Chemical and ionic systems involve complex interactions between species, such as reactions, catalysis, charge repulsion, and electrochemistry. Dynamical system modeling, viewing each of these systems as a interconnected network of differential equations, provides a powerful tool for understanding and simulating these interactions. In the first part of this work, we design a scientific machine learning tool that is able to learn the structure and form of a differential equation system from data. Our design allows for some of the states of the dynamical system to be sparsely measured, as well as incorporating prior knowledge about the structure of chemical reactions. We successfully apply this system to learn the structure of a synthetic chemical reaction network where only some of the chemical concentrations are measured. In the second part of this work we build a continuum model for ionic devices. We use this model to explore the timeseries dynamics of an ionic diode and show which geometrical features control which behaviors. In the third part of this work we derive a lumped-element systems-model for ionic circuitry. We construct a small set of basic elements from first principles, and then use these elements to hierarchically construct a large library of ionic devices, including diodes, transistors, and power supplies. We validate this model against experimental ionic circuitry and demonstrate its utility in both predicting the behavior of complex ionic circuits and as a design tool to create novel circuit functionality.