Methods for scientific simulation, machine learning, and nonlinear control

Abstract

This thesis is concerned with the development and implementation of numerical algorithms within the disciplines of computational science, machine learning, and nonlinear adaptive control. In computational science, we study a remarkable similarity between the dynamics of elastoplastic materials in the framework of hypo-elastoplasticity, a useful modeling toolkit for hard materials, and the incompressible Navier-Stokes equations. We develop a three-dimensional projection algorithm for quasi-static hypo-elastoplasticity based on projection methods in computational fluid dynamics, and then extend the method to a generalized algorithm that solves for the material stress and velocity fields on a fixed grid related to the physical domain through an abstract linear transformation. We apply these approaches to the study of shear banding in amorphous materials. In nonlinear adaptive control, we take insight from recent developments in optimization for machine learning to develop and analyze new adaptive control algorithms inspired by existing methods in optimization. In particular, we design a new class of mirror descent-like methods; by categorizing their implicit bias, we design an approach to incorporate regularization into adaptive algorithms while provably maintaining convergence and stability. We also develop a new analysis technique using approaches from reinforcement learning theory to provide the first finite-time guarantees for stochastic discrete-time nonlinear adaptive control. In machine learning, we apply tools from nonlinear dynamical systems theory to study learning algorithms from the perspective of limiting continuous-time differential equations. We analyze distributed stochastic gradient descent-based algorithms using tools traditionally applied to synchronization phenomena in nonlinear oscillators, and develop new geometric algorithms for learning generalized linear models based on mirror descent.