Scientific progress is contingent upon finding succinct, human-interpretable mathematical models whose predictions match experimental results. Almost universally, governing scientific models are differential equations. Historically, a first principles approach has driven scientific discovery. In this approach, domain experts (e.g., scientists) propose mechanistic extrapolations for experimental data collected from a natural system. At its core, this approach relies on pattern recognition, a task that machine learning excels at. However, conventional ML models often must be trained using large, “clean” data sets. Scientific data, by comparison, can be noisy and difficult to obtain. Even if abundant, clean data is available, ML models are often black boxes, which makes analyzing and interpreting them challenging. Therefore, adapting ML methods to scientific discovery requires new, interpretable models and novel approaches that endow these models with physically meaningful inductive biases. In this dissertation, I present four algorithms — PDE-READ, PDE-LEARN, weak-PDE-LEARN, and DDE-FIND — that adapt machine learning techniques to automate scientific discovery by identifying human-interpretable differential equations from noisy, limited measurements of their solutions.