### Learning Differential Equations

Jesse Bettencourt, University of Toronto, 12:00 ET*Abstract:* Differential equations provide a natural and productive language to describe and manipulate physical systems. As well, the interdisciplinary literature developed toward the study of differential equations is rich with conceptual and technical results. I will discuss the integration of these methods with Machine Learning. I will introduce Neural Ordinary Differential Equations, a class of initial value problems whose dynamics are specified by a neural network. I will describe some methods for learning the differential equation via gradient optimization. I will highlight some areas where this treatment is both conceptually elegant and practically effective. In particular, I will discuss Continuous Normalizing Flows for density estimation and an extension (FFJORD) that demonstrates performance improvement through numerical approximation. I will also discuss recent work to regularize learned differential equations such that their solution can be efficiently approximated by a numerical solver. I will describe recent advances in (Higher-Order) Automatic Differentiation that facilitate these methods and may be a useful tool for future techniques to study the interface of physics and Machine Learning.