Total uncertainty quantification in inverse solutions with deep learning surrogate models

We propose an approximate Bayesian method for quantifying the total uncertainty in inverse partial differential equation (PDE) solutions obtained with machine learning surrogate models, including operator learning models. The proposed method accounts for uncertainty in the observations, 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 nonlinear diffusion equation with an unknown space-dependent diffusion coefficient. Among other applications, this equation describes the flow of groundwater 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. Our results show that, despite inherent uncertainty, surrogate models can be used for parameter and state estimation as an alternative to the inverse methods relying on (more accurate) numerical PDE solvers.