Improved Uncertainty Quantification in Physics-Informed Neural Networks Using Error Bounds and Solution Bundles

TL;DR

This paper enhances uncertainty quantification in Physics-Informed Neural Networks by integrating error bounds and solution bundles, enabling more reliable solutions and parameter estimation in differential equations.

Contribution

It introduces a two-step Bayesian training method for PINNs that incorporates error bounds to improve uncertainty estimates and applies this to inverse problems in cosmology.

Findings

Improved uncertainty estimates over PINN solutions.

Enhanced parameter estimation accuracy in inverse problems.

Effective application to cosmological differential equations.

Abstract

Physics-Informed Neural Networks (PINNs) have been widely used to obtain solutions to various physical phenomena modeled as Differential Equations. As PINNs are not naturally equipped with mechanisms for Uncertainty Quantification, some work has been done to quantify the different uncertainties that arise when dealing with PINNs. In this paper, we use a two-step procedure to train Bayesian Neural Networks that provide uncertainties over the solutions to differential equation systems provided by PINNs. We use available error bounds over PINNs to formulate a heteroscedastic variance that improves the uncertainty estimation. Furthermore, we solve forward problems and utilize the obtained uncertainties when doing parameter estimation in inverse problems in cosmology.