<p dir="ltr">Uncertainty quantification is a science and engineering subject that aims to quantify and analyze the uncertainty arising from mathematical models, simulations, and measurement data. An uncertainty quantification analysis usually consists of conducting experiments to collect data, creating and calibrating mathematical models, predicting through numerical simulation, making decisions using predictive results, and comparing the model prediction with new experimental data.</p><p dir="ltr">The overarching goal of uncertainty quantification is to determine how likely some quantities in this analysis are if some other information is not exactly known and ultimately facilitate decision-making. This dissertation delivers a complete package, including theory, algorithms, and applications of information field theory, a Bayesian uncertainty quantification tool that leverages the state-of-the-art machine learning framework to accelerate solving the classical uncertainty quantification problems specified by differential equations.</p>