  We propose a method for constructing the time-dependent phase space
distribution function (DF) of a collisionless system from an isolated kinematic
snapshot. In general, the problem of mapping a single snapshot to a
time-dependent function is intractable. Here we assume a finite series
representation of the DF, constructed from the spectrum of the system's Koopman
operator. This reduces the original problem to one of mapping a kinematic
snapshot to a discrete spectrum rather than to a time-dependent function. We
implement this mapping with a convolutional neural network (CNN). The method is
demonstrated on two example models: the quantum simple harmonic oscillator, and
a self-gravitating isothermal plane. The latter system exhibits phase space
spiral structure similar to that observed in Gaia Data Release 2.
