The need to rapidly solve PDEs in engineering design workflows has spurred the rise of neural surrogate models. In particular, neural operator models provide a discretization-invariant surrogate by retaining the infinite-dimensional, functional form of their arguments. Despite improved throughput, such methods lack guarantees on accuracy, unlike classical numerical PDE solvers. Optimizing engineering designs under these potentially miscalibrated surrogates thus runs the risk of producing designs that perform poorly upon deployment. In a similar vein, there is growing interest in automated decision-making under black-box predictors in the finite-dimensional setting, where a similar risk of suboptimality exists under poorly calibrated models. For this reason, methods have emerged that produce adversarially robust decisions under uncertainty estimates of the upstream model. One such framework leverages conformal prediction, a distribution-free post-hoc uncertainty quantification method, to provide these estimates due to its natural pairing with black-box predictors. We herein extend this line of conformally robust decision-making to infinite-dimensional function spaces. We first extend the typical conformal prediction guarantees over finite-dimensional spaces to infinite-dimensional Sobolev spaces. We then demonstrate how such uncertainty can be leveraged to robustly formulate engineering design tasks and characterize the suboptimality of the resulting robust optimal designs. We then empirically demonstrate the generality of our functional conformal coverage method across a diverse collection of PDEs, including the Poisson and heat equations, and showcase the significant improvement of such robust design in a quantum state discrimination task.

翻译：工程设计中快速求解偏微分方程的需求推动了神经代理模型的兴起。特别是神经算子模型通过保留其参数的无限维函数形式，提供了离散化不变的代理方法。尽管这些方法提高了计算效率，但与经典数值偏微分方程求解器不同，它们缺乏精度保证。因此，在这些可能未校准的代理模型下进行工程设计优化，存在部署后性能不佳的风险。类似地，在有限维场景下，基于黑盒预测器的自动化决策制定日益受到关注，而在校准不佳的模型下同样存在次优决策风险。为此，涌现出多种方法能在上游模型的不确定性估计下生成对抗性鲁棒决策。其中一个框架利用共形预测——一种分布无关的事后不确定性量化方法——因其与黑盒预测器的天然适配性而提供此类估计。本文将此类共形鲁棒决策框架扩展至无限维函数空间。我们首先将有限维空间上的典型共形预测保证扩展至无限维Sobolev空间，进而论证如何利用此类不确定性鲁棒地构建工程设计任务，并分析所得鲁棒最优设计的次优性特征。最后通过泊松方程、热传导方程等多样化的偏微分方程案例实证验证我们提出的函数共形覆盖方法的普适性，并在量子态判别任务中展示此类鲁棒设计带来的显著性能提升。