PICProp: Physics-Informed Confidence Propagation for Uncertainty Quantification

TL;DR

This paper introduces PICProp, a novel physics-informed method for propagating confidence intervals in PDE solutions, addressing limitations of existing uncertainty quantification approaches in deep learning.

Contribution

The paper presents PICProp, a bi-level optimization-based approach for valid confidence interval estimation in PDEs without heavy assumptions, advancing uncertainty quantification in physics-informed learning.

Findings

PICProp provides valid confidence intervals with probabilistic guarantees.

The method outperforms traditional approaches in computational experiments.

It enables uncertainty propagation across the entire domain of PDEs.

Abstract

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.