  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.
