  The process of transforming observed data into predictive mathematical models
of the physical world has always been paramount in science and engineering.
Although data is currently being collected at an ever-increasing pace, devising
meaningful models out of such observations in an automated fashion still
remains an open problem. In this work, we put forth a machine learning approach
for identifying nonlinear dynamical systems from data. Specifically, we blend
classical tools from numerical analysis, namely the multi-step time-stepping
schemes, with powerful nonlinear function approximators, namely deep neural
networks, to distill the mechanisms that govern the evolution of a given
data-set. We test the effectiveness of our approach for several benchmark
problems involving the identification of complex, nonlinear and chaotic
dynamics, and we demonstrate how this allows us to accurately learn the
dynamics, forecast future states, and identify basins of attraction. In
particular, we study the Lorenz system, the fluid flow behind a cylinder, the
Hopf bifurcation, and the Glycoltic oscillator model as an example of
complicated nonlinear dynamics typical of biological systems.
