Total Uncertainty Quantification in Inverse PDE Solutions Obtained with Reduced-Order Deep Learning Surrogate Models
This work addresses uncertainty quantification for inverse PDE problems in fields like groundwater flow, offering a method that improves upon existing techniques, though it appears incremental as it builds on prior Bayesian and ensemble methods.
The authors tackled the problem of quantifying total uncertainty in inverse PDE solutions using deep learning surrogate models, proposing an approximate Bayesian method that accounts for observation, PDE, and surrogate uncertainties, and found it provides similar or more descriptive posteriors than iterative ensemble smoother, while deep ensembling underestimates uncertainty.
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.