Error and Uncertainty in Differential Equations Using Neural Networks

Abstract

Neural networks have been shown to be a flexible class of models, and there has been a recent surge in research focusing on their applications to solve differential equations. Since the solutions generated by these networks are numerical approximations, the true solutions will not exactly match the functions generated by the networks. In this thesis, we seek to create a framework for visualizing where the true solution to a differential equation might lie, given a network output. We begin by using Bayesian inference methods for visualizing a network's error on deterministic differential equations before extending this process to realizations of stochastic differential equations. We then develop a methodology for modeling the distribution of stochastic differential equations using neural networks by applying the same Bayesian inference methods in a more statistically rigorous context.