Title: Computational Inversion with Wasserstein Distances and Neural Network Induced Loss Functions
Description: This thesis presents a systematic computational investigation of loss functions in solving inverse problems of partial differential equations. The primary efforts are spent on understanding optimization-based computational inversion with loss functions defined with the Wasserstein metrics and with deep learning models. The scientific contributions of the thesis can be summarized in two directions. In the first part of this thesis, we investigate the general impacts of different Wasserstein metrics and the properties of the approximate solutions to inverse problems obtained by minimizing loss functions based on such metrics. We contrast the results to those of classical computational inversion with loss functions based on the 𝐿² and 𝐻⁻ metric. We identify critical parameters, both in the metrics and the inverse problems to be solved, that control the performance of the reconstruction algorithms. We highlight the frequency disparity in the reconstructions with the Wasserstein metrics as well as its consequences, for instance, the pre-conditioning effect, the robustness against high-frequency noise, and the loss of resolution when data used contain random noise.

We examine the impact of mass unbalance and conduct a comparative study on the differences and important factors of various unbalanced Wasserstein metrics. In the second part of the thesis, we propose loss functions formed on a novel offline-online computational strategy for coupling classical least-square computational inversion with modern deep learning approaches for full waveform inversion (FWI) to achieve advantages that can not be achieved with only one component. In a nutshell, we develop an offline learning strategy to construct a robust approximation to the inverse operator and utilize it to produce a viable initial guess and design a new loss function for the online inversion with a new dataset. We demonstrate through both theoretical analysis and numerical simulations that our neural network induced loss functions developed by the coupling strategy improve the loss landscape as well as computational efficiency of FWI with reliable offline training on moderate computational resources in terms of both the size of the training dataset and the computational cost needed.