Efficient data-driven inference and uncertainty quantification for surrogate modeling and inverse problems

In recent years, data-driven methods such as deep neural networks (DNNs) have demonstrated remarkable success across various applications. However, these methods often demand a large amount of data to provide accurate predictions, which may be scarce in scientific and engineering problems. In these applications, however, additional information about the system, including physical, constraint, and statistical information, are frequently available and can be harnessed to enhance estimates and facilitate uncertainty quantification.

This thesis tackles these challenges by investigating several data-driven approaches for inverse and surrogate modeling problems. Specifically, we present novel methods and applications, including Ensemble Kalman inversion for parameter inference and upstream prediction in rainfall-runoff models, Ensemble Kalman inversion for Bayesian Physics-Informed Neural Networks, and nonnegative Gaussian process regression. These approaches capitalize on supplementary information about the systems under study to refine estimates and deliver uncertainty quantification.

Collectively, these techniques offer promising directions for data-driven modeling in scientific and engineering applications where direct measurements may be limited and only partially cover the quantities of interest, and where physics, bound, and other indirect information can be employed to bolster estimates and enable uncertainty quantification. The methods introduced in this thesis hold the potential to augment the reliability and accuracy of data driven models across a diverse range of settings.