Exploring Uncertainty Quantification via Deep Learning in Inverse Problems

Abstract

With advancements in geophysics, there is a concerted effort to identify subsurface properties without invasive measures. Current methods allow the estimation of these properties based on surface measurements, and estimating these properties is known as an inverse problem. However, solving an inverse problem requires knowledge of the associated forward problem, which models the generation of these measurements from subsurface properties. Diverse methodologies, including those leveraging deep learning, are employed to tackle both inverse and forward problems. In this dissertation, we propose three neural network methodologies aimed at reducing the training time or at obtaining more accurate estimations: (a) two novel sampling techniques. First, a gradient-based resampling method that generates subsamples based on cumulative probability functions derived from data gradients, enhancing stochastic gradient descent compared to uniform subsampling. Second, an adaptive sampling method that generates samples in regions where neural network fitting exceeds a predefined maximum error criterion, facilitating training with a smaller initial database that expands as needed; (b) a Memory-based Monte Carlo Quadrature Rule for Deep Ritz to reduce errors in the energy estimation. Given a memory parameter, this technique significantly reduces the variance of the energy estimation with a computational cost comparable to the Deep Ritz method; (c) a method to train Galerkin discretizations to efficiently learn quantities of interest of solutions to a parametric partial differential equation. The central component in our approach is an efficient neuralnetwork- weighted Minimal-Residual formulation, which, after training, provides Galerkin-based approximations in standard discrete spaces that accurately estimate quantities of interest, regardless of the coarseness of the discrete space.