Studies in Monte Carlo Inversion and Generative Deep Learning

Abstract

Geophysical inversion is generally concerned with the inference of Earth's structure from data recorded at its surface. The data are generally sparse, incomplete, noisy, and ambiguous. These characteristics make geophysical inversion difficult and lead to challenges, including non-uniqueness and high uncertainties in the obtained models. A way to address some of these difficulties is through ensemble-generating inversion methods such as Markov chain Monte Carlo. The goal is to build a large collection of suitable models that together characterise uncertainty and non-uniqueness of the properties of interest. Though powerful, Monte Carlo methods have significant drawbacks such as long computational times and a large amount of digital outputs. The generation, handling, and distribution of model ensembles is challenging and it is desirable to find alternative methodologies.

In recent years, generative deep learning algorithms (`generative models') have attracted much attention in the machine learning community and beyond. With their help, it is possible to build a parametric representation from a collection of samples. As we discovered during the course of this project, this representation can be used in two major ways: a) it offers a data compression as the original ensemble can be replaced by the parameters of the generative model; and b) it can be used to sample a large number of samples that follow the original distribution. However, deep learning methods lack the theoretical convergence guarantees that make Monte Carlo methods attractive, and so it is beneficial to combine both.

This thesis explores different ways in which Monte Carlo methods can benefit from generative models. First, we demonstrate the feasibility of generative models to reproduce the posterior distribution of a Bayesian inverse problem, leading to large rates of compression and computational acceleration. Second, this approach can be extended to train a generative model 'on-the-fly' and use it as adaptive proposal distribution in the sampling process itself. Third, we propose a new framework in which an autoencoder can be used to learn a surrogate model of both the forward and inverse operators, with perspectives for subsurface monitoring. The thesis ends with an outlook on the many other possible ways in which Monte Carlo methods and generative models may be combined. Show more