Total Uncertainty Quantification in Inverse PDE Solutions Obtained with Reduced-Order Deep Learning Surrogate Models

TL;DR

This paper introduces a Bayesian approach to quantify total uncertainty in inverse PDE solutions using reduced-order deep learning surrogates, effectively capturing observation, PDE, and surrogate model uncertainties.

Contribution

It presents a novel method that combines surrogate models with Bayesian sampling to accurately quantify uncertainty in inverse PDE problems.

Findings

The proposed method yields comparable or better uncertainty estimates than existing ensemble methods.

Deep ensembling tends to underestimate uncertainty and provides less informative posteriors.

The approach is validated on a groundwater flow model with promising results.

Abstract

We propose an approximate Bayesian method for quantifying the total uncertainty in inverse PDE solutions obtained with machine learning surrogate models, including operator learning models. The proposed method accounts for uncertainty in the observations and PDE and surrogate models. First, we use the surrogate model to formulate a minimization problem in the reduced space for the maximum a posteriori (MAP) inverse solution. Then, we randomize the MAP objective function and obtain samples of the posterior distribution by minimizing different realizations of the objective function. We test the proposed framework by comparing it with the iterative ensemble smoother and deep ensembling methods for a non-linear diffusion equation with an unknown space-dependent diffusion coefficient. Among other problems, this equation describes groundwater flow in an unconfined aquifer. Depending on the training dataset and ensemble sizes, the proposed method provides similar or more descriptive posteriors of the parameters and states than the iterative ensemble smoother method. Deep ensembling underestimates uncertainty and provides less informative posteriors than the other two methods.