PICProp: Physics-Informed Confidence Propagation for Uncertainty Quantification
This addresses uncertainty quantification challenges in physics-informed learning, offering a method with probabilistic guarantees, though it appears incremental as it builds on existing bi-level optimization approaches.
The paper tackles the problem of uncertainty quantification in physics-informed learning by introducing confidence interval estimation for deterministic partial differential equations, proposing PICProp to propagate confidence from data to the entire domain with probabilistic guarantees, and demonstrating its validity through computational experiments.
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.