Standard approaches for uncertainty quantification in deep learning and physics-informed learning have persistent limitations. Indicatively, strong assumptions regarding the data likelihood are required, the performance highly depends on the selection of priors, and the posterior can be sampled only approximately, which leads to poor approximations because of the associated computational cost. This paper introduces and studies confidence interval (CI) estimation for deterministic partial differential equations as a novel problem. That is, to propagate confidence, in the form of CIs, from data locations to the entire domain with probabilistic guarantees. We propose a method, termed Physics-Informed Confidence Propagation (PICProp), based on bi-level optimization to compute a valid CI without making heavy assumptions. We provide a theorem regarding the validity of our method, and computational experiments, where the focus is on physics-informed learning.

翻译：深度学习与物理信息学习中标准的不确定性量化方法存在固有局限性。例如，需要对数据似然函数施加强假设，模型性能高度依赖先验分布的选择，且后验分布仅能近似采样，这导致因相关计算成本过高而产生较差的近似效果。本文首次提出并研究了确定性偏微分方程置信区间估计这一新问题，即从数据位置出发，将置信区间形式的置信度以概率保证传播至整个计算域。我们基于双层优化提出了一种名为物理信息置信传播（PICProp）的方法，无需强假设即可计算有效的置信区间。本文给出了关于方法有效性的理论证明，并开展了以物理信息学习为核心的计算实验。