  The problem of determining the underlying dynamics of a system when only
given data of its state over time has challenged scientists for decades. In
this paper, the approach of using machine learning to model the updates of the
phase space variables is introduced; this is done as a function of the phase
space variables. (More generally, the modeling is done over functions of the
jet space.) This approach (named FJet) allows one to accurately replicate the
dynamics, and is demonstrated on the examples of the damped harmonic
oscillator, the damped pendulum, and the Duffing oscillator; the underlying
differential equation is also accurately recovered for each example. In
addition, the results in no way depend on how the data is sampled over time
(i.e., regularly or irregularly). It is demonstrated that a regression
implementation of FJet is similar to the model resulting from a Taylor series
expansion of the Runge-Kutta (RK) numerical integration scheme. This
identification confers the advantage of explicitly revealing the function space
to use in the modeling, as well as the associated uncertainty quantification
for the updates. Finally, it is shown in the undamped harmonic oscillator
example that the stability of the updates is stable $10^9$ times longer than
with $4$th-order RK (with time step $0.1$).
