Understanding complex data requires some form of model analysis, whether it be with machine learning, statistical, or physical models. Such analysis is useful for producing predictions, identifying dynamics from noise, and understanding the system in question. Complex or "black box" models can accurately predict dynamics of the world around us, and are useful tools when matching data is the primary goal. However, these models fail at increasing understanding of the system of study.

This thesis is dedicated to the analysis of simple models, and when, how, and why they work to represent data. Herein, we analyze the application of the simple SIR epidemic model to complex epidemiological dynamics present in the COVID-19 pandemic in the U.S. We show that, despite the model's simplicity and apparently violated assumptions, it still has a place in matching and predicting real data, and we can learn key intuition from this fact. We also identify key contraction arrangements in the network-like slime mold Physarum polycephalum using a low Reynolds number flow model. These contraction arrangements, or modes, are key to understanding how, exactly, the organism gets its wide array of different behaviors. Our analysis places these modes in a simple, physically-understandable context that will allow researchers to connect measurable physical features to real complex behavior.