Approximation Theory for Neural Networks Applied to Pdes

This thesis studies approximation theory related to neural-network-based methodsfor computing approximate solutions to partial differential equations(PDEs). The methods considered are; Physics-Informed Neural Networks(PINNs), PCA based model reduction(PCA-Net),and Deep Operator Networks(DeepONets). To this end, we introduce the methods and presentthe approximation theory and error analysis that surround them. In doing this, we completeseveral proofs with additional details and add supplementary material, making the theory moreaccessible to readers unfamiliar with the topic. As a model problem, we consider the Darcy flowproblem. For this, we show how to apply the error analysis for the PINN and DeepONet formulations and we make from scratch implementation of each method and use them to solve specificinstances of the Darcy problem. For the PCA-Net and DeepONet we specifically focus on theproblem where the coefficients of the partial differential equation are realizations of a randomfield. We also review the use of Kolmogorov-Arnold networks when integrated with the PINNand DeepONet architectures. Here we first make a review of the theory behind them, discusstheir implementation and proceed by presenting methods for optimizing the training proceduresof these networks, and end with a discussion of their implementation. We end the thesis with adiscussion of the results and mention a few possible directions for future work