  In the context of science, the well-known adage "a picture is worth a
thousand words" might well be "a model is worth a thousand datasets." In this
manuscript we introduce the SciML software ecosystem as a tool for mixing the
information of physical laws and scientific models with data-driven machine
learning approaches. We describe a mathematical object, which we denote
universal differential equations (UDEs), as the unifying framework connecting
the ecosystem. We show how a wide variety of applications, from automatically
discovering biological mechanisms to solving high-dimensional
Hamilton-Jacobi-Bellman equations, can be phrased and efficiently handled
through the UDE formalism and its tooling. We demonstrate the generality of the
software tooling to handle stochasticity, delays, and implicit constraints.
This funnels the wide variety of SciML applications into a core set of training
mechanisms which are highly optimized, stabilized for stiff equations, and
compatible with distributed parallelism and GPU accelerators.
