The solution of PDEs in decision-making tasks is increasingly being undertaken with the help of neural operator surrogate models due to the need for repeated evaluation. Such methods, while significantly more computationally favorable compared to their numerical counterparts, fail to provide any calibrated notions of uncertainty in their predictions. Current methods approach this deficiency typically with ensembling or Bayesian posterior estimation. However, these approaches either require distributional assumptions that fail to hold in practice or lack practical scalability, limiting their applications in practice. We, therefore, propose a novel application of conformal prediction to produce distribution-free uncertainty quantification over the function spaces mapped by neural operators. We then demonstrate how such prediction regions enable a formal regret characterization if leveraged in downstream robust decision-making tasks. We further demonstrate how such posited robust decision-making tasks can be efficiently solved using an infinite-dimensional generalization of Danskin's Theorem and calculus of variations and empirically demonstrate the superior performance of our proposed method over more restrictive modeling paradigms, such as Gaussian Processes, across several engineering tasks.

翻译：在决策任务中求解偏微分方程时，由于需要重复计算，越来越多地借助神经算子代理模型来完成。这类方法虽然计算效率显著优于数值方法，却无法提供经过校准的预测不确定性度量。现有方法通常通过集成学习或贝叶斯后验估计来弥补这一缺陷，但这些方法要么依赖于实践中难以满足的分布假设，要么缺乏实际可扩展性，限制了其实际应用。为此，我们提出一种新颖的共形预测应用方法，为神经算子映射的函数空间提供无需分布假设的不确定性量化。我们进一步证明，若将此类预测区域应用于下游鲁棒决策任务，可实现形式化的遗憾值表征。通过丹斯金定理的无限维推广及变分法，我们展示了如何高效求解此类鲁棒决策问题，并在多个工程任务中通过实证验证了所提方法相对于高斯过程等限制性建模范式的优越性能。