域外学习